diff --git a/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf b/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..0689daf21bf4383decb69f05eaf74aa532dafcdc Binary files /dev/null and b/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf differ diff --git a/-dFAT4oBgHgl3EQfqB3j/content/tmp_files/2301.08645v1.pdf.txt b/-dFAT4oBgHgl3EQfqB3j/content/tmp_files/2301.08645v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..0c806242b6ce44864cb4bef829bbd95d99ddc6f8 --- /dev/null +++ b/-dFAT4oBgHgl3EQfqB3j/content/tmp_files/2301.08645v1.pdf.txt @@ -0,0 +1,255 @@ +arXiv:2301.08645v1 [gr-qc] 20 Jan 2023 +Complex scalar field in κ-Minkowski spacetime +Andrea Bevilacqua1, ∗ +1National Centre for Nuclear Research, Pasteura 7, 02-093 Warszawa, Poland +It is often expected that one cannot treat spacetime as a continuous manifold as the Planck +scale is approached, because of to possible effects due to a quantum theory of gravity. There +have been several proposals to model such a deviation from the classical behaviour, one of +which is noncommutativity of spacetime coordinates. In this context, the non-commutativity +scale is seen as an observer-independent length scale. Of course, such a scale impose a modi- +fication of ordinary relativistic symmetries, which now need to be deformed to accommodate +this fundamental scale. The κ-Poincar´e algebra is an example of this deformation. In what +follows I will briefly describe a construction of a κ-deformed complex scalar field theory, +while at the same time shedding light on the behaviour of discrete and continuous symme- +tries in this formalism. This in turn will open the way to the study of the application of +this formalism to actual physical processes. I will then conclude with some comments and +prospects for the future. +I. +FROM NON-COMMUTATIVE SPACETIME TO THE ACTION +We will present a brief preview of the works in Refs. [1], [2]. For a more detailed introduction +to the framework and the formalism, see for example Ref. [3]. The starting point is k-Minkowski +spacetime, whose coordinates satisfy the an(3) Lie algebra defined as [ˆx0, ˆxi] = iˆxi/κ. +Notice +that one recovers canonical Minkowski spacetime in the formal limit κ → ∞. To have a more +physical understanding of the above commutator, notice that 1/κ has dimensions of length, and it +is sometimes identified with the Plank length. In order to build fields in this spacetime, we need to +define plane waves first. To do so, one can proceed as follows. First pick an explicit representation +of the an(3) Lie algebra in terms of matrices (it turns out that the lowest dimensional one has +dimension 5). Then, one can define a plane wave as an element ˆek of the relative AN(3) Lie group, +i.e. for example1 ˆek = exp(ikiˆxi) exp(ik0ˆx0). Notice that, since the elements of the matrices ˆxµ +∗ andrea.bevilacqua@ncbj.gov.pl +1 A different choice on the ordering results in a change of coordinates in momentum space. + +2 +are dimensionful, so are the parameters kµ. Written explicitly, one has +ˆek = + + + + + + + + + + + +¯p4 +κ +k +κ +p0 +κ +p +κ +1 +p +κ +¯p0 +κ +− k +κ +p4 +κ + + + + + + + + + + + +(1) +where2 +p0 = κ sinh k0 +κ + k2 +2κek0/κ +pi = kiek0/κ +p4 = κ cosh k0 +κ − k2 +2κek0/κ. +(2) +Both kµ and pA(k) (A = 0, 1, 2, 3, 4) are two different coordinates of momentum space. Notice that +the pA are not independent, since −p0 + p2 + p2 +4 = κ2 (notice also that p+ = p0 + p4 > 0). It is +therefore clear that momentum space is curved. This is reflected in a deformed sum of momenta, +which is defined through the group property ˆekˆel =: ˆek⊕l. At the same time, inverse momenta +are defined in terms of group inverses ˆe−1 +k +=: ˆeS(k), and S(k) is called the antipode of k. In order +to simplify the treatment of integrals and derivatives, we can use a3 Weyl map W to send group +elements ˆek into canonical plane waves ep := exp(ipµ(k)xµ). The group law is preserved thanks +to the ⋆ product defined by W(ˆek⊕l) = ep(k)⊕q(l) =: ep(k) ⋆ eq(l). In general, the ⋆ product is not +commutative. Because of this, we choose the following action for a κ-deformed complex scalar +field. +S = 1 +2 +� +R4 d4x[(∂µφ)† ⋆ (∂µφ) + (∂µφ) ⋆ (∂µφ)† − m2(φ† ⋆ φ + φ ⋆ φ†)]. +(3) +To obtain the equations of motion one usually integrates by parts, but in this deformed context +derivatives do not satisfy the Leibniz rule. In fact, if they did, one could get the following contra- +diction. +i(p ⊕ q)µep⊕q = ∂µ(ep ⋆ eq) = (∂µep) ⋆ eq + ep ⋆ ∂µeq = i(p + q)ep⊕q. +(4) +Hence one has to use the appropriate deformations for the Leibniz rules. After some computations, +one can verify that the equations of motion satisfied by the field are the canonical Klein-Gordon +equations, and the field which satisfies them can be written as +φ(x) = +� +d3p +�2ωp +ξ(p)ape−i(ωpt−px) + +� +d3p∗ +� +2|ω∗p| +ξ(p)b† +p∗ei(S(ωp)t−S(p)x). +(5) +2 The explicit expression of ¯p4 and ¯p0 in terms of kµ is not relevant for the present discussion. +3 There are several equivalent ways in which a suitable Weyl map can be chosen, see Ref. +[1] for more details. + +3 +The action of the discrete transformations C, P, T can be defined in the usual way in the deformed +context, i.e. Tφ(t, x)T −1 = φ(−t, x), Pφ(t, x)P −1 = φ(t, −x), and Cφ(t, x)C−1 = φ†(t, x). Notice +that this is due to the presence of the antipode in the on-shell field. Furthermore, the action is +manifestly invariant under both CPT and κ-deformed Lorentz transformations. +II. +CHARGES AND FEATURES OF THE MODEL +We now need to compute the charges for our model. There are two main ways to do so. The +first is by direct computation using the Noether theorem. However, the fact that derivatives do +not follow the Leibniz rule makes this job a prohibitively difficult one, except for the case of +translation charges. The second way is to use the covariant phase-space formalism. Although very +direct, this second method needs to be carefully defined in the deformed context. +We chose a +hybrid approach, namely we derived the translation charges from the Noether theorem, and then +we built a covariant phase space approach which was able to reproduce the translation charges, +allowing then to compute the remaining charges. Given a symplectic form Ω, and a symmetry in +spacetime described by the vector field ξ, the charge Qξ can be defined by −δξ⌟ Ω = δQξ. In this +context δξ is a vector field in phase space describing the variation δξA of any physical quantity +A in phase space under the action of the symmetry ξ in spacetime, δ is the exterior derivative in +phase space, and ⌟ represents a contraction. The translation charges computed directly from the +Noether theorem are the following4: +Pµ = +� +d3p α(p)[−S(p)µa† +pap + pµb† +p∗bp∗]. +(6) +The quantity α(p) is a function of momenta whose explicit expression does not concern us in this +context. To reproduce Eq. (6) in the covariant phase space formalism, we need to assume the +following deformation of the canonical contraction rule between vector fields and forms5: +δv⌟ (A ∧ B) = (δvA)B + A(S(δv)B), +(7) +where S(δv) can be appropriately defined [2]. +Using this definition, one can compute all the +remaining charges, and the creation/annihilation operators algebra (which turns out to be the +canonical one). For example, the boost charge Ni is +Ni=− 1 +2 +� d3p α +� +S(ωp) +� +∂a† +p +∂S(p)i ap −a† +p +∂ap +∂S(p)i +� ++ωp +� +bp +∂b† +p +∂pi − ∂bp +∂pi b† +p +�� +. +(8) +4 There is also a fifth charge P4, see [1] for further details. +5 Recall that the canonical relation is δv⌟ (A ∧ B) = (δvA)B − A(δvB). + +4 +One can then verify that the charges satisfy the usual non-deformed Poincar´e algebra. However, +due to the effects of κ-deformation, one can also verify that, e.g, [Ni, C] ̸= 0, meaning that CPT +symmetry is subtly violated. More explicitly, using the definition +C = +� +d3p (b† +pap + a† +pbp), +(9) +one can show that [Ni, C] is given by +i +2 +� +d3p +� +S(ωp) +� +∂ap +∂S(p)i b† +p − ap +∂b† +p +∂S(p)i + +∂a† +p +∂S(p)i bp − a† +p +∂bp +∂S(p)i +� ++ ++ ωp +� +∂b† +p +∂pi ap − b† +p +∂ap +∂pi + ∂bp +∂pi a† +p − bp +∂a† +p +∂pi +� � +. +(10) +Notice that in the limit κ → ∞ one recovers the canonical result [Ni, C] = 0. +III. +COMMENTS AND CONCLUSIONS +The fact that [Ni, C] ̸= 0 has several important physical consequences. The most apparent one +is a difference in the decay time between particles and antiparticles. Furthermore, we now have a +well defined theory which will allow us to study in details the propagator and the n-point functions +in general. From this point of view, it will be interesting to tackle the loops in this deformed +context. Finally, what has been done for the case of the complex scalar field will be extended to +fields of higher spins, in order to expand the discussion to more realistic phenomena. +ACKNOWLEDGMENTS +Parts of these works were supported by funds provided by the Polish National Science Center, +the project number 2019/33/B/ST2/00050. +[1] M. Arzano, A. Bevilacqua, J. Kowalski-Glikman, G. Rosati, and J. Unger, Phys. Rev. D 103, 106015 +(2021) +[2] A. Bevilacqua, J. Kowalski-Glikman, and W. Wislicki, Phys. Rev. D 105, 105004 (2022) +[3] M. Arzano, and J. Kowalski-Glikman, Deformations of Spacetime Symmetries: Gravity, Group-Valued +Momenta, and Non-Commutative Fields, Lecture Notes in Physics, Springer, 2021. + diff --git a/-dFAT4oBgHgl3EQfqB3j/content/tmp_files/load_file.txt b/-dFAT4oBgHgl3EQfqB3j/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..c7a94fdb072974ee8d15eae7cb49e38702c36da8 --- /dev/null +++ b/-dFAT4oBgHgl3EQfqB3j/content/tmp_files/load_file.txt @@ -0,0 +1,98 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf,len=97 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'} +page_content='08645v1 [gr-qc] 20 Jan 2023 Complex scalar field in κ-Minkowski spacetime Andrea Bevilacqua1, ∗ 1National Centre for Nuclear Research, Pasteura 7, 02-093 Warszawa, Poland It is often expected that one cannot treat spacetime as a continuous manifold as the Planck scale is approached, because of to possible effects due to a quantum theory of gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'} +page_content=' There have been several proposals to model such a deviation from the classical behaviour, one of which is noncommutativity of spacetime coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'} +page_content=' In this context, the non-commutativity scale is seen as an observer-independent length scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'} +page_content=' Of course, such a scale impose a modi- fication of ordinary relativistic symmetries, which now need to be deformed to accommodate this fundamental scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'} +page_content=' The κ-Poincar´e algebra is an example of this deformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'} +page_content=' In what follows I will briefly describe a construction of a κ-deformed complex scalar field theory, while at the same time shedding light on the behaviour of discrete and continuous symme- tries in this formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'} +page_content=' This in turn will open the way to the study of the application of this formalism to actual physical processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'} +page_content=' I will then conclude with some comments and prospects for the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'} +page_content=' FROM NON-COMMUTATIVE SPACETIME TO THE ACTION We will present a brief preview of the works in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'} +page_content=' [1], [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'} +page_content=' For a more detailed introduction to the framework and the formalism, see for example Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'} +page_content=' [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'} +page_content=' The starting point is k-Minkowski spacetime, whose coordinates satisfy the an(3) Lie algebra defined as [ˆx0, ˆxi] = iˆxi/κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'} +page_content=' Notice that one recovers canonical Minkowski spacetime in the formal limit κ → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'} +page_content=' To have a more physical understanding of the above commutator, notice that 1/κ has dimensions of length, and it is sometimes identified with the Plank length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'} +page_content=' In order to build fields in this spacetime, we need to define plane waves first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'} +page_content=' To do so, one can proceed as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'} +page_content=' First pick an explicit representation of the an(3) Lie algebra in terms of matrices (it turns out that the lowest dimensional one has dimension 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'} +page_content=' Then, one can define a plane wave as an element ˆek of the relative AN(3) Lie group, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'} +page_content=' for example1 ˆek = exp(ikiˆxi) exp(ik0ˆx0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'} +page_content=' Notice that, since the elements of the matrices ˆxµ ∗ andrea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'} +page_content='bevilacqua@ncbj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'} +page_content='gov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'} +page_content='pl 1 A different choice on the ordering results in a change of coordinates in momentum space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'} +page_content=' 2 are dimensionful, so are the parameters kµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'} +page_content=' Written explicitly, one has ˆek = \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed ¯p4 κ k κ p0 κ p κ 1 p κ ¯p0 κ − k κ p4 κ \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 (1) where2 p0 = κ sinh k0 κ + k2 2κek0/κ pi = kiek0/κ p4 = κ cosh k0 κ − k2 2κek0/κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'} +page_content=' (2) Both kµ and pA(k) (A = 0, 1, 2, 3, 4) are two different coordinates of momentum space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'} +page_content=' Notice that the pA are not independent, since −p0 + p2 + p2 4 = κ2 (notice also that p+ = p0 + p4 > 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'} +page_content=' It is therefore clear that momentum space is curved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'} +page_content=' This is reflected in a deformed sum of momenta, which is defined through the group property ˆekˆel =: ˆek⊕l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'} +page_content=' At the same time, inverse momenta are defined in terms of group inverses ˆe−1 k =: ˆeS(k), and S(k) is called the antipode of k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'} +page_content=' In order to simplify the treatment of integrals and derivatives, we can use a3 Weyl map W to send group elements ˆek into canonical plane waves ep := exp(ipµ(k)xµ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'} +page_content=' The group law is preserved thanks to the ⋆ product defined by W(ˆek⊕l) = ep(k)⊕q(l) =: ep(k) ⋆ eq(l).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'} +page_content=' In general, the ⋆ product is not commutative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'} +page_content=' Because of this, we choose the following action for a κ-deformed complex scalar field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'} +page_content=' S = 1 2 � R4 d4x[(∂µφ)† ⋆ (∂µφ) + (∂µφ) ⋆ (∂µφ)† − m2(φ† ⋆ φ + φ ⋆ φ†)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'} +page_content=' (3) To obtain the equations of motion one usually integrates by parts, but in this deformed context derivatives do not satisfy the Leibniz rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'} +page_content=' In fact, if they did, one could get the following contra- diction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'} +page_content=' i(p ⊕ q)µep⊕q = ∂µ(ep ⋆ eq) = (∂µep) ⋆ eq + ep ⋆ ∂µeq = i(p + q)ep⊕q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'} +page_content=' (4) Hence one has to use the appropriate deformations for the Leibniz rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'} +page_content=' After some computations, one can verify that the equations of motion satisfied by the field are the canonical Klein-Gordon equations, and the field which satisfies them can be written as φ(x) = � d3p �2ωp ξ(p)ape−i(ωpt−px) + � d3p∗ � 2|ω∗p| ξ(p)b† p∗ei(S(ωp)t−S(p)x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'} +page_content=' (5) 2 The explicit expression of ¯p4 and ¯p0 in terms of kµ is not relevant for the present discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'} +page_content=' 3 There are several equivalent ways in which a suitable Weyl map can be chosen, see Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'} +page_content=' [1] for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'} +page_content=' 3 The action of the discrete transformations C, P, T can be defined in the usual way in the deformed context, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'} +page_content=' Tφ(t, x)T −1 = φ(−t, x), Pφ(t, x)P −1 = φ(t, −x), and Cφ(t, x)C−1 = φ†(t, x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'} +page_content=' Notice that this is due to the presence of the antipode in the on-shell field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'} +page_content=' Furthermore, the action is manifestly invariant under both CPT and κ-deformed Lorentz transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'} +page_content=' CHARGES AND FEATURES OF THE MODEL We now need to compute the charges for our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'} +page_content=' There are two main ways to do so.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'} +page_content=' The first is by direct computation using the Noether theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'} +page_content=' However, the fact that derivatives do not follow the Leibniz rule makes this job a prohibitively difficult one, except for the case of translation charges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'} +page_content=' The second way is to use the covariant phase-space formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'} +page_content=' Although very direct, this second method needs to be carefully defined in the deformed context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'} +page_content=' We chose a hybrid approach, namely we derived the translation charges from the Noether theorem, and then we built a covariant phase space approach which was able to reproduce the translation charges, allowing then to compute the remaining charges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'} +page_content=' Given a symplectic form Ω, and a symmetry in spacetime described by the vector field ξ, the charge Qξ can be defined by −δξ⌟ Ω = δQξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'} +page_content=' In this context δξ is a vector field in phase space describing the variation δξA of any physical quantity A in phase space under the action of the symmetry ξ in spacetime, δ is the exterior derivative in phase space, and ⌟ represents a contraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'} +page_content=' The translation charges computed directly from the Noether theorem are the following4: Pµ = � d3p α(p)[−S(p)µa† pap + pµb† p∗bp∗].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'} +page_content=' (6) The quantity α(p) is a function of momenta whose explicit expression does not concern us in this context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'} +page_content=' To reproduce Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'} +page_content=' (6) in the covariant phase space formalism, we need to assume the following deformation of the canonical contraction rule between vector fields and forms5: δv⌟ (A ∧ B) = (δvA)B + A(S(δv)B), (7) where S(δv) can be appropriately defined [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'} +page_content=' Using this definition, one can compute all the remaining charges, and the creation/annihilation operators algebra (which turns out to be the canonical one).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'} +page_content=' For example, the boost charge Ni is Ni=− 1 2 � d3p α � S(ωp) � ∂a† p ∂S(p)i ap −a† p ∂ap ∂S(p)i � +ωp � bp ∂b† p ∂pi − ∂bp ∂pi b† p �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'} +page_content=' (8) 4 There is also a fifth charge P4, see [1] for further details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'} +page_content=' 5 Recall that the canonical relation is δv⌟ (A ∧ B) = (δvA)B − A(δvB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'} +page_content=' 4 One can then verify that the charges satisfy the usual non-deformed Poincar´e algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'} +page_content=' However, due to the effects of κ-deformation, one can also verify that, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'} +page_content='g, [Ni, C] ̸= 0, meaning that CPT symmetry is subtly violated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'} +page_content=' More explicitly, using the definition C = � d3p (b† pap + a† pbp), (9) one can show that [Ni, C] is given by i 2 � d3p � S(ωp) � ∂ap ∂S(p)i b† p − ap ∂b† p ∂S(p)i + ∂a† p ∂S(p)i bp − a† p ∂bp ∂S(p)i � + + ωp � ∂b† p ∂pi ap − b† p ∂ap ∂pi + ∂bp ∂pi a† p − bp ∂a† p ∂pi � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'} +page_content=' (10) Notice that in the limit κ → ∞ one recovers the canonical result [Ni, C] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'} +page_content=' COMMENTS AND CONCLUSIONS The fact that [Ni, C] ̸= 0 has several important physical consequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'} +page_content=' The most apparent one is a difference in the decay time between particles and antiparticles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'} +page_content=' Furthermore, we now have a well defined theory which will allow us to study in details the propagator and the n-point functions in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'} +page_content=' From this point of view, it will be interesting to tackle the loops in this deformed context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'} +page_content=' Finally, what has been done for the case of the complex scalar field will be extended to fields of higher spins, in order to expand the discussion to more realistic phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'} +page_content=' ACKNOWLEDGMENTS Parts of these works were supported by funds provided by the Polish National Science Center, the project number 2019/33/B/ST2/00050.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'} +page_content=' [1] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'} +page_content=' Arzano, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'} +page_content=' Bevilacqua, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'} +page_content=' Kowalski-Glikman, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'} +page_content=' Rosati, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'} +page_content=' Unger, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'} +page_content=' D 103, 106015 (2021) [2] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'} +page_content=' Bevilacqua, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'} +page_content=' Kowalski-Glikman, and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'} +page_content=' Wislicki, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'} +page_content=' D 105, 105004 (2022) [3] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'} +page_content=' Arzano, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'} +page_content=' Kowalski-Glikman, Deformations of Spacetime Symmetries: Gravity, Group-Valued Momenta, and Non-Commutative Fields, Lecture Notes in Physics, Springer, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFAT4oBgHgl3EQfqB3j/content/2301.08645v1.pdf'} diff --git a/.gitattributes b/.gitattributes index 923fb0e6eb9c88ff3070877b3ba0d949d0951b9a..07ba4d5e08a2628f0a72ca5cb35ea72c492357fa 100644 --- a/.gitattributes +++ b/.gitattributes @@ -5256,3 +5256,58 @@ rdAyT4oBgHgl3EQfZvcN/content/2301.00227v1.pdf filter=lfs diff=lfs merge=lfs -tex XtAzT4oBgHgl3EQfYvw0/content/2301.01339v1.pdf filter=lfs diff=lfs merge=lfs -text INAzT4oBgHgl3EQfxv6_/content/2301.01744v1.pdf filter=lfs diff=lfs merge=lfs -text 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a/09FIT4oBgHgl3EQf3Sv5/content/tmp_files/2301.11381v1.pdf.txt b/09FIT4oBgHgl3EQf3Sv5/content/tmp_files/2301.11381v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..ad2c35185666aa29fde78a1f5b663a7d113cd391 --- /dev/null +++ b/09FIT4oBgHgl3EQf3Sv5/content/tmp_files/2301.11381v1.pdf.txt @@ -0,0 +1,1541 @@ +Imaging anisotropic waveguide exciton polaritons in tin sulfide + +Yilong Luan1,2, Hamidreza Zobeiri3, Xinwei Wang3, Eli Sutter4,5, Peter Sutter6*, Zhe Fei1,2* + +1Department of Physics and Astronomy, Iowa State University, Ames, Iowa 50011, USA +2Ames Laboratory, U. S. Department of Energy, Iowa State University, Ames, Iowa 50011, USA +3Department of Mechanical Engineering, Iowa State University, Ames, IA 50011, USA +4Department of Mechanical and Materials Engineering, University of Nebraska-Lincoln, Lincoln, NE +68588, USA +5Nebraska Center for Materials and Nanoscience, University of Nebraska-Lincoln, Lincoln, NE 68588, +USA. +6Department of Electrical and Computer Engineering, University of Nebraska-Lincoln, Lincoln, NE 68588, +USA + +*Corresponding to: (P.S.) psutter@unl.edu, (Z.F.) zfei@iastate.edu. + +Abstract +In recent years, novel materials supporting in-plane anisotropic polaritons have attracted a lot of +research interest due to their capability of shaping nanoscale field distributions and controlling +nanophotonic energy flows. Here we report a nano-optical imaging study of waveguide exciton polaritons +(EPs) in tin sulfide (SnS) in the near-infrared (IR) region using the scattering-type scanning near-field +optical microscopy (s-SNOM). With s-SNOM, we mapped in real space the propagative EPs in SnS, which +show sensitive dependence on the excitation energy and sample thickness. Moreover, we found that both +the polariton wavelength and propagation length are anisotropic in the sample plane. In particular, in a +narrow spectral range from 1.32 to 1.44 eV, the EPs demonstrate quasi-one-dimensional propagation, which +is rarely seen in natural polaritonic materials. Further analysis indicates that the observed polariton +anisotropy is originated from the different optical bandgaps and exciton binding energies along the two +principal crystal axes of SnS. + +Key Words: Tin sulfide, waveguide, exciton polaritons, s-SNOM, anisotropy, quasi-one-dimensional + +Main text +In-plane anisotropic polaritons1,2 were first studied in metasurfaces3-5 where nanostructuring of the +polaritonic media or substrates breaks the symmetry, thus enabling polaritonic anisotropy. Later, several +natural materials were predicted and/or experimentally confirmed to support in-plane anisotropic +polaritons.6-10 For example, anisotropic plasmon polaritons and hybrid plasmon-phonon polaritons were +observed in black phosphorus carbides with far-field infrared (IR) spectroscopy.8 Anisotropic phonon +polaritons with hyperbolic wavefronts were imaged in MoO3,9-11 which can be conveniently tailored by +controlling the sample thickness and by stack- and twist-engineering.12-18 Compared to nano-engineered +anisotropic metasurfaces, natural materials with intrinsic anisotropic polaritons are generally more +convenient for applications and can avoid potential material quality degradation due to complex nano- +fabrications. Despite these advantages, natural materials supporting in-plane anisotropic polaritons are rare +and are so far mainly studied in the mid-IR range. New materials enabling anisotropic polaritons in other +technologically important spectral regions (e.g., near-IR and visible) are desired. +In this Letter, we report the experimental discovery of strongly-anisotropic exciton polaritons (EPs) +in tin sulfide (SnS) in the technologically-important near-IR region. SnS is a post-transition-metal +monochalcogenide and a van der Waals (vdW) layered semiconductor with an orthorhombic structure, +analogous to that of black phosphorous.6-7 As sketched in Figure 1b, the two in-plane axes of SnS, namely +the a and b axes, are along the zigzag and armchair directions, respectively. SnS has been widely studied +due to its unique anisotropic optoelectronic properties19-23 and potential applications related to +photodetection and solar energy harvesting.24-27 In particular, the energies of excitons or optical bandgaps + +along the a and b axes of SnS are about Ea ≈ 1.39 eV and Eb ≈ 1.66 eV respectively,22,23 which directly +impact the polaritonic responses. Note that EPs have previously been studied in other vdW semiconductors +(e.g. WSe2, MoSe2, etc.) with imaging28-32 and spectroscopic methods,33-37 where the EPs are isotropic in the +sample plane. The samples studied here are SnS microcrystals supported on mica wafers (Figure S1a). As +introduced in detail in the earlier work,19 these microcrystals have a wrap-around layered core-shell +structure: the thick SnS core is coated with a thin crystalline shell (thickness ≈ 3 nm) of layered tin disulfide +(SnS2). A detailed characterization of the wrap-around core-shell structures and their synthesis process were +reported in the earlier work.19 Note that the thin SnS shell is isotropic in the sample plane38,39, so the +observed anisotropic properties of EPs are solely due to the SnS core. The SnS2 shell mainly serves as a +protection layer of the SnS core and the waveguide EPs. Detailed discussions about the effect of the SnS2 +shell are given in the Supporting Information. + +To excite and probe EPs in SnS, we employed a scattering-type scanning near-field optical +microscope (s-SNOM) that was built based on an atomic force microscope (AFM). As illustrated in Figure +1a, the sharp metalized tip in s-SNOM excited by a p-polarized laser beam generates strong evanescent +fields underneath the tip. These evanescent fields with a wide range of wavevectors40 can effectively excite +transverse-magnetic (TM) polaritons inside the sample.29 The excitation source used in the study is a +broadband (1.24-1.77 eV) Ti:sapphire laser that covers the bandgap and exciton energies of SnS (see +discussions below). We used a parabolic mirror to focus the laser beam at the tip apex, and the scattered +photons off the tip/sample system are collected by the same parabolic mirror and then counted by a +photodetector. More introductions about the nano-optical setup are in Supporting Information. + +In Figure 1c, we plot the AFM topography image of a typical SnS microcrystal coated with a thin +SnS2 shell.19 The lateral sizes of the crystal are approximately 6-8 m, and the thickness is about 100 nm +including the SnS2 shell. Here the crystal has a total of eight edges, among which the four short edges are +along the a or b axes.19 We were able to determine the crystal axes of the sample by examining the shape +of the crystal and by Raman spectroscopy (see Supporting Information). Figure 1d plots the near-field +amplitude (s) images taken simultaneously with the topography image (Figure 1c) at the excitation energy +of E = 1.38 eV. Here, the in-plane wavevector of the laser (kin) is along the b axis of SnS. From Figure 1d, +one can see many interference fringes and oscillations inside the sample. We focus on a string of one- +dimensional (1D) oscillations extending from the left edge to the crystal center along the b axis (marked +with a white arrow). According to previous studies,29,30 these oscillations are generated due to the +interference between two major beam paths as sketched in Figure 1a. In the first path (P1), the excitation +photons are scattered back directly by the tip apex. In the second path (P2), the excitation photons are first +transferred into waveguide EPs by the s-SNOM tip. These EPs then propagate toward the sample edge and +get scattered into photons. Photons collected through the two beam paths are coherent with each other, so +they can generate interference. When scanning the tip perpendicular to the sample edge, the distance +between the tip and the sample edge varies, so a string of bright and dark spots forms due to constructive +and destructive interferences, respectively. As sketched in Figure 1a, the left short edge of the crystal is +responsible for the generation of the 1D interference oscillations along the direction of the white arrow in +Figure 1d. Other edges can also scatter EPs into photons and generate interference patterns. For example, +the four long edges that are about 43º relative to the b axis are responsible for the bight fringes parallel to +these edges (see Figure S2b). There are other possible interference mechanisms (e.g., edge excitation of +polaritons), but they are not responsible for the fringes/oscillations observed in our samples. Detailed +discussions of different interference mechanisms are given in Section 3 of the Supporting Information. +From Fig. 1d,f, we also seen fringes on the substrate side, which are generated due to the excitation and +scattering of photons at the air/mica interface as confirmed by dispersion analysis (see Figure S10). +Figure 1e,f plot the AFM and corresponding s-SNOM imaging data of the same crystal as those in +Figure 1c,d but rotated 90º relative to the surface normal (c axis). Here the in-plane wavevector of the +excitation laser is along the a axis (kin // a). Interestingly, we found no interference oscillations in the interior +of the crystal as those seen in Figure 1d, indicating that no waveguide EPs are propagating along the a axis. +To further explore the anisotropic polaritonic responses, we performed energy-dependent s-SNOM imaging. +The results are shown in Figure 2, where we plot the s-SNOM imaging data with kin along both the b axis + +(Figure 2a-e) and a axis (Figure 2f-j) at various excitation energies. Again, we focus on the 1D oscillations +at the crystal center (along the direction of the white arrows) that evolve systematically with E. For kin // b, +the oscillations are clearly seen for photon energies from 1.29 to 1.48 eV, and their periods decrease with +increasing energy. In the case of kin // a, the interference oscillations appear only at energies below 1.32 eV, +and there are no clear 1D oscillations from 1.38 to 1.48 eV. + +The s-SNOM imaging data shown in Figures 1 and 2 provide direct evidence of in-plane anisotropic +EPs of SnS in the near-IR region. To support the experimental data, we performed finite-element +simulations of the waveguide EPs using Comsol Multiphysics. In the model, we placed a vertically +polarized excitation dipole (pz) right above the sample surface. The optical constants of SnS and SnS2 were +obtained from the literature.22,23,38,39 A detailed description of the Comsol model is given in Supporting +Information. The simulation results are shown in Figure 2k-o and Figure S5, where we plot the real-space +images of polariton field amplitude (|Ez|) and polariton field (Ez) of EPs respectively. Here the EPs are +launched by a vertically polarized dipole (pz) located at the center of the image. At E = 1.29 eV (Figure 2k), +the dipole-launched anisotropic EPs propagate at all directions with elliptic wavefronts. As E increases to +1.32 eV (Figure 2l), the EPs show a faster decay along the a axis while keeping a relatively long propagation +distance along the b axis. The propagation along the a axis is even shorter at higher energies E ≥ 1.38 eV +(Figure 2m-o). As a result, the EPs appear to be quasi-1D along the b axis. The polaritonic simulations are +consistent with s-SNOM imaging data in Figures 1 and 2. + +With the s-SNOM imaging data and Comsol simulation results, we were able to perform a +quantitative analysis of the dispersion and propagation properties of the anisotropic EPs. In Figure 3a,c, we +plot the line profiles extracted across the 1D interference oscillations in the energy-dependent s-SNOM +images (Figure 2). We then performed Fourier transforms (FTs) of these profiles to accurately obtain the +periods () of the interference oscillations that are linked to the polariton wavelength (p) in the following +relationship:29,30 + +0/ ≡ k/k0 ≈ 0/p – cos . (1) + +Here, λ0 is the excitation photon wavelength, k0 = 2π/λ0 is the free-space photon wavevector, kρ = 2π/ρ is +the inverse period of the interference oscillations, and  ≈ 30º is the incidence angle of the laser beam +relative to the sample plane (see Figure 1a). The FT profiles for kin // b and kin // a are shown respectively +in Figures 3b and 3d, where the peaks (marked with blue arrows) correspond to k. We then determined the +polariton wavevector (kp = 2π/p) using Eq. (1) for every given excitation energy, based on which we obtain +the energy-momentum dispersion relations of the EPs. + +The experimental dispersion data points of EPs obtained through FT analysis (Figure 3b,d) are +plotted in Figure 4a,b as black squares, which are sitting on the theoretical dispersion colormaps. In the +colormaps, we plot the imaginary part of the reflection coefficients Im(rp) that represents the photonic +density of states (see Supporting Information). Here the TM waveguide modes are visualized as bright +curves (marked with blue dashed curves).29 In addition to the dispersion relations, the Im(rp) colormaps also +reveal the mode broadening (k) that corresponds to the damping (see discussions in the following +paragraph). This method of dispersion calculation has been widely used in the studies of polaritons in a +variety of materials.29,30,40,41 In the dispersion diagrams, we also plot the dispersion data points extracted +from Comsol simulations (Figure 2k-o and Figure S5). The dispersion relations of the EPs from +experimental data, Comsol simulations, and the Im(rp) colormaps are consistent with each other, which +validates our experimental and theoretical approaches. From the dispersion diagrams, we can examine the +light-exciton interactions close to the exciton energies Ea and Eb (marked with white dashed lines in Figure +4a,b). The waveguide mode along the b axis exhibits a clear back-bending behavior that is a signature +behavior of light-exciton interactions.28,29 By fitting the dispersion with the coupled oscillator model, we +were able to determine the Rabi splitting energy (~160 meV), which is larger than the average polariton +linewidth (~105 meV) (see Supporting Information). Therefore, the EPs along the b axis are in the strong +coupling regime. The mode coupling is much weaker along the a axis, likely due to the small exciton + +binding energy. According to literature,22 excitons along the b axis are robust with a binding energy of ~ +50 meV. The binding energy of excitons along the a axis, on the other hand, is much smaller and Ea is close +to the fundamental bandgap.22 The light-exciton coupling is much stronger at lower temperatures (e.g., T = +27 K) with more prominent mode bending features close to the exciton energies (Figure S8). + +In addition to the polariton dispersion, we also extracted the propagation lengths (Lep) of the EPs +(Lep ≡ 1/[2Im(kep)]). The extraction was done by fitting the decay trend of the polariton oscillations from +both the s-SNOM data (Figure 2a-j) and Comsol simulations (Figure 2k-o and Figure S5). A detailed +description of the fitting procedures is given in the Supporting Information. As shown in Figure 4c,d, Lep +along both the a and b axes are over 3 m at E = 1.29 eV. With increasing E, Lep drops systematically along +both directions, but the drop along the a axis is much faster. As E approaches Ea ≈ 1.39 eV, Lep along the a +axis drops below 1 m and becomes unmeasurable. Lep along the b axis, on the other hand, is as high as 2.5 +m at E = 1.38 eV, where quasi-1D EPs were observed (Figure 1d,f). Lep drops to 1 m or below along the +b axis when E gets close to Eb ≈ 1.66 eV. The energy dependence of Lep is fully consistent with the mode +broadening behaviors shown in the theoretical dispersion colormaps in Figure 4. The larger the polariton +width (k), the smaller the propagation length. + +Finally, we explored the dependence of EPs on the thicknesses of SnS crystals. In Figure 5a, we +plot the nano-optical images of SnS microcrystals with various thicknesses (d) taken at an excitation energy +of E = 1.38 eV. Due to the strong damping of EPs along the a axis, we only show in Figure 5 the data +images with the excitation along the b axis (kin // b). We focus on the 1D interference oscillations (Figures +1 and 2), which evolve systematically with varying thicknesses. Figure 5b plots the line profiles taken +directly across the 1D oscillations in Figure 5a. Using Fourier transform and Eq. (1), we extracted the +polariton wavelengths p at different sample thicknesses, which are plotted in Figure 5c. Here one can see +that p decreases systematically with increasing d, which is expected since the crystal thickness determines +both the out-of-plane (kz ~ 1/d) and in-plane wavevectors of the waveguide mode.42 For samples with +thicknesses over 150 nm, p drops below 300 nm that is 3 times smaller than the photon wavelength 0 = +900 nm. The mode confinement is comparable to if not better than waveguide EPs in other materials.29-32 + +In summary, we have performed a comprehensive nano-optical study of SnS microcrystals using +s-SNOM. We found through near-field imaging that SnS supports waveguide EPs in near-IR, which are +sensitively dependent on both the excitation energy and sample thickness. More interestingly, both the +dispersion and transport properties of the EPs are strongly anisotropic in the sample plane. In particular, in +the energy range from 1.35 to 1.55 eV, the EPs show quasi-1D propagation along the b axis, which has not +been reported in other natural polaritonic materials. Future studies with a pump-probe s-SNOM setup31,32 +are expected for the exploration of the ultrafast dynamics of anisotropic EPs in SnS. It is also interesting to +study TE waveguide EPs of SnS that could be seen in atomically thin crystals, where active tunability of +EPs is possible with electrical gating. + +The anisotropic EPs discovered here in SnS are promising for a variety of applications. One +potential application is low-pass waveguide filters for planar photonic circuits. The cut-off energies of the +filters can be chosen by selecting the direction of signals propagating through the waveguide (i.e., along a +or b axes), Another possible application is selective nanophotonic interconnection. A concept device is +sketched in Figure S13, where the signal source is connecting two devices through an SnS interconnector. +The two ports for the two devices are along the a and b axes, respectively. When the incident photonic +signal (e.g., on or off signals) is at energies of E ≤ 1.29 eV, EPs can propagate along all directions in SnS, +so both devices can receive the signal. When the incident photonic signal is at energies of 1.38 eV ≤ E ≤ +1.48 eV, EPs propagate only along the b axis, so device 2 will not receive the signal. Therefore, the signal +interconnection can be controlled selectively by choosing different energies of the incident signals. Such a +directional control of the flow of nanophotonic energy and signals cannot be easily realized in isotropic +polaritonic materials without complicated nano-fabrications. With the potential controllability or tunability +by chemical doping or electrical gating, SnS-based polaritonic devices could play an important role in future +planar optics43 in the technologically important near-IR region. + + +Supporting Information +Nano-optics setup; Determination of the crystal axes of SnS; Interference mechanisms; COMSOL +simulations; Dispersion calculations; Low-temperature dispersion of EPs in SnS; Effect of the SnS2 shell +on waveguide EPs; Photonic mode at the air/mica interface; Extraction of the propagation lengths of EPs; +Coupling strength of EPs + +Corresponding Authors +Peter Sutter, Email: psutter@unl.edu, +Zhe Fei, Email: zfei@iastate.edu. + +Notes +The authors declare no competing interests. + +Acknowledgments + +This work is supported by the National Science Foundation under Grant No. DMR-1945560. The +nano-optics setup used in the work is supported in part by Ames Laboratory. Ames Laboratory is operated +for the U.S. Department of Energy by Iowa State University under Grant No. DE-AC02-07CH11358. +Materials synthesis, electron microscopy, and complementary cathodoluminescence spectroscopy by E.S. +and P.S. were supported by the National Science Foundation, Division of Materials Research, Solid State +and Materials Chemistry Program under Grant No. DMR-1904843. H.Z. and X.W. are grateful for the +support from National Science Foundation under Grant No. CBET-1930866 and CMMI-203246. + +References +(1) Ma, W.; Shabbir, B.; Ou, Q.; Dong, Y.; Chen, H.; Li, P.; Zhang, X.; Lu, Y.; Bao, Q. Anisotropic +polaritons in van der Waals materials. InfoMat. 2020, 2, 777-790. +(2) Wang, C.; Zhang, G.; Huang, S.; Xie, Y.; Yan, H. The Optical Properties and Plasmonics of Anisotropic +2D Materials. Adv. Optical Mater. 2020, 8, 1900996. +(3) Li, P.; Dolado, I.; Alfaro-Mozaz, F. J.; Casanova, F.; Hueso, L. E.; Liu, S.; Edgar, J. H.; Nikitin, A. Y.; +Vélez, S.; Hillenbrand, R. 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S.; Liu, M.; Gannett, W.; Regan, +W.; Watanabe, K.; Taniguchi, T.; Thiemens, M.; Dominguez, G.; Castro Neto, A. H.; Zettl, A.; +Keilmann, F.; Jarillo-Herrero, P.; Fogler, M. M.; Basov, D. N. Tunable phonon polaritons in atomically +thin van der Waals crystals of boron nitride. Science 2014, 343, 1125-1129. +(42) +Hunsperger, R. G. Integrated Optics. 6th ed.; Springer, New York, NY, 2009. +(43) +Genevet, P., Capasso, F., Aieta, F., Khorasaninejad & Devlin, R. Recent advances in planar optics: +from plasmonic to dielectric metasurfaces. Optica 2017, 4, 139-152. + + + +Figure captions + + + +Figure 1. (a) Illustration of the experimental setup and the two beam paths (labeled as ‘P1’ and ‘P2’) +responsible for the formation of the observed interference oscillations. (b) Sketch the crystal structure of +SnS. The Sn and S atoms are in silver and yellow, respectively. (c)-(f) The AFM topography and the +simultaneously-taken nano-IR images of an SnS microcrystal (thickness = 100 nm) with in-plane +wavevector (k//) of the excitation laser along the b (c,d) and a (e,f) axes, respectively. The red arrows in (c) +and (e) mark the direction of k//. The white arrow in (d) marks the 1D oscillations discussed in the main +text. + + +Figure 2. (a)-(e) Energy-dependent imaging data of EPs in SnS with the in-plane laser wavevector kin along +the b axis. (f)-(j) Energy-dependent imaging data of EPs in SnS with kin along the a axis. Here the sample + +a +focusing +cantilever +c +AFM +d +s-SNOM, 1.38 eV +mirror +mica +Kin +Ra +tip +kin // b +sample +sample +EP +d (nm) +s (a.u.) +100 +13.0 +mica +2um +2μm +AFM +s-SNOM, 1.38 eV +b +e +f +mica +-10 +10.5 +Kin//a +sample +a +SnS crystal +2μum +2uma +ki. Il b, E=1.29 eV +b +C +k, II b, E=1.38 eV +d +kn // b, E=1.32 eV +k. I/ b, E=1.44 eV +e +k.// b, E=1.48 eV +s (a.u.) +2um +max +f +k.// a, E=1.29eV +g +k, // a, E=1.32 eV +h +kn // a, E=1.38 eV +kin // a, E=1.44 eV +1 +kn // a, E=1.48 eV +2um +k +[E], E=1.29 eV +[E,], E=1.32 eV +w +IEzl, E=1.38 eV +n +IE21, E=1.44 eV +[Ez, E=1.48 eV +[E2] +(a.u.) +4umis the 100-nm-thick SnS microcrystal shown in Figure 1. (k)-(o) Simulated polariton field amplitude (|Ez|) +maps of waveguide EPs in 100-nm-thick SnS at various energies. The EPs are excited by a point dipole (pz) +located at the center of the image right above the sample surface. + + +Figure 3. Real-space line profiles along the direction of the 1D interference oscillations in Figure 2 and +their Fourier-transformed (FT) profiles with the in-plane laser wavevector kin along both the b axis (a,b) +and the a axis (c,d), respectively. The unit for the horizontal axes of the FT profiles is k0 = 2π/λ0. + + +Figure 4. (a),(b) Dispersion diagrams of the EPs along the b and a axes, respectively. The colormaps plot +the imaginary part of the reflection coefficient Im(rp) that represents the photonic density of states. The blue +dashed curves mark the dispersion of the waveguide EPs. (c),(d) The propagation lengths of EPs along both +b axis and the a axis, respectively. The red curves are drawn to guide the eye. The data points in all panels +were obtained from s-SNOM imaging data (squares) and Comsol simulations (triangles). + + +a +profiles (kin ll b ) +b +FT (kin Il b ) +profiles (kin Il a ) +d +FT (kin ll α) +1.53eV +.53ev +1.50eV +1.50eV +44ev +'n: +'n' +.41ev +41eV +a +1.38eV +1.38eV +s +1.35eV +1.32eV +29eV +0 +2 +4 +1 +2 +3 +0 +2 +4 +1 +2 +3 +x (μm) +k (ko) +x (μm) +k (ko)a +kin Il b +b +kin Il a +c +kin // b +1.8 +1.8 +6 +口 +Data +Data +Simulation +5 +Simulation +1.7 +1.7 +(un) +4 +3 +Eb +2 +1.6 +1.6 +(eV) +(eV) +0 +山 +1.5 +W1.5 +d +kin /l α +口 +Data +△ +Simulation +1.4 +1.4 +4 +Im(rp) +xew +3 +1.3 +1.3 +2 +AH +0 +1.2 +1.2 +0 +1.5 +2.5 +1.3 +1.4 +2 +2.5 +3 +1.5 +2 +3 +1.5 +k (ko) +k (ko) +E(eV) +Figure 5. (a) Nano-optical images of SnS microcrystals with various thicknesses. Here the excitation +energy E = 1.38 eV and the in-plane wavevector is along the b axis. (b) Line profiles taken perpendicular +to the interference oscillations in (a). (c) Extracted polariton wavelength p versus the thickness of SnS +crystals. + +Supporting Information for + +Imaging anisotropic waveguide exciton polaritons in tin sulfide + +Yilong Luan1,2, Hamidreza Zobeiri3, Xinwei Wang3, Eli Sutter4,5, Peter Sutter6*, Zhe Fei1,2* + +1Department of Physics and Astronomy, Iowa State University, Ames, Iowa 50011, USA +2Ames Laboratory, U. S. Department of Energy, Iowa State University, Ames, Iowa 50011, USA +3Department of Mechanical Engineering, Iowa State University, Ames, IA 50011, USA +4Department of Mechanical and Materials Engineering, University of Nebraska-Lincoln, Lincoln, NE +68588, USA +5Nebraska Center for Materials and Nanoscience, University of Nebraska-Lincoln, Lincoln, NE 68588, +USA. +6Department of Electrical and Computer Engineering, University of Nebraska-Lincoln, Lincoln, NE 68588, +USA + +*Corresponding to: (P.S.) psutter@unl.edu, (Z.F.) zfei@iastate.edu. + + + +List of contents: + +1. Nano-optics setup + +2. Determination of the crystal axes of SnS + +3. Interference mechanisms + +4. COMSOL simulations + +5. Dispersion calculations + +6. Low-temperature dispersion of EPs in SnS + + +a +d = 160 nm +b +c +700 +口 +Data +Simulation +d = 140 nm +600 +Theory +s (a.u.) +max +台贝 +d = 120 nm +500 +('n'e) += 120 nm +(nm) +d = 100 nm +400 +d = 100 nm +0 +d = 91 nm +300 +d = 91 nm +E = 1.38 eV +1 +1 +1 +200 +a +2 μm +0 +1 +2 +3 +4 +5 +80 +100 +120 +140 +160 +180 +kin // B +x (μum) +d (nm)7. Effect of the SnS2 shell on waveguide EPs + +8. Photonic mode at the air/mica interface + +9. Extraction of the propagation lengths of EPs + +10. Coupling strength of EPs + + +References for the Supporting Information + +Supporting Figures: Figures S1-S13 + + + + + +1. Nano-optics setup +To image the propagative waveguide EPs in SnS, we applied the s-SNOM from Neaspec GmbH +The s-SNOM was built based on a tapping-mode Atomic Force Microscope (AFM). The AFM tips used in +the study were Pt/Ir-coated silicon tips (Arrow NCPT from Nanoandmore GmbH) with a tapping frequency +of ~270 kHz. The tapping amplitude of the tip was set to be about 50 nm. For optical excitations, we used +a Ti:sapphire laser (Spectra-Physics, Tsunami) operating at the continuous-wave mode with a photon +energy tunable from 1.3 to 1.8 eV that covers the exciton and bandgap energies of SnS. The main observable +of the s-SNOM is the complex near-field scattering signal that is modulated due to the tip tapping. +Demodulating the signal at the nth harmonics (n ≥ 2) of the tapping frequency can effectively suppress the +background signal (n = 2 in the current work). In addition, the pseudo-heterodyne interferometric detection +method is used to extract both the amplitude (s) and phase () of the near-field scattering signal. In the +current work, we mainly discuss the amplitude part of the signal that is sufficient for describing propagative +EPs. All s-SNOM measurements were performed at ambient conditions. + +2. Determination of the crystal axes of SnS +As introduced in the main text, our samples are SnS microcrystals coated with a thin shell of SnS2. +In Figure S1a, we show an optical photo of the sample, where tens of microcrystals are sitting on the mica +substrate. Prior to the s-SNOM imaging measurements, we first determined the in-plane crystal axes (i.e., +a and b axes) of SnS. The most convenient way to determine the crystal orientation is by inspecting the +crystal shape.1,2 As shown in Figure S1b, the crystal is slightly elongated along the b axis. As a result, the +crystal corner angles are 85º and 95º, respectively. Therefore, by measuring the corner angle, we can +determine the crystal axes. Alternatively, we also applied polarization-dependent Raman spectroscopy to +confirm the crystal axes. Here the incident laser beam is polarization-controlled, and the detector collects +photons from all polarizations. Due to the strong anisotropy of SnS, Raman spectra show sensitive +dependence on the polarization direction of the incident laser.1 In Figure S1c, we plot the Raman spectra of +a SnS microcrystal with laser polarization along the a and b axes. When polarization is parallel with the a +axis (the black curve in Figure S1c), there are two outstanding peaks at 159 cm-1 (B3g) and 188 cm-1 (Ag), +respectively. When polarization parallel with b axis (the red curve in Figure S1c), one more prominent peak +at around 93 cm-1 (Ag) emerges in addition to the two peaks at 159 cm-1 and 188 cm-1. The Raman peak at +93 cm-1 was used to distinguish the a and b axes of SnS.1 + +3. Interference mechanisms + +As discussed in the main text, the real-space fringes or oscillations of EPs observed in our s-SNOM +imaging data were formed due to the interference between tip-back-scattered photons and edge-scattered + +EPs (termed as “Mechanism I”). In the main text, we focus on the discussions of the string of 1D oscillations +at the center of the SnS microcrystal. The optical paths responsible for the formation of these interference +oscillations are sketched in Figure 1a of the main text and Figure S2a. In this case, the incident laser beam +is perpendicular to the short edge (labeled as edge I in Figure S2a) and has an incident angle of  ≈ 30º +relative to the sample plane. Upon tip illumination, part of the laser beam is scattered directly by the tip to +the detector (labeled as path P1). The tip also excites in-plane EPs and propagate perpendicular to the short +edge along the a axis (e.g., edge I in Figure S2a). When reaching edge I, the EPs are scattered to be photons +(labeled as path P2) that are collected by the detector. The photons collected from the two paths interfere +with each other and thus generating the 1D interference oscillations at the crystal center. In addition to these +1D oscillations, there are also interference fringes parallel to the relatively long edges (e.g., edge II in Figure +S2b) that are about 43º relative to the b axis. The general mechanism for the fringe formation is similar to +those of the 1D oscillations. The main difference is that the EPs responsible for the fringe formation parallel +to edge II are propagating along a direction off the two principal crystal axes (i.e., a and b axes). With +careful boundary-condition analysis, we found that the propagation direction of the EPs has an angle of  +≈ 20º relative to the b axis to form interference fringes parallel to edge II (see Figure S2b). + +In addition to “mechanism I” discussed above and in the main text, there are also other possible +interference mechanisms. One possible mechanism (termed as “mechanism II”) is related to edge excitation +of EPs followed by tip scattering, and the other involves tip excitation of EPs followed by edge reflection +and tip scattering (termed as “mechanism III”). As discussed below, both mechanisms II and III are not +responsible for the interference oscillations/fringes in the current work. + +Mechanism II (edge excitation → tip scattering) requires that the focused laser spot (radius ~ 1 m) +is at the sample edge, which is only possible when the tip is very close to the sample edge because the laser +is always focused on the tip apex. Therefore, for interference fringes or oscillations 1 m away from the +sample edge, edge excitation has little contributions. In addition, edge excitation followed by tip scattering +is exactly the reverse process of tip excitation followed by edge scattering, so the distance of the optical +paths and hence the interference fringes are expected to be the same in the two cases. Finally, the s-SNOM +tip is in principle more efficient in polariton excitation than the sample edge due to its metallicity. +Considering the above three factors, we believe Mechanism II is not responsible for the interference fringes +or oscillations observed in our work. + +Mechanism III (tip excitation → edge reflection → tip scattering) plays important role when +polaritons or plasmons are highly confined (confinement factor kp/k0 >> 1), which is typical for plasmons +and polaritons in the mid-infrared region (e.g., graphene plasmons).3,4 Here, highly confined polaritons or +plasmons are efficiently reflected at the sample edge due to the large impedance mismatch. In the case of +polaritons or plasmons in the near infrared or visible range (e.g., metal plasmons or exciton polaritons), the +confinement is weaker. As a result, only a small portion of polaritons or plasmons can be reflected. Take +EPs of SnS for example, the confinement factor kp/k0 is in the range of 1.6-2.5 (see Figure 4a in the main +text), therefore the polariton reflectance at the sample edge R ≈ |(kp – k0)/( kp + k0)|2 is in the range of 5% to +18%. Moreover, additional geometric and intrinsic damping during the round-trip propagation (from the tip +to the edge and back to the tip) further weakens the reflected EPs. Therefore, Mechanism III also does not +play an important role in the fringes/oscillations observed in SnS. + +4. COMSOL simulations +To support the experimental study, we performed finite-element simulations of waveguide EPs in +SnS with COMSOL Multiphysics. To excite EPs, we placed a z-polarized point dipole (pz) right above the +sample surface. We used two types of models to simulate the SnS microcrystal sample. The first one is a +realistic model, where the sample was set to be a four-layer heterostructure (SnS2/SnS/SnS2/mica). Due to +the ultra-thin SnS2 shell layer (thickness ~ 3 nm), the realistic model requires ultra-fine messing, so it is +time-consuming and not suitable for the simulations of large samples. The second one is an effective model, +where the SnS layer is set to be in a homogeneous dielectric environment. The effective permittivity (eff) +of the homogeneous dielectric environment can be considered as an average value of air, SnS2, and mica. + +We treated eff as a fitting parameter that was determined by comparing the two models. As shown in Figure +S3, the simulated polariton field (out-of-plane Ez field) maps of the two models are consistent with each +other when using eff = 2.2 for the 100-nm-thick microcrystal sample. The consistency is also checked at all +other excitation energies. Due to the simplicity of the effective model, the simulations are much more +efficient. Moreover, we can simulate very large samples with a size of tens of microns, which is necessary +for the extraction of the propagation lengths of EPs. The main simulation results in this work were produced +with the effective model. The permittivity of SnS along the a and b axes (plotted in Figure S4) used in the +Comsol simulations and dispersion calculations is from previous literature.5,6 The c-axis permittivity of SnS +is adopted from Ref. 7. In Figure S4, we also mark the optical bandgap or exciton energies (blue arrows), +which are 1.66 eV and 1.39 eV along the b and a axes, respectively. The permittivity of SnS2 is set to be +about 10 in the ab plane and 6 along the c axis in our spectral range.8,9 The permittivity of mica is set to be +2.5 according to Ref. 10. +In Figure 2k-o and Figure S5a-e, we plot respectively the polariton field amplitude (|Ez|) and +polariton field (Ez) maps at various excitation energies. Based on these field maps, we were able to +determine the polariton wavelength and propagation length, which match well the experimental results (see +Figure 4, Figure S6 and Figure S7). To better visualize the anisotropy of the EP modes, we performed 2D +Fourier transform of the Ez field maps in Figure S5a-e to generate the isofrequency contours. The results +are shown in Figure S5f-j, where one can see that the EP mode has a clear elliptic shape at E = 1.29 eV and +E = 1.32 eV. As E increases to 1.38 eV and above, the top part of the ellipses (i.e., corresponding to the +mode along the a axis) becomes strongly weakened due to the high damping, so EPs prefer propagating +along the b axis. + +5. Dispersion calculations +In Figure 4 of the main text, we plot the dispersion diagrams of SnS along both the a and b axes, +where the data points obtained from s-SNOM experiments and Comsol simulations are overlaid on the +theoretical dispersion colormaps. In the energy-momentum dispersion colormaps, we plot the imaginary +part of the p-polarization reflection coefficient Im(rp), which represents the photonic density of states. The +bright curves shown in the dispersion colormaps correspond to transverse magnetic (TM) waveguide +modes.11 The transverse-electric waveguide modes, on the other hand, can be revealed when plotting the +imaginary part of the s-polarization reflection coefficient Im(rs).12 In the transfer-matrix calculations, we +considered the entire SnS2/SnS/SnS2/Mica heterostructure. Take the microcrystal sample in Figs. 1 and 2 +in the main text for example, the crystal thickness is ~100 nm. Considering a 3-nm-thick SnS2 shell at the +top and bottom,1 the SnS core has a thickness of ~94 nm. + +6. Low-temperature dispersion of EPs in SnS + +To explore theoretically the low-temperature behavior of waveguide EPs in SnS, we plot in Figure +S8a,b the calculated polariton dispersion of the 100-nm-thick SnS microcrystal at T = 27 K. The low- +temperature permittivity of SnS is from Refs. 5 and 6. For comparison, we plot in Figure S8c,d the +dispersion diagrams of EPs at T = 300 K (replotted from Figure 4 in the main text). Compared to room- +temperature dispersion color plots, the waveguide polariton mode at T = 27 K (Figure S8) is sharper due to +the smaller damping at the lower temperature. Besides, the exciton energies slightly increase at the lower +temperature. Similar temperature dependence of exciton energies has also been seen in other van der Waals +semiconductors.13,14 Furthermore, the light-exciton coupling is much stronger at T = 27 K with more +prominent mode bending features close to the exciton energies. + +7. Effect of the SnS2 shell on waveguide EPs + +As discussed in the main text, the SnS microcrystals studied in this work are coated with a thin +SnS2 shell that has a thickness of ~ 3 nm at the top and bottom surfaces.1 Here we wish to evaluate the +effect of the thin SnS2 shell on the waveguide EPs. In Figure S9, we plot the calculated dispersion diagrams +of waveguide EPs along the two principal axes of SnS with (Figure S9a,c) and without (Figure S9b,d) +consideration of the SnS2 shell. From Figure S9, one can see that the SnS2 shell has a very limited effect on + +the EPs. Close examination indicates that the SnS2 shell only induces a slight (~3-4%) decrease of polariton +wavelengths of EPs propagating along both the a and b axes. The polaritonic anisotropy along the a and b +axes, on the other hand, is solely due to the SnS core. + +8. Photonic mode at the air/mica interface + +The fringes are also seen on the mica substrate (Figures 1,2 and Figure S10a), which are generated +due to the interference of photonic modes propagating at or close to the surface of the mica substrate. The +interference mechanism for the substrate fringes is sketched in Figure S10b. To verify that, we performed +a dispersion analysis of the substrate mode. In Figure S10c, we show the excitation-energy-dependent fringe +profiles extracted along the blue dashed line in Figure S10a. With Fourier transform (Figure S10d), we +were able to determine the fringe period, which can be converted directly into the mode wavevector of the +substrate ks using the following equation: + +0/ ≡ ks/k0 ≈ 0/s + cos ≡ ks/k0 + cos . (S1) + +Note the difference between Eq. S1 with Eq. 1 in the manuscript (‘+’ sign instead of ‘-’ sign). Following +the Fourier transform, we obtained the dispersion data points, which match well the theoretical dispersion +colormap (Figure S10e). From the dispersion diagram, one can see that the wavevector of the substrate +mode is roughly proportional to the free-space photon wavevector k0 indicating their photonic nature (note +that the k axis in the dispersion diagram is normalized to k0). The mode wavevector is between k0 to nk0, +where n ≈ 1.6 is the refractive index of mica. Therefore, we believe the substrate mode measured here +corresponds to in-plane photons at the air/mica interface. + +9. Extraction of the propagation lengths of EPs + +In this section, we describe the extraction processes of the propagation lengths Lep ≡ 1/(2q2), where +q2 is the imaginary component of the polariton wavevector q = q1 + iq2. We extracted Lep from both the +experimental data and COMSOL simulations. The experimental Lep was extracted from the polariton fringe +profiles shown in Figure 3a,c in the main text. We first subtracted the baseline signal of the sample to obtain +the pure EP fringe oscillations. The baseline signal comes mainly from the background signal of the sample +without the generation of the propagative EPs. Detailed introductions about baseline subtraction can be +found in Ref. 12. The baseline-corrected fringe profiles are plotted as black curves in Figs. S6a and S7a, +which show clear decay with distance. We then performed envelop fitting of the profiles with a radial +exponential decay function x-1/2exp(-x/2Lep), which is expected for radially propagating 2D waves. Note +that not all experimental profiles can be fitted due to the high damping. As shown in Figs. S6a, the profile +at E = 1.54 eV cannot be fitted for kin // b. In the case of kin // a (Figure S7a), the profiles at E ≥ 1.38 eV +cannot be fitted. Similar fitting procedures were also applied to extract Lep from the simulated EP +oscillations (see Figs. S6b and S7b). + +10. Coupling strength of EPs + +In this section, we estimate the coupling strength of EPs propagating along the b axis of SnS. The +criterion for strong coupling is that the Rabi splitting energy R ≈ 2hg is larger than the average EP +linewidth (ex + ph)/2, where hg is the coupling energy, ex and ph are the linewidths (full width at half +maximum) of exciton and waveguide photon mode.15 To determine the Rabi splitting energy, we fit the +dispersion relationship of the EP mode along the b axis of SnS using the equation below:15,16 +2 +2 +1 +( +) +4( +) +2 +2 +ph +ex +ph +ex +E +E +E +E +E +hg + ++ + + +− ++ +. (S2) +Note that the fitting is mainly based on the bottom-branch of the EP mode that was verified experimentally. +Similar approach has been adopted in Ref. 16. The fitting result is shown in Figure S11, where the fitting +curves with Eq. S2 match well the dispersion relation of EPs revealed by the colormap. Note that the +dispersion colormap is a replot of Figure 4a without normalization of the k axis to k0. Through the fit, we + +obtain the Rabi splitting energy to be about 160 meV, which is comparable to or even bigger than those +reported in other materials. The exciton linewidth of SnS along the b axis is about 140 meV at room +temperature by fitting the dielectric function from previous literature (see Figure S12a,b).5 The linewidth +of the bare waveguide photon mode is estimated to be 70 meV at the exciton energy (see Figure S12c), so +the average polariton linewidth is about 105 meV, which is smaller than the Rabi splitting energy. Therefore, +we conclude that the EPs of SnS along the b axis is in the strong coupling regime. + +References for the Supporting Information +(44) +Sutter, P.; Wang, J.; Sutter, E. Wrap-Around Core–Shell Heterostructures of Layered Crystals. Adv. +Mater. 2019, 31, 1902166. +(45) +Lin. S.; Carvalho, C.; Yan, S.; Li, R.; Kim, S.; Rodin, A.; Carvalho, L.; Chan, E. M.; Wang, X.; +Castro Neto, A. H.; Yao, J. Accessing valley degree of freedom in bulk Tin(II) sulfide at room +temperature. Nat. Commun. 2018, 9, 1455. +(46) +Fei, Z.; Rodin, A. S.; Andreev, G. O.; Bao, W.; McLeod, A. S.; Wagner, M.; Zhang, L. M.; Zhao, +Z.; Thiemens, M.; Dominguez, G.; Fogler, M. M.; Castro Neto, A. H.; Lau, C. N.; Keilmann, F.; Basov, +D. N. Gate-tuning of graphene plasmons revealed by infrared nano-imaging. Nature 2012, 487, 82−85. +(47) +Chen, J.; Badioli, M.; González, P. A.; Thongrattanasiri, S.; Huth, F.; Osmond, J.; Spasenović, M.; +Centeno, A.; Pesquera, A.; Godignon, P.; Elorza, A. Z.; Camara, N.; García de Abajo, F. J.; Hillenbrand, +R.; Koppens, F. H. L. Optical nano-imaging of gate-tunable graphene plasmons. Nature 2012, 487, +77−80. +(48) +Nguyen, H. T.; Le, Y. L.; Nguyen, T. M. H.; Kim, T. J.; Nguyen, X. A.; Kim, B.; Kim, K.; Lee, +W.; Cho, S.; Kim, Y. D. Temperature dependence of the dielectric function and critical points of α‑SnS +from 27 to 350 K. Sci. Rep. 2020, 10, 18396. +(49) +Le, V. L.; Cuong, D. D.; Nguyen, H. T.; Nguyen, X. A.; Kim, B.; Kim, K.; Lee, W.; Hong, S. C.; +Kim, T. J.; Kim, Y. D. Anisotropic behavior of excitons in single-crystal α-SnS. AIP Adv. 2020, 10, +105003. +(50) +Banai, R. E.; Burton, L. A.; Choi, S. G.; Hofherr, F.; Sorgenfrei, T.; Walsh, A.; To, B.; Gröll, A.; +Brownson, J. R. S. Ellipsometric characterization and density-functional theory analysis of anisotropic +optical properties of single-crystal α-SnS. J. Appl. Phys. 2014, 116, 013511. +(51) +Bertrand, Y.; Leveque, G.; Raisin, C.; Levy, F. Optical properties of SnSe2 and SnS2. J. Phys. C: +Solid State Phys. 1979, 12, 2907-2916. +(52) +Lucovsky, G. Mikkelsen, J. C. Jr. Liang, W. Y.; White, R. M.; Martin, R. M. Optical phonon +anisotropies in the layer crystals SnS2 and SnSe2. Phys. Rev. B 1976, 14, 1663-1669. +(53) +Nitsche, R. & Fritz, T. Precise determination of the complex optical constant of mica. Appl. Opt. +43, 3263-3270 (2004). +(54) +Hu, F.; Luan, Y.; Scott, M. E.; Yan, J.; Mandrus, D. G.; Xu, X.; Fei, Z. Imaging exciton-polariton +transport in MoSe2 waveguides, Nat. Photon. 2017, 11, 356-360. +(55) +Hu, F.; Luan, Y.; Speltz, J.; Zhong, D.; Liu, C. H.; Yan, J.; Mandrus, D. G.; Xu, X.; Fei, Z. "Imaging +propagative exciton polaritons in atomically-thin WSe2 waveguides", Phys. Rev. B 100, 121301(R) +(2019). +(56) +Liu, X.; Bao, W.; Li, Q.; Ropp, C.; Wang, Y.; Zhang, X. Control of Coherently Coupled Exciton +Polaritons in Monolayer Tungsten Disulphide. Phys. Rev. Lett. 2017, 119, 027403. +(57) +Christopher, J. W.; Goldberg, B. B.; Swan, A. K. Long tailed trions in monolayer MoS2: +Temperature dependent asymmetry and resulting red-shift of trion photoluminescence spectra. Sci. Rep. +2017, 7, 14062. +(58) +Hu, F.; Fei, Z.; Recent progress on exciton polaritons in layered transition‐metal dichalcogenides. +Adv. Opt. Mater. 2020, 8, 1901003. +(59) +Dovzhenko, D.; Lednev, M.; Mochalov, K.; Vaskan, I.; Samokhvalov, P.; Rakovich, Y.; Nabiev, I. +Strong excitonphoton coupling with colloidal quantum dots in a tunable microcavity. Appl. Phys. Lett. +2021, 119, 011102. + + + + + + +Supporting Figures + + +Figure S1. (a) A large-area optical photo of SnS microcrystals on a mica substrate. (b) The AFM image of +an SnS microcrystal. The marked angle of the crystal corner is 85º. (c) Polarization-dependent Raman +spectra of SnS along both the a and b axes. Detailed discussions are given in Section 2 of the Supporting +Information. + + +Figure S2. Illustrations of the formation mechanism of 1D interference oscillations at the crystal center (a) +and interference fringes parallel to the long edges (b). Detailed discussions are given in Section 3 of the +Supporting Information. + + +Figure S3. The simulated polariton field (Ez field) maps with the realistic model (a) and the effective model +(b). Detailed discussions are given in Section 4 of the Supporting Information. + + +a +b +C +50 +mica +Raman intensity (a.u.) +aaxis +baxis +40 +30 +SnS +850 +20 +10 +50μm +12μm +0 +50 +100 +150 +200 +250 +300 +Ramanshift(cm)a +b +Laser +Laser +P1 +tip +EPs +tp +EPs +edge I +edge II6 +Ez (a.u.) +800nm +Figure S4. The real part (red) and the imaginary part (black) of the permittivity along b axis (a) and a axis +(b). The blue arrow marks the optical bandgap or exciton energy. + + +Figure S5. (a-e) Simulated out-of-plane field (Ez) maps of waveguide EPs in 100-nm-thick SnS at various +energies. The EPs are excited by a point dipole (pz) located at the center of the image right above the sample +surface. (f-j) Energy-dependent isofrequency contour maps of waveguide EPs generated by the Fourier +transform of the Ez maps in (a-e). Detailed discussions are given in Section 4 of the Supporting Information. + + + + +a +b axis +25 +b +25 +aaxis +20 +20 +15 +15 +3 +3 +10 +10 +5 +5 +0 +0 +1.0 +1.2 +1.4 +1.6 +1.8 +2.0 +1.0 +1.2 +1.4 +1.6 +1.8 +2.0 +E (eV) +E (eV)a +Ez, E=1.29 eV +b +Ez,E=1.32eV +c +Ez, E=1.38 eV +p +Ez, E=1.44 eV +e +Ez, E=1.48 eV +Ez +(a.u.) +4μm +2DFFT,E=1.29eV +g +2DFFT.E=1.32eVh +2DFFT,E=1.38eV +- +2D FFT, E=1.44 eV +2D FFT, E=1.48 eV +3 +FT +ky (105 cm-1) +(a.u.) +max +0 +a +ta +ta +Aa +4a +min +3 +-3 +0 +3 +-3 +0 +3 +-3 +0 +3 +-3 +0 +3 +-3 +0 +3 +kx (105 cm-1) +kx (105 cm-1) +kx (105 cm-1) +kx (105 cm-1) +kx (105 cm-1) +Figure S6. Fitting of the propagation length (Lep) of EPs along the b-axis based on the experimental data +(a) and Comsol simulations (b). + + + + +Figure S7. Fitting of the propagation length (Lep) of EPs along the a-axis based on the experimental data +(a) and Comsol simulations (b). + + + +a +Experimental, kin // b +b +Simulation,ki // b +E=1.29eV,Lep=4.5um +E=1.29eV.Len=5um +E=1.32eV, Lep=4um +E=1.32eV.Len=4um +E=1.35eV,Lep=3.2um +E=1.35eV,Lep=3.5um +E=1.38eV,Lep=2.9um +E=1.38eV,Lep=3.2um +'n' +E=1.41eV,Lep=2.2um +a +S +E +E=1.44eV, Lep=2um +T Z1Zz E=1.44eV Lep=2.1um +E=1.48eV,Len=1.3um +E=1.48eV, Lep=1.5um +E=1.51eV,Lep=1.0um +AZ z= =151eV Lep=0.8um +E=1.54eV +E=1.54eV, Lep=0.4um +0 +2 +4 +0 +2 +4 +x (μm) +x (μm)a +Experimental, kin // a +b +Simulation, k// a +E=1.29eV,Lep=3.2um +E=1.29eV,Lep=3.5um +E=1.32eV,L +-ep=1.7um +=1.32eV,Lep=2.2um +E=1.35eV.L +:1.1um +E=1.35eV,Lep=1.4um +E=1.38eV +E=1.38eV.Len=1um +(a.u.) +E=1.41eV +_E=1.41eV,Lep=0.75um +S +EN +E=1.44eV +E=1.44eV, Lep=0.5um +E=1.48eV +E=1.48eV,Lep=0.45um +E=1.51eV +E=1.51eV,Lep=0.35um +E=1.54eV +E=1.54eV,Lep=0.2um +0 +2 +4 +0 +2 +4 +x (μm) +x (μm) +Figure S8. The dispersion colormaps of SnS along the two principal axes at T = 27 K (a,b) and T = 300 K +(c,d). Detailed discussions are given in Section 6 of the Supporting Information. + + +Figure S9. The calculated dispersion colormaps of waveguide EPs along the two principal axes of SnS +with (a,c) and without (b,d) consideration of the thin SnS2 shell. + + + + +a +kin // b, T = 27 K +b +kin /l a, T = 27 K +c +kin Il b, T = 300K +d +kin // a, T= 300K +1.8 +1.8 +1.8 +1.8 +1.7 +1.7 +1.7 +1.7 +Eb +Eb +1.6 +1.6 +1.6 +1.6 +E +E +E +E +1.4 +1.4 +1.4 +1.4 +Im(p) +max +Ea +Ea +1.3 +1.3 +1.3 +1.3 +0 +1.2 +1.2 +1.2 +1.2 +1.5 +2 +2.5 +3 +3.5 +1.5 +2 +2.5 +3 +3.5 +1.5 +2 +2.5 +3 +3.5 +1.5 +2 +2.5 +3 +3.5 +k (ko) +k (ko) +k (ko) +k (ko)a +kin /l b, with SnS +b +kin /l b, w/o SnS +c +kin /l a, withSnS +d +kin /l a, w/o SnS +1.8 +1.8 +1.8 +1.8 +1.7 +1.7 +1.7 +1.7 +Eb +Eb +1.6 +1.6 +1.6 +1.6 +E +E +E +E +1.4 +1.4 +1.4 +1.4 +Im(rp) +max +Ea +1.3 +1.3 +1.3 +1.3 +0 +1.2 +1.2 +1.2 +1.2 +1.5 +2 +2.5 +3 +3.5 +1.5 +2 +2.5 +3 +3.5 +1.5 +2 +2.5 +3 +3.5 +1.5 +2 +2.5 +3 +3.5 +k (ko) +k (ko) +k (ko) +k (ko) +Figure S10. (a) Nano-optical imaging data of the SnS crystal on mica substrate at E = 1.38 eV. (b) +Illustration of the interference mechanism for the formation of fringes on the substrate. (c) Fringe profiles +at various excitation energies taken along the blue dashed line shown in (a). (d) Fourier transform of the +energy-dependent fringe profiles shown in (c). (e) Experimental and theoretical dispersion diagram of the +substrate photon mode of mica. The two vertical dashed lines mark the photon dispersion in air (k = k0) and +in bulk mica (k=1.6k0). Detailed discussions are given in Section 8 of the Supporting Information. + + + +Figure S11. (a) Fitting the dispersion relation of the EP mode along the b axis of SnS. The white dotted +curves were produced with Eq. S2. (b) Calculated dispersion relation of the bare waveguide photon mode +without coupling with excitons along the b axis of SnS. Detailed discussions are given in Section 10 of the +Supporting Information. + + + + +(a) +s-SNOM, 1.38 eV +(c) +substrate mode profiles +(d) +Fouriertransform +(e) +substrate mode dispersion +1.8 +Ik=1.6ko +mica +1.53 eV +1.53eV +1.50 eV +1.50eV +1.7 +1.47 eV +1.47 eV +1.6 +1.44 eV +1.44 eV +(Cn +(a.u.) +(eV) +口 +200nm +1.41eV +1.41 eV +口 +1.5 +(b) +S +1.38eV +1.38eV +E +P2 +1.35 eV +1.35eV +1.4 +1.32 eV +口口口 +1.32 eV +substrate +1.29eV +1.3 +Sns +1.29 eV +photons +mica +1.2 +0.0 +0.5 +1.0 +1.5 +0 +1 +2 +3 +4 +5 +0.5 +1.5 +2.5 +3.5 +x (um) +k (10° cm) +k (ko)a +2 +b +2 +1.8 +1.8 +日1.4 +日1.4 +1.2 +1.2 +1 +0.5 +1 +1.5 +2 +2.5 +3 +0.5 +1 +1.5 +2 +2.5 +3 +k(105cm-1) +(105cm-1)a +b +200 +c +2.0 +0 +T = 27K +T=100K +exciton linewidth (meV) +T=200K +T=300K +150 +1.5 +(cn'e) ()ul +6 +100 +1.0 +50 +0.5 +2 +0 +0 +0.0 +1.4 +1.5 +1.6 +1.7 +1.8 +1.9 +0 +50 +100 +150 +200 +250 +300 +1.2 +1.4 +1.6 +1.8 +2.0 +E (eV) +T (K) +E (eV)Figure S12. (a) Fitting the exciton linewidth from the b-axis permittivity of SnS from Ref. 5. (b) +Temperature-dependent linewidth of excitons along the b axis of SnS based on the fitting in (a). (c) Line +profile of the photonic mode taken along the vertical dashed line in Figure S11b. Detailed discussions are +given in Section 10 of the Supporting Information. + + +Figure S13. Illustration of a selective nanophotonic interconnector based on anisotropic EPs in SnS. When +the incident photonic signal is at energies of E ≤ 1.29 eV, both devices 1 and 2 can receive the signal. When +the incident photonic signal is at energies of E ≥ 1.38 eV, only device 1 can receive the signal. + + + + + + + + + + + +to device +Incident signal +Sns +interconnector +s \ No newline at end of file diff --git a/09FIT4oBgHgl3EQf3Sv5/content/tmp_files/load_file.txt b/09FIT4oBgHgl3EQf3Sv5/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..29cd9a215d92f6387fe40a8d883efb5945a9f41b --- /dev/null +++ b/09FIT4oBgHgl3EQf3Sv5/content/tmp_files/load_file.txt @@ -0,0 +1,2116 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf,len=2115 +page_content='Imaging anisotropic waveguide exciton polaritons in tin sulfide Yilong Luan1,2, Hamidreza Zobeiri3, Xinwei Wang3, Eli Sutter4,5, Peter Sutter6*, Zhe Fei1,2* 1Department of Physics and Astronomy, Iowa State University, Ames, Iowa 50011, USA 2Ames Laboratory, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Department of Energy, Iowa State University, Ames, Iowa 50011, USA 3Department of Mechanical Engineering, Iowa State University, Ames, IA 50011, USA 4Department of Mechanical and Materials Engineering, University of Nebraska-Lincoln, Lincoln, NE 68588, USA 5Nebraska Center for Materials and Nanoscience, University of Nebraska-Lincoln, Lincoln, NE 68588, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' 6Department of Electrical and Computer Engineering, University of Nebraska-Lincoln, Lincoln, NE 68588, USA Corresponding to: (P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=') psutter@unl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='edu, (Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=') zfei@iastate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Abstract In recent years, novel materials supporting in-plane anisotropic polaritons have attracted a lot of research interest due to their capability of shaping nanoscale field distributions and controlling nanophotonic energy flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Here we report a nano-optical imaging study of waveguide exciton polaritons (EPs) in tin sulfide (SnS) in the near-infrared (IR) region using the scattering-type scanning near-field optical microscopy (s-SNOM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' With s-SNOM, we mapped in real space the propagative EPs in SnS, which show sensitive dependence on the excitation energy and sample thickness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Moreover, we found that both the polariton wavelength and propagation length are anisotropic in the sample plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' In particular, in a narrow spectral range from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='32 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='44 eV, the EPs demonstrate quasi-one-dimensional propagation, which is rarely seen in natural polaritonic materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Further analysis indicates that the observed polariton anisotropy is originated from the different optical bandgaps and exciton binding energies along the two principal crystal axes of SnS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Key Words: Tin sulfide, waveguide, exciton polaritons, s-SNOM, anisotropy, quasi-one-dimensional Main text In-plane anisotropic polaritons1,2 were first studied in metasurfaces3-5 where nanostructuring of the polaritonic media or substrates breaks the symmetry, thus enabling polaritonic anisotropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Later, several natural materials were predicted and/or experimentally confirmed to support in-plane anisotropic polaritons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='6-10 For example, anisotropic plasmon polaritons and hybrid plasmon-phonon polaritons were observed in black phosphorus carbides with far-field infrared (IR) spectroscopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='8 Anisotropic phonon polaritons with hyperbolic wavefronts were imaged in MoO3,9-11 which can be conveniently tailored by controlling the sample thickness and by stack- and twist-engineering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='12-18 Compared to nano-engineered anisotropic metasurfaces, natural materials with intrinsic anisotropic polaritons are generally more convenient for applications and can avoid potential material quality degradation due to complex nano- fabrications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Despite these advantages, natural materials supporting in-plane anisotropic polaritons are rare and are so far mainly studied in the mid-IR range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' New materials enabling anisotropic polaritons in other technologically important spectral regions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=', near-IR and visible) are desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' In this Letter, we report the experimental discovery of strongly-anisotropic exciton polaritons (EPs) in tin sulfide (SnS) in the technologically-important near-IR region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' SnS is a post-transition-metal monochalcogenide and a van der Waals (vdW) layered semiconductor with an orthorhombic structure, analogous to that of black phosphorous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='6-7 As sketched in Figure 1b, the two in-plane axes of SnS, namely the a and b axes, are along the zigzag and armchair directions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' SnS has been widely studied due to its unique anisotropic optoelectronic properties19-23 and potential applications related to photodetection and solar energy harvesting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='24-27 In particular, the energies of excitons or optical bandgaps along the a and b axes of SnS are about Ea ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='39 eV and Eb ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='66 eV respectively,22,23 which directly impact the polaritonic responses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Note that EPs have previously been studied in other vdW semiconductors (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' WSe2, MoSe2, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=') with imaging28-32 and spectroscopic methods,33-37 where the EPs are isotropic in the sample plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' The samples studied here are SnS microcrystals supported on mica wafers (Figure S1a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' As introduced in detail in the earlier work,19 these microcrystals have a wrap-around layered core-shell structure: the thick SnS core is coated with a thin crystalline shell (thickness ≈ 3 nm) of layered tin disulfide (SnS2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' A detailed characterization of the wrap-around core-shell structures and their synthesis process were reported in the earlier work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='19 Note that the thin SnS shell is isotropic in the sample plane38,39, so the observed anisotropic properties of EPs are solely due to the SnS core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' The SnS2 shell mainly serves as a protection layer of the SnS core and the waveguide EPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Detailed discussions about the effect of the SnS2 shell are given in the Supporting Information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' To excite and probe EPs in SnS, we employed a scattering-type scanning near-field optical microscope (s-SNOM) that was built based on an atomic force microscope (AFM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' As illustrated in Figure 1a, the sharp metalized tip in s-SNOM excited by a p-polarized laser beam generates strong evanescent fields underneath the tip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' These evanescent fields with a wide range of wavevectors40 can effectively excite transverse-magnetic (TM) polaritons inside the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='29 The excitation source used in the study is a broadband (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='24-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='77 eV) Ti:sapphire laser that covers the bandgap and exciton energies of SnS (see discussions below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' We used a parabolic mirror to focus the laser beam at the tip apex, and the scattered photons off the tip/sample system are collected by the same parabolic mirror and then counted by a photodetector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' More introductions about the nano-optical setup are in Supporting Information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' In Figure 1c, we plot the AFM topography image of a typical SnS microcrystal coated with a thin SnS2 shell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='19 The lateral sizes of the crystal are approximately 6-8 \uf06dm, and the thickness is about 100 nm including the SnS2 shell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Here the crystal has a total of eight edges, among which the four short edges are along the a or b axes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='19 We were able to determine the crystal axes of the sample by examining the shape of the crystal and by Raman spectroscopy (see Supporting Information).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Figure 1d plots the near-field amplitude (s) images taken simultaneously with the topography image (Figure 1c) at the excitation energy of E = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='38 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Here, the in-plane wavevector of the laser (kin) is along the b axis of SnS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' From Figure 1d, one can see many interference fringes and oscillations inside the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' We focus on a string of one- dimensional (1D) oscillations extending from the left edge to the crystal center along the b axis (marked with a white arrow).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' According to previous studies,29,30 these oscillations are generated due to the interference between two major beam paths as sketched in Figure 1a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' In the first path (P1), the excitation photons are scattered back directly by the tip apex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' In the second path (P2), the excitation photons are first transferred into waveguide EPs by the s-SNOM tip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' These EPs then propagate toward the sample edge and get scattered into photons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Photons collected through the two beam paths are coherent with each other, so they can generate interference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' When scanning the tip perpendicular to the sample edge, the distance between the tip and the sample edge varies, so a string of bright and dark spots forms due to constructive and destructive interferences, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' As sketched in Figure 1a, the left short edge of the crystal is responsible for the generation of the 1D interference oscillations along the direction of the white arrow in Figure 1d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Other edges can also scatter EPs into photons and generate interference patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' For example, the four long edges that are about 43º relative to the b axis are responsible for the bight fringes parallel to these edges (see Figure S2b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' There are other possible interference mechanisms (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=', edge excitation of polaritons), but they are not responsible for the fringes/oscillations observed in our samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Detailed discussions of different interference mechanisms are given in Section 3 of the Supporting Information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' From Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' 1d,f, we also seen fringes on the substrate side, which are generated due to the excitation and scattering of photons at the air/mica interface as confirmed by dispersion analysis (see Figure S10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Figure 1e,f plot the AFM and corresponding s-SNOM imaging data of the same crystal as those in Figure 1c,d but rotated 90º relative to the surface normal (c axis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Here the in-plane wavevector of the excitation laser is along the a axis (kin // a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Interestingly, we found no interference oscillations in the interior of the crystal as those seen in Figure 1d, indicating that no waveguide EPs are propagating along the a axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' To further explore the anisotropic polaritonic responses, we performed energy-dependent s-SNOM imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' The results are shown in Figure 2, where we plot the s-SNOM imaging data with kin along both the b axis (Figure 2a-e) and a axis (Figure 2f-j) at various excitation energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Again, we focus on the 1D oscillations at the crystal center (along the direction of the white arrows) that evolve systematically with E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' For kin // b, the oscillations are clearly seen for photon energies from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='29 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='48 eV, and their periods decrease with increasing energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' In the case of kin // a, the interference oscillations appear only at energies below 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='32 eV, and there are no clear 1D oscillations from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='38 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='48 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' The s-SNOM imaging data shown in Figures 1 and 2 provide direct evidence of in-plane anisotropic EPs of SnS in the near-IR region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' To support the experimental data, we performed finite-element simulations of the waveguide EPs using Comsol Multiphysics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' In the model, we placed a vertically polarized excitation dipole (pz) right above the sample surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' The optical constants of SnS and SnS2 were obtained from the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='22,23,38,39 A detailed description of the Comsol model is given in Supporting Information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' The simulation results are shown in Figure 2k-o and Figure S5, where we plot the real-space images of polariton field amplitude (|Ez|) and polariton field (Ez) of EPs respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Here the EPs are launched by a vertically polarized dipole (pz) located at the center of the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' At E = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='29 eV (Figure 2k), the dipole-launched anisotropic EPs propagate at all directions with elliptic wavefronts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' As E increases to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='32 eV (Figure 2l), the EPs show a faster decay along the a axis while keeping a relatively long propagation distance along the b axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' The propagation along the a axis is even shorter at higher energies E ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='38 eV (Figure 2m-o).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' As a result, the EPs appear to be quasi-1D along the b axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' The polaritonic simulations are consistent with s-SNOM imaging data in Figures 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' With the s-SNOM imaging data and Comsol simulation results, we were able to perform a quantitative analysis of the dispersion and propagation properties of the anisotropic EPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' In Figure 3a,c, we plot the line profiles extracted across the 1D interference oscillations in the energy-dependent s-SNOM images (Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' We then performed Fourier transforms (FTs) of these profiles to accurately obtain the periods (\uf072) of the interference oscillations that are linked to the polariton wavelength (\uf06cp) in the following relationship:29,30 \uf06c0/\uf072 ≡ k\uf072/k0 ≈ \uf06c0/\uf06cp – cos\uf061 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' (1) Here, λ0 is the excitation photon wavelength, k0 = 2π/λ0 is the free-space photon wavevector, kρ = 2π/ρ is the inverse period of the interference oscillations, and \uf061 ≈ 30º is the incidence angle of the laser beam relative to the sample plane (see Figure 1a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' The FT profiles for kin // b and kin // a are shown respectively in Figures 3b and 3d, where the peaks (marked with blue arrows) correspond to k\uf072.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' We then determined the polariton wavevector (kp = 2π/\uf06cp) using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' (1) for every given excitation energy, based on which we obtain the energy-momentum dispersion relations of the EPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' The experimental dispersion data points of EPs obtained through FT analysis (Figure 3b,d) are plotted in Figure 4a,b as black squares, which are sitting on the theoretical dispersion colormaps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' In the colormaps, we plot the imaginary part of the reflection coefficients Im(rp) that represents the photonic density of states (see Supporting Information).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Here the TM waveguide modes are visualized as bright curves (marked with blue dashed curves).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='29 In addition to the dispersion relations, the Im(rp) colormaps also reveal the mode broadening (\uf044k) that corresponds to the damping (see discussions in the following paragraph).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' This method of dispersion calculation has been widely used in the studies of polaritons in a variety of materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='29,30,40,41 In the dispersion diagrams, we also plot the dispersion data points extracted from Comsol simulations (Figure 2k-o and Figure S5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' The dispersion relations of the EPs from experimental data, Comsol simulations, and the Im(rp) colormaps are consistent with each other, which validates our experimental and theoretical approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' From the dispersion diagrams, we can examine the light-exciton interactions close to the exciton energies Ea and Eb (marked with white dashed lines in Figure 4a,b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' The waveguide mode along the b axis exhibits a clear back-bending behavior that is a signature behavior of light-exciton interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='28,29 By fitting the dispersion with the coupled oscillator model, we were able to determine the Rabi splitting energy (~160 meV), which is larger than the average polariton linewidth (~105 meV) (see Supporting Information).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Therefore, the EPs along the b axis are in the strong coupling regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' The mode coupling is much weaker along the a axis, likely due to the small exciton binding energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' According to literature,22 excitons along the b axis are robust with a binding energy of ~ 50 meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' The binding energy of excitons along the a axis, on the other hand, is much smaller and Ea is close to the fundamental bandgap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='22 The light-exciton coupling is much stronger at lower temperatures (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=', T = 27 K) with more prominent mode bending features close to the exciton energies (Figure S8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' In addition to the polariton dispersion, we also extracted the propagation lengths (Lep) of the EPs (Lep ≡ 1/[2Im(kep)]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' The extraction was done by fitting the decay trend of the polariton oscillations from both the s-SNOM data (Figure 2a-j) and Comsol simulations (Figure 2k-o and Figure S5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' A detailed description of the fitting procedures is given in the Supporting Information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' As shown in Figure 4c,d, Lep along both the a and b axes are over 3 \uf06dm at E = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='29 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' With increasing E, Lep drops systematically along both directions, but the drop along the a axis is much faster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' As E approaches Ea ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='39 eV, Lep along the a axis drops below 1 \uf06dm and becomes unmeasurable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Lep along the b axis, on the other hand, is as high as 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='5 \uf06dm at E = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='38 eV, where quasi-1D EPs were observed (Figure 1d,f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Lep drops to 1 \uf06dm or below along the b axis when E gets close to Eb ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='66 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' The energy dependence of Lep is fully consistent with the mode broadening behaviors shown in the theoretical dispersion colormaps in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' The larger the polariton width (\uf044k), the smaller the propagation length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Finally, we explored the dependence of EPs on the thicknesses of SnS crystals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' In Figure 5a, we plot the nano-optical images of SnS microcrystals with various thicknesses (d) taken at an excitation energy of E = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='38 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Due to the strong damping of EPs along the a axis, we only show in Figure 5 the data images with the excitation along the b axis (kin // b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' We focus on the 1D interference oscillations (Figures 1 and 2), which evolve systematically with varying thicknesses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Figure 5b plots the line profiles taken directly across the 1D oscillations in Figure 5a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Using Fourier transform and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' (1), we extracted the polariton wavelengths \uf06cp at different sample thicknesses, which are plotted in Figure 5c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Here one can see that \uf06cp decreases systematically with increasing d, which is expected since the crystal thickness determines both the out-of-plane (kz ~ 1/d) and in-plane wavevectors of the waveguide mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='42 For samples with thicknesses over 150 nm, \uf06cp drops below 300 nm that is 3 times smaller than the photon wavelength \uf06c0 = 900 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' The mode confinement is comparable to if not better than waveguide EPs in other materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='29-32 In summary, we have performed a comprehensive nano-optical study of SnS microcrystals using s-SNOM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' We found through near-field imaging that SnS supports waveguide EPs in near-IR, which are sensitively dependent on both the excitation energy and sample thickness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' More interestingly, both the dispersion and transport properties of the EPs are strongly anisotropic in the sample plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' In particular, in the energy range from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='35 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='55 eV, the EPs show quasi-1D propagation along the b axis, which has not been reported in other natural polaritonic materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Future studies with a pump-probe s-SNOM setup31,32 are expected for the exploration of the ultrafast dynamics of anisotropic EPs in SnS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' It is also interesting to study TE waveguide EPs of SnS that could be seen in atomically thin crystals, where active tunability of EPs is possible with electrical gating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' The anisotropic EPs discovered here in SnS are promising for a variety of applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' One potential application is low-pass waveguide filters for planar photonic circuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' The cut-off energies of the filters can be chosen by selecting the direction of signals propagating through the waveguide (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=', along a or b axes), Another possible application is selective nanophotonic interconnection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' A concept device is sketched in Figure S13, where the signal source is connecting two devices through an SnS interconnector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' The two ports for the two devices are along the a and b axes, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' When the incident photonic signal (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=', on or off signals) is at energies of E ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='29 eV, EPs can propagate along all directions in SnS, so both devices can receive the signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' When the incident photonic signal is at energies of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='38 eV ≤ E ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='48 eV, EPs propagate only along the b axis, so device 2 will not receive the signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Therefore, the signal interconnection can be controlled selectively by choosing different energies of the incident signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Such a directional control of the flow of nanophotonic energy and signals cannot be easily realized in isotropic polaritonic materials without complicated nano-fabrications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' With the potential controllability or tunability by chemical doping or electrical gating, SnS-based polaritonic devices could play an important role in future planar optics43 in the technologically important near-IR region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Supporting Information Nano-optics setup;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Determination of the crystal axes of SnS;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Interference mechanisms;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' COMSOL simulations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Dispersion calculations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Low-temperature dispersion of EPs in SnS;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Effect of the SnS2 shell on waveguide EPs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Photonic mode at the air/mica interface;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Extraction of the propagation lengths of EPs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Coupling strength of EPs Corresponding Authors Peter Sutter, Email: psutter@unl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='edu, Zhe Fei, Email: zfei@iastate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Notes The authors declare no competing interests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Acknowledgments This work is supported by the National Science Foundation under Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' DMR-1945560.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' The nano-optics setup used in the work is supported in part by Ames Laboratory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Ames Laboratory is operated for the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Department of Energy by Iowa State University under Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' DE-AC02-07CH11358.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Materials synthesis, electron microscopy, and complementary cathodoluminescence spectroscopy by E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' were supported by the National Science Foundation, Division of Materials Research, Solid State and Materials Chemistry Program under Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' DMR-1904843.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' are grateful for the support from National Science Foundation under Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' CBET-1930866 and CMMI-203246.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' References (1) Ma, W.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Zhao, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Dominguez, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Thiemens, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Fogler, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Tauber, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Castro-Neto, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Lau, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Keilmann, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Basov, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Infrared Nanoscopy of Dirac Plasmons at the Graphene-SiO2 interface Nano Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' 2011, 11, 4701-4705.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' (41) Dai, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Fei, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Ma, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Rodin, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Wagner, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' McLeod, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Liu, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Gannett, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Regan, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Watanabe, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Taniguchi, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Thiemens, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Dominguez, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Castro Neto, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' H.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Fogler, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Basov, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Tunable phonon polaritons in atomically thin van der Waals crystals of boron nitride.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Science 2014, 343, 1125-1129.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' (42) Hunsperger, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Integrated Optics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' 6th ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Springer, New York, NY, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' (43) Genevet, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=', Capasso, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=', Aieta, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=', Khorasaninejad & Devlin, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Recent advances in planar optics: from plasmonic to dielectric metasurfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Optica 2017, 4, 139-152.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Figure captions Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' (a) Illustration of the experimental setup and the two beam paths (labeled as ‘P1’ and ‘P2’) responsible for the formation of the observed interference oscillations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' (b) Sketch the crystal structure of SnS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' The Sn and S atoms are in silver and yellow, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' (c)-(f) The AFM topography and the simultaneously-taken nano-IR images of an SnS microcrystal (thickness = 100 nm) with in-plane wavevector (k//) of the excitation laser along the b (c,d) and a (e,f) axes, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' The red arrows in (c) and (e) mark the direction of k//.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' The white arrow in (d) marks the 1D oscillations discussed in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' (a)-(e) Energy-dependent imaging data of EPs in SnS with the in-plane laser wavevector kin along the b axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' (f)-(j) Energy-dependent imaging data of EPs in SnS with kin along the a axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Here the sample a focusing cantilever c AFM d s-SNOM, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='38 eV mirror mica Kin Ra tip kin // b sample sample EP d (nm) s (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=') 100 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='0 mica 2um 2μm AFM s-SNOM, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='38 eV b e f mica 10 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='5 Kin//a sample a SnS crystal 2μum 2uma ki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Il b, E=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='29 eV b C k, II b, E=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='38 eV d kn // b, E=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='32 eV k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' I/ b, E=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='44 eV e k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='// b, E=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='48 eV s (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=') 2um max f k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='// a, E=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='29eV g k, // a, E=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='32 eV h kn // a, E=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='38 eV kin // a, E=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='44 eV 1 kn // a, E=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='48 eV 2um k [E], E=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='29 eV [E,], E=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='32 eV w IEzl, E=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='38 eV n IE21, E=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='44 eV [Ez, E=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='48 eV [E2] (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=') 4umis the 100-nm-thick SnS microcrystal shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' (k)-(o) Simulated polariton field amplitude (|Ez|) maps of waveguide EPs in 100-nm-thick SnS at various energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' The EPs are excited by a point dipole (pz) located at the center of the image right above the sample surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Real-space line profiles along the direction of the 1D interference oscillations in Figure 2 and their Fourier-transformed (FT) profiles with the in-plane laser wavevector kin along both the b axis (a,b) and the a axis (c,d), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' The unit for the horizontal axes of the FT profiles is k0 = 2π/λ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' (a),(b) Dispersion diagrams of the EPs along the b and a axes, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' The colormaps plot the imaginary part of the reflection coefficient Im(rp) that represents the photonic density of states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' The blue dashed curves mark the dispersion of the waveguide EPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' (c),(d) The propagation lengths of EPs along both b axis and the a axis, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' The red curves are drawn to guide the eye.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' The data points in all panels were obtained from s-SNOM imaging data (squares) and Comsol simulations (triangles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' a profiles (kin ll b ) b FT (kin Il b ) profiles (kin Il a ) d FT (kin ll α) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='53eV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='53ev 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='50eV 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content="50eV 44ev 'n: 'n' ." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='41ev 41eV a 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='38eV 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='38eV s 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='35eV 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='32eV 29eV 0 2 4 1 2 3 0 2 4 1 2 3 x (μm) k (ko) x (μm) k (ko)a kin Il b b kin Il a c kin // b 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='8 6 口 Data Data Simulation 5 Simulation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='7 (un) 4 3 Eb 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='6 (eV) (eV) 0 山 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='5 W1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='5 d kin /l α 口 Data △ Simulation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='4 4 Im(rp) xew 3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='3 2 AH 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='2 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='4 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='5 3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='5 2 3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='5 k (ko) k (ko) E(eV) Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' (a) Nano-optical images of SnS microcrystals with various thicknesses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Here the excitation energy E = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='38 eV and the in-plane wavevector is along the b axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' (b) Line profiles taken perpendicular to the interference oscillations in (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' (c) Extracted polariton wavelength \uf06cp versus the thickness of SnS crystals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Supporting Information for Imaging anisotropic waveguide exciton polaritons in tin sulfide Yilong Luan1,2, Hamidreza Zobeiri3, Xinwei Wang3, Eli Sutter4,5, Peter Sutter6*, Zhe Fei1,2* 1Department of Physics and Astronomy, Iowa State University, Ames, Iowa 50011, USA 2Ames Laboratory, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Department of Energy, Iowa State University, Ames, Iowa 50011, USA 3Department of Mechanical Engineering, Iowa State University, Ames, IA 50011, USA 4Department of Mechanical and Materials Engineering, University of Nebraska-Lincoln, Lincoln, NE 68588, USA 5Nebraska Center for Materials and Nanoscience, University of Nebraska-Lincoln, Lincoln, NE 68588, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' 6Department of Electrical and Computer Engineering, University of Nebraska-Lincoln, Lincoln, NE 68588, USA Corresponding to: (P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=') psutter@unl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='edu, (Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=') zfei@iastate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' List of contents: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Nano-optics setup 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Determination of the crystal axes of SnS 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Interference mechanisms 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' COMSOL simulations 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Dispersion calculations 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Low-temperature dispersion of EPs in SnS a d = 160 nm b c 700 口 Data Simulation d = 140 nm 600 Theory s (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=") max 台贝 d = 120 nm 500 ('n'e) = 120 nm (nm) d = 100 nm 400 d = 100 nm 0 d = 91 nm 300 d = 91 nm E = 1." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='38 eV 1 1 1 200 a 2 μm 0 1 2 3 4 5 80 100 120 140 160 180 kin // B x (μum) d (nm)7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Effect of the SnS2 shell on waveguide EPs 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Photonic mode at the air/mica interface 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Extraction of the propagation lengths of EPs 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Coupling strength of EPs References for the Supporting Information Supporting Figures: Figures S1-S13 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Nano-optics setup To image the propagative waveguide EPs in SnS, we applied the s-SNOM from Neaspec GmbH The s-SNOM was built based on a tapping-mode Atomic Force Microscope (AFM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' The AFM tips used in the study were Pt/Ir-coated silicon tips (Arrow NCPT from Nanoandmore GmbH) with a tapping frequency of ~270 kHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' The tapping amplitude of the tip was set to be about 50 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' For optical excitations, we used a Ti:sapphire laser (Spectra-Physics, Tsunami) operating at the continuous-wave mode with a photon energy tunable from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='3 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='8 eV that covers the exciton and bandgap energies of SnS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' The main observable of the s-SNOM is the complex near-field scattering signal that is modulated due to the tip tapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Demodulating the signal at the nth harmonics (n ≥ 2) of the tapping frequency can effectively suppress the background signal (n = 2 in the current work).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' In addition, the pseudo-heterodyne interferometric detection method is used to extract both the amplitude (s) and phase (\uf079) of the near-field scattering signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' In the current work, we mainly discuss the amplitude part of the signal that is sufficient for describing propagative EPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' All s-SNOM measurements were performed at ambient conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Determination of the crystal axes of SnS As introduced in the main text, our samples are SnS microcrystals coated with a thin shell of SnS2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' In Figure S1a, we show an optical photo of the sample, where tens of microcrystals are sitting on the mica substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Prior to the s-SNOM imaging measurements, we first determined the in-plane crystal axes (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=', a and b axes) of SnS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' The most convenient way to determine the crystal orientation is by inspecting the crystal shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='1,2 As shown in Figure S1b, the crystal is slightly elongated along the b axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' As a result, the crystal corner angles are 85º and 95º, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Therefore, by measuring the corner angle, we can determine the crystal axes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Alternatively, we also applied polarization-dependent Raman spectroscopy to confirm the crystal axes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Here the incident laser beam is polarization-controlled, and the detector collects photons from all polarizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Due to the strong anisotropy of SnS, Raman spectra show sensitive dependence on the polarization direction of the incident laser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='1 In Figure S1c, we plot the Raman spectra of a SnS microcrystal with laser polarization along the a and b axes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' When polarization is parallel with the a axis (the black curve in Figure S1c), there are two outstanding peaks at 159 cm-1 (B3g) and 188 cm-1 (Ag), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' When polarization parallel with b axis (the red curve in Figure S1c), one more prominent peak at around 93 cm-1 (Ag) emerges in addition to the two peaks at 159 cm-1 and 188 cm-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' The Raman peak at 93 cm-1 was used to distinguish the a and b axes of SnS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Interference mechanisms As discussed in the main text, the real-space fringes or oscillations of EPs observed in our s-SNOM imaging data were formed due to the interference between tip-back-scattered photons and edge-scattered EPs (termed as “Mechanism I”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' In the main text, we focus on the discussions of the string of 1D oscillations at the center of the SnS microcrystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' The optical paths responsible for the formation of these interference oscillations are sketched in Figure 1a of the main text and Figure S2a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' In this case, the incident laser beam is perpendicular to the short edge (labeled as edge I in Figure S2a) and has an incident angle of \uf061 ≈ 30º relative to the sample plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Upon tip illumination, part of the laser beam is scattered directly by the tip to the detector (labeled as path P1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' The tip also excites in-plane EPs and propagate perpendicular to the short edge along the a axis (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=', edge I in Figure S2a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' When reaching edge I, the EPs are scattered to be photons (labeled as path P2) that are collected by the detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' The photons collected from the two paths interfere with each other and thus generating the 1D interference oscillations at the crystal center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' In addition to these 1D oscillations, there are also interference fringes parallel to the relatively long edges (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=', edge II in Figure S2b) that are about 43º relative to the b axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' The general mechanism for the fringe formation is similar to those of the 1D oscillations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' The main difference is that the EPs responsible for the fringe formation parallel to edge II are propagating along a direction off the two principal crystal axes (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=', a and b axes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' With careful boundary-condition analysis, we found that the propagation direction of the EPs has an angle of \uf066 ≈ 20º relative to the b axis to form interference fringes parallel to edge II (see Figure S2b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' In addition to “mechanism I” discussed above and in the main text, there are also other possible interference mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' One possible mechanism (termed as “mechanism II”) is related to edge excitation of EPs followed by tip scattering, and the other involves tip excitation of EPs followed by edge reflection and tip scattering (termed as “mechanism III”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' As discussed below, both mechanisms II and III are not responsible for the interference oscillations/fringes in the current work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Mechanism II (edge excitation → tip scattering) requires that the focused laser spot (radius ~ 1 \uf06dm) is at the sample edge, which is only possible when the tip is very close to the sample edge because the laser is always focused on the tip apex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Therefore, for interference fringes or oscillations 1 \uf06dm away from the sample edge, edge excitation has little contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' In addition, edge excitation followed by tip scattering is exactly the reverse process of tip excitation followed by edge scattering, so the distance of the optical paths and hence the interference fringes are expected to be the same in the two cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Finally, the s-SNOM tip is in principle more efficient in polariton excitation than the sample edge due to its metallicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Considering the above three factors, we believe Mechanism II is not responsible for the interference fringes or oscillations observed in our work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Mechanism III (tip excitation → edge reflection → tip scattering) plays important role when polaritons or plasmons are highly confined (confinement factor kp/k0 >> 1), which is typical for plasmons and polaritons in the mid-infrared region (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=', graphene plasmons).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='3,4 Here, highly confined polaritons or plasmons are efficiently reflected at the sample edge due to the large impedance mismatch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' In the case of polaritons or plasmons in the near infrared or visible range (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=', metal plasmons or exciton polaritons), the confinement is weaker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' As a result, only a small portion of polaritons or plasmons can be reflected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Take EPs of SnS for example, the confinement factor kp/k0 is in the range of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='6-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='5 (see Figure 4a in the main text), therefore the polariton reflectance at the sample edge R ≈ |(kp – k0)/( kp + k0)|2 is in the range of 5% to 18%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Moreover, additional geometric and intrinsic damping during the round-trip propagation (from the tip to the edge and back to the tip) further weakens the reflected EPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Therefore, Mechanism III also does not play an important role in the fringes/oscillations observed in SnS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' COMSOL simulations To support the experimental study, we performed finite-element simulations of waveguide EPs in SnS with COMSOL Multiphysics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' To excite EPs, we placed a z-polarized point dipole (pz) right above the sample surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' We used two types of models to simulate the SnS microcrystal sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' The first one is a realistic model, where the sample was set to be a four-layer heterostructure (SnS2/SnS/SnS2/mica).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Due to the ultra-thin SnS2 shell layer (thickness ~ 3 nm), the realistic model requires ultra-fine messing, so it is time-consuming and not suitable for the simulations of large samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' The second one is an effective model, where the SnS layer is set to be in a homogeneous dielectric environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' The effective permittivity (\uf065eff) of the homogeneous dielectric environment can be considered as an average value of air, SnS2, and mica.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' We treated \uf065eff as a fitting parameter that was determined by comparing the two models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' As shown in Figure S3, the simulated polariton field (out-of-plane Ez field) maps of the two models are consistent with each other when using \uf065eff = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='2 for the 100-nm-thick microcrystal sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' The consistency is also checked at all other excitation energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Due to the simplicity of the effective model, the simulations are much more efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Moreover, we can simulate very large samples with a size of tens of microns, which is necessary for the extraction of the propagation lengths of EPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' The main simulation results in this work were produced with the effective model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' The permittivity of SnS along the a and b axes (plotted in Figure S4) used in the Comsol simulations and dispersion calculations is from previous literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='5,6 The c-axis permittivity of SnS is adopted from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' In Figure S4, we also mark the optical bandgap or exciton energies (blue arrows), which are 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='66 eV and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='39 eV along the b and a axes, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' The permittivity of SnS2 is set to be about 10 in the ab plane and 6 along the c axis in our spectral range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='8,9 The permittivity of mica is set to be 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='5 according to Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' In Figure 2k-o and Figure S5a-e, we plot respectively the polariton field amplitude (|Ez|) and polariton field (Ez) maps at various excitation energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Based on these field maps, we were able to determine the polariton wavelength and propagation length, which match well the experimental results (see Figure 4, Figure S6 and Figure S7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' To better visualize the anisotropy of the EP modes, we performed 2D Fourier transform of the Ez field maps in Figure S5a-e to generate the isofrequency contours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' The results are shown in Figure S5f-j, where one can see that the EP mode has a clear elliptic shape at E = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='29 eV and E = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='32 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' As E increases to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='38 eV and above, the top part of the ellipses (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=', corresponding to the mode along the a axis) becomes strongly weakened due to the high damping, so EPs prefer propagating along the b axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Dispersion calculations In Figure 4 of the main text, we plot the dispersion diagrams of SnS along both the a and b axes, where the data points obtained from s-SNOM experiments and Comsol simulations are overlaid on the theoretical dispersion colormaps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' In the energy-momentum dispersion colormaps, we plot the imaginary part of the p-polarization reflection coefficient Im(rp), which represents the photonic density of states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' The bright curves shown in the dispersion colormaps correspond to transverse magnetic (TM) waveguide modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='11 The transverse-electric waveguide modes, on the other hand, can be revealed when plotting the imaginary part of the s-polarization reflection coefficient Im(rs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='12 In the transfer-matrix calculations, we considered the entire SnS2/SnS/SnS2/Mica heterostructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Take the microcrystal sample in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' 1 and 2 in the main text for example, the crystal thickness is ~100 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Considering a 3-nm-thick SnS2 shell at the top and bottom,1 the SnS core has a thickness of ~94 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Low-temperature dispersion of EPs in SnS To explore theoretically the low-temperature behavior of waveguide EPs in SnS, we plot in Figure S8a,b the calculated polariton dispersion of the 100-nm-thick SnS microcrystal at T = 27 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' The low- temperature permittivity of SnS is from Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' 5 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' For comparison, we plot in Figure S8c,d the dispersion diagrams of EPs at T = 300 K (replotted from Figure 4 in the main text).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Compared to room- temperature dispersion color plots, the waveguide polariton mode at T = 27 K (Figure S8) is sharper due to the smaller damping at the lower temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Besides, the exciton energies slightly increase at the lower temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Similar temperature dependence of exciton energies has also been seen in other van der Waals semiconductors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='13,14 Furthermore, the light-exciton coupling is much stronger at T = 27 K with more prominent mode bending features close to the exciton energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Effect of the SnS2 shell on waveguide EPs As discussed in the main text, the SnS microcrystals studied in this work are coated with a thin SnS2 shell that has a thickness of ~ 3 nm at the top and bottom surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='1 Here we wish to evaluate the effect of the thin SnS2 shell on the waveguide EPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' In Figure S9, we plot the calculated dispersion diagrams of waveguide EPs along the two principal axes of SnS with (Figure S9a,c) and without (Figure S9b,d) consideration of the SnS2 shell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' From Figure S9, one can see that the SnS2 shell has a very limited effect on the EPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Close examination indicates that the SnS2 shell only induces a slight (~3-4%) decrease of polariton wavelengths of EPs propagating along both the a and b axes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' The polaritonic anisotropy along the a and b axes, on the other hand, is solely due to the SnS core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Photonic mode at the air/mica interface The fringes are also seen on the mica substrate (Figures 1,2 and Figure S10a), which are generated due to the interference of photonic modes propagating at or close to the surface of the mica substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' The interference mechanism for the substrate fringes is sketched in Figure S10b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' To verify that, we performed a dispersion analysis of the substrate mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' In Figure S10c, we show the excitation-energy-dependent fringe profiles extracted along the blue dashed line in Figure S10a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' With Fourier transform (Figure S10d), we were able to determine the fringe period, which can be converted directly into the mode wavevector of the substrate ks using the following equation: \uf06c0/\uf072 ≡ ks/k0 ≈ \uf06c0/\uf06cs + cos\uf061 ≡ ks/k0 + cos\uf061 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' (S1) Note the difference between Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' S1 with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' 1 in the manuscript (‘+’ sign instead of ‘-’ sign).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Following the Fourier transform, we obtained the dispersion data points, which match well the theoretical dispersion colormap (Figure S10e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' From the dispersion diagram, one can see that the wavevector of the substrate mode is roughly proportional to the free-space photon wavevector k0 indicating their photonic nature (note that the k axis in the dispersion diagram is normalized to k0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' The mode wavevector is between k0 to nk0, where n ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='6 is the refractive index of mica.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Therefore, we believe the substrate mode measured here corresponds to in-plane photons at the air/mica interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Extraction of the propagation lengths of EPs In this section, we describe the extraction processes of the propagation lengths Lep ≡ 1/(2q2), where q2 is the imaginary component of the polariton wavevector q = q1 + iq2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' We extracted Lep from both the experimental data and COMSOL simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' The experimental Lep was extracted from the polariton fringe profiles shown in Figure 3a,c in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' We first subtracted the baseline signal of the sample to obtain the pure EP fringe oscillations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' The baseline signal comes mainly from the background signal of the sample without the generation of the propagative EPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Detailed introductions about baseline subtraction can be found in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' The baseline-corrected fringe profiles are plotted as black curves in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' S6a and S7a, which show clear decay with distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' We then performed envelop fitting of the profiles with a radial exponential decay function x-1/2exp(-x/2Lep), which is expected for radially propagating 2D waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Note that not all experimental profiles can be fitted due to the high damping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' As shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' S6a, the profile at E = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='54 eV cannot be fitted for kin // b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' In the case of kin // a (Figure S7a), the profiles at E ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='38 eV cannot be fitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Similar fitting procedures were also applied to extract Lep from the simulated EP oscillations (see Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' S6b and S7b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Coupling strength of EPs In this section, we estimate the coupling strength of EPs propagating along the b axis of SnS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' The criterion for strong coupling is that the Rabi splitting energy \uf057R ≈ 2hg is larger than the average EP linewidth (\uf047ex + \uf047ph)/2, where hg is the coupling energy, \uf047ex and \uf047ph are the linewidths (full width at half maximum) of exciton and waveguide photon mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='15 To determine the Rabi splitting energy, we fit the dispersion relationship of the EP mode along the b axis of SnS using the equation below:15,16 2 2 1 ( ) 4( ) 2 2 ph ex ph ex E E E E E hg \uf0b1 + \uf0bb \uf0b1 − + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' (S2) Note that the fitting is mainly based on the bottom-branch of the EP mode that was verified experimentally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Similar approach has been adopted in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' The fitting result is shown in Figure S11, where the fitting curves with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' S2 match well the dispersion relation of EPs revealed by the colormap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Note that the dispersion colormap is a replot of Figure 4a without normalization of the k axis to k0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Through the fit, we obtain the Rabi splitting energy to be about 160 meV, which is comparable to or even bigger than those reported in other materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' The exciton linewidth of SnS along the b axis is about 140 meV at room temperature by fitting the dielectric function from previous literature (see Figure S12a,b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='5 The linewidth of the bare waveguide photon mode is estimated to be 70 meV at the exciton energy (see Figure S12c), so the average polariton linewidth is about 105 meV, which is smaller than the Rabi splitting energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Therefore, we conclude that the EPs of SnS along the b axis is in the strong coupling regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' References for the Supporting Information (44) Sutter, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Wang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Sutter, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Wrap-Around Core–Shell Heterostructures of Layered Crystals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' 2019, 31, 1902166.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' (45) Lin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Carvalho, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Yan, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Li, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Kim, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Rodin, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Carvalho, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=';' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Fei, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' "Imaging propagative exciton polaritons in atomically-thin WSe2 waveguides", Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' B 100, 121301(R) (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' (56) Liu, X.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Wang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Zhang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Control of Coherently Coupled Exciton Polaritons in Monolayer Tungsten Disulphide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' 2017, 119, 027403.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' (57) Christopher, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Goldberg, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Swan, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Long tailed trions in monolayer MoS2: Temperature dependent asymmetry and resulting red-shift of trion photoluminescence spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Rep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' 2017, 7, 14062.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' (58) Hu, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Fei, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Recent progress on exciton polaritons in layered transition‐metal dichalcogenides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' 2020, 8, 1901003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' (59) Dovzhenko, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Lednev, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Mochalov, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Vaskan, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Samokhvalov, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Rakovich, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Nabiev, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Strong excitonphoton coupling with colloidal quantum dots in a tunable microcavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' 2021, 119, 011102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Supporting Figures Figure S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' (a) A large-area optical photo of SnS microcrystals on a mica substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' (b) The AFM image of an SnS microcrystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' The marked angle of the crystal corner is 85º.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' (c) Polarization-dependent Raman spectra of SnS along both the a and b axes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Detailed discussions are given in Section 2 of the Supporting Information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Figure S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Illustrations of the formation mechanism of 1D interference oscillations at the crystal center (a) and interference fringes parallel to the long edges (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Detailed discussions are given in Section 3 of the Supporting Information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Figure S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' The simulated polariton field (Ez field) maps with the realistic model (a) and the effective model (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Detailed discussions are given in Section 4 of the Supporting Information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' a b C 50 mica Raman intensity (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=') aaxis baxis 40 30 SnS 850 20 10 50μm 12μm 0 50 100 150 200 250 300 Ramanshift(cm)a b Laser Laser P1 tip EPs tp EPs edge I edge II6 Ez (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=') 800nm Figure S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' The real part (red) and the imaginary part (black) of the permittivity along b axis (a) and a axis (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' The blue arrow marks the optical bandgap or exciton energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Figure S5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' (a-e) Simulated out-of-plane field (Ez) maps of waveguide EPs in 100-nm-thick SnS at various energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' The EPs are excited by a point dipole (pz) located at the center of the image right above the sample surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' (f-j) Energy-dependent isofrequency contour maps of waveguide EPs generated by the Fourier transform of the Ez maps in (a-e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Detailed discussions are given in Section 4 of the Supporting Information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' a b axis 25 b 25 aaxis 20 20 15 15 3 3 10 10 5 5 0 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='0 E (eV) E (eV)a Ez, E=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='29 eV b Ez,E=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='32eV c Ez, E=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='38 eV p Ez, E=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='44 eV e Ez, E=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='48 eV Ez (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=') 4μm 2DFFT,E=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='29eV g 2DFFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='E=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='32eVh 2DFFT,E=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='38eV 2D FFT, E=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='44 eV 2D FFT, E=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='48 eV 3 FT ky (105 cm-1) (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=') max 0 a ta ta Aa 4a min 3 3 0 3 3 0 3 3 0 3 3 0 3 3 0 3 kx (105 cm-1) kx (105 cm-1) kx (105 cm-1) kx (105 cm-1) kx (105 cm-1) Figure S6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Fitting of the propagation length (Lep) of EPs along the b-axis based on the experimental data (a) and Comsol simulations (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Figure S7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Fitting of the propagation length (Lep) of EPs along the a-axis based on the experimental data (a) and Comsol simulations (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' a Experimental, kin // b b Simulation,ki // b E=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='29eV,Lep=4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='5um E=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='29eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='Len=5um E=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='32eV, Lep=4um E=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='32eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='Len=4um E=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='35eV,Lep=3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='2um E=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='35eV,Lep=3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='5um E=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='38eV,Lep=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='9um E=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='38eV,Lep=3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content="2um 'n' E=1." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='41eV,Lep=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='2um a S E E=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='44eV, Lep=2um T Z1Zz E=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='44eV Lep=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='1um E=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='48eV,Len=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='3um E=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='48eV, Lep=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='5um E=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='51eV,Lep=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='0um AZ z= =151eV Lep=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='8um E=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='54eV E=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='54eV, Lep=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='4um 0 2 4 0 2 4 x (μm) x (μm)a Experimental, kin // a b Simulation, k// a E=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='29eV,Lep=3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='2um E=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='29eV,Lep=3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='5um E=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='32eV,L ep=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='7um =1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='32eV,Lep=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='2um E=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='35eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='L :1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='1um E=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='35eV,Lep=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='4um E=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='38eV E=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='38eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='Len=1um (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=') E=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='41eV _E=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='41eV,Lep=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='75um S EN E=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='44eV E=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='44eV, Lep=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='5um E=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='48eV E=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='48eV,Lep=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='45um E=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='51eV E=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='51eV,Lep=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='35um E=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='54eV E=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='54eV,Lep=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='2um 0 2 4 0 2 4 x (μm) x (μm) Figure S8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' The dispersion colormaps of SnS along the two principal axes at T = 27 K (a,b) and T = 300 K (c,d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Detailed discussions are given in Section 6 of the Supporting Information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Figure S9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' The calculated dispersion colormaps of waveguide EPs along the two principal axes of SnS with (a,c) and without (b,d) consideration of the thin SnS2 shell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' a kin // b, T = 27 K b kin /l a, T = 27 K c kin Il b, T = 300K d kin // a, T= 300K 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='7 Eb Eb 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='6 E E E E 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='4 Im(p) max Ea Ea 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='3 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='5 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='5 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='5 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='5 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='5 k (ko) k (ko) k (ko) k (ko)a kin /l b, with SnS b kin /l b, w/o SnS c kin /l a, withSnS d kin /l a, w/o SnS 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='7 Eb Eb 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='6 E E E E 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='4 Im(rp) max Ea 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='3 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='5 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='5 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='5 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='5 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='5 k (ko) k (ko) k (ko) k (ko) Figure S10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' (a) Nano-optical imaging data of the SnS crystal on mica substrate at E = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='38 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' (b) Illustration of the interference mechanism for the formation of fringes on the substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' (c) Fringe profiles at various excitation energies taken along the blue dashed line shown in (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' (d) Fourier transform of the energy-dependent fringe profiles shown in (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' (e) Experimental and theoretical dispersion diagram of the substrate photon mode of mica.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' The two vertical dashed lines mark the photon dispersion in air (k = k0) and in bulk mica (k=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='6k0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Detailed discussions are given in Section 8 of the Supporting Information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Figure S11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' (a) Fitting the dispersion relation of the EP mode along the b axis of SnS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' The white dotted curves were produced with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' (b) Calculated dispersion relation of the bare waveguide photon mode without coupling with excitons along the b axis of SnS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Detailed discussions are given in Section 10 of the Supporting Information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' (a) s-SNOM, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='38 eV (c) substrate mode profiles (d) Fouriertransform (e) substrate mode dispersion 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='8 Ik=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='6ko mica 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='53 eV 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='53eV 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='50 eV 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='50eV 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='47 eV 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='47 eV 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='44 eV 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='44 eV (Cn (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=') (eV) 口 200nm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='41eV 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='41 eV 口 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='5 (b) S 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='38eV 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='38eV E P2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='35 eV 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='35eV 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='32 eV 口口口 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='32 eV substrate 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='29eV 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='3 Sns 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='29 eV photons mica 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='5 0 1 2 3 4 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='5 x (um) k (10° cm) k (ko)a 2 b 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='8 日1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='4 日1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='2 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='5 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='5 3 k(105cm-1) (105cm-1)a b 200 c 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='0 0 T = 27K T=100K exciton linewidth (meV) T=200K T=300K 150 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content="5 (cn'e) ()ul 6 100 1." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='0 50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='5 2 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='9 0 50 100 150 200 250 300 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='0 E (eV) T (K) E (eV)Figure S12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' (a) Fitting the exciton linewidth from the b-axis permittivity of SnS from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' (b) Temperature-dependent linewidth of excitons along the b axis of SnS based on the fitting in (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' (c) Line profile of the photonic mode taken along the vertical dashed line in Figure S11b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Detailed discussions are given in Section 10 of the Supporting Information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Figure S13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' Illustration of a selective nanophotonic interconnector based on anisotropic EPs in SnS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' When the incident photonic signal is at energies of E ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='29 eV, both devices 1 and 2 can receive the signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' When the incident photonic signal is at energies of E ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content='38 eV, only device 1 can receive the signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} +page_content=' to device Incident signal Sns interconnector s' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FIT4oBgHgl3EQf3Sv5/content/2301.11381v1.pdf'} diff --git a/1dE0T4oBgHgl3EQfdgBe/vector_store/index.faiss b/1dE0T4oBgHgl3EQfdgBe/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..8f4cc6ab8beac338c048c15b0bd91a1e1702ef64 --- /dev/null +++ b/1dE0T4oBgHgl3EQfdgBe/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:578a38fed8de70238324b171903852fe60f11badc513317493b4297419225b0b +size 1179693 diff --git a/1dFRT4oBgHgl3EQfmDfL/content/tmp_files/2301.13600v1.pdf.txt b/1dFRT4oBgHgl3EQfmDfL/content/tmp_files/2301.13600v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..71f6c348646f7fa104c33689c8f931b9f2de478d --- /dev/null +++ b/1dFRT4oBgHgl3EQfmDfL/content/tmp_files/2301.13600v1.pdf.txt @@ -0,0 +1,2199 @@ +arXiv:2301.13600v1 [cs.GT] 31 Jan 2023 +CONSTRAINED PHI-EQUILIBRIA +ARXIV PREPRINT +Martino Bernasconi +Politecnico di Milano +martino.bernasconideluca@polimi.it +Matteo Castiglioni +Politecnico di Milano +matteo.castiglioni@polimi.it +Alberto Marchesi +Politecnico di Milano +alberto.marchesi@polimi.it +Francesco Trov`o +Politecnico di Milano +francesco1.trovo@polimi.it +Nicola Gatti +Politecnico di Milano +nicola.gatti@polimi.it +February 1, 2023 +ABSTRACT +The computational study of equilibria involving constraints on players’ strategies has been largely +neglected. However, in real-world applications, players are usually subject to constraints ruling +out the feasibility of some of their strategies, such as, e.g., safety requirements and budget caps. +Computational studies on constrained versions of the Nash equilibrium have lead to some results +under very stringent assumptions, while finding constrained versions of the correlated equilibrium +(CE) is still unexplored. In this paper, we introduce and computationally characterize constrained +Phi-equilibria—a more general notion than constrained CEs—in normal-form games. We show +that computing such equilibria is in general computationally intractable, and also that the set of the +equilibria may not be convex, providing a sharp divide with unconstrained CEs. Nevertheless, we +provide a polynomial-time algorithm for computing a constrained (approximate) Phi-equilibrium +maximizing a given linear function, when either the number of constraints or that of players’ actions +is fixed. Moreover, in the special case in which a player’s constraints do not depend on other players’ +strategies, we show that an exact, function-maximizing equilibrium can be computed in polynomial +time, while one (approximate) equilibrium can be found with an efficient decentralized no-regret +learning algorithm. +1 +Introduction +Over the last years, equilibrium computation problems have received a terrific attention from AI and ML re- +search (Brown and Sandholm, 2019; Bakhtin et al., 2022), as game-theoretical equilibrium notions provide a prin- +cipled framework to deal with multi-player decision-making problems. Most of the works on equilibrium computation +problems focus on classical solution concepts—such as the well-known Nash equilibrium (NE) (Nash, 1951) and cor- +related equilibrium (CE) (Aumann, 1974)—, thus neglecting the presence of constraints entirely. However, in most +of the real-world applications, the players are usually subject to constraints that rule out the feasibility of some of +their strategies, such as, e.g., safety requirements and budget caps. Thus, addressing equilibrium notions involving +constraints is a crucial step needed for the operationalization of game-theoretic concepts into real-world settings. +The study of equilibrium notions involving constraints was initiated by Arrow and Debreu (1954), who define the +concept of generalized NE (GNE). The GNE can be interpreted as an NE of a game where players’ strategies are +subject to some constraints, which must be satisfied at the equilibrium and also determine which are the feasible +players’ deviations. However, given that computing a GNE is clearly PPAD-hard (Daskalakis et al., 2009), all the +works dealing with the computation of GNEs (see, e.g., (Facchinei and Kanzow, 2010)) provide efficient algorithms +only in specific settings that require very stringent assumptions. +Most of the computationally challenges in finding GNEs are inherited from the NE. In settings in which constrained are +not involved, the computational issues of NEs are usually circumvented by considering weaker equilibrium notions. + +ARXIV PREPRINT - FEBRUARY 1, 2023 +Among them, those that have received most of the attention in the literature are the CE and its variations, which +have been shown to be efficiently computable in several settings of interest (Papadimitriou and Roughgarden, 2008; +Celli et al., 2020). Surprisingly, with the only exception of (Chen et al., 2022) (see Section 1.2 for a detailed discussion +on it), no work has considered the problem of computing CEs in constrained settings. Thus, investigating whether the +CE retains its nice computational properties when adding constraints on players’ strategies is an open interesting +question. +1.1 +Original Contributions +In this paper, we introduce and computationally characterize constrained Phi-equilibria, starting, as it is customary, +from the setting of normal-form games. Our equilibria include the constrained versions of the classical CE and all of its +variations as special cases, by generalizing the notion of Phi-equilibria introduced by Greenwald and Jafari (2003) to +constrained settings. In particular, constrained Phi-equilibria are defined as Phi-equilibria, but in games where players +are subject to some constraints. Such constraints must be satisfied at the equilibrium, and, additionally, players are +only allowed to undertake safe deviations, namely those that are feasible according to the constraints. Crucially, the +set of safe deviations of a player does not only depend on the strategy of that player, but also on those of the others. +We start by showing that one of the most appealing computational properties of Phi-equilibria, namely that the set +of the equilibria of a game is convex, is lost when moving to their constrained version. This raises considerable +computational challenges in computing constrained Phi-equilibria. Indeed, we formally prove a strong intractability +result: for any factor α > 0, it is not possible, unless P = NP, to find in polynomial time a constrained (approximate) +Phi-equilibrium which achieves a multiplicative approximation α of the optimal value of a given linear function. Then, +in the rest of the paper, we show several ways in which such a negative result can be circumvented. +We prove that a constrained approximate Phi-equilibrium which maximizes a given linear function can be found in +polynomial time, when either the number of constraints or that of players’ actions is fixed. Our results are based on a +general algorithm that employs a non-standard “Lagrangification” of the constraints defining the set of safe deviations +of a player. Moreover, the algorithm needs a way of dealing with the non-convexity of the set of the equilibria, which +we provide in the form of a clever discretization of the space of the Lagrange multipliers. +Finally, we focus on the special case in which the constraints defining the safe deviations of a player do not depend +on the the strategies of the other players, but only on the strategy of that player. This includes constrained Phi- +equilibria identifying a particular constrained version of the coarse CE by Moulin and Vial (1978a), in which the +players’ strategies are subject to marginal cost constraints. These arise in several real-world applications in which +the players have bounded resources, such as, e.g., budget-constrained bidding in auctions. In such a special case, we +prove that a constrained (exact) Phi-equilibrium maximizing a given linear function can be computed in polynomial +time, and we provide an efficient decentralized no-regret learning algorithm for finding one constrained (approximate) +Phi-equilibrium. +1.2 +Related Works +GNEs +Rosen (1965) initiated the study of the computational properties of GNEs. After that, several other works +addressed the problem of computing GNEs by mainly exploiting techniques based on quasi-variational inequalities +(see (Facchinei and Kanzow, 2010) for a survey). More recently, some works (Kanzow and Steck, 2016; Bueno et al., +2019; Jordan et al., 2022; Goktas and Greenwald, 2022) also studied the convergence of iterative optimization algo- +rithms to GNEs. In order to provide efficient algorithms, all these works need to introduce very stringent assumptions, +which are even stronger than those required for the efficient computation of NEs. +Constrained Markov Games +Equilibrium notions involving constraints have also been addressed in the literature +on Markov games, with (Altman and Shwartz, 2000; Alvarez-Mena and Hern´andez-Lerma, 2006) being the first works +introducing GNEs in such a field. More recently, Hakami and Dehghan (2015) defined a notion of constrained CE in +Markov games. However, the incentive constraints in their notion of equilibrium only predicate on “pure” deviations, +which, in presence of constraints, may lead to empty sets of safe deviations. Very recently, Chen et al. (2022) gener- +alize the work of Hakami and Dehghan (2015) by considering “mixed” deviations. However, their algorithm provides +rather weak convergence guarantees, as it only ensures that the returned solution satisfies incentive constraints in ex- +pectation. Indeed, as we show in Proposition 3.1, the set of constrained equilibria may not be convex (it is easy to see +that Example 1 also applies to the setting studied by Chen et al. (2022)), and, thus, the fact that incentive constraints are +only satisfied in expectation does not necessarily imply that the “true” incentive constraints defining the equilibrium +are satisfied. We refer the reader to Appendix A for additional details on these aspects. +2 + +ARXIV PREPRINT - FEBRUARY 1, 2023 +2 +Preliminaries +In this section, we introduce all the preliminary definitions and results that are needed in the rest of the paper. +2.1 +Cost-constrained Normal-form Games +In a normal-form game, there is a finite set N := {1, . . . , n} of n players. Each player i ∈ N has a finite set Ai of +actions available, with s := |Ai| for i ∈ N being the number of players’ actions.1 We denote by a ∈ A :=×i∈N Ai +an action profile specifying an action ai for each player i ∈ N. Moreover, for i ∈ N, we let a−i ∈ A−i := +×j̸=i∈N Ai be an action profile of all players other than i, while (a, a−i) is the action profile obtained by adding +a ∈ Ai to a−i. Finally, we let ui : A → [0, 1] be the utility function of player i ∈ N, with ui(a) being the utility +perceived by that player when the action profile a ∈ A is played. +We extend classical normal-form games by considering the case in which each player i ∈ N has mi cost functions, +namely ci,j : A → [−1, 1] for j ∈ [mi].2 Each player i ∈ N is subject to mi constraints, which require that all player +i’s costs are less than or equal to zero.3 For ease of notation, we assume w.l.o.g. that all players have the same number +of constraints, namely m := mi for all i ∈ N. Moreover, we encode the costs of player i ∈ N by a vector-valued +function ci : A → [−1, 1]m such that, for every a ∈ A, the j-th component of the vector ci(a) is ci,j(a). +Correlated Strategies +In this paper, we deal with solution concepts defined by correlated strategies. A correlated +strategy z ∈ ∆A is a probability distribution defined over the set of actions profiles, with z[a] denoting the probability +assigned to a ∈ A.4 With an abuse of notation, for every player i ∈ N, we let ui(z) be player i’s expected utility +when the action profile played by the players is drawn from z ∈ ∆A. In particular, it holds ui(z) := � +a∈A ui(a)z[a]. +Similarly, we let ci(z) := � +a∈A ci(a)z[a] be the vector of player i’s expected costs, so that player i’s constraints +can be compactly written as ci(z) ⪯ 0. Finally, we define S ⊆ ∆A as the set of safe correlated strategies, which are +those satisfying the cost constraints of all players. Formally: +S := {z ∈ ∆A | ci(z) ⪯ 0 +∀i ∈ N} . +In the following, we assume w.l.o.g. that S ̸= ∅. +2.2 +Constrained Phi-equilibria +We generalize the notion of Phi-equilibria (Greenwald and Jafari, 2003) to cost-constrained normal-form games. Such +equilibria are defined as correlated strategies z ∈ ∆A that are robust against a given set Φ of players’ deviations, in +the sense that, if a mediator draws an action profile a ∈ A according to z and recommends to play action ai to each +player i ∈ N, then no player has an incentive to deviate from their recommendation by selecting a deviation in Φ. +For every i ∈ N, we let Φi be the set of player i’s deviations, i.e., linear transformations φi : Ai → ∆Ai that prescribe +a probability distribution over player i’s actions for every possible action recommendation. For ease of notation, we +encode a deviation φi by means of its matrix representation. Formally, an entry φi[b, a] of the matrix represents the +probability assigned to action a ∈ Ai by φi(b). We denote the set of all the possible deviations by Φ := {Φi}i∈N . +Given a correlated strategy z ∈ ∆A and a deviation φi ∈ Φi, we define φi ⋄ z as the modification of z induced by φi, +which is a linear transformation that can be expressed as follows in terms of matrix representation: +(φi ⋄ z)[ai, a−i] := +� +b∈Ai +φi[b, ai]z[b, a−i], +for every ai ∈ Ai and a−i ∈ A−i. Moreover, given a set Φi of deviations of player i ∈ N, in the following we denote +by ΦS +i (z) := {φi ∈ Φi | φi ⋄ z ∈ S} the set of safe deviations for player i at a given correlated strategy z ∈ ∆A. +We are now ready to provide our definition of constrained Phi-equilibria in cost-constrained normal-form games. +Definition 2.1 (Constrained ǫ-Phi-equilibria). Given a set Φ := {Φi}i∈N of deviations and an ǫ > 0, a constrained +ǫ-Phi-equilibrium is a safe correlated strategy z ∈ S such that, for all i ∈ N, the following holds: +ui(z) ≥ ui(φi ⋄ z) − ǫ +∀φi ∈ ΦS +i (z). +1For ease of presentation, in this paper we assume that all the players have the same number of actions. All the results can be +easily generalized to the case of different numbers of actions. +2In this paper, given some x ∈ N>0, we let [x] := {1, . . . , x} be the set of the first x natural numbers. +3Since z ∈ ∆A, we can assume w.l.o.g. that all the constraints are of the form ≤ 0, as any constraint can always be cast in such +a form by suitably manipulating the cost function ci,j. +4In this paper, given a finite set X, we denote by ∆X the set of all the probability distributions defined over the elements of X. +3 + +ARXIV PREPRINT - FEBRUARY 1, 2023 +A constrained Phi-equilibrium is defined for ǫ = 0. +2.3 +Computing Constrained Phi-equilibria +In the following, we formally introduce the computational problem that we study in the rest of the paper. +We denote by I := (Γ, Φ) an instance of the problem, where the tuple Γ := (N, A, {ui}i∈N , {ci,j}i∈N,j∈[m]) is a +cost-constrained normal-form game and Φ := {Φi}i∈N is a set of deviations. Moreover, we let |I| be the size (in +terms of number of bits) of the instance I. We assume that the number n of players is fixed, so that |I| does not grow +exponentially in n.5 We also make the following assumption on how the sets of deviations are represented: +Assumption 1. For every i ∈ N, the set Φi is a polytope encoded by a finite of linear inequalities.6 +Let us remark that, in games without constraints, this assumption is met by all the sets Φ which determine the classical +notions of Phi-equilibria (Greenwald and Jafari, 2003). +Next, we formally define our computational problem: +Definition 2.2 (APXCPE(α, ǫ)). For any α, ǫ > 0, we define APXCPE(α, ǫ) as the problem of finding, given an +instance I := (Γ, Φ) and a linear function ℓ : ∆A → R as input, a constrained ǫ-Phi-equilibrium z ∈ ∆A such that +ℓ(z) ≥ αℓ(z′) for all constrained Phi-equilibria z′ ∈ ∆A. +Intuitively, APXCPE(α, ǫ) asks to compute a constrained ǫ-Phi-equilibrium whose value for the linear function ℓ is at +least a fraction α of the maximum value which can be achieved by an (exact) constrained Phi-equilibrium. +In order to ensure that an instance of our problem is well defined, we make the following “Slater-like” assumption on +how the players’ cost constraints are defined. +Assumption 2. For every correlated strategy z ∈ ∆A, player i ∈ N, and index j ∈ [m], there exists φ◦ +i ∈ ΦS +i (z): +ci,j(φ◦ +i ⋄ z) ≤ −ρ, +where ρ > 0 and 1/ρ is O(poly(|I|)), with poly(|I|) being a polynomial function of the instance size |I|. +In Assumption 2, the condition ρ > 0 is required to guarantee the existence of a constrained Phi-equilibrium (see +Theorem 2.1) and that the sets ΦS +i (z) are non-empty (otherwise our solution concept would be ill defined). Moreover, +the second condition on ρ in Assumption 2 is equivalent to requiring that our algorithms run in time polynomial in 1 +ρ. +Assumption 2 also allows us to prove the existence of our equilibria, by showing that the constrained Nash equilibria +introduced by Altman and Shwartz (2000), which always exist under Assumption 2, are also constrained Phi-equilibria. +Theorem 2.1. Given a cost-constrained normal-form game Γ and a set Φ of deviations, if Assumption 2 is satisfied, +then Γ admits a constrained Phi-equilibrium. +2.4 +Relation with Unconstrained Phi-equilibria +We conclude the section by discussing the relation between our constrained Phi-equilibria and classical equilibrium +concepts for unconstrained games. +Correlated Equilibrium +When there are no constraints, the correlated equilibrium (CE) (Aumann, 1974) is a spe- +cial case of Phi-equilibrium. As shown by Greenwald and Jafari (2003), the CE is obtained when the sets Φi contain +all the possible deviations. Formally, the CE is defined by the set ΦALL := {Φi,ALL} of deviations such that: +Φi,ALL := +� +φi +��� +� +a∈Ai +φi[b, a] = 1 +∀b ∈ Ai +� +. +5Notice that the size of the representation of a normal-form game is O(sn), and, thus, exponential in n. Any algorithm that +runs in time polynomial in such instance size is not computationally appealing, as even its input has size exponential in n. For this +reason, we focus on the case in which n is fixed, and, thus, the instance size does not grow exponentially with n. +6Notice that, since each φi ∈ Φi is represented as a matrix, a linear inequality is expressed as � +b,a∈Ai M[b, a]φi[b, a] ≤ d, +for some matrix M and scalar d. +4 + +ARXIV PREPRINT - FEBRUARY 1, 2023 +Coarse Correlated Equilibrium +The coarse correlated equilibrium (CCE) (Moulin and Vial, 1978b) is a special +(unconstrained) Phi-equilibrium whose set of deviations is ΦCCE := {Φi,CCE}i∈N such that: +Φi,CCE := +� +φi +��� ∃h ∈ ∆Ai : φi[b, a] = h[a] ∀b, a ∈ Ai +� +. +Intuitively, such sets contain all the possible deviations that prescribe the same probability distribution independently +of the received action recommendation. +Thus, our constrained Phi-equilibria include the generalization of CEs and CCEs to cost-constrained games. +Our definition of constrained Phi-equilibrium needs to employ “mixed” deviations that map action recommendations +to probability distributions over actions. This is necessary in presence of constraints. Instead, without them, one +can simply consider “pure” deviations that map recommendations to actions deterministically Greenwald and Jafari +(2003). +3 +Challenges of Constrained Phi-equilibria +In this section, we show that, in cost-constrained normal-form games, Phi-equilibria loose the nice computational +properties that they exhibit in unconstrained settings. This is crucially determined by the fact that the set of constrained +Phi-equilibria may not be convex in general. +Proposition 3.1. Given any instance I := (Γ, Φ), the set of constrained Phi-equilibria may not be convex. +Proposition 3.1 is proved by the following example. +Example 1. Let ΦALL be the set of all the possible deviations in a two-player game in which each player has two +actions, namely A1 = A2 = {a0, a1}. The first player’s utility is such that u1(a, a′) = 0 for all a ∈ A1 and a′ ∈ A2, +while the second player’s utility is such that u2(a0, a1) = 1, and 0 otherwise. Both players share the same single +cost constraint (m = 1). Their cost functions are defined as ci(a0, a1) = 1, ci(a0, a0) = − 1 +2, and ci(a1, a) = −1 +for all a ∈ A2. Notice that the instance defined above satisfies Assumption 2 for ρ = 1/2. It is easy to see that the +correlated strategy z1 ∈ ∆A such that z1[a0, a0] = 2 +3 and z1[a0, a1] = 1 +3 is a constrained Phi-equilibrium. Moreover, +the “pure” correlated strategy z2 ∈ ∆A such that z2[a1, a0] = 1 is also a constrained Phi-equilibrium. However, the +combination z3 = 1 +2(z1 + z2) is not a constrained Phi-equilibrium. Indeed, the second player has an incentive to +deviate by using a deviation φ2 such that φ2[a0, a1] = 1 and φ2[a1, a1] = 1. Such a deviation prescribes to play action +a1 when a0 is recommended, and to play action a1 when the recommendation is a1. Straightforward calculations show +that, for every a ∈ A1: +(φ2 ⋄ z3)[a, a′] = +� 1 +2 +if +a′ = a1 +0 +otherwise, +and u2(φ2 ⋄ z3) = 1 +2 > u2(z3) = 1 +6. Moreover, the deviation is safe, since φ2 ∈ ΦS +2 (z3) as c2(φ2 ⋄ z3) = 0. +In order to formally asses the computational challenges of computing constrained Phi-equilibria, we prove the follow- +ing strong inapproximability result: +Theorem 3.1 (Hardness). For any constant α > 0, the problem APXCPE(α, (α/s)2) is NP-hard, where s is the +number of players’ actions in the instance given as input. +Intuitively, Theorem 3.1 states that, for every multiplicative approximation factor α > 0, it is not possible to find +a constrained ǫ-Phi-equilibrium having value of ℓ at least a fraction α of its optimal value in time polynomial in 1 +ǫ. +Moreover, as a byproduct of Theorem 3.1, we also get the inapproximability up to within any factor of the problem of +computing an optimal constrained (exact) Phi-equilibrium. +Notice that the hardness result in Theorem 3.1 cannot hold for values of ǫ that are independent from the instance size. +Indeed, as we prove in Corollary 4.3 in Section 4, problem APXCPE(1, ǫ) can be solved in quasi-polynomial time +in the instance size whenever ǫ > 0 is a given constant. Thus, any NP-hardness result for APXCPE(α, ǫ) would +contradict the exponential-time hypothesis.7 +4 +Computing Optimal Constrained ǫ-Phi-equilibria Efficiently +In this section, we show how to circumvent the negative result established by Theorem 3.1. In particular, we prove +that, when the number of cost constraints is fixed, problem APXCPE(1, ǫ) can be solved in time polynomial in the +7The exponential-time hypothesis conjectures that solving 3SAT requires at least exponential time. +5 + +ARXIV PREPRINT - FEBRUARY 1, 2023 +instance size and 1 +ǫ for ǫ > 0 (Corollary 4.2). Moreover, we also prove that, in general, for any constant ǫ > 0 problem +APXCPE(1, ǫ) admits a quasi-polynomial-time algorithm, whose running time becomes polynomial when the number +of players’ actions is fixed (Corollary 4.3). +First, in Section 4.1, we provide a general algorithm that is at the core of the two main results of this section. Then, in +Section 4.1, we show how the algorithm can be suitably instantiated in order to prove each result. In the rest of this +section, unless stated otherwise, we always assume that an ǫ > 0 has been fixed, and that I := (Γ, Φ) and ℓ : ∆A → R +are the inputs of a given instance of problem APXCPE(1, ǫ). +4.1 +General Algorithm +The main technical tool that we employ in order to design our algorithm is a “Lagrangification” of the constraints +defining the sets ΦS +i (z) of safe deviations. First, we prove the following preliminary result, which shows that strong +duality holds for the problem maxφi∈ΦS +i (z) ui(φi ⋄ z) of finding the best safe deviation for player i ∈ N at z ∈ ∆A. +Lemma 4.1. For every z ∈ ∆A and i ∈ N, it holds +max +φi∈ΦS +i (z) ui(φi ⋄ z) = sup +φi∈Φi +inf +ηi∈Rm ++ +� +ui(φi ⋄ z) − η⊤ +i ci(φi ⋄ z) +� += +inf +ηi∈Rm ++ +sup +φi∈Φi +� +ui(φi ⋄ z) − η⊤ +i ci(φi ⋄ z) +� +. +Then, by exploiting Lemma 4.1, we can prove that, under Assumption 2, strong duality continues to hold even when +restricting the Lagrange multipliers ηi to have ℓ1-norm less than or equal to 1/ρ. Formally: +Lemma 4.2. Let D := +� +η ∈ Rm ++ | ||η||1 ≤ 1/ρ +� +. Then, for every z ∈ ∆A and i ∈ N, it holds: +max +φi∈ΦS +i (z) ui(φi ⋄ z) = max +φi∈Φi min +ηi∈D +� +ui(φi ⋄ z) − η⊤ +i ci(φi ⋄ z) +� += min +ηi∈D max +φi∈Φi +� +ui(φi ⋄ z) − η⊤ +i ci(φi ⋄ z) +� +. +Lemma 4.2 allows us to write player i’s incentive constraints in the definition of constrained ǫ-Phi-equilibria as +ui(z) ≥ min +ηi∈D max +φi∈Φi +� +ui(φi ⋄ z) − η⊤ +i ci(φi ⋄ z) +� +− ǫ. +(1) +This crucially allows us to show the following result: solving problem APXCPE(1, ǫ) is equivalent to computing +max(η1,...,ηn)∈Dn Fǫ(η1, . . . , ηn), where Fǫ(η1, . . . , ηn) is the optimal value of a suitable maximization problem +parameterized by tuples of Lagrange multipliers ηi ∈ D, one per player i ∈ N. Such a problem asks to compute a safe +correlated strategy maximizing the linear function ℓ subject to players’ incentive constraints that are re-formulated by +means of Lemma 4.2. Formally, we define Fǫ(η1, . . . , ηn) as the maximum of ℓ(z) over those z ∈ S that additionally +satisfy the following constraint for every i ∈ N: +ui(z) ≥ max +φi∈Φi +� +ui(φi ⋄ z) − η⊤ +i ci(φi ⋄ z) +� +− ǫ. +(2) +Notice that the min operator that appears in the right-hand side of Constraints (1) is dropped by adding the outer +maximization over the tuples (η1, . . . , ηn) ∈ Dn, as the maximum of ℓ is always achieved when the right-hand side +of such constraints is as small as possible. +Next, we show that Fǫ(η1, . . . , ηn) can be computed in polynomial time by means of the ellipsoid algorithm. +Lemma 4.3. For every tuple (η1, . . . , ηn) ∈ Dn, the value of Fǫ(η1, . . . , ηn) can be computed in time polynomial in +the instance size |I| and 1 +ǫ. +Proof. We show that Fǫ(η1, . . . , ηn) can be solved in polynomial time by means of the ellipsoid algorithm. Let us +notice that Constraints (2) can be equivalently encoded by a set of linear inequalities, one for each player i ∈ N +and deviation φi ∈ vert(Φi), where vert(Φi) denotes the set of vertexes of the polytope Φi (recall Assumption 1). +Thus, solving Fǫ(η1, . . . , ηn) is equivalent to solving an LP with a (possibly) exponential number of constraints, but +polynomially-many variables. Such an LP can be solved in polynomial time by means of the ellipsoid algorithm, +provided that a polynomial-time separation oracle for the linearized version of Constraints (2) is available. Such an +oracle can be implemented by solving the maximization in the right-hand side of Constraints (2) for a correlated +strategy z ∈ ∆A given as input. Formally, the separation oracle solves the following problem for each player i ∈ N: +φ⋆ +i ∈ arg max +φi∈Φi +� +ui(φi ⋄ z) − η⊤ +i ci(φi ⋄ z) +� +, +6 + +ARXIV PREPRINT - FEBRUARY 1, 2023 +which can be done efficiently thanks to Assumption 1. Then, if the separation oracle finds any φ⋆ +i such that: +ui(z) ≥ ui(φ⋆ +i ⋄ z) − η⊤ +i ci(φ⋆ +i ⋄ z), +it outputs the above inequality as a separating hyperplane to be used in the ellipsoid algorithm. +Lemma 4.3 is not enough to complete our algorithm, since we need an efficient way of optimizing Fǫ(η1, . . . , ηn) +over all the tuples of Lagrange multipliers. This problem is non-trivial, since Fǫ(η1, . . . , ηn) is non-concave in ηi. +Nevertheless, we show that, by restricting the domain D of the Lagrange multipliers to a suitably-defined finite “small” +subset, we can still find a constrained ǫ-Phi-equilibrium whose value of ℓ is at least as large as that of any constrained +(exact) Phi-equilibrium. This is enough to solve APXCPE(1, ǫ). In particular, we need a finite subset of “good” +Lagrange multipliers, in the sense of the following definition. +Definition 4.1. Given any δ > 0, a set ˜D ⊆ D is δ-optimal if, for every z ∈ ∆A and i ∈ N, the following holds: +min +ηi∈ ˜ +D +max +φi∈Φi +� +ui(φi ⋄ z) − η⊤ +i ci(φi ⋄ z) +� +≤ +max +φi∈ΦS +i (z) ui(φi ⋄ z) + δ. +Intuitively, thanks to Lemma 4.2, if we optimize the Lagrange multipliers over a δ-optimal set ˜D ⊆ D, instead of +optimizing them over D, then we are allowing the players to violate incentive constraints by at most δ. +In the following, we assume that a finite δ-optimal set ˜D ⊆ D is available. In Section 4.2, se show how to design two +particular δ-optimal sets that allow to prove our main results. For ease of presentation, we let +L ˜ +D,ǫ := +max +(η1,...,ηn)∈ ˜ +Dn Fǫ(η1, . . . , ηn) +be the optimal value of Fǫ(η1, . . . , ηn) when the Lagrange multipliers are constrained to be in a δ-optimal set ˜D ⊆ D. +Next, we show that, given any δ-optimal set ˜D with δ ≤ ǫ, the value of L ˜ +D,ǫ is at least that achieved by constrained +(exact) Phi-equilibria, namely LD,0. Formally: +Lemma 4.4. Given any 0 < δ ≤ ǫ and a δ-optimal set ˜D ⊆ D, the following holds: L ˜ +D,ǫ ≥ LD,0. +Intuitively, Lemma 4.4 is proved by noticing that, provided that δ ≤ ǫ, the incentive constraints violation introduced +by using ˜D instead of D is at most ǫ. Moreover, the set of feasible correlated strategies can only expand by allowing +incentive constraints to be violated, and, thus, the value of the objective ℓ can only increase. +Lemma 4.4 suggests a way of solving APXCPE(1, ǫ). Indeed, given a finite δ-optimal set ˜D ⊆ D with δ ≤ ǫ, by +enumerating over all the tuples of Lagrange multipliers ηi ∈ ˜D, one per player i ∈ N, we can find the desired +constrained ǫ-Phi-equilibrium. The following theorem shows that this procedure gives an algorithm for APXCPE(1, ǫ) +that runs in time polynomial in the instance size, | ˜D|, and 1 +ǫ. +Theorem 4.1. Given a finite δ-optimal set ˜D ⊆ D with δ ≤ ǫ, there exists an algorithm that solves APXCPE(1, ǫ) +and runs in time polynomial in the instance size |I|, the number | ˜D| of elements in ˜D, and 1 +ǫ for every ǫ > 0. +Proof. The algorithm works by enumerating over all the possible tuples of Lagrange multipliers ηi ∈ ˜D, one per +player i ∈ N. These are polynomially many in the size | ˜D| when the number of players n is fixed. For every tuple +(η1, . . . , ηn) ∈ ˜Dn, the algorithm solves Fǫ(η1, . . . , ηn), which can be done in time polynomial in |I| and 1 +ǫ thanks +to Lemma 4.3. Finally, the algorithm returns the correlated strategy z ∈ ∆A with the highest value of ℓ among those +computed while solving Fǫ(η1, . . . , ηn). It is easy to see that the returned solution solves problem APXCPE(1, ǫ) by +applying Lemma 4.4. This concludes the proof. +4.2 +Instantiating the General Algorithm +Next, we show how to build δ-optimal sets ˜D that, when they are plugged in the algorithm in Theorem 4.1, allow us +to derive our results. In particular, we consider the set: +Dτ := +� +η ∈ D +��� ηj = kτ, k ∈ {0, . . . , ⌊1/τρ⌋} ∀j ∈ [m] +� +, +which is a discretization of D with a regular lattice of step τ ∈ R+ (notice that ηj is the j-th component of η). +By a simple stars and bars combinatorial argument, we have that |Dτ| = +�⌊1/τρ⌋+m +m +� +. Thus, since it holds that +7 + +ARXIV PREPRINT - FEBRUARY 1, 2023 +|Dτ| = O((1/τρ)m), if the number of constraints m is fixed, |Dτ| is bounded by a polynomial in 1/τρ. Moreover, +simple combinatorial arguments show that |Dτ| ≤ (1 + m)⌊1/τρ⌋.8 Thus, it also holds that |Dτ| = O(m +1/τρ). Notice +that the two bounds on |Dτ| are non-comparable, and, thus, they give rise to two distinct results, as we show in the +following. +By using the first bound on |Dτ|, we can show that the set Dτ is δ-optimal for δ = mτ. Formally: +Lemma 4.5. For any τ > 0, the set Dτ is (τm)-optimal. +Thus, whenever the number m of cost constraints is fixed, Lemma 4.5, together with Theorem 4.1, allows us to provide +a polynomial-time algorithm. Indeed, it is sufficient to apply Theorem 4.1 for the (τm)-optimal set Dτ with τ := ǫ/m +to obtain the following first main result: +Corollary 4.2. There exists an algorithm that solves problem APXCPE(1, ǫ) in time polynomial in |I| and 1 +ǫ for every +ǫ > 0, when the number m of cost constraints is fixed. +On the other hand, by using the second bound on |Dτ|, we can show that Dτ is δ-optimal for δ depending logarithmi- +cally on the number of players’ actions. Formally: +Lemma 4.6. For any τ > 0, the set Dτ is δ-optimal for δ = 2 +� +2τ log s/ρ, where s is the number of players’ actions. +Lemma 4.6 (together with Theorem 4.1) immediately gives us a quasi-polynomial-time for solving APXCPE(1, ǫ) for +a given constant ǫ > 0. Moreover, its running time becomes polynomial when the number of players’ actions is fixed. +Corollary 4.3. For any constant ǫ > 0, there exists an algorithm that solves APXCPE(1, ǫ) in time O(|I|log s). +Moreover, when the number s of players’ actions is fixed, the algorithm runs in time polynomial in |I|. +Notice that it is in general not possible to design an algorithm that runs in time polynomial in 1 +ǫ, since this would +contradict the hardness result in Theorem 3.1. +5 +A Special Case: Deviation-dependent Costs +We complete our computational study of constrained Phi-equilibria by considering a special case in which player i’s +costs associated to a deviation φi only depend on φi and not on the (overall) modified correlated strategy φi ⋄ z. +We consider instances satisfying the following assumption. +Assumption 3. For every player i ∈ N and player i’s deviation φi ∈ Φi, there exists a function ˜ci : Φi → [−1, 1]m +such that ˜ci(φi) := ci(φi ⋄ z) for every z ∈ ∆A. +Notice that, whenever Assumption 3 holds, the set ΦS +i (z) of safe deviations does not depend on z. Thus, in the rest of +this section, we write w.l.o.g. ΦS +i rather than ΦS +i (z). +A positive effect of Assumption 3 is that it recovers the convexity of the set of constrained Phi-equilibria, rendering +them more akin to unconstrained ones. Formally: +Proposition 5.1. For instances I := (Γ, Φ) satisfying Assumption 3, the set of constrained ǫ-Phi-equilibria is convex. +Proposition 5.1 suggests that constrained Phi-equilibria are much more computationally appealing under Assumption 3 +than in general, as we indeed show in the rest of this section. +First, in Section 5.1, we show that APXCPE(1, 0) admits a polynomial-time algorithm under Assumption 3. Then, in +Section 5.2, we design a no-regret learning algorithm that efficiently computes one constrained ǫ-Phi equilibrium with +ǫ = O(1/ +√ +T) as the number of rounds T grows. Finally, in Section 5.3, we provide a natural example of constrained +Phi-equilibria satisfying Assumption 3. +5.1 +A Poly-time Algorithm for Optimal Equilibria +We prove that, whenever Assumption 3 holds, the problem of computing an (exact) Phi-equilibrium maximizing a +given linear function can be solved in polynomial time. This is done by formulating the problem as an LP with +polynomially-many variables and exponentially-many constraints, which can be solved by means of the ellipsoid +method, similarly to how we compute Fǫ(η1, . . . ηn) in Section 4 (see the proof of Lemma 4.3). Formally: +Theorem 5.1. Restricted to instances I := (Γ, Φ) which satisfy Assumption 3, APXCPE(1, 0) admits a polynomial- +time algorithm. +8See Appendix D for a formal proof. +8 + +ARXIV PREPRINT - FEBRUARY 1, 2023 +5.2 +An Efficient No-regret Learning Algorithm +Next, we show how Assumption 3 allows us to find a constrained ǫ-Phi-equilibrium by means of a polynomial-time +decentralized no-regret learning algorithm. Our algorithm is based on the Phi-regret minimization framework intro- +duced by Greenwald and Jafari (2003), which needs to be extended in order to be able to work with polytopal sets ΦS +i +of safe deviations, rather than finite sets of “pure” deviations. +Algorithm 1 Learning a Constrained ǫ-Phi-equilibria +Require: Regret minimizers Ri for the sets ΦS +i , for i ∈ N +1: Initialize the regret minimizers Ri +2: for t = 1, . . . , T do +3: +for each player i ∈ N do +4: +φi,t ← Ri.RECOMMEND() +5: +Play according to a distribution xi,t ∈ ∆Ai s.t. +xi,t[a] = +� +b∈Ai +φi,t[b, a]xi,t[b] +∀a ∈ Ai +6: +end for +7: +zt ← ⊗i∈N xi,t +8: +Ri.OBSERVE(φi �→ ui(φi ⋄ zt)) +9: end for +10: return ¯zT := 1 +T +�T +t=1 zt +Algorithm 1 outlines our no-regret algorithm. It instantiates a regret minimizer Ri for the polytope ΦS +i for each i ∈ N. +Ri is an object that, at each round t ∈ [T ], recommends a safe deviation φi,t ∈ ΦS +i to player i (Line 4 of Algorithm 1), +and, then, observes a function φi �→ ui(φi ⋄ zt) that specifies the utility that would have been obtained by selecting +any safe deviation φi ∈ ΦS +i at round t (Line 8 of Algorithm 1). Ri guarantees that the regret RT +i cumulated by player +i over [T ] grows sublinearly, i.e., RT +i = o(T ), where: +RT +i := max +φi∈Φi +T +� +t=1 +ui(φi ⋄ zt) − +T +� +t=1 +ui(φi,t ⋄ zt), +which is how much player i loses by selecting φi,t at each t rather than choosing the same best-in-hindsight deviation +at all rounds. Notice that, by taking inspiration from the Phi-regret framework (Greenwald and Jafari, 2003), given +a recommended deviation φi,t, player i actually plays according to a probability distribution xi,t ∈ ∆Ai, which +is a stationary distribution of the matrix representing φi,t. This is crucial in order to implement the algorithm in a +decentralized fashion and to provide convergence guarantees to constrained ǫ-Phi-equilibria (see Theorem 5.2). All the +distributions xi,t jointly determine a correlated strategy zt ∈ ∆A at each round t ∈ [T ], defined as zt := ⊗i∈N xi,t, +where ⊗ denotes the product among distributions; formally, zt[a] := � +i∈N xi,t[ai] for all a ∈ A. +Algorithm 1 provides the following guarantees: +Theorem 5.2. Given an instance I := (Γ, Φ) satisfying Assumption 3, after T ∈ N>0 rounds, Algorithm 1 returns a +correlated strategy ¯zT ∈ ∆A that is a constrained ǫT -Phi-equilibrium with ǫT = O(1/ +√ +T). Moreover, each round of +Algorithm 1 runs in polynomial time. +Let us remark that the crucial property which allows us to design Algorithm 1 is that the sets ΦS +i of safe deviations do +not depend on players other than i. Finally, from Theorem 5.2, the following result follows: +Corollary 5.3. In instances I := (Γ, Φ) satisfying Assumption 3, a constrained ǫ-Phi-equilibrium can be computed +in time polynomial in the instance size and 1 +ǫ by means of a decentralized learning algorithm. +5.3 +Marginally-constrained CCE +We conclude the section by introducing a particular (natural) notion of constrained ǫ-Phi-equilibrium for which As- +sumption 3 is satisfied. This is a constrained version of the classical CCE in cost-constrained normal-form games +where a player’s costs only depend on the action of that player. We call it marginally-constrained ǫ-CCE. Formally, +such an equilibrium is defined for games in which, for every player i ∈ N, it holds ci(a) = ci(a′) for all a, a′ ∈ A +such that ai = a′ +i, and for the set ΦCCE of CCE deviations that we have previously introduced in Section 2.4. Next, +we prove that, with the definition above, Assumption 3 is satisfied. +9 + +ARXIV PREPRINT - FEBRUARY 1, 2023 +Theorem 5.4. For instances I := (Γ, ΦCCE) such that ci(a) = ci(a′) for every player i ∈ N and action profiles +a, a′ ∈ A : ai = a′ +i, Assumption 3 holds. +Thanks to Theorem 5.4, we readily obtain the two following corollaries of Theorems 5.1 and 5.1. +Corollary 5.5. The problem of computing a marginally-constrained (exact) CCE that maximizes a linear function +ℓ : ∆A → R can be solved in polynomial time. +Corollary 5.6. A marginally-constrained ǫ-CCE can be computed in time polynomial in the instance size and 1 +ǫ by +means of a decentralized learning algorithm. +6 +Discussion and Open Problems +The main positive results that we provide in this paper (Corollaries 4.2 and 4.3) show that a constrained ǫ-Phi equilib- +rium maximizing a given linear function can be computed in time polynomial in the instance size and 1 +ǫ, when either +the number of constraints or that of players’ actions is fixed. Clearly, this implies that, under the same assumptions, +a constrained ǫ-Phi-equilibrium can be found efficiently. Moreover, in Section 5, we designed an efficient no-regret +learning algorithm that finds a constrained ǫ-Phi-equilibrium in settings enjoying special properties (Corollary 5.3). +However, the problem of efficiently computing a constrained ǫ-Phi-equilibrium remains open in general. Formally: +Definition 6.1 (Open Problem). Given any instance I := (Γ, Φ), find a constrained ǫ-Phi-equilibrium in time polyno- +mial in the instance size and 1 +ǫ. +Solving the problem above is non-trivial. Proposition 3.1 in Section 3 proves that the set of constrained ǫ-Phi-equilibria +is non-convex, and, thus, solving the problem in Definition 6.1 is out of scope for most of the known equilibrium +computation techniques. On the other hand, it is unlikely that such a problem is NP-hard. Indeed, a constrained +ǫ-Phi-equilibrium always exists and, given any z ∈ ∆A, it is possible to verify whether z is an equilibrium or not +in polynomial time. Formally, such a problem is said to belong to the TFNP complexity class, and, thus, standard +arguments show that, if the problem is NP-hard, then NP = coNP (Megiddo and Papadimitriou, 1991). Thus, one +should try to reduce the problem in Definition 6.1 to problems in TFNP, such as that of computing a Nash equilibrium. +However, while the problem in Definition 6.1 shares some properties with that of computing a Nash equilibrium, such +as the non-convexity of the set of the equilibria, the former is fundamentally different from the latter, since it exhibits +correlation among the players. Thus, a reduction from such a problem to that of computing Nash equilibria would +require a gadget to break the correlation among the players, and doing that is highly non-trivial as cost constraints are +expressed by linear functions. +10 + +ARXIV PREPRINT - FEBRUARY 1, 2023 +References +Eitan Altman and Adam Shwartz. 2000. Constrained markov games: Nash equilibria. In Advances in dynamic games +and applications. Springer, 213–221. +Jorge Alvarez-Mena and On´esimo Hern´andez-Lerma. 2006. Existence of Nash equilibria for constrained stochastic +games. Mathematical Methods of Operations Research 63, 2 (2006), 261–285. +Kenneth J Arrow and Gerard Debreu. 1954. Existence of an equilibrium for a competitive economy. Econometrica: +Journal of the Econometric Society (1954), 265–290. +Robert J Aumann. 1974. Subjectivity and correlation in randomized strategies. 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Finding Correlated Equilibrium of Constrained Markov Game: A +Primal-Dual Approach. In Advances in Neural Information Processing Systems. +Constantinos Daskalakis, Paul W Goldberg, and Christos H Papadimitriou. 2009. The complexity of computing a +Nash equilibrium. SIAM J. Comput. 39, 1 (2009), 195–259. +Ivar Ekeland and Roger Temam. 1999. Convex analysis and variational problems. SIAM. +Francisco Facchinei and Christian Kanzow. 2010. Generalized Nash equilibrium problems. Annals of Operations +Research 175, 1 (2010), 177–211. +Denizalp Goktas and Amy Greenwald. 2022. Exploitability Minimization in Games and Beyond. Advances in Neural +Information Processing Systems (2022). +Amy Greenwald and Amir Jafari. 2003. A general class of no-regret learning algorithms and game-theoretic equilibria. +In Learning theory and kernel machines. Springer, 2–12. +Vesal Hakami and Mehdi Dehghan. 2015. Learning stationary correlated equilibria in constrained general-sum stochas- +tic games. IEEE Transactions on Cybernetics 46, 7 (2015), 1640–1654. +Johan H˚astad. 1999. Clique is hard to approximate within n1−ǫ. Acta Mathematica 182, 1 (1999), 105–142. +Elad Hazan et al. 2016. Introduction to online convex optimization. Foundations and Trends® in Optimization 2, 3-4 +(2016), 157–325. +Michael I Jordan, Tianyi Lin, and Manolis Zampetakis. 2022. First-Order Algorithms for Nonlinear Generalized Nash +Equilibrium Problems. arXiv preprint arXiv:2204.03132 (2022). +Christian Kanzow and Daniel Steck. 2016. Augmented Lagrangian methods for the solution of generalized Nash +equilibrium problems. SIAM Journal on Optimization 26, 4 (2016), 2034–2058. +Nimrod Megiddo and Christos H Papadimitriou. 1991. On total functions, existence theorems and computational +complexity. Theoretical Computer Science 81, 2 (1991), 317–324. +H. Moulin and J-P Vial. 1978a. Strategically zero-sum games: the class of games whose completely mixed equilibria +cannot be improved upon. INT J GAME THEORY 7, 3 (1978), 201–221. +Herv´e Moulin and J-P Vial. 1978b. +Strategically zero-sum games: the class of games whose completely mixed +equilibria cannot be improved upon. International Journal of Game Theory 7, 3 (1978), 201–221. +John Nash. 1951. Non-cooperative games. Annals of mathematics (1951), 286–295. +Christos H Papadimitriou and Tim Roughgarden. 2008. Computing correlated equilibria in multi-player games. Jour- +nal of the ACM (JACM) 55, 3 (2008), 1–29. +J Ben Rosen. 1965. Existence and uniqueness of equilibrium points for concave n-person games. Econometrica: +Journal of the Econometric Society (1965), 520–534. +David Zuckerman. 2007. Linear Degree Extractors and the Inapproximability of Max Clique and Chromatic Number. +Theory of Computing 3, 6 (2007), 103–128. +11 + +ARXIV PREPRINT - FEBRUARY 1, 2023 +A +On the Weaknesses of the Guarantees of the Algorithm of Chen et al. (2022) +The Algorithm of Chen et al. (2022) finds a distribution µ over correlated strategies ∆A such that: +Ez∼µ +� +max +φi∈ΦS +i (z) ui(φi ⋄ z) − ui(z) +� +≤ 0. +(3) +However, here we claim that this solution concept inherits some weaknesses from the non-convexity of the equilibria +set that we proved in Theorem 5.1. Indeed, consider the same instance of Theorem 5.1 and consider the uniform +distribution µ over {z1, z2}. In Theorem 5.1 we proved that maxφi∈ΦS +i (z1) ui(φi ⋄z1)−ui(z1) ≤ 0 for all i ∈ {1, 2} +and maxφi∈ΦS +i (z2) ui(φi ⋄ z2) − ui(z2) ≤ 0 for all i ∈ {1, 2} and thus Equation (3) holds over the distribution µ. +However we show that the expected correlated strategy z3 derived from distribution µ, i.e., z3 = Ez∼µ[z] = 1 +2z1 + +1 +2z2, it is not a feasible equilibrium, or an approximate one. +Indeed, in Theorem 5.1, we proved that maxφ2∈ΦS +2 (z3) u2(φ2 ⋄z3)−u2(z3) ≥ 1 +3, showing that the average correlated +strategies returned by their Algorithm is not an equilibrium nor close to it. +This comes from the peculiar fact about Constrained Phi-equilibria that exhibit non-convex set of solutions, which is +in striking contrast with the unconstrained case. Indeed the guarantees of Equilibria (3) would imply that Ez∼µ[z] is +a equilibrium in the unconstrained case in which the set of equilibria is convex. +B +Proofs Omitted from Section 2 +Theorem 2.1. Given a cost-constrained normal-form game Γ and a set Φ of deviations, if Assumption 2 is satisfied, +then Γ admits a constrained Phi-equilibrium. +Proof. With assumption 2 Altman and Shwartz (2000, Theorem 2.1) proves the existence of a constrained Nash equi- +librium. In our setting this is equivalent to a product distribution z = ⊗i∈[N]xi so that it is a Constrained Phi- +equilibrium for any set of deviations Φi.9 This is easily seen by observing that a Constrained Nash Equilibria is +defined as: +� +a∈A +ui + + � +j∈[N] +xj(aj) + + ≥ +� +a∈A +ui + +˜xi(aj) +� +j∈[N]\{i} +xi(ai) + + +for all ˜xi ∈ ∆(Ai) s.t. xi ⊗ x−i ∈ S. +On the other hand it easily seen that for all φi ∈ Φi(z) there exists some ˜xi ∈ ∆(Ai) such that +φi ⋄ +� +⊗j∈[N]xj +� += ˜xi ⊗ x−i +and ˜xi ⊗ x−i ∈ S. +This is proved by the following calculations: +φi ⋄ +� +⊗j∈[N]xj +� +[ai, a−i] := +� +b∈Ai +φi[b, ai]xi(b)x−i(a−i) +(4) += ˜xi(ai) ⊗ x−i(a−i), +(5) +where ˜xi(ai) := � +b∈Ai φi[b, ai]xi(b) and ˜xi ∈ ∆(Ai) since, by definition, � +ai∈Ai φi[b, ai] = 1 for all b ∈ Ai. +This proves that a Constrained Nash Equilibrium is a Phi-Constrained Equilibrium for all Φ. +C +Proofs Omitted from Section 3 +Theorem 3.1 (Hardness). For any constant α > 0, the problem APXCPE(α, (α/s)2) is NP-hard, where s is the +number of players’ actions in the instance given as input. +9As common in the normal form game literature, for any distribution x ∈ ∆(X) and y ∈ ∆(Y ), x ⊗ y ∈ ∆(X × Y ) is the +product distribution defined as (x ⊗ y)[a, b] = x[a]y[b] for a ∈ X and b ∈ Y . +12 + +ARXIV PREPRINT - FEBRUARY 1, 2023 +Proof. We reduce from GAP-INDEPENDENT-SET, which is a promise problem that formally reads as follows: +given an δ > 0 and a graph G = (V, E), with set of nodes V and set of edges E, determine whether G admits +an independent set of size at least |V |1−δ or all the independent sets of G have size smaller than |V |δ. +GAP- +INDEPENDENT-SET is NP-hard for every δ > 0 (H˚astad, 1999; Zuckerman, 2007). +Let ℓ = |V | and α > 0 be the desired approximation factor. Given an instance of GAP-INDEPENDENT-SET, +we build an instance such that if there exists an independent set of size ℓ1−δ, then there exists a Constrained Phi- +equilibrium with social welfare 1. Otherwise, if all the independent sets have size at most ℓδ, all the Constrained +ǫ-Phi-equilibria have social welfare at most α/2. We can use any δ > 0, since we simply need ℓδ < ℓ1−δ. Moreover, +we take ǫ = +α2 +128ℓ2 . As we will see, ℓ will be smaller than the number of action of the players, satisfying the condition +in the statement. +Construction. +The first player has a set of actions A1 that includes actions a0, a1, a2 and an action av for each +v ∈ V . Moreover, the first player has an action aF .10 The second player has a set of actions A2 that includes actions +av and ¯av for each v ∈ V . Moreover, the second player has an action aF . Let γ = η = α/8. The utility of the first +agent is as follows: +• u1(a0, a) = γ + 1 +2η for all a ∈ A2 \ {aF}, +• u1(a1, av) = γ + η and u1(a2, av) = γ for all v ∈ V . +• u1(a1, ¯av) = γ and u1(a2, ¯av) = γ + η for all v ∈ V . +• u1(av, av) = u1(av, ¯av) = γ for all v ∈ V +• u1(av, av′) = γ and u1(av, ¯av′) = γ + +ℓ−ℓ1−δ +ℓ−ℓ1−δ−1η for all v′ ̸= v. +• u1(aF , a) = 0 for each a ∈ A2. +• u1(a, aF ) = 0 for each a ∈ A1. +The utility of the second agent is u2(a0, a) = 1 for each a ∈ A2 \ {aF} and 0 otherwise. +There is a cost function cv for each v ∈ V , which is common to both the agents. For each v ∈ V , the cost function cv +is such that +• cv(av, av′) = −1 for each v′ ̸= v, (v, v′) ∈ E, +• cv(av, av′) = 0 for each v′ ̸= v, (v, v′) /∈ E, +• cv(av, av) = 1 for each v ∈ V . +• cv(aF , a) = − 1 +4ℓ2 for each a ∈ A2. +• cv(a, aF ) = − 1 +4ℓ2 for each a ∈ A1. +• For every other action profile the cost is 0. +We dropped the player index from the cost functions c as they are equal to both players. +Moreover, we set of deviations Φi = Φi,ALL for both players i ∈ {1, 2}. +Notice that the instance satisfies Assumption 2. Indeed, the deviation φi such that φi[a, aF ] = 1 for all a ∈ Ai for +i ∈ {1, 2}, that deviates deterministically to aF is always strictly feasible for both player 1 and player 2. Moreover, its +cost is polynomial in the instance size. +Completeness. +We show that if there exists an independent set of size ℓ1−δ, then the social welfare of an optimal +Constrained Phi-equilibria is at least 1. Let V ∗ be an independent set of size ℓ1−δ. We build a Constrained Phi- +equilibria z with social welfare at least 1. Consider the correlated strategy such that z[a0, av] = +1 +2ℓ1−δ for all v ∈ V ∗, +while z[a0, ¯av] = +1 +2(ℓ−ℓ1−δ) for all v /∈ V ∗. All the other action profiles have probability 0. +10This action is needed only to satisfy the strictly feasibility assumption. +13 + +ARXIV PREPRINT - FEBRUARY 1, 2023 +It is easy to see that the correlated strategy has social welfare at least 1 since player 1 always plays action a0 and +u2(a0, a) = 1 for all a ∈ A2. Moreover, it is easy to verify that it is safe since cv(a0, a) ≤ 0 for each a ∈ A2. Hence, +to show that z is an Constrained Phi-equilibria we only need to prove that it satisfies the incentive constraints. The +incentive constraints of the second player are satisfied since they obtain the maximum possible utility, i.e., 1. +Consider now a possible deviation of the first player φ1 ∈ Φ1. As a first step, we compute the expected utility of a +strategy φ1. Let us define the following quantities: +• T 1 = � +v∈V ∗ φ1[a0, av] +�� +z[a0, av] + z[a0, ¯av] + � +v′̸=v z[a0, av′] +� +γ + +� +γ + +ℓ−ℓ1−δ +ℓ−ℓ1−δ−1η +� � +v′̸=v z[a0, ¯av] +� +• T 2 = � +v /∈V ∗ φ1[a0, av] +�� +z[a0, av] + z[a0, ¯av] + � +v′̸=v z[a0, av′] +� +γ + +� +γ + +ℓ−ℓ1−δ +ℓ−ℓ1−δ−1η +� � +v′̸=v z[a0, ¯av] +� +• T 3 = +� +γ + η +2 +� +φ1[a0, a0] + γ+η +2 (φ1[a0, a1] + φ1[a0, a2]) + γ +2(φ1[a0, a1] + φ1[a0, a2]) +We bound each component individually. +T 1 = +� +v∈V ∗ +φ1[a0, av] + + + +z[a0, av] + z[a0, ¯av] + +� +v′̸=v +z[a0, av′] + + γ + +� +γ + η +ℓ − ℓ1−δ +ℓ − ℓ1−δ − 1 +� � +v′̸=v +z[a0, ¯av] + + += +� +v∈V ∗ +φ1[a0, av] +�1 +2γ + 1 +2 +� +γ + η +ℓ − ℓ1−δ +ℓ − ℓ1−δ − 1 +�� += +� +v∈V ∗ +φ1[a0, av] +� +γ + η +2 +ℓ − ℓ1−δ +ℓ − ℓ1−δ − 1 +� +≤ +� +v∈V ∗ +φ1[a0, av](γ + η), +where in the last inequality we use +ℓ−ℓ1−δ +ℓ−ℓ1−δ−1 ≤ 2 for ℓ large enough. while +T 2 = +� +v /∈V ∗ +φ1[a0, av] + + + +z[a0, av] + z[a0, ¯av] + +� +v′̸=v +z[a0, av′] + + γ + +� +γ + η +ℓ − ℓ1−δ +ℓ − ℓ1−δ − 1 +� � +v′̸=v +z[a0, ¯av] + + += +� +v /∈V ∗ +φ1[a0, av] +��1 +2 + +1 +2(ℓ − ℓ1−δ) +� +γ + +�1 +2 − +1 +2(ℓ − ℓ1−δ) +� � +γ + η +ℓ − ℓ1−δ +ℓ − ℓ1−δ − 1 +�� += +� +v /∈V ∗ +φ1[a0, av] +� +γ + η +2 +� +ℓ − ℓ1−δ +ℓ − ℓ1−δ − 1 − +1 +ℓ − ℓ1−δ − 1 +�� += +� +v /∈V ∗ +φ1[a0, av] +� +γ + η +2 +� +. +Finally, +T 3 = [a0, a0] +� +γ + η +2 +� ++ γ + η +2 +([a0, a1] + [a0, a2]) + γ +2 ([a0, a1] + [a0, a2]) += +� +γ + η +2 +� +([a0, a0] + φ1[a0, a1] + φ1[a0, a2]) +Finally, the utility of a deviation φ1 is +� +a1∈A1,a2∈A2 +� +a∈A1 +φ1[a1, a]z[a1, a2]u1(a, a2) += +� +a∈A1,a2∈A2 +φ1[a0, a]z[a0, a2]u1(a, a2) += T 1 + T 2 + T 3 +≤ (γ + η) +� +v∈V ∗ +φ1[a0, av] + +� +γ + η +2 +� � +v /∈V ∗ +φ1[a0, av] + +� +γ + η +2 +� +(φ[a0, a0] + φ1[a0, a1] + φ1[a0, a2]) +14 + +ARXIV PREPRINT - FEBRUARY 1, 2023 += η +2 +� +v∈V ∗ +φ1[a0, av] + +� +γ + η +2 +� +(1 − φ1[a0, aF]) +Now, we show that no deviation φ1 ∈ Φ1 is both safe and increases player 1 utility. In particular, we show that if a +strategy φ1 increases the utility than it is not safe. Indeed, if φ1 increases the utility, then +� +a1∈A1, +a2∈A2 +� +a∈A1 +φ1[a1, a]z[a1, a2]u1(a, a2) > γ + η +2 +This implies that +η +2 +� +v∈V ∗ +φ1[a0, av] + +� +γ + η +2 +� +(1 − φ1[a0, aF ]) > γ + η +2 +and +� +v∈V ∗ +φ1[a0, av] > 1 +2φ1[a0, aF ] +(6) +Next, we show that any φ1 that increases the utility (and hence that satisfies Eq (6)) is not a feasible deviation. First, +notice that equation (6) implies that there is a ¯v ∈ V ∗ such that +φ1[a0, a¯v] > 1 +2ℓφ1[a0, aF ]. +(7) +Then, we show that the deviation φ1 violates the constraint c¯v. In particular, +� +a1∈A1,a2∈A2 +� +a∈A1 +φ1[a1, a]z[a1, a2]cv(a, a2) = φ1[a0, a¯v]z[a0, a¯v]1 − 1 +4ℓ2 φ1[a0, aF ] − +� +v∈V ∗:(v,¯v)∈E +φ1[a0, a¯v]z[a0, av]1 += φ1[a0, a¯v]z[a0, a¯v] − 1 +4ℓ2 φ1[a0, aF ] += +1 +2ℓ1− 1 +ℓ φ1[a0, a¯v] − 1 +4ℓ2 φ1[a0, aF ] +> φ1[a0, a¯v] +� +1 +2ℓ1− 1 +ℓ − 1 +2ℓ +� +≥ 0, +where the second inequality holds since V ∗ is an independent set, and the second-to-last inequality by Equation (7). +Hence, there is no deviation φ1 that increases players 1 utility and that is safe. This concludes the first part of the +proof. +Soundness. We show that if there exists a Constrained w-Phi-equilibria with social welfare α/2, then there exists an +independent set of size strictly larger than ℓδ, reaching a contradiction. Suppose by contradiction that there exists a +Constrained ǫ-Phi-equilibrium z with social welfare strictly greater than α/2. Thus, +� +a′∈A2\{aF } +z[a0, a′] · 1 + +� +a∈A1,a′∈A2 +(γ + η) ≥ +� +a∈A1,a′∈A2 +z[a, a′](u1(a, a′) + u2(a, a′)) ≥ α/2, +where the first inequality comes from u2(a0, a′) = 1 for each a′ ∈ A2 \ {aF } and 0 otherwise, and u1(a, a′) ≤ γ + η +for each a ∈ A1 and a′ ∈ A2. This implies +� +a′∈A2 +z[a0, a′] ≥ α/4. +(8) +Then, we show that z assigns similar probabilities on the set of action profiles {a0, av}v∈V and {a0, ¯av}v∈V Given +an a ∈ A1, let φa ∈ Φ1 be a deviation of the first player such that φa[a0, a] = 1 and φa[a′, a′] = 1 for each a′ ̸= a0. +Since z is an Constrained ǫ-Phi-equilibrium there is no feasible deviation φa that increases the utility of player 1 by +more than ǫ. This implies that +����� +� +v∈V +z[a0, av] − +� +v∈V +z[a0, ¯av] +����� ≤ 2ǫ +η . +(9) +15 + +ARXIV PREPRINT - FEBRUARY 1, 2023 +Indeed, if +� +v∈V +z[a0, av] > +� +v∈V +z[a0, ¯av] + 2ǫ +η , +(10) +then the deviation φa1 has utility at least +� +v∈V +z[a0, av]φa1[a0, a1](γ + η) + z[a0, ¯av]φa1[a0, a1]γ + +� +a∈A1\{a0} +� +a′∈A2 +z[a, a′]φa1[a, a]ui(a, a′) += η +� +v∈V +z[a0, av] + γ +� +v∈V +(z[a0, av] + z[a0, ¯av]) + +� +a∈A1\{a0} +� +a′∈A2 +z[a, a′]φa1[a, a]ui(a, a′) +> η +2 +� +2ǫ +η + +� +v∈V +(z[a0, av] + z[a0, ¯av]) +� ++ γ +� +v∈V +(z[a0, av] + z[a0, ¯av]) ++ +� +a∈A1\{a0} +� +a′∈A2 +z[a, a′]φa1[a, a]ui(a, a′) +≥ ǫ + +�η +2 + γ +� � +v∈V +(z[a0, av] + z[a0, ¯av]) + +� +a∈A1\{a0} +� +a′∈A2 +z[a, a′]φa1[a, a]ui(a, a′) +≥ u1(z) + ǫ, +where the first inequality comes from adding � +v∈V z[a0, av] to both sides of Equation (10). Moreover, φa1 is feasible +since for each constraint c¯v, ¯v ∈ V , it has cost +� +v∈V +(z[a0, av]φa1[a0, a1]c¯v(a1, av) + z[a0, ¯av]φa1[a0, a1]c¯v(a1, av)) ++ +� +a∈A1\{a0} +� +a′∈A2 +z[a, a′]φa1[a, a]c¯v(a, a′) += +� +v∈V +(z[a0, av]φa1[a0, a1]c¯v(a0, av) + z[a0, ¯av]φa1[a0, a1]c¯v(a0, ¯av)) ++ +� +a∈A1\{a0} +� +a′∈A2 +z[a, a′]φa1[a, a]c¯v(a, a′) += c¯v(z) ≤ 0. +A similar argument shows that if � +v∈V z[a0, av] < � +v∈V z[a0, ¯av] − 2ǫ +η then the deviation φa2 is safe and increases +the utility. As a consequence of Equation (9), it holds +2 +� +v∈V +z[a0, ¯av] ≥ +� +v∈V +(z[a0, av] + z[a0, ¯av]) − ǫ +η = +� +a∈A2\{aF } +z[a0, a] − 2ǫ +η , +(11) +where the first inequality comes from adding � +v∈V z[a0, ¯av] to both sides of � +v∈V z[a0, ¯av]| ≥ � +v∈V z[a0, av]− 2ǫ +η +The next step is to show that it is if there is no safe deviation φav, v ∈ V , that increases the utility, then there exists +an independent set of size larger than ℓδ. Since z is an Constrained ǫ-Phi-equilibrium, for each av, v ∈ V one of the +following two conditions holds: i) φav /∈ ΦS +1 (z) or ii) u1(φav ⋄ z) ≤ u1(z) + ǫ. Let V 1 ⊆ V be the set of vertexes +v such that φav is not safe, i.e., φav /∈ ΦS +1 (z), and V 2 = V \ V 1 be the set of v such that φav does not increase the +utility by more than ǫ and are not in V 1, i.e., u1(φav ⋄ z) ≤ u1(z) and φav ∈ ΦS +1 (z). We show that |V 2| ≤ ℓ − ℓ1−δ. +Indeed, for each v ∈ V 2, deviation φav does not increase the utility and hence it holds: +γ +� +a∈A2\{aF } +z[a0, a] + η +ℓ − ℓ1−δ +ℓ − ℓ1−δ − 1 +� +v′̸=v +z[a0, ¯av′] + +� +a∈A1\{a0} +� +a′∈A2 +z[a, a′]φa1[a, a]ui(a, a′) += +� � +v′∈V +z[a0, a] + z[a0, ¯av] +� +φ[a0, av]γ + +� +v′̸=v +φ[a0, av]z[a0, ¯av′] +� +γ + η +ℓ − ℓ1−δ +ℓ − ℓ1−δ − 1 +� ++ +� +a∈A1\{a0} +� +a′∈A2 +z[a, a′]φa1[a, a]ui(a, a′) +16 + +ARXIV PREPRINT - FEBRUARY 1, 2023 +≤ u1(z) + ǫ += +� +γ + η +2 +� +� +a∈A2\{aF } +z[a0, a] + +� +a∈A1\{a0} +� +a′∈A2 +z[a, a′]ui(a, a′) + ǫ, +where the inequality holds since the lhs is the utility of the deviation φav. +This implies +�� +v′ +z[a0, ¯av′] − z[a0, ¯av] +� +η +ℓ − ℓ1−δ +ℓ − ℓ1−δ − 1 ≤ η +2 +� +a∈A2\{aF } +z[a0, a] + ǫ ≤ η +� +v∈V +z[a0, ¯av] + 2ǫ, +where the last inequality holds by Equation (11). Hence, +z[a0, ¯av] +ℓ − ℓ1−δ +ℓ − ℓ1−δ − 1 ≥ +� +ℓ − ℓ1−δ +ℓ − ℓ1−δ − 1 − 1 +� � +v′ +z[a0, ¯av′] − 2ǫ/η, +and +¯z[a0, av] ≥ +1 +ℓ − ℓ1−δ +� +v′ +z[a0, ¯av′] − 2ǫ/η. +(12) +Suppose that |V 2| > ℓ − ℓ1−δ, and hence Equation (12) is satisfied by at least |V 2| ≥ ℓ − ℓ1−δ + 1 vertexes. We need +the following inequality. +1 +ℓ +� +v′ +z[a0, ¯av′] ≥ 1 +ℓ +� +a∈A2\{aF } +z[a0, a] − 2ǫ +ℓη ≥ α +4ℓ − 2ǫ +ℓη = α +4ℓ − α +8ℓ3 ≥ α +8ℓ = 2ℓ +η +� α2 +16ℓ2 +� += 2ℓ +η ǫ +(13) +where the first inequality comes from Equation (11), and the second one by Equation (8). Then, summing over the +|V 2| inequalities we get +� +v∈V 2 +¯z[a0, av] ≥ (ℓ − ℓ1−δ + 1) +� +1 +ℓ − ℓ1−δ +� +v′ +z[a0, ¯av′] − 2ǫ/η +� +≥ +� +v′ +z[a0, ¯av′] + 1 +ℓ +� +v′ +z[a0, ¯av′] − 2ℓ ǫ +η +> +� +v′ +z[a0, ¯av′], +where the last inequality follows from equation (13). Hence, we reach a contradiction and |V 2| ≤ ℓ − ℓ1−δ. +To conclude the proof, we show that V 1 is an independent set. Since |V 1| ≥ |V | − |V 2| = ℓ1−δ we reach a +contradiction. Let v and v′ be two nodes in V 1 and w.l.o.g. let z[a0, av] ≥ z[a0, av′]. We show that (v, v′) /∈ E. +Since v′ ∈ V 1, φav is not a safe deviation for player 1 with respect to constraint cv′. if (v, v′) ∈ E, then +� +a1∈A1,a2∈A2 +� +a∈ A1 +φ[a1, a]z[a1, a2]cv(a, a2) += z[a0, a′ +v] − +� +v′′:(v′′,v′)∈E +z[a0, av′′] − 1 +4ℓz[a0, aF]cv(a, a2) ++ +� +a1∈A1\{a0},a2∈A2 +� +a∈A1 +φ[a1, a]z[a1, a2]cv(a, a2) +≤ z[a0, a′ +v] − z[a0, av] − 1 +4ℓz[a0, aF ]cv(a, a2)+ ++ +� +a1∈A1\{a0},a2∈A2 +� +a∈A1 +φ[a1, a]z[a1, a2]cv(a, a2) +≤ − 1 +4ℓz[a0, aF ]cv(a, a2) + +� +a1∈A1\{a0},a2∈A2 +� +a∈A1 +φ[a1, a]z[a1, a2]cv(a, a2) +17 + +ARXIV PREPRINT - FEBRUARY 1, 2023 += cv(z) ≤ 0. +Hence, (v, v′) /∈ E. Since V 1 is an independent set of size at least ℓ1−δ we reach a contradiction. This concludes the +proof. +D +Proofs Omitted from Section 4 +Lemma 4.1. For every z ∈ ∆A and i ∈ N, it holds +max +φi∈ΦS +i (z) ui(φi ⋄ z) = sup +φi∈Φi +inf +ηi∈Rm ++ +� +ui(φi ⋄ z) − η⊤ +i ci(φi ⋄ z) +� += +inf +ηi∈Rm ++ +sup +φi∈Φi +� +ui(φi ⋄ z) − η⊤ +i ci(φi ⋄ z) +� +. +Proof. First, it is easy to see that +sup +φi∈ΦS +i (z) +ui(φi ⋄ z) = sup +φi∈Φi +inf +ηi∈Rm ++ +� +ui(φi ⋄ z) − η⊤ +i ci(φi ⋄ z) +� +. +Indeed, for every φi /∈ ΦS +i (z), it holds that the vector ci(φi ⋄ z) has at least one positive component, and, thus, the +vector of Lagrange multipliers ηi can be selected so that ui(φi ⋄ z) − η⊤ +i ci(φi ⋄ z) goes to −∞. This implies that +the supremum over Φi cannot be attained in ΦS +i (z). On the other hand, for every φi ∈ ΦS +i (z), all the components of +ci(φi ⋄ z) are negative, and, thus, the inf is achieved by ηi = 0. This proves the first equality. +Then, the second equality directly follows from the generalization of the max-min theorem for unbounded domains +(see (Ekeland and Temam, 1999, Proposition 2.3)), which allows us to swap the sup and the inf. +Lemma D.1. For any two real-valued functions f(x) and g(x) with g(x) ≤ c then min(f(x), g(x)) ≤ min(f(x), c). +Proof. We can identify three sets I1, I2 and I3 defined as follows: +I1 := {x s.t. f(x) ≥ c} +I2 := {x s.t. g(x) ≤ f(x) ≤ c} +I3 := {x s.t. f(x) ≤ g(x) ≤ c}. +Then for all x ∈ I1 we have that min(f(x), c) = c ≥ min(f(x), g(x)) = g(x), while for all x ∈ I2 we have +that min(f(x), c) = f(x) ≥ min(f(x), g(x)) = g(x). Finally for all x ∈ I3 we have min(f(x), c) = f(x) = +min(f(x), g(x)) = f(x). In all three sets we have that min(f(x), c) ≥ min(f(x), g(x)). +Lemma D.2. For all ηi ∈ Dc it holds that +sup +φi∈Φi +� +ui(φi ⋄ z) − η⊤ +i ci(φi ⋄ z) +� +≥ 1 +Proof. Thanks to Assumption 2 we have that for all z ∈ ∆(A) we have that there exists ˜φi ∈ ΦS +i (z) such that +ci(˜φi ⋄ z) ⪯ −ρ1. Then, for all ηi ∈ Dc we have: +η⊤ +i ci(˜φi ⋄ z) ≤ −ρ∥ηi∥1 ≤ −1. +This easily concludes the proof of the statement +sup +φi∈Φi +� +ui(φi ⋄ z) − η⊤ +i ci(φi ⋄ z) +� +≥ ui(˜φi ⋄ z) − η⊤ +i ci(˜φi ⋄ z) ≥ 1, +as ui is positive. +Lemma D.3. For all ηi ∈ D we have +inf +ηi∈D sup +φi∈Φi +� +ui(φi ⋄ z) − η⊤ +i ci(φi ⋄ z) +� +≤ 1 +18 + +ARXIV PREPRINT - FEBRUARY 1, 2023 +Proof. Since ui ≤ 1 we have that +inf +ηi∈D sup +φi∈Φi +� +ui(φi ⋄ z) − η⊤ +i ci(φi ⋄ z) +� +≤ 1 − sup +ηi∈D +inf +φi∈Φi η⊤ +i ci(φi ⋄ z). +Next we claim that sup +ηi∈D +inf +φi∈Φi η⊤ +i ci(φi ⋄ z) ≥ 0. This follows from the fact that for all negative components of +ci(φi ⋄ z) then the corresponding components of ηi will be 0. This concludes the statement. +Lemma 4.2. Let D := +� +η ∈ Rm ++ | ||η||1 ≤ 1/ρ +� +. Then, for every z ∈ ∆A and i ∈ N, it holds: +max +φi∈ΦS +i (z) ui(φi ⋄ z) = max +φi∈Φi min +ηi∈D +� +ui(φi ⋄ z) − η⊤ +i ci(φi ⋄ z) +� += min +ηi∈D max +φi∈Φi +� +ui(φi ⋄ z) − η⊤ +i ci(φi ⋄ z) +� +. +Proof. In Lemma 4.1 we already showed that: +sup +φi∈ΦS +i (z) +ui(φi ⋄ z) = sup +φi∈Φi +inf +ηi∈Rm ++ +� +ui(φi ⋄ z) − η⊤ +i ci(φi ⋄ z) +� += +inf +ηi∈Rm ++ +sup +φi∈Φi +� +ui(φi ⋄ z) − η⊤ +i ci(φi ⋄ z) +� +. +Note that to prove the statement it is enough to prove that: +inf +ηi∈Rm ++ +sup +φi∈Φi +� +ui(φi ⋄ z) − η⊤ +i ci(φi ⋄ z) +� += inf +ηi∈D sup +φi∈Φi +� +ui(φi ⋄ z) − η⊤ +i ci(φi ⋄ z) +� +and more specifically that: +inf +ηi∈Rm ++ +sup +φi∈Φi +� +ui(φi ⋄ z) − η⊤ +i ci(φi ⋄ z) +� +≥ inf +ηi∈D sup +φi∈Φi +� +ui(φi ⋄ z) − η⊤ +i ci(φi ⋄ z) +� +since the reverse inequality holds trivially. We can show this by the following inequalities: +inf +ηi∈Rm ++ +sup +φi∈Φi +� +ui(φi ⋄ z) − η⊤ +i ci(φi ⋄ z) +� += min +� +inf +ηi∈D sup +φi∈Φi +� +ui(φi ⋄ z) − η⊤ +i ci(φi ⋄ z) +� +, inf +ηi∈Dc sup +φi∈Φi +� +ui(φi ⋄ z) − η⊤ +i ci(φi ⋄ z) +� +� +≥ min +� +inf +ηi∈D sup +φi∈Φi +� +ui(φi ⋄ z) − η⊤ +i ci(φi ⋄ z) +� +, 1 +� += inf +ηi∈D sup +φi∈Φi +� +ui(φi ⋄ z) − η⊤ +i ci(φi ⋄ z) +� +, +where the first inequality hold thanks to Lemma D.1 and Lemma D.2, while that last equation follows from Lemma D.3. +Lemma 4.4. Given any 0 < δ ≤ ǫ and a δ-optimal set ˜D ⊆ D, the following holds: L ˜ +D,ǫ ≥ LD,0. +Proof. By definition we have that: L ¯ +D,ǫ = ℓ(˜z⋆), where ˜z⋆ is a solution to the problem +P1 := + + + + + + + + + +˜z⋆ ∈ arg max +z∈S +ℓ(z) s.t. +ǫ + ui(˜z⋆) ≥ max +φi∈Φi +� +ui(φi ⋄ ˜z⋆) − ˜η⋆,⊤ +i +ci(φi ⋄ ˜z⋆) +� +˜η⋆ +i ∈ arg inf +ηi∈ ¯ +D sup +φi∈Φi +� +ui(φi ⋄ ˜z⋆) − η⊤ +i ci(φi ⋄ ˜z⋆) +� +On the other hand, call z⋆ the optimal Constrained Phi-equilibrium. This is a solution to the problem: +P2 := + + + + + + + + + +z⋆ ∈ arg max +z∈S +ℓ(z) s.t. +ui(z⋆) ≥ max +φi∈Φi +� +ui(φi ⋄ z⋆) − η⋆,⊤ +i +ci(φi ⋄ z⋆) +� +η⋆ +i ∈ arg inf +ηi∈D sup +φi∈Φi +� +ui(φi ⋄ z⋆) − η⊤ +i ci(φi ⋄ z⋆) +� +19 + +ARXIV PREPRINT - FEBRUARY 1, 2023 +which has value LD,0 = ℓ(z⋆). +Moreover, thanks to Lemma 4.2 and since ¯D is δ-optimal we have that: +max +φi∈Φi +� +ui(φi ⋄ ˜z⋆) − ˜η⋆,⊤ +i +ci(φi ⋄ ˜z⋆) +� +≤ max +φi∈Φi +� +ui(φi ⋄ z⋆) − η⋆,⊤ +i +ci(φi ⋄ z⋆) +� ++ δ +which implies that feasible correlated strategies of problem P2 are feasible correlated strategies of problem P1, and +thus problem P1 as long as δ ≥ ǫ. Thus problem P1 is the problem of maximizing the same objective function over a +larger set then problem P2 and thus L ¯ +D,ǫ ≥ LD,0. +Lemma 4.5. For any τ > 0, the set Dτ is (τm)-optimal. +Proof. By Lemma 4.2, we know that for each player there exists an η⋆ +i ∈ D such that maxφ∈ΦS +i (z) ui(φi ⋄ z) = +max +φi∈Φi +� +ui(φi ⋄ z) − η⋆,⊤ +i +ci(φi ⋄ z) +� +. By construction of Dǫ there exists a ¯ηi ∈ Dǫ such that ||¯ηi − η⋆ +i ||∞ ≤ ǫ. Thus +max +φ∈ΦS +i (z) ui(φi ⋄ z) = max +φi∈Φi +� +ui(φi ⋄ z) − η⋆,⊤ +i +ci(φi ⋄ z) +� +≤ max +φi∈Φi +� +ui(φi ⋄ z) − ¯η⊤ +i ci(φi ⋄ z) +� ++ mǫ, +where the last inequality comes the fact that: +|(η⋆ +i − ¯ηi)⊤ci(φi ⋄ z)| ≤ ∥ci(φi ⋄ z)∥1∥η⋆ +i − ¯ηi∥∞ ≤ mǫ +as ci ∈ [−1, 1]m. +Lemma 4.6. For any τ > 0, the set Dτ is δ-optimal for δ = 2 +� +2τ log s/ρ, where s is the number of players’ actions. +Proof. The proof exploits a probability interpretation of the Lagrange multipliers. Let η⋆ be the optimal multipliers, +i.e., , η⋆ ∈ argminη∈D maxφi∈Φi +� +ui(φi ⋄ z) − η⊤ci(φi ⋄ z) +� +. Now consider a basis Γ = { 1 +ρej}j∈[m] ∪ {0} for D. +By Carathoedory’s theorem there exists a distribution γ ∈ ∆(Γ) such that η⋆ = � +η∈Γ γηη. Assume that ǫ and ρ are +such that 1/ǫρ is an integer and take 1/ρǫ samples from the distribution γ and call ˜η the resulting empirical mean. +First, we argue that ˜η ∈ Dǫ. Indeed ˜ηj = +kj +1/ρǫ +1 +ρ = ǫ +� +kj +1/ρǫ +1 +ρǫ +� += ǫkj where kj ∈ N and thus we have that ˜η ∈ Dǫ.11 +Now we show that with high probability ˜η ∈ Dǫ is close (in terms of utilities) to the optimal multiplier η⋆. First +observe that: +η⋆,⊤ +i +ci(φi ⋄ z) := +� +ai∈Ai,bi∈Ai + +φi[b, ai] +� +a−i∈A−i +η⋆,⊤ci(ai, a−i)z[b, a−i] + + +(14a) +≤ +� +ai∈Ai,bi∈Ai + +φi[b, ai] + +δai,b + +� +a−i∈A−i +˜η⊤ci(ai, a−i)z[b, a−i] + + + + +(14b) += +� +ai∈Ai,bi∈Ai + +φi[b, ai] +� +a−i∈A−i +˜η⊤ci(ai, a−i)z[b, a−i] + + + +� +ai∈Ai,bi∈Ai +φi[b, ai]δai,b +(14c) += ˜η⊤ +i ci(φi ⋄ z) + +� +ai∈Ai,bi∈Ai +φi[b, ai]δai,b +(14d) +where the inequality comes from applying the Hoeffeding’s inequality to every ai, b ∈ Ai: +������ +� +a−i∈A−i +(˜η − η⋆)⊤ ci(ai, a−i)z[b, a−i] +������ +≤ δai,b +11If ǫ if not such that 1/ρǫ ∈ N then the one can take ⌈1/ρǫ⌉ samples from γ ∈ ∆(Γ) and then the statement hold for a slightly +smaller ǫ′ < ǫ defined as ǫ′ := +1 +⌈1/ρǫ⌉ +1 +ρ. +20 + +ARXIV PREPRINT - FEBRUARY 1, 2023 +where +δai,b += +2 +ρ +� +2 +1/ρǫ log +� +2 +pai,b +� � +� +a−i∈A−i z[b, a−i] +� +since +the +range +of +the +each +sample +is +1 +ρ +�� +a−i∈A−i z[b, a−i] +� +. +Moreover, for Hoeffeding’s inequality, for every ai, b ∈ Ai the above inequality holds with probability at least +1 − pai,b and thus holds for all the ai, b ∈ Ai simultaneously, with probability at least p := � +ai,b∈Ai pai,b. +If then we take pai,b := +1 +2|Ai|2 for all ai, b ∈ Ai, then we have that p = 1/2 > 0 and δ := δai,b = +2 +ρ +� +2 +1/ρǫ log (|Ai|) +� +� +a−i∈A−i z[b, a−i] +� +Now the following holds with probability at lest 1/2: +������ +� +a−i∈A−i +(˜η − η⋆)⊤ ci(ai, a−i)z[b, a−i] +������ +≤ δ + + +� +a−i∈A−i +z[b, a−i] + + , +∀ai, b ∈ Ai +The proof is concluded by observing plugging this definition of δ += +δai,b in Equation (14) yields +� +ai∈Ai,bi∈Ai φi[b, ai]δai,b = δ, and we can conclude that: +η⋆,⊤ +i +ci(φi ⋄ z) ≤ ˜η⊤ +i ci(φi ⋄ z) + δ. +This holds with positive probability, and thus shows the existence of such ˜η ∈ Dǫ for which the above inequality holds +and thus Dǫ is +� +2 +� +2ǫ +ρ log(|Ai|) +� +-optimal. +E +Proofs Omitted from Section 5 +Proposition 5.1. For instances I := (Γ, Φ) satisfying Assumption 3, the set of constrained ǫ-Phi-equilibria is convex. +Proof. Let z′ and z′′ be Constrained ǫ-Phi-equilibria that is for all i ∈ [N]: +ǫ + ui(z′) ≥ ui(φ′ +i ⋄ z′) +for φ′ ∈ arg max +φi∈ΦS +i +ui(φi ⋄ z′). Equivalently it holds for all i ∈ [N] that: +ǫ + ui(z′′) ≥ ui(φ′′ +i ⋄ z′′) +where φ′′ ∈ arg max +φi∈ΦS +i +ui(φi ⋄ z′′). For any z := αz′ + (1 − α)z′′ we have that: +ǫ + ui(z) = α (ǫ + ui(z′)) + (1 − α) (ǫ + ui(z′′)) +≥ αui(φ′ +i ⋄ z′) + (1 − α)ui(φ′′ +i ⋄ z′′) +≥ max +φi∈ΦS +i +ui(φi ⋄ z), +where the inequality holds for the linearity of ui, the first inequality as both z′ and z′′ are Constrained ǫ-Phi-equilibria +and the last inequality holds since the max is a convex operator. +Theorem 5.1. Restricted to instances I := (Γ, Φ) which satisfy Assumption 3, APXCPE(1, 0) admits a polynomial- +time algorithm. +Proof. APXCPE(1, 0) amounts to solving the following problem: +max +z∈S ℓ(z) +s.t. +(15a) +ui(z) ≥ max +φi∈ΦS +i +ui(φi ⋄ z) +∀i ∈ N, +(15b) +which can be written as an LP with (possibly) exponentially-many constraints, by writing a constraint for each vertex +of ΦS +i . We can find an exact solution to such an LP in polynomial time by means of the ellipsoid algorithm that uses +suitable separation oracle. Such an oracle solves the following optimization problem for every i ∈ N: +φ⋆ +i ∈ arg max +φi∈ΦS +i +ui(φi ⋄ z). +21 + +ARXIV PREPRINT - FEBRUARY 1, 2023 +Then, the oracle returns as a separating hyperplane the incentive constraint corresponding to a φ⋆ +i (if any) such that +ui(z) ≥ ui(φ⋆ +i ⋄ z). Since all the steps of the separation oracle can be implemented in polynomial time, the ellipsoid +algorithm runs in polynomial time, concluding the proof. +Theorem 5.2. Given an instance I := (Γ, Φ) satisfying Assumption 3, after T ∈ N>0 rounds, Algorithm 1 returns a +correlated strategy ¯zT ∈ ∆A that is a constrained ǫT -Phi-equilibrium with ǫT = O(1/ +√ +T). Moreover, each round of +Algorithm 1 runs in polynomial time. +Proof. Any regret minimizer Ri for ΦS +i guarantees that, for every φi ∈ ΦS +i : +T +� +t=1 +ui(φi ⋄ zt) − +T +� +t=1 +ui(φi,t ⋄ zt) ≤ ǫi,T T, +(16) +where ǫi,T = o(T ). Since xi,t[a] = � +b∈Ai φi,t[b, a]xi,t[b] for all a ∈ Ai, for every t ∈ [T ] and a = (ai, a−i) ∈ A: +(φi,t ⋄ zt)[ai, a−i] = +� +b∈Ai +φi,t[b, ai]z[b, a−i] += +� +b∈Ai +φi,t[b, ai] +� +xi,t[b] ⊗ x−i,t[a−i] +� += +� � +b∈Ai +φi,t[b, ai]xi,t[b] +� +⊗ x−i,t[a−i] += xi,t[ai] ⊗ x−i,t[a−i] += zt[ai, a−i], +Plugging the equation above into Equation (16), we get: +T +� +t=1 +ui(φi ⋄ zt) − +T +� +t=1 +ui(zt) ≤ ǫi,T T. +Now, since ¯zT := �T +t=1 zt and ui(z) is linear in z, we can conclude that, for every i ∈ N and φi ∈ ΦS +i : +ui(zT ) ≥ ui(φi ⋄ ¯zT ) − ǫi,T , +and, thus, by letting ǫT := maxi∈N ǫi,T we get that ¯zT satisfies the incentivize constrained for being a constrained +ǫT -Phi-equilibrium. We are left to verify that ¯zT ∈ S, namely ci(¯zT ) ≤ 0 for all i ∈ N. This readily proved as: +ci(¯zT ) = 1 +T +T +� +t=1 +ci(zt) += 1 +T +T +� +t=1 +ci(φi,t ⋄ zt) += 1 +T +T +� +t=1 +˜ci(φi,t) +≤ 0, +where the first equality holds since ci is linear, the second equality holds thanks to zt = φi,t ⋄ zt, the third one by +Assumption 3, while the inequality holds since φi,t ∈ ΦS +i . This concludes the proof of the first part of the statement. +In conclusion, Algorithm 1 runs in polynomial time as finding xi,t[a] = � +b∈Ai φi,t[b, ai]xi,t[b] for all a ∈ Ai is +equivalent to finding a stationary distribution of a Markov Chain, which can be done in polynomial time. Moreover, +we can implement the regret minimizers Ri over the polytopes ΦS +i so that their operations run in polynomial time, +such as, e.g., online gradient descent; see (Hazan et al., 2016). +Theorem 5.4. For instances I := (Γ, ΦCCE) such that ci(a) = ci(a′) for every player i ∈ N and action profiles +a, a′ ∈ A : ai = a′ +i, Assumption 3 holds. +22 + +ARXIV PREPRINT - FEBRUARY 1, 2023 +Proof. Since the costs ci(a) of player i ∈ N only depends on player i’s action ai and not on the actions of other +players, it is possible to show that there exists ˜ci : ΦCCE → [−1, 1]m such that the following holds for every z ∈ ∆A: +˜ci(φi) := ci(φi ⋄ z). +Indeed, for every φi ∈ ΦCCE, by definition of ΦCCE there exists a probability distribution h ∈ ∆Ai : φi[b, a] = h[a] +for all b, a ∈ Ai. Then, for every ai ∈ Ai and a−i ∈ A−i, we can write: +(φi ⋄ z)[ai, a−i] = +� +b∈Ai +φi[b, ai]z[b, a−i] += +� +b∈Ai +h[ai]z[b, a−i] += h[ai] +� +b∈Ai +z[b, a−i]. +Moreover, it holds: +ci(φi ⋄ z)[ai, a−i] = +� +a∈A +ci(a)(φi ⋄ z)[ai, a−i] += +� +a∈A +ci(a)h[ai] +� +b∈Ai +z[b, a−i] += +� +ai∈Ai +ci(ai, ·)h[ai] +� +a−i∈A−i +� +b∈Ai +z[b, a−i] += +� +ai∈Ai +ci(ai, ·)h[ai], +which only depends on φi, as desired. Notice that, in the equations above, for every a ∈ Ai we let ci(a, ·) be the +(unique) value of ci(a) for all a ∈ A : ai = a. +23 + diff --git a/1dFRT4oBgHgl3EQfmDfL/content/tmp_files/load_file.txt b/1dFRT4oBgHgl3EQfmDfL/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..ad1b3b885ce6d4c5968bc64c2b009aa9ab6ca38f --- /dev/null +++ b/1dFRT4oBgHgl3EQfmDfL/content/tmp_files/load_file.txt @@ -0,0 +1,983 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf,len=982 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='13600v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='GT] 31 Jan 2023 CONSTRAINED PHI-EQUILIBRIA ARXIV PREPRINT Martino Bernasconi Politecnico di Milano martino.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='bernasconideluca@polimi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='it Matteo Castiglioni Politecnico di Milano matteo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='castiglioni@polimi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='it Alberto Marchesi Politecnico di Milano alberto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='marchesi@polimi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='it Francesco Trov`o Politecnico di Milano francesco1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='trovo@polimi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='it Nicola Gatti Politecnico di Milano nicola.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='gatti@polimi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='it February 1, 2023 ABSTRACT The computational study of equilibria involving constraints on players’ strategies has been largely neglected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' However, in real-world applications, players are usually subject to constraints ruling out the feasibility of some of their strategies, such as, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=', safety requirements and budget caps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Computational studies on constrained versions of the Nash equilibrium have lead to some results under very stringent assumptions, while finding constrained versions of the correlated equilibrium (CE) is still unexplored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' In this paper, we introduce and computationally characterize constrained Phi-equilibria—a more general notion than constrained CEs—in normal-form games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' We show that computing such equilibria is in general computationally intractable, and also that the set of the equilibria may not be convex, providing a sharp divide with unconstrained CEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Nevertheless, we provide a polynomial-time algorithm for computing a constrained (approximate) Phi-equilibrium maximizing a given linear function, when either the number of constraints or that of players’ actions is fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Moreover, in the special case in which a player’s constraints do not depend on other players’ strategies, we show that an exact, function-maximizing equilibrium can be computed in polynomial time, while one (approximate) equilibrium can be found with an efficient decentralized no-regret learning algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' 1 Introduction Over the last years, equilibrium computation problems have received a terrific attention from AI and ML re- search (Brown and Sandholm, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Bakhtin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=', 2022), as game-theoretical equilibrium notions provide a prin- cipled framework to deal with multi-player decision-making problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Most of the works on equilibrium computation problems focus on classical solution concepts—such as the well-known Nash equilibrium (NE) (Nash, 1951) and cor- related equilibrium (CE) (Aumann, 1974)—, thus neglecting the presence of constraints entirely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' However, in most of the real-world applications, the players are usually subject to constraints that rule out the feasibility of some of their strategies, such as, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=', safety requirements and budget caps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Thus, addressing equilibrium notions involving constraints is a crucial step needed for the operationalization of game-theoretic concepts into real-world settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' The study of equilibrium notions involving constraints was initiated by Arrow and Debreu (1954), who define the concept of generalized NE (GNE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' The GNE can be interpreted as an NE of a game where players’ strategies are subject to some constraints, which must be satisfied at the equilibrium and also determine which are the feasible players’ deviations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' However, given that computing a GNE is clearly PPAD-hard (Daskalakis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=', 2009), all the works dealing with the computation of GNEs (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=', (Facchinei and Kanzow, 2010)) provide efficient algorithms only in specific settings that require very stringent assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Most of the computationally challenges in finding GNEs are inherited from the NE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' In settings in which constrained are not involved, the computational issues of NEs are usually circumvented by considering weaker equilibrium notions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' ARXIV PREPRINT - FEBRUARY 1, 2023 Among them, those that have received most of the attention in the literature are the CE and its variations, which have been shown to be efficiently computable in several settings of interest (Papadimitriou and Roughgarden, 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Celli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Surprisingly, with the only exception of (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=', 2022) (see Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='2 for a detailed discussion on it), no work has considered the problem of computing CEs in constrained settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Thus, investigating whether the CE retains its nice computational properties when adding constraints on players’ strategies is an open interesting question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='1 Original Contributions In this paper, we introduce and computationally characterize constrained Phi-equilibria, starting, as it is customary, from the setting of normal-form games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Our equilibria include the constrained versions of the classical CE and all of its variations as special cases, by generalizing the notion of Phi-equilibria introduced by Greenwald and Jafari (2003) to constrained settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' In particular, constrained Phi-equilibria are defined as Phi-equilibria, but in games where players are subject to some constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Such constraints must be satisfied at the equilibrium, and, additionally, players are only allowed to undertake safe deviations, namely those that are feasible according to the constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Crucially, the set of safe deviations of a player does not only depend on the strategy of that player, but also on those of the others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' We start by showing that one of the most appealing computational properties of Phi-equilibria, namely that the set of the equilibria of a game is convex, is lost when moving to their constrained version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' This raises considerable computational challenges in computing constrained Phi-equilibria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Indeed, we formally prove a strong intractability result: for any factor α > 0, it is not possible, unless P = NP, to find in polynomial time a constrained (approximate) Phi-equilibrium which achieves a multiplicative approximation α of the optimal value of a given linear function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Then, in the rest of the paper, we show several ways in which such a negative result can be circumvented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' We prove that a constrained approximate Phi-equilibrium which maximizes a given linear function can be found in polynomial time, when either the number of constraints or that of players’ actions is fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Our results are based on a general algorithm that employs a non-standard “Lagrangification” of the constraints defining the set of safe deviations of a player.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Moreover, the algorithm needs a way of dealing with the non-convexity of the set of the equilibria, which we provide in the form of a clever discretization of the space of the Lagrange multipliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Finally, we focus on the special case in which the constraints defining the safe deviations of a player do not depend on the the strategies of the other players, but only on the strategy of that player.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' This includes constrained Phi- equilibria identifying a particular constrained version of the coarse CE by Moulin and Vial (1978a), in which the players’ strategies are subject to marginal cost constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' These arise in several real-world applications in which the players have bounded resources, such as, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=', budget-constrained bidding in auctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' In such a special case, we prove that a constrained (exact) Phi-equilibrium maximizing a given linear function can be computed in polynomial time, and we provide an efficient decentralized no-regret learning algorithm for finding one constrained (approximate) Phi-equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='2 Related Works GNEs Rosen (1965) initiated the study of the computational properties of GNEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' After that, several other works addressed the problem of computing GNEs by mainly exploiting techniques based on quasi-variational inequalities (see (Facchinei and Kanzow, 2010) for a survey).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' More recently, some works (Kanzow and Steck, 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Bueno et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Jordan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Goktas and Greenwald, 2022) also studied the convergence of iterative optimization algo- rithms to GNEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' In order to provide efficient algorithms, all these works need to introduce very stringent assumptions, which are even stronger than those required for the efficient computation of NEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Constrained Markov Games Equilibrium notions involving constraints have also been addressed in the literature on Markov games, with (Altman and Shwartz, 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Alvarez-Mena and Hern´andez-Lerma, 2006) being the first works introducing GNEs in such a field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' More recently, Hakami and Dehghan (2015) defined a notion of constrained CE in Markov games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' However, the incentive constraints in their notion of equilibrium only predicate on “pure” deviations, which, in presence of constraints, may lead to empty sets of safe deviations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Very recently, Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' (2022) gener- alize the work of Hakami and Dehghan (2015) by considering “mixed” deviations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' However, their algorithm provides rather weak convergence guarantees, as it only ensures that the returned solution satisfies incentive constraints in ex- pectation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Indeed, as we show in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='1, the set of constrained equilibria may not be convex (it is easy to see that Example 1 also applies to the setting studied by Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' (2022)), and, thus, the fact that incentive constraints are only satisfied in expectation does not necessarily imply that the “true” incentive constraints defining the equilibrium are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' We refer the reader to Appendix A for additional details on these aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' 2 ARXIV PREPRINT - FEBRUARY 1, 2023 2 Preliminaries In this section, we introduce all the preliminary definitions and results that are needed in the rest of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='1 Cost-constrained Normal-form Games In a normal-form game, there is a finite set N := {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' , n} of n players.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Each player i ∈ N has a finite set Ai of actions available, with s := |Ai| for i ∈ N being the number of players’ actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='1 We denote by a ∈ A :=×i∈N Ai an action profile specifying an action ai for each player i ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Moreover, for i ∈ N, we let a−i ∈ A−i := ×j̸=i∈N Ai be an action profile of all players other than i, while (a, a−i) is the action profile obtained by adding a ∈ Ai to a−i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Finally, we let ui : A → [0, 1] be the utility function of player i ∈ N, with ui(a) being the utility perceived by that player when the action profile a ∈ A is played.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' We extend classical normal-form games by considering the case in which each player i ∈ N has mi cost functions, namely ci,j : A → [−1, 1] for j ∈ [mi].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='2 Each player i ∈ N is subject to mi constraints, which require that all player i’s costs are less than or equal to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='3 For ease of notation, we assume w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' that all players have the same number of constraints, namely m := mi for all i ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Moreover, we encode the costs of player i ∈ N by a vector-valued function ci : A → [−1, 1]m such that, for every a ∈ A, the j-th component of the vector ci(a) is ci,j(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Correlated Strategies In this paper, we deal with solution concepts defined by correlated strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' A correlated strategy z ∈ ∆A is a probability distribution defined over the set of actions profiles, with z[a] denoting the probability assigned to a ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='4 With an abuse of notation, for every player i ∈ N, we let ui(z) be player i’s expected utility when the action profile played by the players is drawn from z ∈ ∆A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' In particular, it holds ui(z) := � a∈A ui(a)z[a].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Similarly, we let ci(z) := � a∈A ci(a)z[a] be the vector of player i’s expected costs, so that player i’s constraints can be compactly written as ci(z) ⪯ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Finally, we define S ⊆ ∆A as the set of safe correlated strategies, which are those satisfying the cost constraints of all players.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Formally: S := {z ∈ ∆A | ci(z) ⪯ 0 ∀i ∈ N} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' In the following, we assume w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' that S ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='2 Constrained Phi-equilibria We generalize the notion of Phi-equilibria (Greenwald and Jafari, 2003) to cost-constrained normal-form games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Such equilibria are defined as correlated strategies z ∈ ∆A that are robust against a given set Φ of players’ deviations, in the sense that, if a mediator draws an action profile a ∈ A according to z and recommends to play action ai to each player i ∈ N, then no player has an incentive to deviate from their recommendation by selecting a deviation in Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' For every i ∈ N, we let Φi be the set of player i’s deviations, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=', linear transformations φi : Ai → ∆Ai that prescribe a probability distribution over player i’s actions for every possible action recommendation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' For ease of notation, we encode a deviation φi by means of its matrix representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Formally, an entry φi[b, a] of the matrix represents the probability assigned to action a ∈ Ai by φi(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' We denote the set of all the possible deviations by Φ := {Φi}i∈N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Given a correlated strategy z ∈ ∆A and a deviation φi ∈ Φi, we define φi ⋄ z as the modification of z induced by φi, which is a linear transformation that can be expressed as follows in terms of matrix representation: (φi ⋄ z)[ai, a−i] := � b∈Ai φi[b, ai]z[b, a−i], for every ai ∈ Ai and a−i ∈ A−i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Moreover, given a set Φi of deviations of player i ∈ N, in the following we denote by ΦS i (z) := {φi ∈ Φi | φi ⋄ z ∈ S} the set of safe deviations for player i at a given correlated strategy z ∈ ∆A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' We are now ready to provide our definition of constrained Phi-equilibria in cost-constrained normal-form games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='1 (Constrained ǫ-Phi-equilibria).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Given a set Φ := {Φi}i∈N of deviations and an ǫ > 0, a constrained ǫ-Phi-equilibrium is a safe correlated strategy z ∈ S such that, for all i ∈ N, the following holds: ui(z) ≥ ui(φi ⋄ z) − ǫ ∀φi ∈ ΦS i (z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' 1For ease of presentation, in this paper we assume that all the players have the same number of actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' All the results can be easily generalized to the case of different numbers of actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' 2In this paper, given some x ∈ N>0, we let [x] := {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' , x} be the set of the first x natural numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' 3Since z ∈ ∆A, we can assume w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' that all the constraints are of the form ≤ 0, as any constraint can always be cast in such a form by suitably manipulating the cost function ci,j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' 4In this paper, given a finite set X, we denote by ∆X the set of all the probability distributions defined over the elements of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' 3 ARXIV PREPRINT - FEBRUARY 1, 2023 A constrained Phi-equilibrium is defined for ǫ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='3 Computing Constrained Phi-equilibria In the following, we formally introduce the computational problem that we study in the rest of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' We denote by I := (Γ, Φ) an instance of the problem, where the tuple Γ := (N, A, {ui}i∈N , {ci,j}i∈N,j∈[m]) is a cost-constrained normal-form game and Φ := {Φi}i∈N is a set of deviations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Moreover, we let |I| be the size (in terms of number of bits) of the instance I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' We assume that the number n of players is fixed, so that |I| does not grow exponentially in n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='5 We also make the following assumption on how the sets of deviations are represented: Assumption 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' For every i ∈ N, the set Φi is a polytope encoded by a finite of linear inequalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='6 Let us remark that, in games without constraints, this assumption is met by all the sets Φ which determine the classical notions of Phi-equilibria (Greenwald and Jafari, 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Next, we formally define our computational problem: Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='2 (APXCPE(α, ǫ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' For any α, ǫ > 0, we define APXCPE(α, ǫ) as the problem of finding, given an instance I := (Γ, Φ) and a linear function ℓ : ∆A → R as input, a constrained ǫ-Phi-equilibrium z ∈ ∆A such that ℓ(z) ≥ αℓ(z′) for all constrained Phi-equilibria z′ ∈ ∆A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Intuitively, APXCPE(α, ǫ) asks to compute a constrained ǫ-Phi-equilibrium whose value for the linear function ℓ is at least a fraction α of the maximum value which can be achieved by an (exact) constrained Phi-equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' In order to ensure that an instance of our problem is well defined, we make the following “Slater-like” assumption on how the players’ cost constraints are defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' For every correlated strategy z ∈ ∆A, player i ∈ N, and index j ∈ [m], there exists φ◦ i ∈ ΦS i (z): ci,j(φ◦ i ⋄ z) ≤ −ρ, where ρ > 0 and 1/ρ is O(poly(|I|)), with poly(|I|) being a polynomial function of the instance size |I|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' In Assumption 2, the condition ρ > 0 is required to guarantee the existence of a constrained Phi-equilibrium (see Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='1) and that the sets ΦS i (z) are non-empty (otherwise our solution concept would be ill defined).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Moreover, the second condition on ρ in Assumption 2 is equivalent to requiring that our algorithms run in time polynomial in 1 ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Assumption 2 also allows us to prove the existence of our equilibria, by showing that the constrained Nash equilibria introduced by Altman and Shwartz (2000), which always exist under Assumption 2, are also constrained Phi-equilibria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Given a cost-constrained normal-form game Γ and a set Φ of deviations, if Assumption 2 is satisfied, then Γ admits a constrained Phi-equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='4 Relation with Unconstrained Phi-equilibria We conclude the section by discussing the relation between our constrained Phi-equilibria and classical equilibrium concepts for unconstrained games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Correlated Equilibrium When there are no constraints, the correlated equilibrium (CE) (Aumann, 1974) is a spe- cial case of Phi-equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' As shown by Greenwald and Jafari (2003), the CE is obtained when the sets Φi contain all the possible deviations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Formally, the CE is defined by the set ΦALL := {Φi,ALL} of deviations such that: Φi,ALL := � φi ��� � a∈Ai φi[b, a] = 1 ∀b ∈ Ai � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' 5Notice that the size of the representation of a normal-form game is O(sn), and, thus, exponential in n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Any algorithm that runs in time polynomial in such instance size is not computationally appealing, as even its input has size exponential in n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' For this reason, we focus on the case in which n is fixed, and, thus, the instance size does not grow exponentially with n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' 6Notice that, since each φi ∈ Φi is represented as a matrix, a linear inequality is expressed as � b,a∈Ai M[b, a]φi[b, a] ≤ d, for some matrix M and scalar d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' 4 ARXIV PREPRINT - FEBRUARY 1, 2023 Coarse Correlated Equilibrium The coarse correlated equilibrium (CCE) (Moulin and Vial, 1978b) is a special (unconstrained) Phi-equilibrium whose set of deviations is ΦCCE := {Φi,CCE}i∈N such that: Φi,CCE := � φi ��� ∃h ∈ ∆Ai : φi[b, a] = h[a] ∀b, a ∈ Ai � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Intuitively, such sets contain all the possible deviations that prescribe the same probability distribution independently of the received action recommendation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Thus, our constrained Phi-equilibria include the generalization of CEs and CCEs to cost-constrained games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Our definition of constrained Phi-equilibrium needs to employ “mixed” deviations that map action recommendations to probability distributions over actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' This is necessary in presence of constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Instead, without them, one can simply consider “pure” deviations that map recommendations to actions deterministically Greenwald and Jafari (2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' 3 Challenges of Constrained Phi-equilibria In this section, we show that, in cost-constrained normal-form games, Phi-equilibria loose the nice computational properties that they exhibit in unconstrained settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' This is crucially determined by the fact that the set of constrained Phi-equilibria may not be convex in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Given any instance I := (Γ, Φ), the set of constrained Phi-equilibria may not be convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='1 is proved by the following example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Let ΦALL be the set of all the possible deviations in a two-player game in which each player has two actions, namely A1 = A2 = {a0, a1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' The first player’s utility is such that u1(a, a′) = 0 for all a ∈ A1 and a′ ∈ A2, while the second player’s utility is such that u2(a0, a1) = 1, and 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Both players share the same single cost constraint (m = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Their cost functions are defined as ci(a0, a1) = 1, ci(a0, a0) = − 1 2, and ci(a1, a) = −1 for all a ∈ A2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Notice that the instance defined above satisfies Assumption 2 for ρ = 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' It is easy to see that the correlated strategy z1 ∈ ∆A such that z1[a0, a0] = 2 3 and z1[a0, a1] = 1 3 is a constrained Phi-equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Moreover, the “pure” correlated strategy z2 ∈ ∆A such that z2[a1, a0] = 1 is also a constrained Phi-equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' However, the combination z3 = 1 2(z1 + z2) is not a constrained Phi-equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Indeed, the second player has an incentive to deviate by using a deviation φ2 such that φ2[a0, a1] = 1 and φ2[a1, a1] = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Such a deviation prescribes to play action a1 when a0 is recommended, and to play action a1 when the recommendation is a1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Straightforward calculations show that, for every a ∈ A1: (φ2 ⋄ z3)[a, a′] = � 1 2 if a′ = a1 0 otherwise, and u2(φ2 ⋄ z3) = 1 2 > u2(z3) = 1 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Moreover, the deviation is safe, since φ2 ∈ ΦS 2 (z3) as c2(φ2 ⋄ z3) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' In order to formally asses the computational challenges of computing constrained Phi-equilibria, we prove the follow- ing strong inapproximability result: Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='1 (Hardness).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' For any constant α > 0, the problem APXCPE(α, (α/s)2) is NP-hard, where s is the number of players’ actions in the instance given as input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Intuitively, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='1 states that, for every multiplicative approximation factor α > 0, it is not possible to find a constrained ǫ-Phi-equilibrium having value of ℓ at least a fraction α of its optimal value in time polynomial in 1 ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Moreover, as a byproduct of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='1, we also get the inapproximability up to within any factor of the problem of computing an optimal constrained (exact) Phi-equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Notice that the hardness result in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='1 cannot hold for values of ǫ that are independent from the instance size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Indeed, as we prove in Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='3 in Section 4, problem APXCPE(1, ǫ) can be solved in quasi-polynomial time in the instance size whenever ǫ > 0 is a given constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Thus, any NP-hardness result for APXCPE(α, ǫ) would contradict the exponential-time hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='7 4 Computing Optimal Constrained ǫ-Phi-equilibria Efficiently In this section, we show how to circumvent the negative result established by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' In particular, we prove that, when the number of cost constraints is fixed, problem APXCPE(1, ǫ) can be solved in time polynomial in the 7The exponential-time hypothesis conjectures that solving 3SAT requires at least exponential time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' 5 ARXIV PREPRINT - FEBRUARY 1, 2023 instance size and 1 ǫ for ǫ > 0 (Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Moreover, we also prove that, in general, for any constant ǫ > 0 problem APXCPE(1, ǫ) admits a quasi-polynomial-time algorithm, whose running time becomes polynomial when the number of players’ actions is fixed (Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' First, in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='1, we provide a general algorithm that is at the core of the two main results of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Then, in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='1, we show how the algorithm can be suitably instantiated in order to prove each result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' In the rest of this section, unless stated otherwise, we always assume that an ǫ > 0 has been fixed, and that I := (Γ, Φ) and ℓ : ∆A → R are the inputs of a given instance of problem APXCPE(1, ǫ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='1 General Algorithm The main technical tool that we employ in order to design our algorithm is a “Lagrangification” of the constraints defining the sets ΦS i (z) of safe deviations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' First, we prove the following preliminary result, which shows that strong duality holds for the problem maxφi∈ΦS i (z) ui(φi ⋄ z) of finding the best safe deviation for player i ∈ N at z ∈ ∆A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' For every z ∈ ∆A and i ∈ N, it holds max φi∈ΦS i (z) ui(φi ⋄ z) = sup φi∈Φi inf ηi∈Rm + � ui(φi ⋄ z) − η⊤ i ci(φi ⋄ z) � = inf ηi∈Rm + sup φi∈Φi � ui(φi ⋄ z) − η⊤ i ci(φi ⋄ z) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Then, by exploiting Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='1, we can prove that, under Assumption 2, strong duality continues to hold even when restricting the Lagrange multipliers ηi to have ℓ1-norm less than or equal to 1/ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Formally: Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Let D := � η ∈ Rm + | ||η||1 ≤ 1/ρ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Then, for every z ∈ ∆A and i ∈ N, it holds: max φi∈ΦS i (z) ui(φi ⋄ z) = max φi∈Φi min ηi∈D � ui(φi ⋄ z) − η⊤ i ci(φi ⋄ z) � = min ηi∈D max φi∈Φi � ui(φi ⋄ z) − η⊤ i ci(φi ⋄ z) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='2 allows us to write player i’s incentive constraints in the definition of constrained ǫ-Phi-equilibria as ui(z) ≥ min ηi∈D max φi∈Φi � ui(φi ⋄ z) − η⊤ i ci(φi ⋄ z) � − ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' (1) This crucially allows us to show the following result: solving problem APXCPE(1, ǫ) is equivalent to computing max(η1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=',ηn)∈Dn Fǫ(η1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' , ηn), where Fǫ(η1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' , ηn) is the optimal value of a suitable maximization problem parameterized by tuples of Lagrange multipliers ηi ∈ D, one per player i ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Such a problem asks to compute a safe correlated strategy maximizing the linear function ℓ subject to players’ incentive constraints that are re-formulated by means of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Formally, we define Fǫ(η1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' , ηn) as the maximum of ℓ(z) over those z ∈ S that additionally satisfy the following constraint for every i ∈ N: ui(z) ≥ max φi∈Φi � ui(φi ⋄ z) − η⊤ i ci(φi ⋄ z) � − ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' (2) Notice that the min operator that appears in the right-hand side of Constraints (1) is dropped by adding the outer maximization over the tuples (η1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' , ηn) ∈ Dn, as the maximum of ℓ is always achieved when the right-hand side of such constraints is as small as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Next, we show that Fǫ(η1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' , ηn) can be computed in polynomial time by means of the ellipsoid algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' For every tuple (η1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' , ηn) ∈ Dn, the value of Fǫ(η1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' , ηn) can be computed in time polynomial in the instance size |I| and 1 ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' We show that Fǫ(η1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' , ηn) can be solved in polynomial time by means of the ellipsoid algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Let us notice that Constraints (2) can be equivalently encoded by a set of linear inequalities, one for each player i ∈ N and deviation φi ∈ vert(Φi), where vert(Φi) denotes the set of vertexes of the polytope Φi (recall Assumption 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Thus, solving Fǫ(η1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' , ηn) is equivalent to solving an LP with a (possibly) exponential number of constraints, but polynomially-many variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Such an LP can be solved in polynomial time by means of the ellipsoid algorithm, provided that a polynomial-time separation oracle for the linearized version of Constraints (2) is available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Such an oracle can be implemented by solving the maximization in the right-hand side of Constraints (2) for a correlated strategy z ∈ ∆A given as input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Formally, the separation oracle solves the following problem for each player i ∈ N: φ⋆ i ∈ arg max φi∈Φi � ui(φi ⋄ z) − η⊤ i ci(φi ⋄ z) � , 6 ARXIV PREPRINT - FEBRUARY 1, 2023 which can be done efficiently thanks to Assumption 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Then, if the separation oracle finds any φ⋆ i such that: ui(z) ≥ ui(φ⋆ i ⋄ z) − η⊤ i ci(φ⋆ i ⋄ z), it outputs the above inequality as a separating hyperplane to be used in the ellipsoid algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='3 is not enough to complete our algorithm, since we need an efficient way of optimizing Fǫ(η1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' , ηn) over all the tuples of Lagrange multipliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' This problem is non-trivial, since Fǫ(η1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' , ηn) is non-concave in ηi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Nevertheless, we show that, by restricting the domain D of the Lagrange multipliers to a suitably-defined finite “small” subset, we can still find a constrained ǫ-Phi-equilibrium whose value of ℓ is at least as large as that of any constrained (exact) Phi-equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' This is enough to solve APXCPE(1, ǫ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' In particular, we need a finite subset of “good” Lagrange multipliers, in the sense of the following definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Given any δ > 0, a set ˜D ⊆ D is δ-optimal if, for every z ∈ ∆A and i ∈ N, the following holds: min ηi∈ ˜ D max φi∈Φi � ui(φi ⋄ z) − η⊤ i ci(φi ⋄ z) � ≤ max φi∈ΦS i (z) ui(φi ⋄ z) + δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Intuitively, thanks to Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='2, if we optimize the Lagrange multipliers over a δ-optimal set ˜D ⊆ D, instead of optimizing them over D, then we are allowing the players to violate incentive constraints by at most δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' In the following, we assume that a finite δ-optimal set ˜D ⊆ D is available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' In Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='2, se show how to design two particular δ-optimal sets that allow to prove our main results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' For ease of presentation, we let L ˜ D,ǫ := max (η1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=',ηn)∈ ˜ Dn Fǫ(η1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' , ηn) be the optimal value of Fǫ(η1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' , ηn) when the Lagrange multipliers are constrained to be in a δ-optimal set ˜D ⊆ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Next, we show that, given any δ-optimal set ˜D with δ ≤ ǫ, the value of L ˜ D,ǫ is at least that achieved by constrained (exact) Phi-equilibria, namely LD,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Formally: Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Given any 0 < δ ≤ ǫ and a δ-optimal set ˜D ⊆ D, the following holds: L ˜ D,ǫ ≥ LD,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Intuitively, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='4 is proved by noticing that, provided that δ ≤ ǫ, the incentive constraints violation introduced by using ˜D instead of D is at most ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Moreover, the set of feasible correlated strategies can only expand by allowing incentive constraints to be violated, and, thus, the value of the objective ℓ can only increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='4 suggests a way of solving APXCPE(1, ǫ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Indeed, given a finite δ-optimal set ˜D ⊆ D with δ ≤ ǫ, by enumerating over all the tuples of Lagrange multipliers ηi ∈ ˜D, one per player i ∈ N, we can find the desired constrained ǫ-Phi-equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' The following theorem shows that this procedure gives an algorithm for APXCPE(1, ǫ) that runs in time polynomial in the instance size, | ˜D|, and 1 ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Given a finite δ-optimal set ˜D ⊆ D with δ ≤ ǫ, there exists an algorithm that solves APXCPE(1, ǫ) and runs in time polynomial in the instance size |I|, the number | ˜D| of elements in ˜D, and 1 ǫ for every ǫ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' The algorithm works by enumerating over all the possible tuples of Lagrange multipliers ηi ∈ ˜D, one per player i ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' These are polynomially many in the size | ˜D| when the number of players n is fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' For every tuple (η1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' , ηn) ∈ ˜Dn, the algorithm solves Fǫ(η1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' , ηn), which can be done in time polynomial in |I| and 1 ǫ thanks to Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Finally, the algorithm returns the correlated strategy z ∈ ∆A with the highest value of ℓ among those computed while solving Fǫ(η1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' , ηn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' It is easy to see that the returned solution solves problem APXCPE(1, ǫ) by applying Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' This concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='2 Instantiating the General Algorithm Next, we show how to build δ-optimal sets ˜D that, when they are plugged in the algorithm in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='1, allow us to derive our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' In particular, we consider the set: Dτ := � η ∈ D ��� ηj = kτ, k ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' , ⌊1/τρ⌋} ∀j ∈ [m] � , which is a discretization of D with a regular lattice of step τ ∈ R+ (notice that ηj is the j-th component of η).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' By a simple stars and bars combinatorial argument, we have that |Dτ| = �⌊1/τρ⌋+m m � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Thus, since it holds that 7 ARXIV PREPRINT - FEBRUARY 1, 2023 |Dτ| = O((1/τρ)m), if the number of constraints m is fixed, |Dτ| is bounded by a polynomial in 1/τρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Moreover, simple combinatorial arguments show that |Dτ| ≤ (1 + m)⌊1/τρ⌋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='8 Thus, it also holds that |Dτ| = O(m 1/τρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Notice that the two bounds on |Dτ| are non-comparable, and, thus, they give rise to two distinct results, as we show in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' By using the first bound on |Dτ|, we can show that the set Dτ is δ-optimal for δ = mτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Formally: Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' For any τ > 0, the set Dτ is (τm)-optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Thus, whenever the number m of cost constraints is fixed, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='5, together with Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='1, allows us to provide a polynomial-time algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Indeed, it is sufficient to apply Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='1 for the (τm)-optimal set Dτ with τ := ǫ/m to obtain the following first main result: Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' There exists an algorithm that solves problem APXCPE(1, ǫ) in time polynomial in |I| and 1 ǫ for every ǫ > 0, when the number m of cost constraints is fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' On the other hand, by using the second bound on |Dτ|, we can show that Dτ is δ-optimal for δ depending logarithmi- cally on the number of players’ actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Formally: Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' For any τ > 0, the set Dτ is δ-optimal for δ = 2 � 2τ log s/ρ, where s is the number of players’ actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='6 (together with Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='1) immediately gives us a quasi-polynomial-time for solving APXCPE(1, ǫ) for a given constant ǫ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Moreover, its running time becomes polynomial when the number of players’ actions is fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' For any constant ǫ > 0, there exists an algorithm that solves APXCPE(1, ǫ) in time O(|I|log s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Moreover, when the number s of players’ actions is fixed, the algorithm runs in time polynomial in |I|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Notice that it is in general not possible to design an algorithm that runs in time polynomial in 1 ǫ, since this would contradict the hardness result in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' 5 A Special Case: Deviation-dependent Costs We complete our computational study of constrained Phi-equilibria by considering a special case in which player i’s costs associated to a deviation φi only depend on φi and not on the (overall) modified correlated strategy φi ⋄ z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' We consider instances satisfying the following assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' For every player i ∈ N and player i’s deviation φi ∈ Φi, there exists a function ˜ci : Φi → [−1, 1]m such that ˜ci(φi) := ci(φi ⋄ z) for every z ∈ ∆A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Notice that, whenever Assumption 3 holds, the set ΦS i (z) of safe deviations does not depend on z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Thus, in the rest of this section, we write w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' ΦS i rather than ΦS i (z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' A positive effect of Assumption 3 is that it recovers the convexity of the set of constrained Phi-equilibria, rendering them more akin to unconstrained ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Formally: Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' For instances I := (Γ, Φ) satisfying Assumption 3, the set of constrained ǫ-Phi-equilibria is convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='1 suggests that constrained Phi-equilibria are much more computationally appealing under Assumption 3 than in general, as we indeed show in the rest of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' First, in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='1, we show that APXCPE(1, 0) admits a polynomial-time algorithm under Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Then, in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='2, we design a no-regret learning algorithm that efficiently computes one constrained ǫ-Phi equilibrium with ǫ = O(1/ √ T) as the number of rounds T grows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Finally, in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='3, we provide a natural example of constrained Phi-equilibria satisfying Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='1 A Poly-time Algorithm for Optimal Equilibria We prove that, whenever Assumption 3 holds, the problem of computing an (exact) Phi-equilibrium maximizing a given linear function can be solved in polynomial time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' This is done by formulating the problem as an LP with polynomially-many variables and exponentially-many constraints, which can be solved by means of the ellipsoid method, similarly to how we compute Fǫ(η1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' ηn) in Section 4 (see the proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Formally: Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Restricted to instances I := (Γ, Φ) which satisfy Assumption 3, APXCPE(1, 0) admits a polynomial- time algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' 8See Appendix D for a formal proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' 8 ARXIV PREPRINT - FEBRUARY 1, 2023 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='2 An Efficient No-regret Learning Algorithm Next, we show how Assumption 3 allows us to find a constrained ǫ-Phi-equilibrium by means of a polynomial-time decentralized no-regret learning algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Our algorithm is based on the Phi-regret minimization framework intro- duced by Greenwald and Jafari (2003), which needs to be extended in order to be able to work with polytopal sets ΦS i of safe deviations, rather than finite sets of “pure” deviations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Algorithm 1 Learning a Constrained ǫ-Phi-equilibria Require: Regret minimizers Ri for the sets ΦS i , for i ∈ N 1: Initialize the regret minimizers Ri 2: for t = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' , T do 3: for each player i ∈ N do 4: φi,t ← Ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='RECOMMEND() 5: Play according to a distribution xi,t ∈ ∆Ai s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' xi,t[a] = � b∈Ai φi,t[b, a]xi,t[b] ∀a ∈ Ai 6: end for 7: zt ← ⊗i∈N xi,t 8: Ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='OBSERVE(φi �→ ui(φi ⋄ zt)) 9: end for 10: return ¯zT := 1 T �T t=1 zt Algorithm 1 outlines our no-regret algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' It instantiates a regret minimizer Ri for the polytope ΦS i for each i ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Ri is an object that, at each round t ∈ [T ], recommends a safe deviation φi,t ∈ ΦS i to player i (Line 4 of Algorithm 1), and, then, observes a function φi �→ ui(φi ⋄ zt) that specifies the utility that would have been obtained by selecting any safe deviation φi ∈ ΦS i at round t (Line 8 of Algorithm 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Ri guarantees that the regret RT i cumulated by player i over [T ] grows sublinearly, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=', RT i = o(T ), where: RT i := max φi∈Φi T � t=1 ui(φi ⋄ zt) − T � t=1 ui(φi,t ⋄ zt), which is how much player i loses by selecting φi,t at each t rather than choosing the same best-in-hindsight deviation at all rounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Notice that, by taking inspiration from the Phi-regret framework (Greenwald and Jafari, 2003), given a recommended deviation φi,t, player i actually plays according to a probability distribution xi,t ∈ ∆Ai, which is a stationary distribution of the matrix representing φi,t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' This is crucial in order to implement the algorithm in a decentralized fashion and to provide convergence guarantees to constrained ǫ-Phi-equilibria (see Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' All the distributions xi,t jointly determine a correlated strategy zt ∈ ∆A at each round t ∈ [T ], defined as zt := ⊗i∈N xi,t, where ⊗ denotes the product among distributions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' formally, zt[a] := � i∈N xi,t[ai] for all a ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Algorithm 1 provides the following guarantees: Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Given an instance I := (Γ, Φ) satisfying Assumption 3, after T ∈ N>0 rounds, Algorithm 1 returns a correlated strategy ¯zT ∈ ∆A that is a constrained ǫT -Phi-equilibrium with ǫT = O(1/ √ T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Moreover, each round of Algorithm 1 runs in polynomial time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Let us remark that the crucial property which allows us to design Algorithm 1 is that the sets ΦS i of safe deviations do not depend on players other than i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Finally, from Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='2, the following result follows: Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' In instances I := (Γ, Φ) satisfying Assumption 3, a constrained ǫ-Phi-equilibrium can be computed in time polynomial in the instance size and 1 ǫ by means of a decentralized learning algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='3 Marginally-constrained CCE We conclude the section by introducing a particular (natural) notion of constrained ǫ-Phi-equilibrium for which As- sumption 3 is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' This is a constrained version of the classical CCE in cost-constrained normal-form games where a player’s costs only depend on the action of that player.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' We call it marginally-constrained ǫ-CCE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Formally, such an equilibrium is defined for games in which, for every player i ∈ N, it holds ci(a) = ci(a′) for all a, a′ ∈ A such that ai = a′ i, and for the set ΦCCE of CCE deviations that we have previously introduced in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Next, we prove that, with the definition above, Assumption 3 is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' 9 ARXIV PREPRINT - FEBRUARY 1, 2023 Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' For instances I := (Γ, ΦCCE) such that ci(a) = ci(a′) for every player i ∈ N and action profiles a, a′ ∈ A : ai = a′ i, Assumption 3 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Thanks to Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='4, we readily obtain the two following corollaries of Theorems 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='1 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' The problem of computing a marginally-constrained (exact) CCE that maximizes a linear function ℓ : ∆A → R can be solved in polynomial time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' A marginally-constrained ǫ-CCE can be computed in time polynomial in the instance size and 1 ǫ by means of a decentralized learning algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' 6 Discussion and Open Problems The main positive results that we provide in this paper (Corollaries 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='2 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='3) show that a constrained ǫ-Phi equilib- rium maximizing a given linear function can be computed in time polynomial in the instance size and 1 ǫ, when either the number of constraints or that of players’ actions is fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Clearly, this implies that, under the same assumptions, a constrained ǫ-Phi-equilibrium can be found efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Moreover, in Section 5, we designed an efficient no-regret learning algorithm that finds a constrained ǫ-Phi-equilibrium in settings enjoying special properties (Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' However, the problem of efficiently computing a constrained ǫ-Phi-equilibrium remains open in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Formally: Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='1 (Open Problem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Given any instance I := (Γ, Φ), find a constrained ǫ-Phi-equilibrium in time polyno- mial in the instance size and 1 ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Solving the problem above is non-trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='1 in Section 3 proves that the set of constrained ǫ-Phi-equilibria is non-convex, and, thus, solving the problem in Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='1 is out of scope for most of the known equilibrium computation techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' On the other hand, it is unlikely that such a problem is NP-hard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Indeed, a constrained ǫ-Phi-equilibrium always exists and, given any z ∈ ∆A, it is possible to verify whether z is an equilibrium or not in polynomial time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Formally, such a problem is said to belong to the TFNP complexity class, and, thus, standard arguments show that, if the problem is NP-hard, then NP = coNP (Megiddo and Papadimitriou, 1991).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Thus, one should try to reduce the problem in Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='1 to problems in TFNP, such as that of computing a Nash equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' However, while the problem in Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='1 shares some properties with that of computing a Nash equilibrium, such as the non-convexity of the set of the equilibria, the former is fundamentally different from the latter, since it exhibits correlation among the players.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Thus, a reduction from such a problem to that of computing Nash equilibria would require a gadget to break the correlation among the players, and doing that is highly non-trivial as cost constraints are expressed by linear functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' 10 ARXIV PREPRINT - FEBRUARY 1, 2023 References Eitan Altman and Adam Shwartz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' 2000.' metadata={'source': 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Inapproximability of Max Clique and Chromatic Number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Theory of Computing 3, 6 (2007), 103–128.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' 11 ARXIV PREPRINT - FEBRUARY 1, 2023 A On the Weaknesses of the Guarantees of the Algorithm of Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' (2022) The Algorithm of Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' (2022) finds a distribution µ over correlated strategies ∆A such that: Ez∼µ � max φi∈ΦS i (z) ui(φi ⋄ z) − ui(z) � ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' (3) However, here we claim that this solution concept inherits some weaknesses from the non-convexity of the equilibria set that we proved in Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Indeed, consider the same instance of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='1 and consider the uniform distribution µ over {z1, z2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' In Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='1 we proved that maxφi∈ΦS i (z1) ui(φi ⋄z1)−ui(z1) ≤ 0 for all i ∈ {1, 2} and maxφi∈ΦS i (z2) ui(φi ⋄ z2) − ui(z2) ≤ 0 for all i ∈ {1, 2} and thus Equation (3) holds over the distribution µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' However we show that the expected correlated strategy z3 derived from distribution µ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=', z3 = Ez∼µ[z] = 1 2z1 + 1 2z2, it is not a feasible equilibrium, or an approximate one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Indeed, in Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='1, we proved that maxφ2∈ΦS 2 (z3) u2(φ2 ⋄z3)−u2(z3) ≥ 1 3, showing that the average correlated strategies returned by their Algorithm is not an equilibrium nor close to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' This comes from the peculiar fact about Constrained Phi-equilibria that exhibit non-convex set of solutions, which is in striking contrast with the unconstrained case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Indeed the guarantees of Equilibria (3) would imply that Ez∼µ[z] is a equilibrium in the unconstrained case in which the set of equilibria is convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' B Proofs Omitted from Section 2 Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Given a cost-constrained normal-form game Γ and a set Φ of deviations, if Assumption 2 is satisfied, then Γ admits a constrained Phi-equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' With assumption 2 Altman and Shwartz (2000, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='1) proves the existence of a constrained Nash equi- librium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' In our setting this is equivalent to a product distribution z = ⊗i∈[N]xi so that it is a Constrained Phi- equilibrium for any set of deviations Φi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='9 This is easily seen by observing that a Constrained Nash Equilibria is defined as: � a∈A ui \uf8eb \uf8ed � j∈[N] xj(aj) \uf8f6 \uf8f8 ≥ � a∈A ui \uf8eb \uf8ed˜xi(aj) � j∈[N]\\{i} xi(ai) \uf8f6 \uf8f8 for all ˜xi ∈ ∆(Ai) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' xi ⊗ x−i ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' On the other hand it easily seen that for all φi ∈ Φi(z) there exists some ˜xi ∈ ∆(Ai) such that φi ⋄ � ⊗j∈[N]xj � = ˜xi ⊗ x−i and ˜xi ⊗ x−i ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' This is proved by the following calculations: φi ⋄ � ⊗j∈[N]xj � [ai, a−i] := � b∈Ai φi[b, ai]xi(b)x−i(a−i) (4) = ˜xi(ai) ⊗ x−i(a−i), (5) where ˜xi(ai) := � b∈Ai φi[b, ai]xi(b) and ˜xi ∈ ∆(Ai) since, by definition, � ai∈Ai φi[b, ai] = 1 for all b ∈ Ai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' This proves that a Constrained Nash Equilibrium is a Phi-Constrained Equilibrium for all Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' C Proofs Omitted from Section 3 Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='1 (Hardness).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' For any constant α > 0, the problem APXCPE(α, (α/s)2) is NP-hard, where s is the number of players’ actions in the instance given as input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' 9As common in the normal form game literature, for any distribution x ∈ ∆(X) and y ∈ ∆(Y ), x ⊗ y ∈ ∆(X × Y ) is the product distribution defined as (x ⊗ y)[a, b] = x[a]y[b] for a ∈ X and b ∈ Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' 12 ARXIV PREPRINT - FEBRUARY 1, 2023 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' We reduce from GAP-INDEPENDENT-SET, which is a promise problem that formally reads as follows: given an δ > 0 and a graph G = (V, E), with set of nodes V and set of edges E, determine whether G admits an independent set of size at least |V |1−δ or all the independent sets of G have size smaller than |V |δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' GAP- INDEPENDENT-SET is NP-hard for every δ > 0 (H˚astad, 1999;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Zuckerman, 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Let ℓ = |V | and α > 0 be the desired approximation factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Given an instance of GAP-INDEPENDENT-SET, we build an instance such that if there exists an independent set of size ℓ1−δ, then there exists a Constrained Phi- equilibrium with social welfare 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Otherwise, if all the independent sets have size at most ℓδ, all the Constrained ǫ-Phi-equilibria have social welfare at most α/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' We can use any δ > 0, since we simply need ℓδ < ℓ1−δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Moreover, we take ǫ = α2 128ℓ2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' As we will see, ℓ will be smaller than the number of action of the players, satisfying the condition in the statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' The first player has a set of actions A1 that includes actions a0, a1, a2 and an action av for each v ∈ V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Moreover, the first player has an action aF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='10 The second player has a set of actions A2 that includes actions av and ¯av for each v ∈ V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Moreover, the second player has an action aF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Let γ = η = α/8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' The utility of the first agent is as follows: u1(a0, a) = γ + 1 2η for all a ∈ A2 \\ {aF}, u1(a1, av) = γ + η and u1(a2, av) = γ for all v ∈ V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' u1(a1, ¯av) = γ and u1(a2, ¯av) = γ + η for all v ∈ V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' u1(av, av) = u1(av, ¯av) = γ for all v ∈ V u1(av, av′) = γ and u1(av, ¯av′) = γ + ℓ−ℓ1−δ ℓ−ℓ1−δ−1η for all v′ ̸= v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' u1(aF , a) = 0 for each a ∈ A2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' u1(a, aF ) = 0 for each a ∈ A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' The utility of the second agent is u2(a0, a) = 1 for each a ∈ A2 \\ {aF} and 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' There is a cost function cv for each v ∈ V , which is common to both the agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' For each v ∈ V , the cost function cv is such that cv(av, av′) = −1 for each v′ ̸= v, (v, v′) ∈ E, cv(av, av′) = 0 for each v′ ̸= v, (v, v′) /∈ E, cv(av, av) = 1 for each v ∈ V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' cv(aF , a) = − 1 4ℓ2 for each a ∈ A2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' cv(a, aF ) = − 1 4ℓ2 for each a ∈ A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' For every other action profile the cost is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' We dropped the player index from the cost functions c as they are equal to both players.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Moreover, we set of deviations Φi = Φi,ALL for both players i ∈ {1, 2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Notice that the instance satisfies Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Indeed, the deviation φi such that φi[a, aF ] = 1 for all a ∈ Ai for i ∈ {1, 2}, that deviates deterministically to aF is always strictly feasible for both player 1 and player 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Moreover, its cost is polynomial in the instance size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Completeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' We show that if there exists an independent set of size ℓ1−δ, then the social welfare of an optimal Constrained Phi-equilibria is at least 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Let V ∗ be an independent set of size ℓ1−δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' We build a Constrained Phi- equilibria z with social welfare at least 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Consider the correlated strategy such that z[a0, av] = 1 2ℓ1−δ for all v ∈ V ∗, while z[a0, ¯av] = 1 2(ℓ−ℓ1−δ) for all v /∈ V ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' All the other action profiles have probability 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' 10This action is needed only to satisfy the strictly feasibility assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' 13 ARXIV PREPRINT - FEBRUARY 1, 2023 It is easy to see that the correlated strategy has social welfare at least 1 since player 1 always plays action a0 and u2(a0, a) = 1 for all a ∈ A2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Moreover, it is easy to verify that it is safe since cv(a0, a) ≤ 0 for each a ∈ A2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Hence, to show that z is an Constrained Phi-equilibria we only need to prove that it satisfies the incentive constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' The incentive constraints of the second player are satisfied since they obtain the maximum possible utility, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=', 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Consider now a possible deviation of the first player φ1 ∈ Φ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' As a first step, we compute the expected utility of a strategy φ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Let us define the following quantities: T 1 = � v∈V ∗ φ1[a0, av] �� z[a0, av] + z[a0, ¯av] + � v′̸=v z[a0, av′] � γ + � γ + ℓ−ℓ1−δ ℓ−ℓ1−δ−1η � � v′̸=v z[a0, ¯av] � T 2 = � v /∈V ∗ φ1[a0, av] �� z[a0, av] + z[a0, ¯av] + � v′̸=v z[a0, av′] � γ + � γ + ℓ−ℓ1−δ ℓ−ℓ1−δ−1η � � v′̸=v z[a0, ¯av] � T 3 = � γ + η 2 � φ1[a0, a0] + γ+η 2 (φ1[a0, a1] + φ1[a0, a2]) + γ 2(φ1[a0, a1] + φ1[a0, a2]) We bound each component individually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' T 1 = � v∈V ∗ φ1[a0, av] \uf8ee \uf8f0 \uf8eb \uf8edz[a0, av] + z[a0, ¯av] + � v′̸=v z[a0, av′] \uf8f6 \uf8f8 γ + � γ + η ℓ − ℓ1−δ ℓ − ℓ1−δ − 1 � � v′̸=v z[a0, ¯av] \uf8f9 \uf8fb = � v∈V ∗ φ1[a0, av] �1 2γ + 1 2 � γ + η ℓ − ℓ1−δ ℓ − ℓ1−δ − 1 �� = � v∈V ∗ φ1[a0, av] � γ + η 2 ℓ − ℓ1−δ ℓ − ℓ1−δ − 1 � ≤ � v∈V ∗ φ1[a0, av](γ + η), where in the last inequality we use ℓ−ℓ1−δ ℓ−ℓ1−δ−1 ≤ 2 for ℓ large enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' while T 2 = � v /∈V ∗ φ1[a0, av] \uf8ee \uf8f0 \uf8eb \uf8edz[a0, av] + z[a0, ¯av] + � v′̸=v z[a0, av′] \uf8f6 \uf8f8 γ + � γ + η ℓ − ℓ1−δ ℓ − ℓ1−δ − 1 � � v′̸=v z[a0, ¯av] \uf8f9 \uf8fb = � v /∈V ∗ φ1[a0, av] ��1 2 + 1 2(ℓ − ℓ1−δ) � γ + �1 2 − 1 2(ℓ − ℓ1−δ) � � γ + η ℓ − ℓ1−δ ℓ − ℓ1−δ − 1 �� = � v /∈V ∗ φ1[a0, av] � γ + η 2 � ℓ − ℓ1−δ ℓ − ℓ1−δ − 1 − 1 ℓ − ℓ1−δ − 1 �� = � v /∈V ∗ φ1[a0, av] � γ + η 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Finally,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' T 3 = [a0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' a0] � γ + η 2 � + γ + η 2 ([a0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' a1] + [a0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' a2]) + γ 2 ([a0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' a1] + [a0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' a2]) = � γ + η 2 � ([a0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' a0] + φ1[a0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' a1] + φ1[a0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' a2]) Finally,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' the utility of a deviation φ1 is � a1∈A1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='a2∈A2 � a∈A1 φ1[a1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' a]z[a1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' a2]u1(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' a2) = � a∈A1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='a2∈A2 φ1[a0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' a]z[a0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' a2]u1(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' a2) = T 1 + T 2 + T 3 ≤ (γ + η) � v∈V ∗ φ1[a0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' av] + � γ + η 2 � � v /∈V ∗ φ1[a0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' av] + � γ + η 2 � (φ[a0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' a0] + φ1[a0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' a1] + φ1[a0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' a2]) 14 ARXIV PREPRINT - FEBRUARY 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' 2023 = η 2 � v∈V ∗ φ1[a0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' av] + � γ + η 2 � (1 − φ1[a0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' aF]) Now,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' we show that no deviation φ1 ∈ Φ1 is both safe and increases player 1 utility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' In particular, we show that if a strategy φ1 increases the utility than it is not safe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Indeed, if φ1 increases the utility, then � a1∈A1, a2∈A2 � a∈A1 φ1[a1, a]z[a1, a2]u1(a, a2) > γ + η 2 This implies that η 2 � v∈V ∗ φ1[a0, av] + � γ + η 2 � (1 − φ1[a0, aF ]) > γ + η 2 and � v∈V ∗ φ1[a0, av] > 1 2φ1[a0, aF ] (6) Next, we show that any φ1 that increases the utility (and hence that satisfies Eq (6)) is not a feasible deviation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' First, notice that equation (6) implies that there is a ¯v ∈ V ∗ such that φ1[a0, a¯v] > 1 2ℓφ1[a0, aF ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' (7) Then, we show that the deviation φ1 violates the constraint c¯v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' In particular, � a1∈A1,a2∈A2 � a∈A1 φ1[a1, a]z[a1, a2]cv(a, a2) = φ1[a0, a¯v]z[a0, a¯v]1 − 1 4ℓ2 φ1[a0, aF ] − � v∈V ∗:(v,¯v)∈E φ1[a0, a¯v]z[a0, av]1 = φ1[a0, a¯v]z[a0, a¯v] − 1 4ℓ2 φ1[a0, aF ] = 1 2ℓ1− 1 ℓ φ1[a0, a¯v] − 1 4ℓ2 φ1[a0, aF ] > φ1[a0, a¯v] � 1 2ℓ1− 1 ℓ − 1 2ℓ � ≥ 0, where the second inequality holds since V ∗ is an independent set, and the second-to-last inequality by Equation (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Hence, there is no deviation φ1 that increases players 1 utility and that is safe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' This concludes the first part of the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Soundness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' We show that if there exists a Constrained w-Phi-equilibria with social welfare α/2, then there exists an independent set of size strictly larger than ℓδ, reaching a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Suppose by contradiction that there exists a Constrained ǫ-Phi-equilibrium z with social welfare strictly greater than α/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Thus, � a′∈A2\\{aF } z[a0, a′] · 1 + � a∈A1,a′∈A2 (γ + η) ≥ � a∈A1,a′∈A2 z[a, a′](u1(a, a′) + u2(a, a′)) ≥ α/2, where the first inequality comes from u2(a0, a′) = 1 for each a′ ∈ A2 \\ {aF } and 0 otherwise, and u1(a, a′) ≤ γ + η for each a ∈ A1 and a′ ∈ A2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' This implies � a′∈A2 z[a0, a′] ≥ α/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' (8) Then, we show that z assigns similar probabilities on the set of action profiles {a0, av}v∈V and {a0, ¯av}v∈V Given an a ∈ A1, let φa ∈ Φ1 be a deviation of the first player such that φa[a0, a] = 1 and φa[a′, a′] = 1 for each a′ ̸= a0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Since z is an Constrained ǫ-Phi-equilibrium there is no feasible deviation φa that increases the utility of player 1 by more than ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' This implies that ����� � v∈V z[a0, av] − � v∈V z[a0, ¯av] ����� ≤ 2ǫ η .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' (9) 15 ARXIV PREPRINT - FEBRUARY 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' 2023 Indeed,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' if � v∈V z[a0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' av] > � v∈V z[a0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' ¯av] + 2ǫ η ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' (10) then the deviation φa1 has utility at least � v∈V z[a0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' av]φa1[a0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' a1](γ + η) + z[a0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' ¯av]φa1[a0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' a1]γ + � a∈A1\\{a0} � a′∈A2 z[a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' a′]φa1[a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' a]ui(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' a′) = η � v∈V z[a0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' av] + γ � v∈V (z[a0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' av] + z[a0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' ¯av]) + � a∈A1\\{a0} � a′∈A2 z[a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' a′]φa1[a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' a]ui(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' a′) > η 2 � 2ǫ η + � v∈V (z[a0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' av] + z[a0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' ¯av]) � + γ � v∈V (z[a0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' av] + z[a0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' ¯av]) + � a∈A1\\{a0} � a′∈A2 z[a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' a′]φa1[a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' a]ui(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' a′) ≥ ǫ + �η 2 + γ � � v∈V (z[a0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' av] + z[a0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' ¯av]) + � a∈A1\\{a0} � a′∈A2 z[a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' a′]φa1[a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' a]ui(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' a′) ≥ u1(z) + ǫ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' where the first inequality comes from adding � v∈V z[a0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' av] to both sides of Equation (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Moreover, φa1 is feasible since for each constraint c¯v, ¯v ∈ V , it has cost � v∈V (z[a0, av]φa1[a0, a1]c¯v(a1, av) + z[a0, ¯av]φa1[a0, a1]c¯v(a1, av)) + � a∈A1\\{a0} � a′∈A2 z[a, a′]φa1[a, a]c¯v(a, a′) = � v∈V (z[a0, av]φa1[a0, a1]c¯v(a0, av) + z[a0, ¯av]φa1[a0, a1]c¯v(a0, ¯av)) + � a∈A1\\{a0} � a′∈A2 z[a, a′]φa1[a, a]c¯v(a, a′) = c¯v(z) ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' A similar argument shows that if � v∈V z[a0, av] < � v∈V z[a0, ¯av] − 2ǫ η then the deviation φa2 is safe and increases the utility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' As a consequence of Equation (9), it holds 2 � v∈V z[a0, ¯av] ≥ � v∈V (z[a0, av] + z[a0, ¯av]) − ǫ η = � a∈A2\\{aF } z[a0, a] − 2ǫ η , (11) where the first inequality comes from adding � v∈V z[a0, ¯av] to both sides of � v∈V z[a0, ¯av]| ≥ � v∈V z[a0, av]− 2ǫ η The next step is to show that it is if there is no safe deviation φav, v ∈ V , that increases the utility, then there exists an independent set of size larger than ℓδ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Since z is an Constrained ǫ-Phi-equilibrium, for each av, v ∈ V one of the following two conditions holds: i) φav /∈ ΦS 1 (z) or ii) u1(φav ⋄ z) ≤ u1(z) + ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Let V 1 ⊆ V be the set of vertexes v such that φav is not safe, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=', φav /∈ ΦS 1 (z), and V 2 = V \\ V 1 be the set of v such that φav does not increase the utility by more than ǫ and are not in V 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=', u1(φav ⋄ z) ≤ u1(z) and φav ∈ ΦS 1 (z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' We show that |V 2| ≤ ℓ − ℓ1−δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Indeed,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' for each v ∈ V 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' deviation φav does not increase the utility and hence it holds: γ � a∈A2\\{aF } z[a0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' a] + η ℓ − ℓ1−δ ℓ − ℓ1−δ − 1 � v′̸=v z[a0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' ¯av′] + � a∈A1\\{a0} � a′∈A2 z[a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' a′]φa1[a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' a]ui(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' a′) = � � v′∈V z[a0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' a] + z[a0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' ¯av] � φ[a0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' av]γ + � v′̸=v φ[a0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' av]z[a0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' ¯av′] � γ + η ℓ − ℓ1−δ ℓ − ℓ1−δ − 1 � + � a∈A1\\{a0} � a′∈A2 z[a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' a′]φa1[a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' a]ui(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' a′) 16 ARXIV PREPRINT - FEBRUARY 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' 2023 ≤ u1(z) + ǫ = � γ + η 2 � � a∈A2\\{aF } z[a0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' a] + � a∈A1\\{a0} � a′∈A2 z[a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' a′]ui(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' a′) + ǫ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' where the inequality holds since the lhs is the utility of the deviation φav.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' This implies �� v′ z[a0, ¯av′] − z[a0, ¯av] � η ℓ − ℓ1−δ ℓ − ℓ1−δ − 1 ≤ η 2 � a∈A2\\{aF } z[a0, a] + ǫ ≤ η � v∈V z[a0, ¯av] + 2ǫ, where the last inequality holds by Equation (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Hence, z[a0, ¯av] ℓ − ℓ1−δ ℓ − ℓ1−δ − 1 ≥ � ℓ − ℓ1−δ ℓ − ℓ1−δ − 1 − 1 � � v′ z[a0, ¯av′] − 2ǫ/η, and ¯z[a0, av] ≥ 1 ℓ − ℓ1−δ � v′ z[a0, ¯av′] − 2ǫ/η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' (12) Suppose that |V 2| > ℓ − ℓ1−δ, and hence Equation (12) is satisfied by at least |V 2| ≥ ℓ − ℓ1−δ + 1 vertexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' We need the following inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' 1 ℓ � v′ z[a0, ¯av′] ≥ 1 ℓ � a∈A2\\{aF } z[a0, a] − 2ǫ ℓη ≥ α 4ℓ − 2ǫ ℓη = α 4ℓ − α 8ℓ3 ≥ α 8ℓ = 2ℓ η � α2 16ℓ2 � = 2ℓ η ǫ (13) where the first inequality comes from Equation (11), and the second one by Equation (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Then, summing over the |V 2| inequalities we get � v∈V 2 ¯z[a0, av] ≥ (ℓ − ℓ1−δ + 1) � 1 ℓ − ℓ1−δ � v′ z[a0, ¯av′] − 2ǫ/η � ≥ � v′ z[a0, ¯av′] + 1 ℓ � v′ z[a0, ¯av′] − 2ℓ ǫ η > � v′ z[a0, ¯av′], where the last inequality follows from equation (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Hence, we reach a contradiction and |V 2| ≤ ℓ − ℓ1−δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' To conclude the proof, we show that V 1 is an independent set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Since |V 1| ≥ |V | − |V 2| = ℓ1−δ we reach a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Let v and v′ be two nodes in V 1 and w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' let z[a0, av] ≥ z[a0, av′].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' We show that (v, v′) /∈ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Since v′ ∈ V 1, φav is not a safe deviation for player 1 with respect to constraint cv′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' if (v, v′) ∈ E, then � a1∈A1,a2∈A2 � a∈ A1 φ[a1, a]z[a1, a2]cv(a, a2) = z[a0, a′ v] − � v′′:(v′′,v′)∈E z[a0, av′′] − 1 4ℓz[a0, aF]cv(a, a2) + � a1∈A1\\{a0},a2∈A2 � a∈A1 φ[a1, a]z[a1, a2]cv(a, a2) ≤ z[a0, a′ v] − z[a0, av] − 1 4ℓz[a0, aF ]cv(a, a2)+ + � a1∈A1\\{a0},a2∈A2 � a∈A1 φ[a1, a]z[a1, a2]cv(a, a2) ≤ − 1 4ℓz[a0, aF ]cv(a, a2) + � a1∈A1\\{a0},a2∈A2 � a∈A1 φ[a1, a]z[a1, a2]cv(a, a2) 17 ARXIV PREPRINT - FEBRUARY 1, 2023 = cv(z) ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Hence, (v, v′) /∈ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Since V 1 is an independent set of size at least ℓ1−δ we reach a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' This concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' D Proofs Omitted from Section 4 Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' For every z ∈ ∆A and i ∈ N, it holds max φi∈ΦS i (z) ui(φi ⋄ z) = sup φi∈Φi inf ηi∈Rm + � ui(φi ⋄ z) − η⊤ i ci(φi ⋄ z) � = inf ηi∈Rm + sup φi∈Φi � ui(φi ⋄ z) − η⊤ i ci(φi ⋄ z) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' First, it is easy to see that sup φi∈ΦS i (z) ui(φi ⋄ z) = sup φi∈Φi inf ηi∈Rm + � ui(φi ⋄ z) − η⊤ i ci(φi ⋄ z) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Indeed, for every φi /∈ ΦS i (z), it holds that the vector ci(φi ⋄ z) has at least one positive component, and, thus, the vector of Lagrange multipliers ηi can be selected so that ui(φi ⋄ z) − η⊤ i ci(φi ⋄ z) goes to −∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' This implies that the supremum over Φi cannot be attained in ΦS i (z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' On the other hand, for every φi ∈ ΦS i (z), all the components of ci(φi ⋄ z) are negative, and, thus, the inf is achieved by ηi = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' This proves the first equality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Then, the second equality directly follows from the generalization of the max-min theorem for unbounded domains (see (Ekeland and Temam, 1999, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='3)), which allows us to swap the sup and the inf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Lemma D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' For any two real-valued functions f(x) and g(x) with g(x) ≤ c then min(f(x), g(x)) ≤ min(f(x), c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' We can identify three sets I1, I2 and I3 defined as follows: I1 := {x s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' f(x) ≥ c} I2 := {x s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' g(x) ≤ f(x) ≤ c} I3 := {x s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' f(x) ≤ g(x) ≤ c}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Then for all x ∈ I1 we have that min(f(x), c) = c ≥ min(f(x), g(x)) = g(x), while for all x ∈ I2 we have that min(f(x), c) = f(x) ≥ min(f(x), g(x)) = g(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Finally for all x ∈ I3 we have min(f(x), c) = f(x) = min(f(x), g(x)) = f(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' In all three sets we have that min(f(x), c) ≥ min(f(x), g(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Lemma D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' For all ηi ∈ Dc it holds that sup φi∈Φi � ui(φi ⋄ z) − η⊤ i ci(φi ⋄ z) � ≥ 1 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Thanks to Assumption 2 we have that for all z ∈ ∆(A) we have that there exists ˜φi ∈ ΦS i (z) such that ci(˜φi ⋄ z) ⪯ −ρ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Then, for all ηi ∈ Dc we have: η⊤ i ci(˜φi ⋄ z) ≤ −ρ∥ηi∥1 ≤ −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' This easily concludes the proof of the statement sup φi∈Φi � ui(φi ⋄ z) − η⊤ i ci(φi ⋄ z) � ≥ ui(˜φi ⋄ z) − η⊤ i ci(˜φi ⋄ z) ≥ 1, as ui is positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Lemma D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' For all ηi ∈ D we have inf ηi∈D sup φi∈Φi � ui(φi ⋄ z) − η⊤ i ci(φi ⋄ z) � ≤ 1 18 ARXIV PREPRINT - FEBRUARY 1, 2023 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Since ui ≤ 1 we have that inf ηi∈D sup φi∈Φi � ui(φi ⋄ z) − η⊤ i ci(φi ⋄ z) � ≤ 1 − sup ηi∈D inf φi∈Φi η⊤ i ci(φi ⋄ z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Next we claim that sup ηi∈D inf φi∈Φi η⊤ i ci(φi ⋄ z) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' This follows from the fact that for all negative components of ci(φi ⋄ z) then the corresponding components of ηi will be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' This concludes the statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Let D := � η ∈ Rm + | ||η||1 ≤ 1/ρ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Then, for every z ∈ ∆A and i ∈ N, it holds: max φi∈ΦS i (z) ui(φi ⋄ z) = max φi∈Φi min ηi∈D � ui(φi ⋄ z) − η⊤ i ci(φi ⋄ z) � = min ηi∈D max φi∈Φi � ui(φi ⋄ z) − η⊤ i ci(φi ⋄ z) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' In Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='1 we already showed that: sup φi∈ΦS i (z) ui(φi ⋄ z) = sup φi∈Φi inf ηi∈Rm + � ui(φi ⋄ z) − η⊤ i ci(φi ⋄ z) � = inf ηi∈Rm + sup φi∈Φi � ui(φi ⋄ z) − η⊤ i ci(φi ⋄ z) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Note that to prove the statement it is enough to prove that: inf ηi∈Rm + sup φi∈Φi � ui(φi ⋄ z) − η⊤ i ci(φi ⋄ z) � = inf ηi∈D sup φi∈Φi � ui(φi ⋄ z) − η⊤ i ci(φi ⋄ z) � and more specifically that: inf ηi∈Rm + sup φi∈Φi � ui(φi ⋄ z) − η⊤ i ci(φi ⋄ z) � ≥ inf ηi∈D sup φi∈Φi � ui(φi ⋄ z) − η⊤ i ci(φi ⋄ z) � since the reverse inequality holds trivially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' We can show this by the following inequalities: inf ηi∈Rm + sup φi∈Φi � ui(φi ⋄ z) − η⊤ i ci(φi ⋄ z) � = min � inf ηi∈D sup φi∈Φi � ui(φi ⋄ z) − η⊤ i ci(φi ⋄ z) � , inf ηi∈Dc sup φi∈Φi � ui(φi ⋄ z) − η⊤ i ci(φi ⋄ z) � � ≥ min � inf ηi∈D sup φi∈Φi � ui(φi ⋄ z) − η⊤ i ci(φi ⋄ z) � , 1 � = inf ηi∈D sup φi∈Φi � ui(φi ⋄ z) − η⊤ i ci(φi ⋄ z) � , where the first inequality hold thanks to Lemma D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='1 and Lemma D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='2, while that last equation follows from Lemma D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Given any 0 < δ ≤ ǫ and a δ-optimal set ˜D ⊆ D, the following holds: L ˜ D,ǫ ≥ LD,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' By definition we have that: L ¯ D,ǫ = ℓ(˜z⋆), where ˜z⋆ is a solution to the problem P1 := \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f3 ˜z⋆ ∈ arg max z∈S ℓ(z) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' ǫ + ui(˜z⋆) ≥ max φi∈Φi � ui(φi ⋄ ˜z⋆) − ˜η⋆,⊤ i ci(φi ⋄ ˜z⋆) � ˜η⋆ i ∈ arg inf ηi∈ ¯ D sup φi∈Φi � ui(φi ⋄ ˜z⋆) − η⊤ i ci(φi ⋄ ˜z⋆) � On the other hand, call z⋆ the optimal Constrained Phi-equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' This is a solution to the problem: P2 := \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f3 z⋆ ∈ arg max z∈S ℓ(z) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' ui(z⋆) ≥ max φi∈Φi � ui(φi ⋄ z⋆) − η⋆,⊤ i ci(φi ⋄ z⋆) � η⋆ i ∈ arg inf ηi∈D sup φi∈Φi � ui(φi ⋄ z⋆) − η⊤ i ci(φi ⋄ z⋆) � 19 ARXIV PREPRINT - FEBRUARY 1, 2023 which has value LD,0 = ℓ(z⋆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Moreover, thanks to Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='2 and since ¯D is δ-optimal we have that: max φi∈Φi � ui(φi ⋄ ˜z⋆) − ˜η⋆,⊤ i ci(φi ⋄ ˜z⋆) � ≤ max φi∈Φi � ui(φi ⋄ z⋆) − η⋆,⊤ i ci(φi ⋄ z⋆) � + δ which implies that feasible correlated strategies of problem P2 are feasible correlated strategies of problem P1, and thus problem P1 as long as δ ≥ ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Thus problem P1 is the problem of maximizing the same objective function over a larger set then problem P2 and thus L ¯ D,ǫ ≥ LD,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' For any τ > 0, the set Dτ is (τm)-optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='2, we know that for each player there exists an η⋆ i ∈ D such that maxφ∈ΦS i (z) ui(φi ⋄ z) = max φi∈Φi � ui(φi ⋄ z) − η⋆,⊤ i ci(φi ⋄ z) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' By construction of Dǫ there exists a ¯ηi ∈ Dǫ such that ||¯ηi − η⋆ i ||∞ ≤ ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Thus max φ∈ΦS i (z) ui(φi ⋄ z) = max φi∈Φi � ui(φi ⋄ z) − η⋆,⊤ i ci(φi ⋄ z) � ≤ max φi∈Φi � ui(φi ⋄ z) − ¯η⊤ i ci(φi ⋄ z) � + mǫ, where the last inequality comes the fact that: |(η⋆ i − ¯ηi)⊤ci(φi ⋄ z)| ≤ ∥ci(φi ⋄ z)∥1∥η⋆ i − ¯ηi∥∞ ≤ mǫ as ci ∈ [−1, 1]m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' For any τ > 0, the set Dτ is δ-optimal for δ = 2 � 2τ log s/ρ, where s is the number of players’ actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' The proof exploits a probability interpretation of the Lagrange multipliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Let η⋆ be the optimal multipliers, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=', , η⋆ ∈ argminη∈D maxφi∈Φi � ui(φi ⋄ z) − η⊤ci(φi ⋄ z) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Now consider a basis Γ = { 1 ρej}j∈[m] ∪ {0} for D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' By Carathoedory’s theorem there exists a distribution γ ∈ ∆(Γ) such that η⋆ = � η∈Γ γηη.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Assume that ǫ and ρ are such that 1/ǫρ is an integer and take 1/ρǫ samples from the distribution γ and call ˜η the resulting empirical mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' First, we argue that ˜η ∈ Dǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Indeed ˜ηj = kj 1/ρǫ 1 ρ = ǫ � kj 1/ρǫ 1 ρǫ � = ǫkj where kj ∈ N and thus we have that ˜η ∈ Dǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='11 Now we show that with high probability ˜η ∈ Dǫ is close (in terms of utilities) to the optimal multiplier η⋆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' First observe that: η⋆,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='⊤ i ci(φi ⋄ z) := � ai∈Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='bi∈Ai \uf8eb \uf8edφi[b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' ai] � a−i∈A−i η⋆,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='⊤ci(ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' a−i)z[b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' a−i] \uf8f6 \uf8f8 (14a) ≤ � ai∈Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='bi∈Ai \uf8eb \uf8edφi[b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' ai] \uf8eb \uf8edδai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='b + � a−i∈A−i ˜η⊤ci(ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' a−i)z[b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' a−i] \uf8f6 \uf8f8 \uf8f6 \uf8f8 (14b) = � ai∈Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='bi∈Ai \uf8eb \uf8edφi[b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' ai] � a−i∈A−i ˜η⊤ci(ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' a−i)z[b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' a−i] \uf8f6 \uf8f8 + � ai∈Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='bi∈Ai φi[b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' ai]δai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='b (14c) = ˜η⊤ i ci(φi ⋄ z) + � ai∈Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='bi∈Ai φi[b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' ai]δai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='b (14d) where the inequality comes from applying the Hoeffeding’s inequality to every ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' b ∈ Ai: ������ � a−i∈A−i (˜η − η⋆)⊤ ci(ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' a−i)z[b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' a−i] ������ ≤ δai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='b 11If ǫ if not such that 1/ρǫ ∈ N then the one can take ⌈1/ρǫ⌉ samples from γ ∈ ∆(Γ) and then the statement hold for a slightly smaller ǫ′ < ǫ defined as ǫ′ := 1 ⌈1/ρǫ⌉ 1 ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' 20 ARXIV PREPRINT - FEBRUARY 1, 2023 where δai,b = 2 ρ � 2 1/ρǫ log � 2 pai,b � � � a−i∈A−i z[b, a−i] � since the range of the each sample is 1 ρ �� a−i∈A−i z[b, a−i] � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Moreover, for Hoeffeding’s inequality, for every ai, b ∈ Ai the above inequality holds with probability at least 1 − pai,b and thus holds for all the ai, b ∈ Ai simultaneously, with probability at least p := � ai,b∈Ai pai,b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' If then we take pai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='b := 1 2|Ai|2 for all ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' b ∈ Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' then we have that p = 1/2 > 0 and δ := δai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='b = 2 ρ � 2 1/ρǫ log (|Ai|) � � a−i∈A−i z[b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' a−i] � Now the following holds with probability at lest 1/2: ������ � a−i∈A−i (˜η − η⋆)⊤ ci(ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' a−i)z[b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' a−i] ������ ≤ δ \uf8eb \uf8ed � a−i∈A−i z[b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' a−i] \uf8f6 \uf8f8 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' ∀ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' b ∈ Ai The proof is concluded by observing plugging this definition of δ = δai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='b in Equation (14) yields � ai∈Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='bi∈Ai φi[b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' ai]δai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='b = δ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' and we can conclude that: η⋆,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='⊤ i ci(φi ⋄ z) ≤ ˜η⊤ i ci(φi ⋄ z) + δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' This holds with positive probability, and thus shows the existence of such ˜η ∈ Dǫ for which the above inequality holds and thus Dǫ is � 2 � 2ǫ ρ log(|Ai|) � optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' E Proofs Omitted from Section 5 Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' For instances I := (Γ, Φ) satisfying Assumption 3, the set of constrained ǫ-Phi-equilibria is convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Let z′ and z′′ be Constrained ǫ-Phi-equilibria that is for all i ∈ [N]: ǫ + ui(z′) ≥ ui(φ′ i ⋄ z′) for φ′ ∈ arg max φi∈ΦS i ui(φi ⋄ z′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Equivalently it holds for all i ∈ [N] that: ǫ + ui(z′′) ≥ ui(φ′′ i ⋄ z′′) where φ′′ ∈ arg max φi∈ΦS i ui(φi ⋄ z′′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' For any z := αz′ + (1 − α)z′′ we have that: ǫ + ui(z) = α (ǫ + ui(z′)) + (1 − α) (ǫ + ui(z′′)) ≥ αui(φ′ i ⋄ z′) + (1 − α)ui(φ′′ i ⋄ z′′) ≥ max φi∈ΦS i ui(φi ⋄ z), where the inequality holds for the linearity of ui, the first inequality as both z′ and z′′ are Constrained ǫ-Phi-equilibria and the last inequality holds since the max is a convex operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Restricted to instances I := (Γ, Φ) which satisfy Assumption 3, APXCPE(1, 0) admits a polynomial- time algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' APXCPE(1, 0) amounts to solving the following problem: max z∈S ℓ(z) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' (15a) ui(z) ≥ max φi∈ΦS i ui(φi ⋄ z) ∀i ∈ N, (15b) which can be written as an LP with (possibly) exponentially-many constraints, by writing a constraint for each vertex of ΦS i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' We can find an exact solution to such an LP in polynomial time by means of the ellipsoid algorithm that uses suitable separation oracle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Such an oracle solves the following optimization problem for every i ∈ N: φ⋆ i ∈ arg max φi∈ΦS i ui(φi ⋄ z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' 21 ARXIV PREPRINT - FEBRUARY 1, 2023 Then, the oracle returns as a separating hyperplane the incentive constraint corresponding to a φ⋆ i (if any) such that ui(z) ≥ ui(φ⋆ i ⋄ z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Since all the steps of the separation oracle can be implemented in polynomial time, the ellipsoid algorithm runs in polynomial time, concluding the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Given an instance I := (Γ, Φ) satisfying Assumption 3, after T ∈ N>0 rounds, Algorithm 1 returns a correlated strategy ¯zT ∈ ∆A that is a constrained ǫT -Phi-equilibrium with ǫT = O(1/ √ T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Moreover, each round of Algorithm 1 runs in polynomial time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Any regret minimizer Ri for ΦS i guarantees that, for every φi ∈ ΦS i : T � t=1 ui(φi ⋄ zt) − T � t=1 ui(φi,t ⋄ zt) ≤ ǫi,T T, (16) where ǫi,T = o(T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Since xi,t[a] = � b∈Ai φi,t[b, a]xi,t[b] for all a ∈ Ai, for every t ∈ [T ] and a = (ai, a−i) ∈ A: (φi,t ⋄ zt)[ai, a−i] = � b∈Ai φi,t[b, ai]z[b, a−i] = � b∈Ai φi,t[b, ai] � xi,t[b] ⊗ x−i,t[a−i] � = � � b∈Ai φi,t[b, ai]xi,t[b] � ⊗ x−i,t[a−i] = xi,t[ai] ⊗ x−i,t[a−i] = zt[ai, a−i], Plugging the equation above into Equation (16), we get: T � t=1 ui(φi ⋄ zt) − T � t=1 ui(zt) ≤ ǫi,T T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Now, since ¯zT := �T t=1 zt and ui(z) is linear in z, we can conclude that, for every i ∈ N and φi ∈ ΦS i : ui(zT ) ≥ ui(φi ⋄ ¯zT ) − ǫi,T , and, thus, by letting ǫT := maxi∈N ǫi,T we get that ¯zT satisfies the incentivize constrained for being a constrained ǫT -Phi-equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' We are left to verify that ¯zT ∈ S, namely ci(¯zT ) ≤ 0 for all i ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' This readily proved as: ci(¯zT ) = 1 T T � t=1 ci(zt) = 1 T T � t=1 ci(φi,t ⋄ zt) = 1 T T � t=1 ˜ci(φi,t) ≤ 0, where the first equality holds since ci is linear, the second equality holds thanks to zt = φi,t ⋄ zt, the third one by Assumption 3, while the inequality holds since φi,t ∈ ΦS i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' This concludes the proof of the first part of the statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' In conclusion, Algorithm 1 runs in polynomial time as finding xi,t[a] = � b∈Ai φi,t[b, ai]xi,t[b] for all a ∈ Ai is equivalent to finding a stationary distribution of a Markov Chain, which can be done in polynomial time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Moreover, we can implement the regret minimizers Ri over the polytopes ΦS i so that their operations run in polynomial time, such as, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=', online gradient descent;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' see (Hazan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=', 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' For instances I := (Γ, ΦCCE) such that ci(a) = ci(a′) for every player i ∈ N and action profiles a, a′ ∈ A : ai = a′ i, Assumption 3 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' 22 ARXIV PREPRINT - FEBRUARY 1, 2023 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Since the costs ci(a) of player i ∈ N only depends on player i’s action ai and not on the actions of other players, it is possible to show that there exists ˜ci : ΦCCE → [−1, 1]m such that the following holds for every z ∈ ∆A: ˜ci(φi) := ci(φi ⋄ z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Indeed, for every φi ∈ ΦCCE, by definition of ΦCCE there exists a probability distribution h ∈ ∆Ai : φi[b, a] = h[a] for all b, a ∈ Ai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Then, for every ai ∈ Ai and a−i ∈ A−i, we can write: (φi ⋄ z)[ai, a−i] = � b∈Ai φi[b, ai]z[b, a−i] = � b∈Ai h[ai]z[b, a−i] = h[ai] � b∈Ai z[b, a−i].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Moreover, it holds: ci(φi ⋄ z)[ai, a−i] = � a∈A ci(a)(φi ⋄ z)[ai, a−i] = � a∈A ci(a)h[ai] � b∈Ai z[b, a−i] = � ai∈Ai ci(ai, ·)h[ai] � a−i∈A−i � b∈Ai z[b, a−i] = � ai∈Ai ci(ai, ·)h[ai], which only depends on φi, as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' Notice that, in the equations above, for every a ∈ Ai we let ci(a, ·) be the (unique) value of ci(a) for all a ∈ A : ai = a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} +page_content=' 23' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFRT4oBgHgl3EQfmDfL/content/2301.13600v1.pdf'} diff --git a/1tE2T4oBgHgl3EQfNQaE/content/tmp_files/2301.03735v1.pdf.txt b/1tE2T4oBgHgl3EQfNQaE/content/tmp_files/2301.03735v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..66b54f1b32b4f272d5a8b092ea6279c774108589 --- /dev/null +++ b/1tE2T4oBgHgl3EQfNQaE/content/tmp_files/2301.03735v1.pdf.txt @@ -0,0 +1,2223 @@ +arXiv:2301.03735v1 [math.RA] 10 Jan 2023 +Multipliers and weak multipliers of algebras +Yuji Kobayashi and Sin-Ei Takahasi +Laboratory of Mathematics and Games +(https://math-game-labo.com) +2020 MSC Numbers: Primary 43A22; Secondary 17A99, 46J10. +Keywords: (weak) multiplier, (non)associative algebra, Jordan algebra, zeropotent algebra, +annihilator, nihil decomposition, matrix representation. +Abstract +We study general properties of multipliers and weak multipliers of +algebras. +We apply the results to determine the (weak) multipliers of +associative algebras and zeropotent algebras of dimension 3 over an alge- +braically closed field. +1 +Introduction +Multipliers of algebras, in particular, multipliers of Banach algebras, have been +discussed in analysis. In this paper we will discuss them in a purely algebraic +manner. +Let B be a Banach algebra. A mapping T : B → B is called a multiplier of +B, if it satisfies the condition (I) xT (y) = T (xy) = T (x)y (x, y ∈ B). Let M(B) +denote the collection of all multipliers of B, and let B(B) be the collection of all +bounded linear operators on B. Then M(B) forms an algebra and B(B) forms +a Banach algebra. B is called without order if it has no nonzero left or right +annihilator. +If B is without order, then M(B) forms a commutative closed +subalgebra of B(B) (see [2], Proposition 1.4.11). In 1953, Wendel [7] proved +an important result that the multiplier algebra of L1(G) on a locally compact +abelian group G is isometrically isomorphic to the measure algebra on G. The +general theory of multipliers of Banach algebras has been developed by Johnson +[1]. A good reference to the theory of multipliers of Banach algebra is given in +Larsen [5]. +When B is without order, T is a multiplier if it satisfies the condition (II) +xT (y) = T (x)y (x, y ∈ B). Many researchers had been unaware of difference +between conditions (I) and (II) until Zivari-Kazempour [8] (see also [9]) recently +clearly stated the difference. We call a mapping T satisfying (II) a weak mul- +tiplier and denote the set of weak multipliers of B by M ′(B). Then, M(B) +is in general a proper subset of M ′(B). Furthermore, (weak) multipliers can +1 + +be defined for an algebra A not necessarily associative, and they are not lin- +ear mappings in general. We denote the spaces of linear multipliers and linear +weak multipliers of A by LM(A) and LM ′(A) respectively. M(A) and LM(A) +are subalgebras of the algebra AA consisting of all mappings from A to itself. +Meanwhile, M ′(A) and LM ′(A) are closed under the operation ◦ defined by +T ◦ S = T S + ST , and they form a Jordan algebra. +In Sections 2 - 5 we study general properties of (weak) multipliers. In par- +ticular, in sections 3 and 4 we give a decomposition theorem (Theorem 3.1), and +a matrix equation (Theorem 4.2) for (weak) multipliers. They play an essential +role to analyze (weak) multipliers. +Complete classifications of associative algebras and zeropotent algebras of +dimension 3 over an algebraically closed field of characteristic not equal to 2 +were given in Kobayashi et al, [3] and [4]. In Sections 6 and 7 we completely +determine the (linear) (weak) multipliers of those algebras. +2 +Multipliers and weak multipliers +Let K be a field and A be a (not necessarily associative) algebra over K. The +set AA of all mappings from A to A forms an associative algebra over K in the +usual manner. Let L(A) denotes the subalgebra of AA of all linear mappings +from A to A. +A mapping T : A → A is a weak multiplier of A, if +xT (y) = T (x)y +(1) +holds for any x, y ∈ A, and T is a multiplier, if +xT (y) = T (xy) = T (x)y +(2) +for any x, y ∈ A. Let M(A) (resp. M ′(A)) denote the set of all multipliers +(resp. weak multipliers) of A. Define +LM(A) +def += M(A) ∩ L(A) and LM ′(A) +def += M ′(A) ∩ L(A). +Proposition 2.1. M(A) (resp. LM(A)) is a unital subalgebra of AA (resp. +L(A)), and M ′(A) (resp. LM ′(A)) is a Jordan subalgebra of AA (resp. L(A)). +Proof. First, the zero mapping is a multiplier because all the three terms in (2) +are zero. Secondly, the identity mapping is also a multiplier because the three +terms in (2) are xy. Let T, U ∈ M(A). Then we have +x(T +U)(y) = xT (y)+xU(y) = T (xy)+U(xy) = T (x)y+U(x)y = (T +U)(x)y +(3) +and +x(T U)(y) = xT (U(y)) = T (xU(y)) = T U(xy) = T (U(x)y) = (T U)(x)y +(4) +2 + +for any x, y ∈ A. Hence, T + U, T U ∈ M(A). Finally let k ∈ K, then +x(kT )(y) = kxT (y) = kT (xy) = kT (x)y = (kT )(x)y, +(5) +and so kT ∈ M(A). +Therefore, M(A) is a unital subalgebra of AA, and +LM(A) = M(A) ∩ L(A) is a unital subalgebra of L(A). +Next, let T, U ∈ M ′(A). Then, the equalities in (3) and (5) hold removing +the center terms T (xy) + U(xy) and kT (xy) respectively. Hence, M ′(A) is a +subspace of AA. Moreover, we have +x(T U)(y) = xT (U(y)) = T (x)U(y) = U(T (x))y = UT (x)y +and similarly x(UT )(y) = T U(x)y for any x, y ∈ A. Hence, +x(T U + UT )(y) = (T U + UT )(x)y. +It follows that T U + UT ∈ M ′(A).1 +The opposite Aop of A is the algebra with the same elements and the addition +as A, but the multiplication ∗ in it is reversed, that is, x∗y = yx for all x, y ∈ A. +Proposition 2.2. A and Aop have the same multipliers and weak multiplies, +that is, +M(Aop) = M(A) and M ′(Aop) = M ′(A). +Proof. Let T ∈ AA. Then, T ∈ M ′(A), if and only if +x ∗ T (y) = T (y)x = yT (x) = T (x) ∗ y +for any x, y ∈ A, if and only if T ∈ M ′(Aop). Further, T ∈ M(A), if and only if +x ∗ T (y) = T (y)x = T (yx) = T (x ∗ y) = yT (x) = T (x) ∗ y +for any x, y ∈ A, if and only if T ∈ M(Aop). +Let Annl(A) (resp. Annr(A)) be the left (resp. right) annihilator of A and +let A0 be their intersection, that is, +Annl(A) = {a ∈ A | ax = 0 for all x ∈ A}, +Annr(A) = {a ∈ A | xa = 0 for all x ∈ A} +and +A0 = Annl(A) ∩ Annr(A). +They are all subspaces of A, and when A is an associative algebra, they are +two-sided ideals. For a subset X of A, ⟨X⟩ denotes the subspace of A generated +by X. +1In general, for an associative algebra A over a field K of characteristic ̸= 2, the Jordan +product ◦ on A is defined by x ◦ y = (xy + yx)/2 for x, y ∈ A. +3 + +Proposition 2.3. A weak multiplier T of A such that ⟨T (A)⟩ ∩ A0 = {0} is a +linear mapping. +Proof. Let x, y, z ∈ A and a, b ∈ K, and let T be a weak multiplier. We have +T (ax + by)z = (ax + by)T (z) = axT (z) + byT (z) = aT (x)z + bT (y)z += (aT (x) + bT (y))z. +Because z is arbitrary, we have w = T (ax + by) − aT (x) − bT (y) ∈ Annl(A). +Similarly, we can show w ∈ Annr(A), and so w ∈ A0. Hence, if ⟨T (A)⟩ ∩ A0 = +{0}, then w = 0 because w ∈ ⟨T (A)⟩. Since a, b, x, y are arbitrary, T is a linear +mapping. +Corollary 2.4. If A0 = {0}, then any weak multiplier is a linear mapping over +K, that is, M ′(A) = LM ′(A) and M(A) = LM(A). +Proposition 2.5. If T is a weak multiplier, then T (Annl(A)) ⊆ Annl(A), +T (Annr(A)) ⊆ Annr(A) and T (A0) ⊆ A0 . +Proof. Let x ∈ Annl(A), then for any y ∈ A we have +0 = xT (y) = T (x)y. +Hence, T (x) ∈ Annl(A). The other cases are similar. +In this paper we denote the subset {xy | x, y ∈ A} of A by A2, though usually +A2 denotes the subspace of A generated by this set. +Proposition 2.6. Any mapping T : A → A such that T (A) ⊆ A0 is a weak +multiplier. Such a mapping T is a multiplier if and only if T (A2) = {0}. In +particular, if A is the zero algebra, every mapping T is a weak multiplier, and +it is a multiplier if only if T (0) = 0. +Proof. If T (A) ⊆ A0, the both sides are 0 in (1) and T is a weak multiplier. +This T is a multiplier, if only if the term T (xy) in the middle of (2) is 0 for all +x, y ∈ A, that is, T (A2) = {0}. If A is the zero algebra, then A = A0 and A2 += {0}. Hence, any T is a weak multiplier and it is a multiplier if and only if +T (0) = 0. +3 +Nihil decomposition +Let A1 be a subspace of A such that +A = A1 ⊕ A0. +(6) +Here, A1 is not unique, but choosing an appropriate A1 will become important +later. When A1 is fixed, any mapping T ∈ AA is uniquely decomposed as +T = T1 + T0 +(7) +4 + +with T1(A) ⊆ A1 and T0(A) ⊆ A0. We call (6) and (7) a nihil decompositions +of A and T respectively. Let π : A → A1 be the projection and µ : A1 → A be +the embedding, that is, π(x1 + x0) = µ(x1) = x1 for x1 ∈ A1 and x0 ∈ A0. +Let M1(A) (resp. +M0(A)) denote the set of all multipliers T of A with +T (A) ⊆ A1 (resp. T (A) ⊆ A0). Similarly, the sets M ′ +1(A) and M ′ +0(A) of weak +multipliers of A are defined. Also, set +LMi(A) = Mi(A) ∩ L(A) and LM ′ +i(A) = M ′ +i(A) ∩ L(A) +for i = 0, 1. By Proposition 2.3 we see +M ′ +1(A) = LM ′ +1(A) and M1(A) = LM1(A), +and by Proposition 2.6 we have +M ′ +0(A) = AA +0 , M0(A) = {T ∈ AA +0 | T (A2) = {0}}. +(8) +Theorem 3.1. Let A = A1 ⊕ A0 and T = T1 + T0 be nihil decompositions of A +and T ∈ AA respectively. +(i) T is a weak multiplier, if and only if T1 is a weak multiplier. If T is a +weak multiplier, T1 is a linear mapping satisfying T1(A0) = {0}. +(ii) If T1 is a multiplier and T0(A2) = {0}, then T is a multiplier. If A1 is +a subalgebra of A, the converse is also true. +Suppose that A1 is a subalgebra of A, and let Φ be a mapping sending R ∈ +(A1)A1 to µ ◦ R ◦ π ∈ AA. Then, +(iii) Φ gives an algebra isomorphism from M(A1) onto M1(A) and a Jordan +isomorphism from M ′(A1) onto M ′ +1(A). +Proof. Let x, y ∈ A. +(i) If T is a weak multiplier, then +xT1(y) = x(T (y) − T0(y)) = xT (y) = T (x)y = T1(x)y. +Thus, T1 is also a weak multiplier. Moreover, T1 is a linear mapping by Propo- +sition 2.3 and T1(A0) ⊆ A1 ∩ A0 = {0} by Proposition 2.5. Conversely, if T1 is +a weak multiplier, then +xT (y) = xT1(y) = T1(x)y = T (x)y, +and so T is a weak multiplier. +(ii) If T1 is a multiplier and T0(A2) = 0, then T is a multiplier because +xT (y) = xT1(y) = T1(xy) = T (xy) − T0(xy) = T (xy) = T1(x)y = T (x)y. +Next suppose that A1 is a subalgebra. If T is a multiplier, then for any +x, y ∈ A we have +T1(xy) + T0(xy) = T (xy) = xT (y) = x1T1(y), +(9) +5 + +where x = x1 + x0 with x1 ∈ A1 and x0 ∈ A0. Here, x1T1(y) ∈ A1 because A1 +is a subalgebra, and thus, we have T0(xy) = x1T1(y) − T1(xy) ∈ A0 ∩ A1 = {0}. +Since x, y are arbitrary, we get T0(A2) = {0}. Moreover, T1 is a multiplier +because T1(xy) = x1T1(y) = xT1(y) by (9) and similarly T1(xy) = T1(x)y. The +converse is already proved above. +(iii) Let S ∈ (A1)A1 and x = x1 + x0, y = y1 + y0 ∈ A with x1, y1 ∈ A1 and +x0, y0 ∈ A0. Then, π(x) = µ(x1) = x1, π(y) = µ(y1) = y1 and +Φ(S)(x) = µ(S(π(x))) = µ(S(x1)) = S(x1). +Thus, if S ∈ M ′(A1), we have +xΦ(S)(y) = xS(y1) = x1S(y1) = S(x1)y1 = Φ(S)(x)y1 = Φ(S)(x)y. +Hence, Φ(S) ∈ M ′ +1(A). Moreover, if S ∈ M(A1), then because π is a homomor- +phism, we have +Φ(S)(xy) = S(π(xy)) = S(x1y1) = x1S(y1) = xΦ(S)(y), +and hence Φ(S) ∈ M1(A). Conversely, let T ∈ M ′ +1(A), then because T is a +linear mapping satisfying T (A0) = {0}, there is a linear mapping S ∈ L(A1) on +A1 such that Φ(S) = T , that is, S(x1) = T (x) = T (x1). We have +x1S(y1) = x1T (y1) = T (x1)y1 = S(x1)y1, +and hence S ∈ M ′(A1). Therefore, Φ is a linear isomorphism from M ′(A1) +to M ′ +1(A). +Similarly, Φ gives a linear isomorphism from M(A1) to M1(A). +Moreover, for T, U ∈ M ′(A1), we have +Φ(T U) = µ ◦ T ◦ U ◦ π = µ ◦ T ◦ π ◦ µ ◦ U ◦ π = Φ(T )Φ(U). +Thus, Φ gives an isomorphism of algebras from M(A1) to M1(A) and a Jordan +isomorphism from M ′(A1) to M ′ +1(A). +Theorem 3.1 implies +M ′(A) = M ′ +1(A) ⊕ M ′ +0(A), M1(A) ⊕ M0(A) ⊆ M(A), +where M ′ +0(A) and M0(A) are given as (8). Moreover, if A1 is a subalgebra, we +have +M ′(A) ∼= M ′(A1)⊕(A0)A, M(A) ∼= M(A1)⊕{T ∈ (A0)A | T (A2) = {0}}. (10) +Corollary 3.2. Any weak multiplier T is written as +T = T1 + R +(11) +with T1 ∈ LM ′ +1(A) and R ∈ (A0)A, and it is a multiplier if and only if +R(x1y1) = x1T1(y1) − T1(x1y1) +(12) +for any x1, y1 ∈ A1. +6 + +Proof. As stated above T is written as (11). Let x = x1 + x0, y = y1 + y0 ∈ A +with x1, y1 ∈ A1 and x0, y0 ∈ A0 be arbitrary, then we have +xT (y) = x1(T1(y) + R(y)) = x1T1(y) = x1T1(y1) +(13) +because R(A) ⊆ A0 and T1(A0) = {0}. The last term in (13) is also equal +to T1(x1)y1 = T (x)y. Hence, T is a multiplier, if and only if it is equal to +T (xy) = T (x1y1) = T1(x1y1) + R(x1y1), if and only if (12) holds. +4 +Linear multipliers and matrix equation +In this section, A is a finite dimensional algebra over K. We drive a matrix +equation for a linear mapping on A to be a (weak) multiplier. Suppose that A +is n-dimensional with basis E = {e1, e2, . . . , en}. +Lemma 4.1. A linear mapping T : A → A is a weak multiplier if and only if +eiT (ej) = T (ei)ej, +(14) +and it is a multiplier if and only if +T (eiej) = eiT (ej) = T (ei)ej, +(15) +for all ei, ej ∈ E. +Proof. The necessity of the conditions (14) and (15) is obvious. Let x = x1e1 + +x2e2+· · ·+xnen, y = y1e1+y2e2+· · ·+ynen ∈ A with x1, x2, . . . , xn, y1, y2, . . . , yn ∈ +K. If T satisfies (14), then we have +xT (y) += +( +� +i +xiei)T ( +� +j +yjej) = ( +� +i +xiei)( +� +j +yjT (ej)) = +� +i,j +xiyjeiT (ej) += +� +i,j +xiyjT (ei)ej = ( +� +i +xiT (ei))( +� +j +yjej) = T (x)y. +Hence, T is a weak multiplier. Moreover, if T satisfies (15), it is a multiplier in +a similar way +Let A (we use the bold character) be the multiplication table of A on E. A +is a matrix whose elements are from A defined by +A = EtE, +(16) +where E = (e1, e2, . . . , en) (we again use the bold face E) is the row vector +consisting the basis elements. For a linear mapping T on A and a matrix B +over A, T (B) denotes the matrix obtained by applying T component-wise, that +is, the (i, j)-element of T (B) is T (bij) for the (i, j)-element bij of B.2 We use +the same character T for the representation matrix of T on E, that is, +T (E) = ET. +(17) +. +2This is called a broadcasting (cf. [6]). +7 + +Theorem 4.2. A linear mapping T is a weak multiplier of A if and only if +AT = T tA, +(18) +and T is a multiplier if and only if +T (A) = AT = T tA. +(19) +Proof. By (16) and (17) we have +(e1, e2, . . . , en)t(T (e1), T (e2), . . . , T (en)) = EtT (E) = EtET = AT +(20) +and +(T (e1), T (e2), . . . , T (e2))t(e1, e2, . . . , en) = T (E)tE = T tEtE = T tA. +(21) +By Lemma 4.1, T is a weak multiplier, if and only if (20) and (21) are equal, +if and only if (18) holds. Moreover, T is multiplier if and only if, the leftmost +sides of (20) and (21) are equal to (T (eiej))i,j=1,2,...,n = T (A), if and only if +(19) holds. +The multiplication table of the opposite algebra Aop of A is At. So, the alge- +bras with multiplication tables transposed to each other have the same (weak) +multipliers. +5 +Associative algebras +In this section A is an associative algebra over K. +Proposition 5.1. If A0 = {0}, then we have +M(A) = M ′(A) = LM(A) = LM ′(A). +Proof. Let T ∈ M ′(A), then we have +T (xy)z = xyT (z) = xT (y)z and zT (xy) = T (z)xy = zT (x)y +for any x, y, z ∈ A. It follows that +T (xy) − xT (y) ∈ Annl(A) ∩ Annr(A) = A0 = {0}. +Hence, T (xy) = xT (y) and we see T ∈ M(A). +Moreover, T ∈ LM(A) by +Proposition 2.3. +Let a ∈ A. If xay = axy (resp. xay = xya) for any x, y ∈ A, a is called a +left (resp. right) central element, and a is called a central element if ax = xa +for any x ∈ A. Let Zl(A), (resp. Zr(A), Z(A)) denotes the set of all left central +(resp. right central, central) elements. +8 + +Lemma 5.2. Zl(A) (resp. Zr(A), Z(A)) are subalgebra of A containing Annl(A) +(resp. Annr(A), A0). +Proof. Straightforward. +For a ∈ A, la (resp. ra) denotes the left (resp. right) multiplication by a, +that is, +la(x) = ax, +ra(x) = xa +for x ∈ A. They are linear mappings. +Lemma 5.3. For a ∈ A the following statements are equivalent. +(i) la (resp. ra) is a multiplier, +(ii) la (resp. ra) is a weak multiplier, +(iii) a is left (resp. right) central. +Proof. If la is a weak multiplier, then +xay = xla(y) = la(x)y = axy +for any x, y ∈ A. Hence, a is left central. Because la(x)y = axy = la(xy) and +xla(y) = xay for any x, y ∈ A, la is a multiplier if a is left central. The other +case is similar, and we see that the three statements are equivalent. +Lemma 5.4. Suppose that A has a right (resp. left) identity. Then, a left (resp. +right) central element is central, and Annl(A) = {0} (resp. Annr(A) = {0}). +Proof. Easy. +Because Annl(A) ⊆ Zl and Annr(A) ⊆ Zr by Lemma 5.2, we can make the +quotient algebras ¯Zl(A) = Zl(A)/Annl(A) and ¯Zr(A) = Zr(A)/Annr(A). +Theorem 5.5. Suppose that A has a left (resp. right) identity e. Then, any +multiplier is a left (resp. right) multiplication by a left (resp. right) central +element and is a linear multiplier. The mapping φ : Zl(A) → M(A) sending +a ∈ Zl(A) to la induces an isomorphism ¯φ : Zl(A) → M(A) of algebras. In +particular, if A is unital, M(A) is isomorphic to Z(A). +Proof. Suppose that A has a left identity e. Let T ∈ M ′(A) and set a = T (e). +Then we have +T (x) = eT (x) = T (e)x = ax +for any x ∈ A. Hence, T = la, where a ∈ Zl(A) and T is a linear multiplier +by Lemma 5.3. Therefore, M ′(A) = LM(A) and φ is surjective. Moreover, +for a ∈ Zl(A), φ(a) = 0, if and only if ax = 0 for any x ∈ A, if and only +if a ∈ Annl(A). Thus we have Ker(φ) = Annl(A), and φ induces the desired +isomorphism. Similarly, if A has a right identity, M(A) is isomorphic to Zr(A). +Finally, if A has the identity, then Annl(A) = Annr(A) = {0} and hence M(A) +is isomorphic to Z(A). +9 + +6 +3-dimensional associative algebras +Over an algebraically closed field K of characteristic not equal to 2, we have, +up to isomorphism, 24 families of associative algebras of dimension 3. They are +5 unital algebras U0, U1, U2, U3, U4 defined on basis E = {e, f, g} by +�e +f +g +f +0 +0 +g +0 +0 +� +, +�e +f +g +f +0 +f +g +−f +e +� +, +�e +0 +0 +0 +f +0 +0 +0 +g +� +, +�e +0 +0 +0 +f +g +0 +g +0 +� +, +�e +f +g +f +g +0 +g +0 +0 +� +, +5 curled algebras C0, C1, C2, C3, C4 defined by +�0 +0 +0 +0 +0 +0 +0 +0 +0 +� +, +�0 +0 +0 +0 +0 +e +0 +−e +0 +� +, +�0 +0 +0 +e +f +0 +0 +g +0 +� +, +�0 +0 +0 +0 +0 +0 +e +f +g +� +, +�0 +0 +e +0 +0 +f +0 +0 +g +� +, +non-unital 4 straight algebras S1, S2, S3, S4 defined by +�f +g +0 +g +0 +0 +0 +0 +0 +� +, +�e +0 +0 +0 +g +0 +0 +0 +0 +� +, +�e +0 +0 +0 +f +0 +0 +0 +0 +� +, +�e +f +0 +f +0 +0 +0 +0 +0 +� +, +and non-unital 10 families of waved algebras W1, W2, W4, W5, W6, W7, W8, +W9, W10 and +� +W3(k) +� +k∈H defined by +�0 +0 +0 +0 +0 +0 +0 +0 +e +� +, +�0 +0 +0 +0 +0 +0 +0 +e +0 +� +, +�e +0 +0 +0 +0 +0 +0 +0 +0 +� +, +�0 +0 +0 +0 +0 +0 +0 +f +g +� +, +�0 +0 +0 +0 +0 +f +0 +0 +g +� +, +�e +0 +0 +0 +0 +0 +0 +f +g +� +, +�e +0 +0 +0 +0 +f +0 +0 +g +� +, +�0 +e +0 +e +f +0 +0 +g +0 +� +, +�0 +e +0 +e +f +g +0 +0 +0 +� +and +�0 +0 +0 +0 +e +0 +0 +ke +e +� +, +respectively, where H is a subset of K such that K = H∪−H and H∩−H = {0} +(see [3] for details). We determine the (weak) multipliers of them below. +(0) A = C0 is the zero algebra, so by Proposition 2.6, we have +M ′(A) = AA, M(A) = {T ∈ AA | T (0) = 0} +and +LM(A) = LM ′(A) = L(A). +(i) The unital algebras U0, U2, U3, U4 are commutative, so for such A we have +M(A) = LM(A) = M ′(A) = LM ′(A) = {lx|x ∈ A} ∼= A +by Theorem 5.5. For A = U1, we have +M(A) = LM(A) = M ′(A) = LM ′(A) ∼= Z(A) = Ke. +(ii) For A = C1, We have A0 = Annl(A) = Annr(A) = Ke, and a nihil decompo- +sition A = A1 ⊕ A0 with A1 = Kf + Kg. Let T1 ∈ M ′ +1(A), then by Theorem 3.1, T1 +is a linear mapping such that T1(Ke) = {0}. Let +T1 = + + +0 +0 +0 +0 +q +r +0 +t +u + + +(22) +10 + +with q, r, t, u ∈ K be the representation matrix of T1 on E. By Theorem 4.2, T1 is a +weak multiplier, if and only if + + +0 +0 +0 +0 +te +ue +0 +−qe +−re + + = AT1 = T t +1A = + + +0 +0 +0 +0 +−te +qe +0 +−ue +re + + , +if and only if r = t = 0 and q = u. Hence, M ′ +1(A) = {Tq +�� q ∈ K}, where Tq = + + +0 +0 +0 +0 +q +0 +0 +0 +q + +. By Theorem 3.1 we see +M ′(A) = {Tq +�� q ∈ K} ⊕ (Ke)A, +and +LM ′(A) = + + + + + +a +b +c +0 +q +0 +0 +0 +q + + +��� a, b, c, q ∈ K + + + . +By the multiplication table of A, we have αβ = (xv − yz)e +for α = xf + yg, β = +zf + vg ∈ A1 with x, y, z, v ∈ K. By Corollary 3.2, T ∈ M ′(A) is given by T = Tq + R +with R ∈ (Ke)A and this T is a multiplier, if and only if +R((xv − yz)e) += +R(αβ) = αTq(β) − Tq(αβ) += +α(qβ) − Tq((xv − yz)e) = q(xv − yz)e +for any α and β, if and only if R(xe) = qxe for all x ∈ K. Let Sq = + + +q +0 +0 +0 +q +0 +0 +0 +q + + be +the scalar multiplication by q ∈ K. Then, we see (T − Sq)(A) ⊆ A0 = Ke and +(T − Sq)(xe) = Tq(xe) + R(xe) − Sq(xe) = 0 + qxe − qxe = 0, +for any x ∈ K, that is, (T − Sq)(Ke) = {0}. Thus, we have +M(A) = {Sq +�� q ∈ K} ⊕ {R ∈ (Ke)A | R(Ke) = {0}}, +and +LM(A) = + + + + + +a +b +c +0 +a +0 +0 +0 +a + + +��� a, b, c ∈ K + + + . +(iii) A = C2: Because Annl(A) = Ke and Annr(A) = Kg, we see A0 = {0}. +Hence, any weak multiplier T is a linear multiplier by Proposition 5.1. By Theorem +4.2, +T = + + +a +b +c +p +q +r +s +t +u + + +(23) +is a (weak) multiplier, if and only if + + +0 +0 +0 +ae + pf +be + qf +ce + rf +pg +qg +rg + + = AT = T tA = + + +pe +pf + sg +0 +qe +qf + tg +0 +re +rf + ug +0 + + , +11 + +if and only if b = c = p = r = s = t = 0 and a = q = u, that is, T is the scalar +multiplication Sa by a. Consequently, +M(A) = M ′(A) = LM(A) = LM ′(A) = {Sa +�� a ∈ K} ∼= K. +(iv) C3 and C4 are opposed to each other, and have the same (weak) multipliers +by Proposition 2.2. +Let A = C3, then, A has a left identity g, Zl(A) = A and +Annl(A) = Ke + Kf. Hence, by Theorem 5.4, +M(A) = M ′(A) = LM(A) = LM ′(A) = A/(Ke + Kg) = {Sa +�� a ∈ K}. +(v) A = S1: We have A0 = Annl(A) = Annr(A) = Kg, and A = A1 ⊕ A0 with +A1 = Ke + Kf. Then, T1 ∈ M ′ +1(A) is a linear mapping with T (Kg) = {0}. Let +T1 = + + +a +b +0 +p +q +0 +0 +0 +0 + + +(24) +be its representation on E. T1 is a weak multiplier, if and only if + + +af + pg +bf + qg +0 +ag +bg +0 +0 +0 +0 + + = AT1 = T t +1A = + + +af + pg +ag +0 +bf + qg +bg +0 +0 +0 +0 + + , +if and only if b = 0 and a = q. Hence, +M ′(A) = {T a,p +1 +| a, p ∈ K} ⊕ (Kg)A, +where T a,p +1 += + + +a +0 +0 +p +a +0 +0 +0 +0 + +. So, T ∈ M ′(A) is written as T = T a,p +1 ++R with R ∈ (Kg)A, +and this T is multiplier, if and only if +R(xzf + (xv + yz)g) += +R(αβ) = αT a,p +1 +(β) − T a,p +1 +(αβ) += +α(aze + (pz + av)f) − T a,p +1 +(xzf + (xv + yz)g) += +axzf + (pxz + axv + ayz)g − axzf += +(pxz + a(xv + yz))g +for any α = xe + yf, β = ze + vf ∈ A1 with x, y, z, v ∈ K, if and only if R(xf + yg) = +(px + ay)g for all x, y ∈ K. Let T a,p = + + +a +0 +0 +p +a +0 +0 +p +a + +, then (T − T a,p)(A) ⊆ Kg, and +(T − T a,p)(xf + yg) += +(T a,p +1 ++ R − T a,p) (xf + yg) += +axf + (px + ay)g − (axf + pxg + ayg) = 0. +for any x, y ∈ K. Thus, (T − T a,p)(Kf + Kg) = {0}, and hence +M(A) = {T a,p | a, p ∈ K} ⊕ {R ∈ (Kg)A | R(Kf + Kg) = {0}}. +Taking the intersections of M ′(A) and M(A) with L(A), we obtain +LM ′(A) = + + + + + +a +0 +0 +p +a +0 +s +t +u + + +��� a, p, s, t, u ∈ K + + + +12 + +and +LM(A) = + + + + + +a +0 +0 +p +a +0 +s +p +a + + +��� a, p, s ∈ K + + + . +(vi) A = S2: We have A0 = Annl(A) = Annr(A) = Kg, and A = A1 ⊕ A0 with +A1 = Ke + Kf. Let a linear mapping T1 ∈ M ′ +1(A) be represented as (24), then T1 is +a weak multiplier, if and only if + + +ae +be +0 +pg +qg +0 +0 +0 +0 + + = AT = T tA = + + +ae +pg +0 +be +qg +0 +0 +0 +0 + + , +if and only if b = p = 0. Hence, +M ′(A) = {T a,q +1 +�� a, q ∈ K} ⊕ (Kg)A, +where T a,q +1 += + + +a +0 +0 +0 +q +0 +0 +0 +0 + +, and +LM ′(A) = + + + + + +a +0 +0 +0 +q +0 +s +t +u + + +��� a, q, s, t, u ∈ K + + + . +By Corollary 3.2, a weak multiplier T written as T = T a,q +1 ++ R for a, q ∈ K and +R ∈ (Kg)A is multiplier, if and only if +R(xze + yvg) += +R(αβ) = αT a,q +1 +(β) − T a,q +1 +(xze + yvg) += +(xe + yf)(aze + qvf) − axze += +yqvg, +for any α = xe + yf, β = ze + vf ∈ A1 with x, y, z, v ∈ K, if only if R(xe + yg) = +qyg for all x, y ∈ K. +Let T a,q = + + +a +0 +0 +0 +q +0 +0 +0 +q + +, then T a,q ∈ M(A) and we have +(T − T a,q)(xe + yg) = 0 for any x, y ∈ K in the same way as (v) above. +Hence, +(T − T a,q)(Ke + Kg) = {0}, and we have +M(A) = {T a,p | a, p ∈ K} ⊕ {R ∈ (Kg)A | R(Ke + Kg) = {0}} +and +LM(A) = + + + + + +a +0 +0 +0 +p +0 +0 +t +0 + + +��� a, p, t ∈ K + + + . +(vii) A = S3: We have A0 = Kg and A = A1 ⊕ A0 with A1 = Ke + Kf. Since +A1 is a subalgebra of A, by Theorem 3.1 we obtain the equalities (10) in Section 3. +Because A1 is a commutative unital algebra, +M(A1) = M ′(A1) = A1 = +�� +a +0 +0 +b +� ��� a, b ∈ K +� +by Theorem 5.5. Hence, +M ′(A) = A1 ⊕ (Kg)A and M(A) = A1 ⊕ {T ∈ (Kg)A | T (Ke + Kf) = 0}. +13 + +Intersecting with L(A) we have +LM ′(A) = + + + + + +a +0 +0 +0 +b +0 +s +t +u + + +��� a, b, s, t, u ∈ K + + + and LM(A) = + + + + + +a +0 +0 +0 +b +0 +0 +0 +u + + +��� a, b, u ∈ K + + + . +(viii) A = S4: We have A = A1 ⊕ A0 with A0 = Kg and A1 = Ke + Kf. Because +A1 is a commutative unital subalgebra of A, similarly to above we have +M ′(A) = A1 ⊕ (Kg)A = +��a +0 +b +a +� ��� a, b ∈ K +� +⊕ (Kg)A, +M(A) += +A1 ⊕ +� +T ∈ (Kg)A | T (A2) = 0 +� += +��a +0 +b +a +� ��� a, b ∈ K +� +⊕ {T ∈ (Kg)A | T (Ke + Kf) = 0}, +LM ′(A) = + + + + + +a +0 +0 +b +a +0 +s +t +u + + +��� a, b, s, t, u ∈ K + + + and LM(A) = + + + + + +a +0 +0 +b +a +0 +0 +0 +u + + +��� a, b, u ∈ K + + + . +(ix) A = W1 : We have A = A1 ⊕ A0 with A0 = Ke + Kf and A1 = Kg. Let +T1 ∈ M ′ +1(A), then T1 is a linear mapping with T1(A0) = {0}. So T1 is determined by +T1(g) = ag with a ∈ K. Let denote this T1 by T a +1 . We have +M ′(A) = {T a +1 | a ∈ K} ⊕ (Ke + Kf)A. +A weak multiplier T = T a +1 + R with R ∈ (Ke + Kf)A is a multiplier, if and only if +R(xye) = R((xg)(yg)) = xgT a +1 (yg) − T a +1 (xye) = axye +for all x, y ∈ K, if and only if R(xe) = axe for any x ∈ K. Let Ta = + + +a +0 +0 +0 +0 +0 +0 +0 +a + +. +Then, (T − Ta)(Ke) = {0} and it follows that +M(A) = {Ta +�� a ∈ K} ⊕ {R ∈ (Ke + Kf)A �� R(Ke) = {0}}. +Also we have +LM ′(A) = + + + + + +a +b +c +p +q +r +0 +0 +u + + +��� a, b, c, p, q, r, u ∈ K + + + +and +LM(A) = + + + + + +a +b +c +0 +q +r +0 +0 +a + + +��� a, b, c, q, r ∈ K + + + . +(x) A = W2 +3: We have A = A1 ⊕ A0 with A0 = Ke and A1 = Kf + Kg. +T ∈ M ′ +1(A) is a linear mapping with T (Ke) = {0}. Let T be represented as (22), then +T is a weak multiplier if and only if + + +0 +0 +0 +0 +0 +0 +0 +qe +re + + = AT = T tA = + + +0 +0 +0 +0 +te +0 +0 +ue +0 + + , +3This is the algebra taken up in [8] +14 + +if and only if r = t = 0, q = u. Hence, +M ′(A) = {Tq | q ∈ K} ⊕ (Ke)A, +where Tq = + + +0 +0 +0 +0 +q +0 +0 +0 +q + +. So, +LM ′(A) = + + + + + +a +b +c +0 +q +0 +0 +0 +q + + +��� a, b, c, q ∈ K + + + . +A weak multiplier T = Tq + R with R ∈ (Ke)A is a multiplier, if and only if +R(yze) = R(αβ) = αTq(β) − Tq(yze) = α(qβ) = qyze +for any α = xf +yg, β = zf +vg ∈ A1 with x, y, z, v ∈ K, if and only if R(xe) = qxe for +all x ∈ K. Let Sa be the scalar multiplication by a ∈ K. Then, (T − Sa)(Ke) = {0}, +and hence, +M(A) = {Sa | a ∈ K} ⊕ {R ∈ (Ke)A | R(Ke) = {0}} +and +LM(A) = + + + + + +a +b +c +0 +a +0 +0 +0 +a + + +��� a, b, c ∈ K + + + . +(xi) A = W4: We have A = A1 ⊕ A0 with A0 = Kf + Kg and A1 = Ke. Because +A1 is a subalgbra isomorphic to the base field K, for T a +1 ∈ M(A1) with a ∈ K given +by T a +1 (e) = ae, we see +M ′(A) = {T a +1 | a ∈ K} ⊕ (fK + gK)A +and +M(A) = {T a +1 | a ∈ K} ⊕ {R ∈ (fK + gK)A | R(Ke) = 0} +by Theorem 3.1. Taking the intersection with L(A) we have +LM ′(A) = + + + + + +a +0 +0 +p +q +r +s +t +u + + +��� a, p, q, r, s, t, u ∈ K + + + +and +LM(A) = + + + + + +a +0 +0 +0 +q +r +0 +t +u + + +��� a, q, r, t, u ∈ K + + + . +(xii) W5 and W6 are opposed. Let A = W5, then A = A1 ⊕ A0 with A0 = Ke and +A1 = Kf + Kg. Since A1 is a subalgebra of A, we have the equalities (10). Because +A1 has a left identity g, we have +M(A1) = LM(A1) = M ′(A1) = LM ′(A1) ∼= (A1)/Kf ∼= Kg +by Theorem 5.5. So, any element in M(A1) is a scalar multiplication Sq +1 in A1 by +q ∈ K. By Theorem 3.1 we have +M ′(A) = {Sq +1 | q ∈ K} ⊕ (Ke)A, +15 + +M(A) = {Sq +1 | q ∈ K} ⊕ {R ∈ (Ke)A | R(Kf + Kg) = 0}, +LM ′(A) = + + + + + +a +b +c +0 +q +0 +0 +0 +q + + +��� a, b, c, q ∈ K + + + and LM(A) = + + + + + +a +0 +0 +0 +q +0 +0 +0 +q + + +��� a, q ∈ K + + + . +(xiii) W7 and W8 are opposed. Let A = W7. We see A0 = Annr(A) = {0}. Hence, +any weak multiplier is a linear multiplier by Proposition 5.1, and T represented as (23) +is a weak multiplier, if and if + + +ae +be +ce +0 +0 +0 +pf + sg +qf + tg +rf + ug + + = AT = T tA = + + +ae +sf +sg +be +tf +tg +ce +uf +ug + + , +if and only if b = c = p = r = s = t = 0, q = u. Therefore, +M(A) = LM(A) = M ′(A) = LM ′(A) = LM ′(A) = + + + + + +a +0 +0 +0 +q +0 +0 +0 +q + + +��� a, q ∈ K + + + . +(xiv) W9 and W10 are opposed. Let A = W9. Then, because A0 = Annl(A) = {0}, +any weak multiplier is a linear multiplier and a linear mapping T represented as (23) +is a weak multiplier if and only if + + +pe +qe +re +ae + pf +be + qf +ce + rf +pg +qg +rg + + = AT = T tA = + + +pe +ae + pf + sg +0 +qe +be + qf + tg +0 +re +ce + rf + ug +0 + + +c = p = r = s = t = 0, a = q = u. Therefore, +LM(A) = M(A) = LM ′(A) = M ′(A) = + + + + + +a +b +0 +0 +a +0 +0 +0 +a + + +��� a, b ∈ K + + + . +(xv) A = W3(k). We have A = A1 ⊕ A0 with A0 = Ke and A1 = Kf + Kg. +T ∈ M ′ +1(A) is a linear mapping with T (Ke) = {0}. Let T be represented as (22), then +T is a weak multiplier if and only if + + +0 +0 +0 +0 +qe +re +0 +(kq + t)e +(kr + u)e + + = AT = T tA = + + +0 +0 +0 +0 +(q + kt)e +te +0 +(r + ku)e +ue + + . +If k = 0, then the above holds if and only if r = t, and otherwise it holds if and only +if r = t = 0, q = u. Thus, +M ′(A) = {T q,r,u +1 +�� q, r, u ∈ K}⊕(Ke)A, LM ′(A) = + + + + + +a +b +c +0 +q +r +0 +r +u + + +��� a, b, c, q, r, u ∈ K + + + +if k = 0, and +M ′(A) = {T q +1 +�� q ∈ K} ⊕ (Ke)A, LM ′(A) = + + + + + +a +b +c +0 +q +0 +0 +0 +q + + +��� a, b, c, q ∈ K + + + +16 + +if k ̸= 0, where +T q,r,u +1 += + + +0 +0 +0 +0 +q +r +0 +r +u + + and T q +1 = + + +0 +0 +0 +0 +q +0 +0 +0 +q + +. +When k = 0, T = T q,r,u +1 ++ R with R ∈ (Ke)A is multiplier, if and only if +R((xz + yv)e) += +R(αβ) = αT q,r,u +1 +(β) − T q,r,u +1 +((xz + yv)e) += +α((qz + rv)f + (rz + uv)g) = (qxz + r(xv + yz) + uyv)e +for any α = xf + yg, β = zf + vg ∈ A1 with x, y, z, v ∈ K, if and only if q = u, r = 0 +and R(xe) = qxe for all x ∈ K. While, when k ̸= 0, T = T q +1 + R with R ∈ (Ke)A is a +multiplier, if and only if +R((xz + y(kz + v))e) += +R(αβ) = αT q +1 (β) − T q +1 (xue + y(kz + v)e) += +α(qβ) = q(xz + y(kz + v))e +for any α, β, if and only if R(xe) = qxe for any x ∈ K. +In the both cases, with +the scalar multiplication Sa by a ∈ K, we have (T − Sa)(Ke) = {0}. Therefore, for +arbitrary k (either k is zero or nonzero) we see +M(A) = {Sa +�� a ∈ K} ⊕ {R ∈ (Ke)A | R(Ke) = {0}} +and +LM(A) = + + + + + +a +b +c +0 +a +0 +0 +0 +a + + +��� a, b, c ∈ K + + + . +7 +3-dimensional zeropotent algebras +A is a zeropotent algebra if x2 = 0 for all x ∈ A. The zeropotent algebra A over K is +anti-commutative, that is, xy = −yx for all x, y ∈ A. Thus we see +A0 = Annl(A) = Annr(A). +Let A be a zeropotent algebras of dimension 3 over K with char(K) ̸= 2. Let +E = {e, f, g} be a basis of A. Because A is anti-commutative, the multiplication table +A of A on E is given as +A = + + +0 +α +−β +−α +0 +γ +β +−γ +0 + + , +where + + + + + +γ = fg = a11e + a12f + a13g +β = ge = a21e + a22f + a23g +α = ef = a31e + a32f + a33g +for aij ∈ K. We call A = + + +a11 +a12 +a13 +a21 +a22 +a23 +a31 +a32 +a33. + + the structural matrix of A (we use the +same symbol A for the algebra and its structural matrix). +Lemma 7.1. If rank(A) ≥ 2, then A0 = {0}. +17 + +Proof. If rank(A) ≥ 2, at least two of fg, ge, ef are linearly independent. Suppose +that α = ef and β = ge are linearly independent (the other cases are similar). If +x = ae + bf + cg with a, b.c ∈ K is in Annl(A), then xe = −bα + cβ, xf = aα − cγ +and xg = −aβ + bγ are all zero. It follows that a = b = c = 0. Hence, we have +Annl(A) = {0} and A0 = Annl(A) = {0}. +Theorem 7.2. Let A be a zeropotent algebra A of dimension 3 with rank(A) ≥ 2 over +K. Then, any weak multiplier of A is the scalar multiplication Sa for some a ∈ K, +that is, +M(A) = M ′(A) = LM(A) = LM ′(A) = {Sa | a ∈ K}. +Proof. By Lemma 7.1 and Corollary 2.4, any weak multiplier T is a linear mapping. +Let T ∈ L(A) be represented as (23). By Theorem 4.2, T is a weak multiplier, if and +only if AT = T tA, if and only if + + +pα − sβ +qα − tβ +rα − uβ +−aα + sγ +−bα + tγ +−cα + uγ +aβ − pγ +bβ − qγ +cβ − rγ + + = + + +−pα + sβ +aα − sγ +−aβ + pγ +−qα + tβ +bα − tγ +−bβ + qγ +−rα + uβ +cα − uγ +−cβ + rγ + + +(25) +holds. Suppose that α = ef, β = ge are linearly independent (the other cases are +similar). Then, because pα − sβ = −pα + sβ by comparing the (1,1)-elements of two +matrices in (25), we have p = s = 0. Comparing (1,2)-elements and (1,3)-elements, +we have qα − tβ = aα − sγ = aα and rα − uβ = −aβ + pγ = −aβ respectively. It +follows that a = q = u and r = t = 0. Comparing (2,2)-elements and (3,3)-elements, +we see b = c = 0. Consequently, (23) holds if and only if +b = c = p = r = s = t = +0 and a = q = u, that is, T = Sa. +In [4] we classify the zeropotent algebras of dimension 3 over an algebraically +field K of characteristic not equal to 2. Up to isomorphism, we have 10 families of +zeropotent algebras. They are +Z0, Z1, Z2, Z3, {Z4(a)}a∈H, Z5, Z6, {Z7(a)}a∈H, Z8 and Z9 +defined by the structural matrices + + +0 +0 +0 +0 +0 +0 +0 +0 +0 + + , + + +0 +0 +0 +0 +0 +0 +0 +0 +1 + + , + + +0 +0 +1 +0 +0 +0 +0 +0 +1 + + , + + +0 +1 +0 +−1 +0 +0 +0 +0 +0 + + , + + +0 +0 +0 +0 +1 +a +0 +0 +1 + + , + + +0 +1 +0 +0 +0 +0 +0 +0 +1 + + , + + +0 +1 +1 +0 +0 +1 +0 +0 +1 + + , + + +1 +a +0 +0 +1 +0 +0 +0 +1 + + , + + +1 +2 +2 +0 +1 +2 +0 +0 +1 + + and + + +1 +3 +3 +0 +1 +3 +0 +0 +1 + + , +respectively. +Z0 is the zero algebra, and Z1 is isomorphic to the 3-dimensional associative algebra +C1, and their (weak) multipliers are already determined in Section 6. The algebras Z3 +to Z9 have rank greater or equal to 2, which are covered by Theorem 7.2. +Thus, only A = Z2 is left to be analyzed. The multiplication table A of A is + + +0 +g +0 +−g +0 +g +0 +−g +0 + +. We see A0 = Annl(A) = Annr(A) = K(e + g), and we have the nihil +decomposition A = A0 ⊕ A1 with A1 = Ke + Kf. A weak multiplier T ∈ M ′ +1(A) is a +linear mapping represented by + + +a +b +c +p +q +r +0 +0 +0 + + satisfying +18 + + + +pg +qg +rg +−ag +−bg +−cg +−pg +−qg +−rg + + = AT = T tA = + + +−pg +ag +pg +−qg +bg +qg +−rg +cg +rg + + . +Hence, a = −c = q and b = p = r = 0. Let Ta be this linear mapping, then by +Theorem 3.1 we have +M ′(A) = {Ta | a ∈ K} ⊕ (K(f + g))A +and +LM ′(A) = + + + + + +a + s +t +−a + u +0 +a +0 +s +t +u + + +��� a, s, t, u ∈ K + + + . +By Corollary 3.2, a weak multiplier T = Ta+R with R ∈ (K(e+g))A is a multiplier +if and only if for any ζ = xe + yf and η = ze + vf with x, y, z, v ∈ K, +R((xv − yz)g) += +R(ζη) = ζTa(η) − Ta((xv − yz)g) += +ζ(aη) + a(xv − yz)e = a(xv − yz)(e + g) +holds. It follows that R(xg) = ax(e + g) for all x ∈ K. Let Sa be the scalar multi- +plication by a, then (T − Sa)(A) ⊆ K(e + g) and (T − Sa)(Kg) = {0}. Hence, we +obtain +M(A) = {Sa | a ∈ K} ⊕ {R ∈ (K(e + g))A | R(Kg) = 0}, +and +LM(A) = + + + + + +a + s +t +0 +0 +a +0 +s +t +a + + +��� a, s, t ∈ K + + + = + + + + + +a +b +0 +0 +c +0 +a − c +b +c + + +��� a, b, c ∈ K + + + . +References +[1] B. E. Johnson, An introduction to the theory of centralizers, Proc. London Math. +Soc., 14 (1964), 299–320. +[2] E. Kaniuth, A Course in commutative Banach algebras, Springer, 2008. +[3] Y. Kobayashi, K. Shirayanagi, M. Tsukada and S.-E. Takahasi, A complete clas- +sification of three-dimensional algebras over R and C , Asian-European J. Math., +14 (2021) 2150131. +[4] Y. Kobayashi, K. Shirayanagi, M. Tsukada and S.-E. Takahasi, Classification of +three dimensional zeropotent algebras over an algebraically closed field, Commu- +nication in Algebra, vol. 45, 2017, 5037—5052. +[5] R. Larsen, An introduction to the theory of multipliers, Berlin, New York, +Springer–Verlag, 1971. +[6] T. Tsukada and et al,, Linear algebra with Python, Theory and Applications, to +be published in Springer. +[7] J. G. Wendel, Left Centralizers and Isomorphisms on group algebras, Pacific J. +Math., 2 (1952), 251–261. +[8] A. Zivari-Kazempour, Almost multipliers of Frechet algebras, The J. Anal., 28(4) +(2020), 1075-1084 +[9] A. Zivari-Kazempour, Approximate θ-multipliers on Banach algebras, Surv. Math. +Appl., 77 (2022), 79–88. +19 + diff --git a/1tE2T4oBgHgl3EQfNQaE/content/tmp_files/load_file.txt b/1tE2T4oBgHgl3EQfNQaE/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..1c246037df9d93edfe6e62d2d35366b39e239267 --- /dev/null +++ b/1tE2T4oBgHgl3EQfNQaE/content/tmp_files/load_file.txt @@ -0,0 +1,560 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf,len=559 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content='03735v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content='RA] 10 Jan 2023 Multipliers and weak multipliers of algebras Yuji Kobayashi and Sin-Ei Takahasi Laboratory of Mathematics and Games (https://math-game-labo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content='com) 2020 MSC Numbers: Primary 43A22;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Secondary 17A99, 46J10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Keywords: (weak) multiplier, (non)associative algebra, Jordan algebra, zeropotent algebra, annihilator, nihil decomposition, matrix representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Abstract We study general properties of multipliers and weak multipliers of algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' We apply the results to determine the (weak) multipliers of associative algebras and zeropotent algebras of dimension 3 over an alge- braically closed field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' 1 Introduction Multipliers of algebras, in particular, multipliers of Banach algebras, have been discussed in analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' In this paper we will discuss them in a purely algebraic manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Let B be a Banach algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' A mapping T : B → B is called a multiplier of B, if it satisfies the condition (I) xT (y) = T (xy) = T (x)y (x, y ∈ B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Let M(B) denote the collection of all multipliers of B, and let B(B) be the collection of all bounded linear operators on B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Then M(B) forms an algebra and B(B) forms a Banach algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' B is called without order if it has no nonzero left or right annihilator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' If B is without order, then M(B) forms a commutative closed subalgebra of B(B) (see [2], Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' In 1953, Wendel [7] proved an important result that the multiplier algebra of L1(G) on a locally compact abelian group G is isometrically isomorphic to the measure algebra on G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' The general theory of multipliers of Banach algebras has been developed by Johnson [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' A good reference to the theory of multipliers of Banach algebra is given in Larsen [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' When B is without order, T is a multiplier if it satisfies the condition (II) xT (y) = T (x)y (x, y ∈ B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Many researchers had been unaware of difference between conditions (I) and (II) until Zivari-Kazempour [8] (see also [9]) recently clearly stated the difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' We call a mapping T satisfying (II) a weak mul- tiplier and denote the set of weak multipliers of B by M ′(B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Then, M(B) is in general a proper subset of M ′(B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Furthermore, (weak) multipliers can 1 be defined for an algebra A not necessarily associative, and they are not lin- ear mappings in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' We denote the spaces of linear multipliers and linear weak multipliers of A by LM(A) and LM ′(A) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' M(A) and LM(A) are subalgebras of the algebra AA consisting of all mappings from A to itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Meanwhile, M ′(A) and LM ′(A) are closed under the operation ◦ defined by T ◦ S = T S + ST , and they form a Jordan algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' In Sections 2 - 5 we study general properties of (weak) multipliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' In par- ticular, in sections 3 and 4 we give a decomposition theorem (Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content='1), and a matrix equation (Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content='2) for (weak) multipliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' They play an essential role to analyze (weak) multipliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Complete classifications of associative algebras and zeropotent algebras of dimension 3 over an algebraically closed field of characteristic not equal to 2 were given in Kobayashi et al, [3] and [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' In Sections 6 and 7 we completely determine the (linear) (weak) multipliers of those algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' 2 Multipliers and weak multipliers Let K be a field and A be a (not necessarily associative) algebra over K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' The set AA of all mappings from A to A forms an associative algebra over K in the usual manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Let L(A) denotes the subalgebra of AA of all linear mappings from A to A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' A mapping T : A → A is a weak multiplier of A, if xT (y) = T (x)y (1) holds for any x, y ∈ A, and T is a multiplier, if xT (y) = T (xy) = T (x)y (2) for any x, y ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Let M(A) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' M ′(A)) denote the set of all multipliers (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' weak multipliers) of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Define LM(A) def = M(A) ∩ L(A) and LM ′(A) def = M ′(A) ∩ L(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' M(A) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' LM(A)) is a unital subalgebra of AA (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' L(A)), and M ′(A) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' LM ′(A)) is a Jordan subalgebra of AA (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' L(A)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' First, the zero mapping is a multiplier because all the three terms in (2) are zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Secondly, the identity mapping is also a multiplier because the three terms in (2) are xy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Let T, U ∈ M(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Then we have x(T +U)(y) = xT (y)+xU(y) = T (xy)+U(xy) = T (x)y+U(x)y = (T +U)(x)y (3) and x(T U)(y) = xT (U(y)) = T (xU(y)) = T U(xy) = T (U(x)y) = (T U)(x)y (4) 2 for any x, y ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Hence, T + U, T U ∈ M(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Finally let k ∈ K, then x(kT )(y) = kxT (y) = kT (xy) = kT (x)y = (kT )(x)y, (5) and so kT ∈ M(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Therefore, M(A) is a unital subalgebra of AA, and LM(A) = M(A) ∩ L(A) is a unital subalgebra of L(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Next, let T, U ∈ M ′(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Then, the equalities in (3) and (5) hold removing the center terms T (xy) + U(xy) and kT (xy) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Hence, M ′(A) is a subspace of AA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Moreover, we have x(T U)(y) = xT (U(y)) = T (x)U(y) = U(T (x))y = UT (x)y and similarly x(UT )(y) = T U(x)y for any x, y ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Hence, x(T U + UT )(y) = (T U + UT )(x)y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' It follows that T U + UT ∈ M ′(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content='1 The opposite Aop of A is the algebra with the same elements and the addition as A, but the multiplication ∗ in it is reversed, that is, x∗y = yx for all x, y ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' A and Aop have the same multipliers and weak multiplies, that is, M(Aop) = M(A) and M ′(Aop) = M ′(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Let T ∈ AA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Then, T ∈ M ′(A), if and only if x ∗ T (y) = T (y)x = yT (x) = T (x) ∗ y for any x, y ∈ A, if and only if T ∈ M ′(Aop).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Further, T ∈ M(A), if and only if x ∗ T (y) = T (y)x = T (yx) = T (x ∗ y) = yT (x) = T (x) ∗ y for any x, y ∈ A, if and only if T ∈ M(Aop).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Let Annl(A) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Annr(A)) be the left (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' right) annihilator of A and let A0 be their intersection, that is, Annl(A) = {a ∈ A | ax = 0 for all x ∈ A}, Annr(A) = {a ∈ A | xa = 0 for all x ∈ A} and A0 = Annl(A) ∩ Annr(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' They are all subspaces of A, and when A is an associative algebra, they are two-sided ideals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' For a subset X of A, ⟨X⟩ denotes the subspace of A generated by X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' 1In general, for an associative algebra A over a field K of characteristic ̸= 2, the Jordan product ◦ on A is defined by x ◦ y = (xy + yx)/2 for x, y ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' 3 Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' A weak multiplier T of A such that ⟨T (A)⟩ ∩ A0 = {0} is a linear mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Let x, y, z ∈ A and a, b ∈ K, and let T be a weak multiplier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' We have T (ax + by)z = (ax + by)T (z) = axT (z) + byT (z) = aT (x)z + bT (y)z = (aT (x) + bT (y))z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Because z is arbitrary, we have w = T (ax + by) − aT (x) − bT (y) ∈ Annl(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Similarly, we can show w ∈ Annr(A), and so w ∈ A0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Hence, if ⟨T (A)⟩ ∩ A0 = {0}, then w = 0 because w ∈ ⟨T (A)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Since a, b, x, y are arbitrary, T is a linear mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' If A0 = {0}, then any weak multiplier is a linear mapping over K, that is, M ′(A) = LM ′(A) and M(A) = LM(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' If T is a weak multiplier, then T (Annl(A)) ⊆ Annl(A), T (Annr(A)) ⊆ Annr(A) and T (A0) ⊆ A0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Let x ∈ Annl(A), then for any y ∈ A we have 0 = xT (y) = T (x)y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Hence, T (x) ∈ Annl(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' The other cases are similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' In this paper we denote the subset {xy | x, y ∈ A} of A by A2, though usually A2 denotes the subspace of A generated by this set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Any mapping T : A → A such that T (A) ⊆ A0 is a weak multiplier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Such a mapping T is a multiplier if and only if T (A2) = {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' In particular, if A is the zero algebra, every mapping T is a weak multiplier, and it is a multiplier if only if T (0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' If T (A) ⊆ A0, the both sides are 0 in (1) and T is a weak multiplier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' This T is a multiplier, if only if the term T (xy) in the middle of (2) is 0 for all x, y ∈ A, that is, T (A2) = {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' If A is the zero algebra, then A = A0 and A2 = {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Hence, any T is a weak multiplier and it is a multiplier if and only if T (0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' 3 Nihil decomposition Let A1 be a subspace of A such that A = A1 ⊕ A0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' (6) Here, A1 is not unique, but choosing an appropriate A1 will become important later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' When A1 is fixed, any mapping T ∈ AA is uniquely decomposed as T = T1 + T0 (7) 4 with T1(A) ⊆ A1 and T0(A) ⊆ A0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' We call (6) and (7) a nihil decompositions of A and T respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Let π : A → A1 be the projection and µ : A1 → A be the embedding, that is, π(x1 + x0) = µ(x1) = x1 for x1 ∈ A1 and x0 ∈ A0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Let M1(A) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' M0(A)) denote the set of all multipliers T of A with T (A) ⊆ A1 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' T (A) ⊆ A0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Similarly, the sets M ′ 1(A) and M ′ 0(A) of weak multipliers of A are defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Also, set LMi(A) = Mi(A) ∩ L(A) and LM ′ i(A) = M ′ i(A) ∩ L(A) for i = 0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' By Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content='3 we see M ′ 1(A) = LM ′ 1(A) and M1(A) = LM1(A), and by Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content='6 we have M ′ 0(A) = AA 0 , M0(A) = {T ∈ AA 0 | T (A2) = {0}}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' (8) Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Let A = A1 ⊕ A0 and T = T1 + T0 be nihil decompositions of A and T ∈ AA respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' (i) T is a weak multiplier, if and only if T1 is a weak multiplier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' If T is a weak multiplier, T1 is a linear mapping satisfying T1(A0) = {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' (ii) If T1 is a multiplier and T0(A2) = {0}, then T is a multiplier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' If A1 is a subalgebra of A, the converse is also true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Suppose that A1 is a subalgebra of A, and let Φ be a mapping sending R ∈ (A1)A1 to µ ◦ R ◦ π ∈ AA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Then, (iii) Φ gives an algebra isomorphism from M(A1) onto M1(A) and a Jordan isomorphism from M ′(A1) onto M ′ 1(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Let x, y ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' (i) If T is a weak multiplier, then xT1(y) = x(T (y) − T0(y)) = xT (y) = T (x)y = T1(x)y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Thus, T1 is also a weak multiplier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Moreover, T1 is a linear mapping by Propo- sition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content='3 and T1(A0) ⊆ A1 ∩ A0 = {0} by Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Conversely, if T1 is a weak multiplier, then xT (y) = xT1(y) = T1(x)y = T (x)y, and so T is a weak multiplier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' (ii) If T1 is a multiplier and T0(A2) = 0, then T is a multiplier because xT (y) = xT1(y) = T1(xy) = T (xy) − T0(xy) = T (xy) = T1(x)y = T (x)y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Next suppose that A1 is a subalgebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' If T is a multiplier, then for any x, y ∈ A we have T1(xy) + T0(xy) = T (xy) = xT (y) = x1T1(y), (9) 5 where x = x1 + x0 with x1 ∈ A1 and x0 ∈ A0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Here, x1T1(y) ∈ A1 because A1 is a subalgebra, and thus, we have T0(xy) = x1T1(y) − T1(xy) ∈ A0 ∩ A1 = {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Since x, y are arbitrary, we get T0(A2) = {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Moreover, T1 is a multiplier because T1(xy) = x1T1(y) = xT1(y) by (9) and similarly T1(xy) = T1(x)y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' The converse is already proved above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' (iii) Let S ∈ (A1)A1 and x = x1 + x0, y = y1 + y0 ∈ A with x1, y1 ∈ A1 and x0, y0 ∈ A0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Then, π(x) = µ(x1) = x1, π(y) = µ(y1) = y1 and Φ(S)(x) = µ(S(π(x))) = µ(S(x1)) = S(x1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Thus, if S ∈ M ′(A1), we have xΦ(S)(y) = xS(y1) = x1S(y1) = S(x1)y1 = Φ(S)(x)y1 = Φ(S)(x)y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Hence, Φ(S) ∈ M ′ 1(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Moreover, if S ∈ M(A1), then because π is a homomor- phism, we have Φ(S)(xy) = S(π(xy)) = S(x1y1) = x1S(y1) = xΦ(S)(y), and hence Φ(S) ∈ M1(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Conversely, let T ∈ M ′ 1(A), then because T is a linear mapping satisfying T (A0) = {0}, there is a linear mapping S ∈ L(A1) on A1 such that Φ(S) = T , that is, S(x1) = T (x) = T (x1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' We have x1S(y1) = x1T (y1) = T (x1)y1 = S(x1)y1, and hence S ∈ M ′(A1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Therefore, Φ is a linear isomorphism from M ′(A1) to M ′ 1(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Similarly, Φ gives a linear isomorphism from M(A1) to M1(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Moreover, for T, U ∈ M ′(A1), we have Φ(T U) = µ ◦ T ◦ U ◦ π = µ ◦ T ◦ π ◦ µ ◦ U ◦ π = Φ(T )Φ(U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Thus, Φ gives an isomorphism of algebras from M(A1) to M1(A) and a Jordan isomorphism from M ′(A1) to M ′ 1(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content='1 implies M ′(A) = M ′ 1(A) ⊕ M ′ 0(A), M1(A) ⊕ M0(A) ⊆ M(A), where M ′ 0(A) and M0(A) are given as (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Moreover, if A1 is a subalgebra, we have M ′(A) ∼= M ′(A1)⊕(A0)A, M(A) ∼= M(A1)⊕{T ∈ (A0)A | T (A2) = {0}}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' (10) Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Any weak multiplier T is written as T = T1 + R (11) with T1 ∈ LM ′ 1(A) and R ∈ (A0)A, and it is a multiplier if and only if R(x1y1) = x1T1(y1) − T1(x1y1) (12) for any x1, y1 ∈ A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' 6 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' As stated above T is written as (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Let x = x1 + x0, y = y1 + y0 ∈ A with x1, y1 ∈ A1 and x0, y0 ∈ A0 be arbitrary, then we have xT (y) = x1(T1(y) + R(y)) = x1T1(y) = x1T1(y1) (13) because R(A) ⊆ A0 and T1(A0) = {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' The last term in (13) is also equal to T1(x1)y1 = T (x)y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Hence, T is a multiplier, if and only if it is equal to T (xy) = T (x1y1) = T1(x1y1) + R(x1y1), if and only if (12) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' 4 Linear multipliers and matrix equation In this section, A is a finite dimensional algebra over K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' We drive a matrix equation for a linear mapping on A to be a (weak) multiplier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Suppose that A is n-dimensional with basis E = {e1, e2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' , en}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' A linear mapping T : A → A is a weak multiplier if and only if eiT (ej) = T (ei)ej, (14) and it is a multiplier if and only if T (eiej) = eiT (ej) = T (ei)ej, (15) for all ei, ej ∈ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' The necessity of the conditions (14) and (15) is obvious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Let x = x1e1 + x2e2+· · ·+xnen, y = y1e1+y2e2+· · ·+ynen ∈ A with x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' , xn, y1, y2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' , yn ∈ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' If T satisfies (14), then we have xT (y) = ( � i xiei)T ( � j yjej) = ( � i xiei)( � j yjT (ej)) = � i,j xiyjeiT (ej) = � i,j xiyjT (ei)ej = ( � i xiT (ei))( � j yjej) = T (x)y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Hence, T is a weak multiplier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Moreover, if T satisfies (15), it is a multiplier in a similar way Let A (we use the bold character) be the multiplication table of A on E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' A is a matrix whose elements are from A defined by A = EtE, (16) where E = (e1, e2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' , en) (we again use the bold face E) is the row vector consisting the basis elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' For a linear mapping T on A and a matrix B over A, T (B) denotes the matrix obtained by applying T component-wise, that is, the (i, j)-element of T (B) is T (bij) for the (i, j)-element bij of B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content='2 We use the same character T for the representation matrix of T on E, that is, T (E) = ET.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' (17) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' 2This is called a broadcasting (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' [6]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' 7 Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' A linear mapping T is a weak multiplier of A if and only if AT = T tA, (18) and T is a multiplier if and only if T (A) = AT = T tA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' (19) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' By (16) and (17) we have (e1, e2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' , en)t(T (e1), T (e2), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' , T (en)) = EtT (E) = EtET = AT (20) and (T (e1), T (e2), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' , T (e2))t(e1, e2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' , en) = T (E)tE = T tEtE = T tA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' (21) By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content='1, T is a weak multiplier, if and only if (20) and (21) are equal, if and only if (18) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Moreover, T is multiplier if and only if, the leftmost sides of (20) and (21) are equal to (T (eiej))i,j=1,2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=',n = T (A), if and only if (19) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' The multiplication table of the opposite algebra Aop of A is At.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' So, the alge- bras with multiplication tables transposed to each other have the same (weak) multipliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' 5 Associative algebras In this section A is an associative algebra over K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' If A0 = {0}, then we have M(A) = M ′(A) = LM(A) = LM ′(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Let T ∈ M ′(A), then we have T (xy)z = xyT (z) = xT (y)z and zT (xy) = T (z)xy = zT (x)y for any x, y, z ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' It follows that T (xy) − xT (y) ∈ Annl(A) ∩ Annr(A) = A0 = {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Hence, T (xy) = xT (y) and we see T ∈ M(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Moreover, T ∈ LM(A) by Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Let a ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' If xay = axy (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' xay = xya) for any x, y ∈ A, a is called a left (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' right) central element, and a is called a central element if ax = xa for any x ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Let Zl(A), (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Zr(A), Z(A)) denotes the set of all left central (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' right central, central) elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' 8 Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Zl(A) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Zr(A), Z(A)) are subalgebra of A containing Annl(A) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Annr(A), A0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' For a ∈ A, la (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' ra) denotes the left (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' right) multiplication by a, that is, la(x) = ax, ra(x) = xa for x ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' They are linear mappings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' For a ∈ A the following statements are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' (i) la (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' ra) is a multiplier, (ii) la (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' ra) is a weak multiplier, (iii) a is left (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' right) central.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' If la is a weak multiplier, then xay = xla(y) = la(x)y = axy for any x, y ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Hence, a is left central.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Because la(x)y = axy = la(xy) and xla(y) = xay for any x, y ∈ A, la is a multiplier if a is left central.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' The other case is similar, and we see that the three statements are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Suppose that A has a right (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' left) identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Then, a left (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' right) central element is central, and Annl(A) = {0} (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Annr(A) = {0}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Easy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Because Annl(A) ⊆ Zl and Annr(A) ⊆ Zr by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content='2, we can make the quotient algebras ¯Zl(A) = Zl(A)/Annl(A) and ¯Zr(A) = Zr(A)/Annr(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Suppose that A has a left (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' right) identity e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Then, any multiplier is a left (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' right) multiplication by a left (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' right) central element and is a linear multiplier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' The mapping φ : Zl(A) → M(A) sending a ∈ Zl(A) to la induces an isomorphism ¯φ : Zl(A) → M(A) of algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' In particular, if A is unital, M(A) is isomorphic to Z(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Suppose that A has a left identity e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Let T ∈ M ′(A) and set a = T (e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Then we have T (x) = eT (x) = T (e)x = ax for any x ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Hence, T = la, where a ∈ Zl(A) and T is a linear multiplier by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Therefore, M ′(A) = LM(A) and φ is surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Moreover, for a ∈ Zl(A), φ(a) = 0, if and only if ax = 0 for any x ∈ A, if and only if a ∈ Annl(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Thus we have Ker(φ) = Annl(A), and φ induces the desired isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Similarly, if A has a right identity, M(A) is isomorphic to Zr(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Finally, if A has the identity, then Annl(A) = Annr(A) = {0} and hence M(A) is isomorphic to Z(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' 9 6 3-dimensional associative algebras Over an algebraically closed field K of characteristic not equal to 2, we have, up to isomorphism, 24 families of associative algebras of dimension 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' They are 5 unital algebras U0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' U1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' U2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' U3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' U4 defined on basis E = {e,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' g} by �e f g f 0 0 g 0 0 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' �e f g f 0 f g −f e � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' �e 0 0 0 f 0 0 0 g � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' �e 0 0 0 f g 0 g 0 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' �e f g f g 0 g 0 0 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' 5 curled algebras C0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' C1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' C2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' C3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' C4 defined by �0 0 0 0 0 0 0 0 0 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' �0 0 0 0 0 e 0 −e 0 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' �0 0 0 e f 0 0 g 0 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' �0 0 0 0 0 0 e f g � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' �0 0 e 0 0 f 0 0 g � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' non-unital 4 straight algebras S1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' S2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' S3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' S4 defined by �f g 0 g 0 0 0 0 0 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' �e 0 0 0 g 0 0 0 0 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' �e 0 0 0 f 0 0 0 0 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' �e f 0 f 0 0 0 0 0 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' and non-unital 10 families of waved algebras W1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' W2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' W4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' W5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' W6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' W7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' W8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' W9,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' W10 and � W3(k) � k∈H defined by �0 0 0 0 0 0 0 0 e � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' �0 0 0 0 0 0 0 e 0 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' �e 0 0 0 0 0 0 0 0 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' �0 0 0 0 0 0 0 f g � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' �0 0 0 0 0 f 0 0 g � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' �e 0 0 0 0 0 0 f g � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' �e 0 0 0 0 f 0 0 g � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' �0 e 0 e f 0 0 g 0 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' �0 e 0 e f g 0 0 0 � and �0 0 0 0 e 0 0 ke e � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' respectively,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' where H is a subset of K such that K = H∪−H and H∩−H = {0} (see [3] for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' We determine the (weak) multipliers of them below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' (0) A = C0 is the zero algebra, so by Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content='6, we have M ′(A) = AA, M(A) = {T ∈ AA | T (0) = 0} and LM(A) = LM ′(A) = L(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' (i) The unital algebras U0, U2, U3, U4 are commutative, so for such A we have M(A) = LM(A) = M ′(A) = LM ′(A) = {lx|x ∈ A} ∼= A by Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' For A = U1, we have M(A) = LM(A) = M ′(A) = LM ′(A) ∼= Z(A) = Ke.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' (ii) For A = C1, We have A0 = Annl(A) = Annr(A) = Ke, and a nihil decompo- sition A = A1 ⊕ A0 with A1 = Kf + Kg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Let T1 ∈ M ′ 1(A), then by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content='1, T1 is a linear mapping such that T1(Ke) = {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Let T1 = \uf8eb \uf8ed 0 0 0 0 q r 0 t u \uf8f6 \uf8f8 (22) 10 with q, r, t, u ∈ K be the representation matrix of T1 on E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' By Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content='2, T1 is a weak multiplier, if and only if \uf8eb \uf8ed 0 0 0 0 te ue 0 −qe −re \uf8f6 \uf8f8 = AT1 = T t 1A = \uf8eb \uf8ed 0 0 0 0 −te qe 0 −ue re \uf8f6 \uf8f8 , if and only if r = t = 0 and q = u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Hence, M ′ 1(A) = {Tq �� q ∈ K}, where Tq = \uf8eb \uf8ed 0 0 0 0 q 0 0 0 q \uf8f6 \uf8f8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' By Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content='1 we see M ′(A) = {Tq �� q ∈ K} ⊕ (Ke)A, and LM ′(A) = \uf8f1 \uf8f2 \uf8f3 \uf8eb \uf8ed a b c 0 q 0 0 0 q \uf8f6 \uf8f8 ��� a, b, c, q ∈ K \uf8fc \uf8fd \uf8fe .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' By the multiplication table of A, we have αβ = (xv − yz)e for α = xf + yg, β = zf + vg ∈ A1 with x, y, z, v ∈ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' By Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content='2, T ∈ M ′(A) is given by T = Tq + R with R ∈ (Ke)A and this T is a multiplier, if and only if R((xv − yz)e) = R(αβ) = αTq(β) − Tq(αβ) = α(qβ) − Tq((xv − yz)e) = q(xv − yz)e for any α and β, if and only if R(xe) = qxe for all x ∈ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Let Sq = \uf8eb \uf8ed q 0 0 0 q 0 0 0 q \uf8f6 \uf8f8 be the scalar multiplication by q ∈ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Then, we see (T − Sq)(A) ⊆ A0 = Ke and (T − Sq)(xe) = Tq(xe) + R(xe) − Sq(xe) = 0 + qxe − qxe = 0, for any x ∈ K, that is, (T − Sq)(Ke) = {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Thus, we have M(A) = {Sq �� q ∈ K} ⊕ {R ∈ (Ke)A | R(Ke) = {0}}, and LM(A) = \uf8f1 \uf8f2 \uf8f3 \uf8eb \uf8ed a b c 0 a 0 0 0 a \uf8f6 \uf8f8 ��� a, b, c ∈ K \uf8fc \uf8fd \uf8fe .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' (iii) A = C2: Because Annl(A) = Ke and Annr(A) = Kg, we see A0 = {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Hence, any weak multiplier T is a linear multiplier by Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' By Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content='2, T = \uf8eb \uf8ed a b c p q r s t u \uf8f6 \uf8f8 (23) is a (weak) multiplier, if and only if \uf8eb \uf8ed 0 0 0 ae + pf be + qf ce + rf pg qg rg \uf8f6 \uf8f8 = AT = T tA = \uf8eb \uf8ed pe pf + sg 0 qe qf + tg 0 re rf + ug 0 \uf8f6 \uf8f8 , 11 if and only if b = c = p = r = s = t = 0 and a = q = u, that is, T is the scalar multiplication Sa by a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Consequently, M(A) = M ′(A) = LM(A) = LM ′(A) = {Sa �� a ∈ K} ∼= K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' (iv) C3 and C4 are opposed to each other, and have the same (weak) multipliers by Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Let A = C3, then, A has a left identity g, Zl(A) = A and Annl(A) = Ke + Kf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Hence, by Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content='4, M(A) = M ′(A) = LM(A) = LM ′(A) = A/(Ke + Kg) = {Sa �� a ∈ K}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' (v) A = S1: We have A0 = Annl(A) = Annr(A) = Kg, and A = A1 ⊕ A0 with A1 = Ke + Kf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Then, T1 ∈ M ′ 1(A) is a linear mapping with T (Kg) = {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Let T1 = \uf8eb \uf8ed a b 0 p q 0 0 0 0 \uf8f6 \uf8f8 (24) be its representation on E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' T1 is a weak multiplier, if and only if \uf8eb \uf8ed af + pg bf + qg 0 ag bg 0 0 0 0 \uf8f6 \uf8f8 = AT1 = T t 1A = \uf8eb \uf8ed af + pg ag 0 bf + qg bg 0 0 0 0 \uf8f6 \uf8f8 , if and only if b = 0 and a = q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Hence, M ′(A) = {T a,p 1 | a, p ∈ K} ⊕ (Kg)A, where T a,p 1 = \uf8eb \uf8ed a 0 0 p a 0 0 0 0 \uf8f6 \uf8f8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' So, T ∈ M ′(A) is written as T = T a,p 1 +R with R ∈ (Kg)A, and this T is multiplier, if and only if R(xzf + (xv + yz)g) = R(αβ) = αT a,p 1 (β) − T a,p 1 (αβ) = α(aze + (pz + av)f) − T a,p 1 (xzf + (xv + yz)g) = axzf + (pxz + axv + ayz)g − axzf = (pxz + a(xv + yz))g for any α = xe + yf, β = ze + vf ∈ A1 with x, y, z, v ∈ K, if and only if R(xf + yg) = (px + ay)g for all x, y ∈ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Let T a,p = \uf8eb \uf8ed a 0 0 p a 0 0 p a \uf8f6 \uf8f8, then (T − T a,p)(A) ⊆ Kg, and (T − T a,p)(xf + yg) = (T a,p 1 + R − T a,p) (xf + yg) = axf + (px + ay)g − (axf + pxg + ayg) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' for any x, y ∈ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Thus, (T − T a,p)(Kf + Kg) = {0}, and hence M(A) = {T a,p | a, p ∈ K} ⊕ {R ∈ (Kg)A | R(Kf + Kg) = {0}}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Taking the intersections of M ′(A) and M(A) with L(A), we obtain LM ′(A) = \uf8f1 \uf8f2 \uf8f3 \uf8eb \uf8ed a 0 0 p a 0 s t u \uf8f6 \uf8f8 ��� a, p, s, t, u ∈ K \uf8fc \uf8fd \uf8fe 12 and LM(A) = \uf8f1 \uf8f2 \uf8f3 \uf8eb \uf8ed a 0 0 p a 0 s p a \uf8f6 \uf8f8 ��� a, p, s ∈ K \uf8fc \uf8fd \uf8fe .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' (vi) A = S2: We have A0 = Annl(A) = Annr(A) = Kg, and A = A1 ⊕ A0 with A1 = Ke + Kf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Let a linear mapping T1 ∈ M ′ 1(A) be represented as (24), then T1 is a weak multiplier, if and only if \uf8eb \uf8ed ae be 0 pg qg 0 0 0 0 \uf8f6 \uf8f8 = AT = T tA = \uf8eb \uf8ed ae pg 0 be qg 0 0 0 0 \uf8f6 \uf8f8 , if and only if b = p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Hence, M ′(A) = {T a,q 1 �� a, q ∈ K} ⊕ (Kg)A, where T a,q 1 = \uf8eb \uf8ed a 0 0 0 q 0 0 0 0 \uf8f6 \uf8f8, and LM ′(A) = \uf8f1 \uf8f2 \uf8f3 \uf8eb \uf8ed a 0 0 0 q 0 s t u \uf8f6 \uf8f8 ��� a, q, s, t, u ∈ K \uf8fc \uf8fd \uf8fe .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' By Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content='2, a weak multiplier T written as T = T a,q 1 + R for a, q ∈ K and R ∈ (Kg)A is multiplier, if and only if R(xze + yvg) = R(αβ) = αT a,q 1 (β) − T a,q 1 (xze + yvg) = (xe + yf)(aze + qvf) − axze = yqvg, for any α = xe + yf, β = ze + vf ∈ A1 with x, y, z, v ∈ K, if only if R(xe + yg) = qyg for all x, y ∈ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Let T a,q = \uf8eb \uf8ed a 0 0 0 q 0 0 0 q \uf8f6 \uf8f8, then T a,q ∈ M(A) and we have (T − T a,q)(xe + yg) = 0 for any x, y ∈ K in the same way as (v) above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Hence, (T − T a,q)(Ke + Kg) = {0}, and we have M(A) = {T a,p | a, p ∈ K} ⊕ {R ∈ (Kg)A | R(Ke + Kg) = {0}} and LM(A) = \uf8f1 \uf8f2 \uf8f3 \uf8eb \uf8ed a 0 0 0 p 0 0 t 0 \uf8f6 \uf8f8 ��� a, p, t ∈ K \uf8fc \uf8fd \uf8fe .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' (vii) A = S3: We have A0 = Kg and A = A1 ⊕ A0 with A1 = Ke + Kf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Since A1 is a subalgebra of A, by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content='1 we obtain the equalities (10) in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Because A1 is a commutative unital algebra, M(A1) = M ′(A1) = A1 = �� a 0 0 b � ��� a, b ∈ K � by Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Hence, M ′(A) = A1 ⊕ (Kg)A and M(A) = A1 ⊕ {T ∈ (Kg)A | T (Ke + Kf) = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' 13 Intersecting with L(A) we have LM ′(A) = \uf8f1 \uf8f2 \uf8f3 \uf8eb \uf8ed a 0 0 0 b 0 s t u \uf8f6 \uf8f8 ��� a, b, s, t, u ∈ K \uf8fc \uf8fd \uf8fe and LM(A) = \uf8f1 \uf8f2 \uf8f3 \uf8eb \uf8ed a 0 0 0 b 0 0 0 u \uf8f6 \uf8f8 ��� a, b, u ∈ K \uf8fc \uf8fd \uf8fe .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' (viii) A = S4: We have A = A1 ⊕ A0 with A0 = Kg and A1 = Ke + Kf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Because A1 is a commutative unital subalgebra of A, similarly to above we have M ′(A) = A1 ⊕ (Kg)A = ��a 0 b a � ��� a, b ∈ K � ⊕ (Kg)A, M(A) = A1 ⊕ � T ∈ (Kg)A | T (A2) = 0 � = ��a 0 b a � ��� a, b ∈ K � ⊕ {T ∈ (Kg)A | T (Ke + Kf) = 0}, LM ′(A) = \uf8f1 \uf8f2 \uf8f3 \uf8eb \uf8ed a 0 0 b a 0 s t u \uf8f6 \uf8f8 ��� a, b, s, t, u ∈ K \uf8fc \uf8fd \uf8fe and LM(A) = \uf8f1 \uf8f2 \uf8f3 \uf8eb \uf8ed a 0 0 b a 0 0 0 u \uf8f6 \uf8f8 ��� a, b, u ∈ K \uf8fc \uf8fd \uf8fe .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' (ix) A = W1 : We have A = A1 ⊕ A0 with A0 = Ke + Kf and A1 = Kg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Let T1 ∈ M ′ 1(A), then T1 is a linear mapping with T1(A0) = {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' So T1 is determined by T1(g) = ag with a ∈ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Let denote this T1 by T a 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' We have M ′(A) = {T a 1 | a ∈ K} ⊕ (Ke + Kf)A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' A weak multiplier T = T a 1 + R with R ∈ (Ke + Kf)A is a multiplier, if and only if R(xye) = R((xg)(yg)) = xgT a 1 (yg) − T a 1 (xye) = axye for all x, y ∈ K, if and only if R(xe) = axe for any x ∈ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Let Ta = \uf8eb \uf8ed a 0 0 0 0 0 0 0 a \uf8f6 \uf8f8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Then, (T − Ta)(Ke) = {0} and it follows that M(A) = {Ta �� a ∈ K} ⊕ {R ∈ (Ke + Kf)A �� R(Ke) = {0}}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Also we have LM ′(A) = \uf8f1 \uf8f2 \uf8f3 \uf8eb \uf8ed a b c p q r 0 0 u \uf8f6 \uf8f8 ��� a, b, c, p, q, r, u ∈ K \uf8fc \uf8fd \uf8fe and LM(A) = \uf8f1 \uf8f2 \uf8f3 \uf8eb \uf8ed a b c 0 q r 0 0 a \uf8f6 \uf8f8 ��� a, b, c, q, r ∈ K \uf8fc \uf8fd \uf8fe .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' (x) A = W2 3: We have A = A1 ⊕ A0 with A0 = Ke and A1 = Kf + Kg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' T ∈ M ′ 1(A) is a linear mapping with T (Ke) = {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Let T be represented as (22), then T is a weak multiplier if and only if \uf8eb \uf8ed 0 0 0 0 0 0 0 qe re \uf8f6 \uf8f8 = AT = T tA = \uf8eb \uf8ed 0 0 0 0 te 0 0 ue 0 \uf8f6 \uf8f8 , 3This is the algebra taken up in [8] 14 if and only if r = t = 0, q = u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Hence, M ′(A) = {Tq | q ∈ K} ⊕ (Ke)A, where Tq = \uf8eb \uf8ed 0 0 0 0 q 0 0 0 q \uf8f6 \uf8f8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' So, LM ′(A) = \uf8f1 \uf8f2 \uf8f3 \uf8eb \uf8ed a b c 0 q 0 0 0 q \uf8f6 \uf8f8 ��� a, b, c, q ∈ K \uf8fc \uf8fd \uf8fe .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' A weak multiplier T = Tq + R with R ∈ (Ke)A is a multiplier, if and only if R(yze) = R(αβ) = αTq(β) − Tq(yze) = α(qβ) = qyze for any α = xf +yg, β = zf +vg ∈ A1 with x, y, z, v ∈ K, if and only if R(xe) = qxe for all x ∈ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Let Sa be the scalar multiplication by a ∈ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Then, (T − Sa)(Ke) = {0}, and hence, M(A) = {Sa | a ∈ K} ⊕ {R ∈ (Ke)A | R(Ke) = {0}} and LM(A) = \uf8f1 \uf8f2 \uf8f3 \uf8eb \uf8ed a b c 0 a 0 0 0 a \uf8f6 \uf8f8 ��� a, b, c ∈ K \uf8fc \uf8fd \uf8fe .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' (xi) A = W4: We have A = A1 ⊕ A0 with A0 = Kf + Kg and A1 = Ke.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Because A1 is a subalgbra isomorphic to the base field K, for T a 1 ∈ M(A1) with a ∈ K given by T a 1 (e) = ae, we see M ′(A) = {T a 1 | a ∈ K} ⊕ (fK + gK)A and M(A) = {T a 1 | a ∈ K} ⊕ {R ∈ (fK + gK)A | R(Ke) = 0} by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Taking the intersection with L(A) we have LM ′(A) = \uf8f1 \uf8f2 \uf8f3 \uf8eb \uf8ed a 0 0 p q r s t u \uf8f6 \uf8f8 ��� a, p, q, r, s, t, u ∈ K \uf8fc \uf8fd \uf8fe and LM(A) = \uf8f1 \uf8f2 \uf8f3 \uf8eb \uf8ed a 0 0 0 q r 0 t u \uf8f6 \uf8f8 ��� a, q, r, t, u ∈ K \uf8fc \uf8fd \uf8fe .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' (xii) W5 and W6 are opposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Let A = W5, then A = A1 ⊕ A0 with A0 = Ke and A1 = Kf + Kg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Since A1 is a subalgebra of A, we have the equalities (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Because A1 has a left identity g, we have M(A1) = LM(A1) = M ′(A1) = LM ′(A1) ∼= (A1)/Kf ∼= Kg by Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' So, any element in M(A1) is a scalar multiplication Sq 1 in A1 by q ∈ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' By Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content='1 we have M ′(A) = {Sq 1 | q ∈ K} ⊕ (Ke)A, 15 M(A) = {Sq 1 | q ∈ K} ⊕ {R ∈ (Ke)A | R(Kf + Kg) = 0}, LM ′(A) = \uf8f1 \uf8f2 \uf8f3 \uf8eb \uf8ed a b c 0 q 0 0 0 q \uf8f6 \uf8f8 ��� a, b, c, q ∈ K \uf8fc \uf8fd \uf8fe and LM(A) = \uf8f1 \uf8f2 \uf8f3 \uf8eb \uf8ed a 0 0 0 q 0 0 0 q \uf8f6 \uf8f8 ��� a, q ∈ K \uf8fc \uf8fd \uf8fe .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' (xiii) W7 and W8 are opposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Let A = W7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' We see A0 = Annr(A) = {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Hence, any weak multiplier is a linear multiplier by Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content='1, and T represented as (23) is a weak multiplier, if and if \uf8eb \uf8ed ae be ce 0 0 0 pf + sg qf + tg rf + ug \uf8f6 \uf8f8 = AT = T tA = \uf8eb \uf8ed ae sf sg be tf tg ce uf ug \uf8f6 \uf8f8 , if and only if b = c = p = r = s = t = 0, q = u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Therefore, M(A) = LM(A) = M ′(A) = LM ′(A) = LM ′(A) = \uf8f1 \uf8f2 \uf8f3 \uf8eb \uf8ed a 0 0 0 q 0 0 0 q \uf8f6 \uf8f8 ��� a, q ∈ K \uf8fc \uf8fd \uf8fe .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' (xiv) W9 and W10 are opposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Let A = W9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Then, because A0 = Annl(A) = {0}, any weak multiplier is a linear multiplier and a linear mapping T represented as (23) is a weak multiplier if and only if \uf8eb \uf8ed pe qe re ae + pf be + qf ce + rf pg qg rg \uf8f6 \uf8f8 = AT = T tA = \uf8eb \uf8ed pe ae + pf + sg 0 qe be + qf + tg 0 re ce + rf + ug 0 \uf8f6 \uf8f8 c = p = r = s = t = 0, a = q = u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Therefore, LM(A) = M(A) = LM ′(A) = M ′(A) = \uf8f1 \uf8f2 \uf8f3 \uf8eb \uf8ed a b 0 0 a 0 0 0 a \uf8f6 \uf8f8 ��� a, b ∈ K \uf8fc \uf8fd \uf8fe .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' (xv) A = W3(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' We have A = A1 ⊕ A0 with A0 = Ke and A1 = Kf + Kg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' T ∈ M ′ 1(A) is a linear mapping with T (Ke) = {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Let T be represented as (22), then T is a weak multiplier if and only if \uf8eb \uf8ed 0 0 0 0 qe re 0 (kq + t)e (kr + u)e \uf8f6 \uf8f8 = AT = T tA = \uf8eb \uf8ed 0 0 0 0 (q + kt)e te 0 (r + ku)e ue \uf8f6 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' If k = 0, then the above holds if and only if r = t, and otherwise it holds if and only if r = t = 0, q = u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Thus, M ′(A) = {T q,r,u 1 �� q, r, u ∈ K}⊕(Ke)A, LM ′(A) = \uf8f1 \uf8f2 \uf8f3 \uf8eb \uf8ed a b c 0 q r 0 r u \uf8f6 \uf8f8 ��� a, b, c, q, r, u ∈ K \uf8fc \uf8fd \uf8fe if k = 0, and M ′(A) = {T q 1 �� q ∈ K} ⊕ (Ke)A, LM ′(A) = \uf8f1 \uf8f2 \uf8f3 \uf8eb \uf8ed a b c 0 q 0 0 0 q \uf8f6 \uf8f8 ��� a, b, c, q ∈ K \uf8fc \uf8fd \uf8fe 16 if k ̸= 0, where T q,r,u 1 = \uf8eb \uf8ed 0 0 0 0 q r 0 r u \uf8f6 \uf8f8 and T q 1 = \uf8eb \uf8ed 0 0 0 0 q 0 0 0 q \uf8f6 \uf8f8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' When k = 0, T = T q,r,u 1 + R with R ∈ (Ke)A is multiplier, if and only if R((xz + yv)e) = R(αβ) = αT q,r,u 1 (β) − T q,r,u 1 ((xz + yv)e) = α((qz + rv)f + (rz + uv)g) = (qxz + r(xv + yz) + uyv)e for any α = xf + yg, β = zf + vg ∈ A1 with x, y, z, v ∈ K, if and only if q = u, r = 0 and R(xe) = qxe for all x ∈ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' While, when k ̸= 0, T = T q 1 + R with R ∈ (Ke)A is a multiplier, if and only if R((xz + y(kz + v))e) = R(αβ) = αT q 1 (β) − T q 1 (xue + y(kz + v)e) = α(qβ) = q(xz + y(kz + v))e for any α, β, if and only if R(xe) = qxe for any x ∈ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' In the both cases, with the scalar multiplication Sa by a ∈ K, we have (T − Sa)(Ke) = {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Therefore, for arbitrary k (either k is zero or nonzero) we see M(A) = {Sa �� a ∈ K} ⊕ {R ∈ (Ke)A | R(Ke) = {0}} and LM(A) = \uf8f1 \uf8f2 \uf8f3 \uf8eb \uf8ed a b c 0 a 0 0 0 a \uf8f6 \uf8f8 ��� a, b, c ∈ K \uf8fc \uf8fd \uf8fe .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' 7 3-dimensional zeropotent algebras A is a zeropotent algebra if x2 = 0 for all x ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' The zeropotent algebra A over K is anti-commutative, that is, xy = −yx for all x, y ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Thus we see A0 = Annl(A) = Annr(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Let A be a zeropotent algebras of dimension 3 over K with char(K) ̸= 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Let E = {e, f, g} be a basis of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Because A is anti-commutative, the multiplication table A of A on E is given as A = \uf8eb \uf8ed 0 α −β −α 0 γ β −γ 0 \uf8f6 \uf8f8 , where \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 γ = fg = a11e + a12f + a13g β = ge = a21e + a22f + a23g α = ef = a31e + a32f + a33g for aij ∈ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' We call A = \uf8eb \uf8ed a11 a12 a13 a21 a22 a23 a31 a32 a33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' \uf8f6 \uf8f8 the structural matrix of A (we use the same symbol A for the algebra and its structural matrix).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' If rank(A) ≥ 2, then A0 = {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' 17 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' If rank(A) ≥ 2, at least two of fg, ge, ef are linearly independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Suppose that α = ef and β = ge are linearly independent (the other cases are similar).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' If x = ae + bf + cg with a, b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content='c ∈ K is in Annl(A), then xe = −bα + cβ, xf = aα − cγ and xg = −aβ + bγ are all zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' It follows that a = b = c = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Hence, we have Annl(A) = {0} and A0 = Annl(A) = {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Let A be a zeropotent algebra A of dimension 3 with rank(A) ≥ 2 over K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Then, any weak multiplier of A is the scalar multiplication Sa for some a ∈ K, that is, M(A) = M ′(A) = LM(A) = LM ′(A) = {Sa | a ∈ K}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' By Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content='1 and Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content='4, any weak multiplier T is a linear mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Let T ∈ L(A) be represented as (23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' By Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content='2, T is a weak multiplier, if and only if AT = T tA, if and only if \uf8eb \uf8ed pα − sβ qα − tβ rα − uβ −aα + sγ −bα + tγ −cα + uγ aβ − pγ bβ − qγ cβ − rγ \uf8f6 \uf8f8 = \uf8eb \uf8ed −pα + sβ aα − sγ −aβ + pγ −qα + tβ bα − tγ −bβ + qγ −rα + uβ cα − uγ −cβ + rγ \uf8f6 \uf8f8 (25) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Suppose that α = ef, β = ge are linearly independent (the other cases are similar).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Then, because pα − sβ = −pα + sβ by comparing the (1,1)-elements of two matrices in (25), we have p = s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Comparing (1,2)-elements and (1,3)-elements, we have qα − tβ = aα − sγ = aα and rα − uβ = −aβ + pγ = −aβ respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' It follows that a = q = u and r = t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Comparing (2,2)-elements and (3,3)-elements, we see b = c = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Consequently, (23) holds if and only if b = c = p = r = s = t = 0 and a = q = u, that is, T = Sa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' In [4] we classify the zeropotent algebras of dimension 3 over an algebraically field K of characteristic not equal to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Up to isomorphism, we have 10 families of zeropotent algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' They are Z0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Z1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Z2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Z3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' {Z4(a)}a∈H,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Z5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Z6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' {Z7(a)}a∈H,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Z8 and Z9 defined by the structural matrices \uf8eb \uf8ed 0 0 0 0 0 0 0 0 0 \uf8f6 \uf8f8 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' \uf8eb \uf8ed 0 0 0 0 0 0 0 0 1 \uf8f6 \uf8f8 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' \uf8eb \uf8ed 0 0 1 0 0 0 0 0 1 \uf8f6 \uf8f8 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' \uf8eb \uf8ed 0 1 0 −1 0 0 0 0 0 \uf8f6 \uf8f8 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' \uf8eb \uf8ed 0 0 0 0 1 a 0 0 1 \uf8f6 \uf8f8 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' \uf8eb \uf8ed 0 1 0 0 0 0 0 0 1 \uf8f6 \uf8f8 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' \uf8eb \uf8ed 0 1 1 0 0 1 0 0 1 \uf8f6 \uf8f8 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' \uf8eb \uf8ed 1 a 0 0 1 0 0 0 1 \uf8f6 \uf8f8 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' \uf8eb \uf8ed 1 2 2 0 1 2 0 0 1 \uf8f6 \uf8f8 and \uf8eb \uf8ed 1 3 3 0 1 3 0 0 1 \uf8f6 \uf8f8 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Z0 is the zero algebra, and Z1 is isomorphic to the 3-dimensional associative algebra C1, and their (weak) multipliers are already determined in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' The algebras Z3 to Z9 have rank greater or equal to 2, which are covered by Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Thus, only A = Z2 is left to be analyzed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' The multiplication table A of A is \uf8eb \uf8ed 0 g 0 −g 0 g 0 −g 0 \uf8f6 \uf8f8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' We see A0 = Annl(A) = Annr(A) = K(e + g), and we have the nihil decomposition A = A0 ⊕ A1 with A1 = Ke + Kf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' A weak multiplier T ∈ M ′ 1(A) is a linear mapping represented by \uf8eb \uf8ed a b c p q r 0 0 0 \uf8f6 \uf8f8 satisfying 18 \uf8eb \uf8ed pg qg rg −ag −bg −cg −pg −qg −rg \uf8f6 \uf8f8 = AT = T tA = \uf8eb \uf8ed −pg ag pg −qg bg qg −rg cg rg \uf8f6 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Hence, a = −c = q and b = p = r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Let Ta be this linear mapping, then by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content='1 we have M ′(A) = {Ta | a ∈ K} ⊕ (K(f + g))A and LM ′(A) = \uf8f1 \uf8f2 \uf8f3 \uf8eb \uf8ed a + s t −a + u 0 a 0 s t u \uf8f6 \uf8f8 ��� a, s, t, u ∈ K \uf8fc \uf8fd \uf8fe .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' By Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content='2, a weak multiplier T = Ta+R with R ∈ (K(e+g))A is a multiplier if and only if for any ζ = xe + yf and η = ze + vf with x, y, z, v ∈ K, R((xv − yz)g) = R(ζη) = ζTa(η) − Ta((xv − yz)g) = ζ(aη) + a(xv − yz)e = a(xv − yz)(e + g) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' It follows that R(xg) = ax(e + g) for all x ∈ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Let Sa be the scalar multi- plication by a, then (T − Sa)(A) ⊆ K(e + g) and (T − Sa)(Kg) = {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Hence, we obtain M(A) = {Sa | a ∈ K} ⊕ {R ∈ (K(e + g))A | R(Kg) = 0}, and LM(A) = \uf8f1 \uf8f2 \uf8f3 \uf8eb \uf8ed a + s t 0 0 a 0 s t a \uf8f6 \uf8f8 ��� a, s, t ∈ K \uf8fc \uf8fd \uf8fe = \uf8f1 \uf8f2 \uf8f3 \uf8eb \uf8ed a b 0 0 c 0 a − c b c \uf8f6 \uf8f8 ��� a, b, c ∈ K \uf8fc \uf8fd \uf8fe .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' References [1] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Johnson, An introduction to the theory of centralizers, Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' London Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQfNQaE/content/2301.03735v1.pdf'} 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0000000000000000000000000000000000000000..b479d31202ad921583b8a53a6d9fe33f0a36f23d --- /dev/null +++ b/7tE1T4oBgHgl3EQf7QXS/content/tmp_files/2301.03533v1.pdf.txt @@ -0,0 +1,889 @@ +arXiv:2301.03533v1 [hep-th] 9 Jan 2023 +FIAN/TD/16/2022 +Unfolded Point Particle as a Field in Minkowski Space +A.A. Tarusov1,2 and M.A. Vasiliev1,2 +1 I.E. Tamm Department of Theoretical Physics, Lebedev Physical Institute, +Leninsky prospect 53, 119991, Moscow, Russia +2 Moscow Institute of Physics and Technology, Institutsky pereulok 9, 141701, +Dolgoprudny, Moscow region, Russia +Abstract +Point-particle dynamics is reformulated as a field theory. This is achieved by using +the unfolded dynamics approach that makes it possible to give dynamical interpretation +to the concept of physical dimension which is 1 for a point particle in the d-dimensional +space-time. The main idea for the description of a k-dimensional on-shell system in +the d-dimensional space is to keep the evolution along d − k dimensions off-shell or, +alternatively, restrict it in a specific way respecting the compatibility conditions of the +resulting unfolded system. The developed approach gives some hints how a non-linear +realization of the symmetry G of a larger-dimensional space in a lower-dimensional +system can emerge from a geometrical realization on the fields in an appropriate G- +invariant space. +For the example of a relativistic point particle considered in this +paper, G is the Poincar´e group. The proposed general scheme is illustrated by simple +examples that reproduce conventional results. +1 + +Table of contents +1 +Introduction +3 +2 +Unfolding +3 +3 +Free particle +6 +4 +Off-shell system +8 +4.1 +General setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +8 +4.2 +Covariant constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +10 +5 +Examples of on-shell systems +11 +5.1 +Lorentz force in a constant field . . . . . . . . . . . . . . . . . . . . . . . . . +12 +5.2 +Gravitational interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +12 +5.3 +Interaction with higher spins . . . . . . . . . . . . . . . . . . . . . . . . . . . +14 +6 +Conclusion +14 +2 + +1 +Introduction +Among different approaches to relativistic theories, one can distinguish between the +world-line particle approach and the field-theoretic one. In this article, we unify these ap- +proaches within the unfolded formulation of dynamical systems [1, 2], which will allow us to +ascribe a dynamical sense to physical dimensions of a system [3, 4]. +An example of this phenomenon was given in [5], where, following Fronsdal’s prescription +[6], an infinite system of massless fields of all spins in four-dimensional space has been +described by a single field in the ten-dimensional space (Analogous results were achieved in +the particle approach somewhat earlier in [7]). +In this approach space-time geometry in which the dynamical equations are formulated is +determined by the symmetry acting on the space, while the physical dimension is associated +with the set of initial data, that determine the evolution of the system [3]. In the papers +[5, 3, 7] (see also [8]) this idea was realized for the symmetry acting linearly on the fields in +both the four- and ten-dimensional space. +The description of the dynamics of point particles elaborated in this article assumes a +non-linear realization of the Lorentz symmetry on the dynamic variables of the point particle +as a consequence of the Einstein constraint on the velocity four-vector +unun = 1 . +(By selecting any un, that obeys this constraint, the Lorentz symmetry gets spontaneously +broken.) The application of the unfolded formalism in this case differs significantly from the +linear case. It requires the introduction of additional fields which encode unconstrained or +specific evolution along ”extra” dimensions of space-time as is explained in this paper. +The paper is organised as follows: Section 2 recalls the unfolded formalism used in the +paper, illustrated by the well-known scalar field example. Section 3 details the application +of that formalism to the case of a free point particle. In Section 4 an off-shell formulation +of the unfolded particle dynamics is presented both in terms of component fields and in +terms of generating functions. +In Section 5 it is explained how an external force to the +equations of motion can be introduced and a number of simple examples of the on-shell +systems is considered. +Section 6 contains brief conclusions with some emphasize on the +further applications and open problems. +2 +Unfolding +The possibility of decreasing the order of a differential equation by introducing new +variables and transitioning to an equivalent system of differential equations is well known. +Extension of this approach to partial differential equations is based on the jet formalism [9]. +Also it was elaborated in the framework of BV-BRST formalism e.g. in [10, 11, 12, 13]. +Unfolded dynamics approach [1, 2] (see also [14]), that is most appropriate for the gauge +theories in the framework of gravity, is a generalization of the first-order formulation of a +system via replacing a partial derivative by de Rham derivative d := ξn +∂ +∂xn and dynamical +3 + +variables by space-time differential forms W(x), which allows one to rewrite the system of +equations in the form +dW Ω(ξn, x) = GΩ(W(ξn, x)) , +(2.1) +where ξn is the anticommuting differential used as a placeholder for dxn, and GΩ(W(ξ, x)) +is some function of W containing only exterior products of the differential forms W(ξ, x) at +the same x (no space-time derivatives in GΩ(W(ξ, x)); wedge products are implicit). The +functions GΩ(W(ξ, x)) cannot be arbitrary, as the de Rham derivative is nilpotent and thus +the compatibility condition dGΩ = 0 must hold. This demands +GΛ(W)∂GΩ(W) +∂W Λ += 0 . +(2.2) +This constraint allows one to show that system (2.1) is manifestly invariant under the fol- +lowing gauge transformations: +δgaugeW Ω(ξn, x) = dǫΩ(ξn, x) + ǫΛ(ξn, x)∂GΩ(W(ξn, x)) +∂W Λ(ξn, x) +, +(2.3) +where +deg ǫΛ(ξn, x) = deg W Λ(ξn, x) − 1 , +with deg ω being a differential form degree of ω. +Generally speaking, gauge invariance only takes place in so-called universal systems [15, +16], in which the compatibility conditions hold as a consequence of the system itself without +taking into account the number of space-time dimensions, i.e. the fact that d+1-forms vanish +in d-dimensional space. Indeed, for non-universal systems partial derivative ∂GΩ +∂W Λ might have +no sense, leading to a non-zero derivative of zero represented by a d + 1-form. +In other terms, the fact that GΛ(ξn, x) is a function of W Ω(ξn, x) can be written as +GΛ = +∞ +� +n=1 +f Λ +Ω1,...ΩnW Ω1...W Ωn , +f Λ +Ω1,...Ωk,Ωk+1,...Ωn = (−1)degΩkdegΩk+1f Λ +Ω1,...Ωk+1,Ωk,...Ωn . +(2.4) +(From now on we omit arguments of GΛ and W Ω if it does not lead to misunderstandings.) +Compatibility condition (2.2) then yields generalized Jacobi identities on the structure con- +stants f +m +� +n=0 +(n + 1)f Λ +[Ω1,...Ωm−nf Φ +Λ,Ωm−n+1,...Ωm} = 0 , +(2.5) +where [} indicates an appropriate (anti)symmetrisation of indices. From this point of view, +the universality of a system means that the generalized Jacobi identities hold true regardless +of the dimension d. The underlying mathematical structure is called the strong homotopy +L∞ algebra [17]. +For universal systems it is also possible to introduce a Q-differential (homological vector +field) of the form [16] +Q = GΩ +∂ +∂W Ω , +(2.6) +4 + +which turns out to be nilpotent, +Q2 = 0 , +as a consequence of compatibility conditions (2.2). In these terms, any universal unfolded +system can be rewritten as +dF(W) = QF(W) , +where F(W) is an arbitrary function. +This way of describing the system as a so-called +Q-manifold relates the de Rham derivation on the world-sheet with coordinates xn to the +derivation Q on the target space with coordinates W Λ. +Unfolded formalism allows for a natural way of description of background geometry via +Maurer–Cartan equations which have the unfolded form. Indeed, let g be a Lie algebra. +Setting W = w and G = 1 +2[w , w] for a one-form w ∈ g, one observes that equation (2.1) +yields the zero-curvature condition +dw + 1 +2[w , w] = 0 . +(2.7) +For g being Poincar´e algebra with the one-form gauge fields (connection) en and wnm, the +unfolded system (2.7) yields the coordinate-independent description of Minkowski space in +the form +Rn = 0 , +Rnm = 0 . +(2.8) +Consider a system of equations +DCA(x) = 0 , +where the fields CA(x) in Minkowski space are valued in a Poincar´e-module V while D is +the exteriour covariant derivative in V with the flat connection w = enPn +ωnmLnm obeying +(2.7): D = d + w. (For instance, Lorentz transformations act on the tensor indices.) This +system is invariant under the following gauge transformations: +δCA = −εA +BCB, +(2.9) +δw(x) = Dε(x). +(2.10) +From here it follows that for a fixed w = w0 the gauge symmetry parameters are restricted +by the condition +D0ε(x) = 0 , +D0 = d + w0. +(2.11) +Since D2 +0 = 0, in the topologically trivial case the zero-form parameter εBA, can be recon- +structed from any point hence generating global symmetries of the system. +Minkowski space in Cartesian coordinates is described by the connection +en = dxn , +ωnm = 0 , +(2.12) +in which case (2.11) yields +∂nεn − εn +men +m = 0 , +∂nεnm = 0 , +(2.13) +which can be solved as εnm = −εmn = const, εn = εnmxm + ǫn, where ǫn is x-independent. +This obviously forms the Poincar´e transformations. +5 + +3 +Free particle +The classical point particle is conventionally described by generalized coordinates qi(s) +depending on the evolution parameter s. In this paper, we also use the generalized coor- +dinates qi = qi(x) which, however, will be treated as space-time fields, i.e., functions of all +space-time coordinates xn. Let us, for simplicity, work with Cartesian coordinates xn of a +flat Minkowski space, which will allow us to omit the Lorentz connection dependent terms. +We start with a first-step equation +Dqi(x) = ejqi +j(x) , +(3.1) +where qij(x) is an arbitrary (for now) matrix and D is the Lorentz covariant derivative. In +Cartesian coordinates this yields +dqi(x) = ejqi +j(x). +(3.2) +At j = 0 one arrives at a differential equation with respect to time t = x0 with an +arbitrary right hand side. +However, in contrast with classical dynamics, other values of +j also produce non-trivial equations, which means that the equations of motion involve +all space-time variables. This unusual modification makes sense when treating generalized +coordinates as embedding functions from our laboratory system to some other. In this case +the space sector of the right hand side is just a Jacobian of that transformation. +It is +convenient to assume that the dimensions of the original and target spaces are the same, +thus the space sector of our matrix has to be non-degenerate. Apart from the dependence +on ”extra” variables, this is similar to the description of a classical particle in terms of the +transformation from a chosen reference frame to the one in which the particle is at rest. +The analysis does not stop here though, since qij must satisfy the compatibility condition +ejDqi +j(x) = 0 . +(3.3) +The general solution to this condition has the form of a one-form with tensor coefficients that +are symmetric in lower indices, which solves (3.3) due to anticommutativity of the one-forms +en, eiej = −ejei, +Dqi +j(x) = ekqi +jk(x) , +qi +jk(x) = qi +kj(x) . +(3.4) +Equation (3.4) also produces the compatibility condition which has the analogous form +Dqi +jk(x) = elqi +(jkl)(x) . +(3.5) +The process can be continued resulting in the infinite set of equations +Dqi +(j1...jn)(x) = ekqi +(j1...jnk)(x) . +(3.6) +This system provides an example of an off-shell unfolded system that does not describe +any non-trivial equations of motion. It is fully analogous to that described in [18] for the +scalar field case. (Supersymmetric extensions of the off-shell unfolded systems were recently +6 + +considered in [19].) Every equation expresses the compatibility of the previous one, but +involves a new object that requires its own compatibility condition. To get nontrivial dy- +namics, however, one has to introduce additional conditions on the coefficients qi(j1...jnk) +describing higher derivatives of the field qi. Imposing constraints on the fields qi(j1...jnk) is +equivalent to imposing some differential equations on the fields qi. +The equations (3.6) admit a more compact form using auxiliary variables yi, so that the +right hand side of the equations results from differentiation of the generating functions with +respect to yi, +Dqi +(j1...jn)(x, y) = ek d +dykqi +(j1...jn)(x, y) , +(3.7) +where qi(x, y) is the generating function +qi(x, y) = +∞ +� +n=0 +1 +n!qi +j1,...jn(x)yj1...yjn , +(3.8) +with the original field qi(x) recovered at y = 0. +To describe a free relativistic particle this way we introduce a time-like “velocity 4-vector” +V i(x) as a new variable, imposing the equations +Dqi(x) = ejVj(x)V i(x), +(3.9) +DV i(x) = 0, +(3.10) +V i(x)Vi(x) = 1. +(3.11) +Let us show that this system indeed describes a free relativistic particle. In Cartesian +coordinates the system takes the form +ej ∂ +∂xj qi(x) = ejVj(x)V i(x), +(3.12) +ej ∂ +∂xj V i(x) = 0. +(3.13) +Here the second equation implies that V i is a constant while the first one contains a one-form +κ = eiVi , +(3.14) +which, in a sense, serves as a projector on the world line of the particle. +There is some freedom in the parametrization of the world line. For example, to take +time x0 as the evolution parameter, one has to reduce dxn to dx0 (equivalently, en → dxnδ0 +n). +This reproduces the familiar equations of motion. Indeed, after such a reduction, the velocity +vector produces a factor of V0, which is just a relativistic gamma-factor γ = (1 − v2 +c2 )−1/2. +This is not surprising, since the differentiation on the left hand side is over laboratory time, +˙qi(x) = γV i(x), +(3.15) +˙V i(x) = 0. +(3.16) +7 + +Thus, equations (3.9)-(3.11) indeed describe propagation of a free point particle with the +4-velocity V i(x). +Since our unfolded system can be easily extended to include the Lorentz connection by +appending (2.7), it inherits the full Poincar´e symmetries as outlined at the end of Section +2. +The unfolded formalism allows us to straightforwardly derive the symmetries of the +system. In this case (2.3) generates the background Poincar´e transformation as well as a +transformation of qi +δqi = 0ǫj ∂ekVkV i +∂ej += 0ǫjVjV i . +(3.17) +Note that in this formalism nontrivial particle dynamics is only along the direction asso- +ciated with κ. In other (transversal) directions the dynamics is trivial with no dependence +on the other coordinates. +4 +Off-shell system +4.1 +General setup +The world-line one-form κ (3.14) makes it possible to formulate the off-shell unfolded +system of a specific form distinguishing between the directions along κ and transversal ones. +The evolution of the system in the transversal directions is necessary for consistency. Indeed, +the naive system +Dqi(x) = κV i(x) , +DV i(x) = κF i(x) , +(4.1) +is inconsistent for arbitrary F i because now κ is not closed +Dκ = −eiDVi = −eiκFi . +(4.2) +While the classical behavior of the system is defined by the terms aligned with V i, to +achieve compatibility in all directions one has to adjust the evolution along the transversal +directions appropriately. +To achieve this it is convenient to introduce the transversal one-forms +ηi := ei − κV i +V 2 , +V iηi = 0 , +(4.3) +where an additional normalization is introduced, since the condition (3.11) is relaxed, as it +is not necessarily true off-shell. (Still we assume that V 2(x) := V i(x)Vi(x) ̸= 0.) The system +then takes the form +Dqi = κV i(x) + ηjHi +j(x), +(4.4) +DV i = κF i(x) + ηjGi +j(x). +(4.5) +Since ηi is V i-transversal, this system is invariant under the “gauge” transformations +H +′i +j(x) = Hi +j(x) + φi(x)Vj(x), +(4.6) +G +′i +j(x) = Gi +j(x) + ψi(x)Vj(x) , +(4.7) +8 + +with arbitrary functions φi(x), ψi(x), that can be gauge fixed by demanding +Hi +jV j = 0 , +Gi +jV j = 0 . +(4.8) +Firstly, let us note that the system is indeed off-shell as long as the condition V iVi = 1 +is not enforced. Indeed, the left hand sides contain d2 components of first derivatives of +qi(x) (or V i(x) for the second equation). On the right hand side, the Hij (Gij) contain +d(d−1) components due to the transversality condition while the V i (F i) span the d leftover +components. +To check the compatibility conditions of this system, one has to act by D on the both +sides of the equations then solving them with respect to Gij and Hij. We analyze the system +in an arbitrary torsion free geometry with Dei = 0, which yields +Dκ = −eiκFi − eiηjGij, +(4.9) +Dηi = 1 +V 4 +� +(ekκFkV i + ekηjGkjV i + κηjGi +j)V 2 − 2κV iVkηjGk +j +� +. +(4.10) +The compatibility of (4.4) yields using DDAi = RikAk = elejRik,ljAk . +DDqi = (Dκ)V i(x) − κDV i(x) + D(ηj)Hi +j(x) − ηjDHi +j(x) = += −V iejκFj − V iekηjGkj − κηjGi +j + 1 +V 2κηlGj +lHi +j − ηjDHi +j = elejRi +k,ljqk . +(4.11) +Expanding the last equation in the basis two-forms κηi and ηiηj, we obtain +(4.12) +ηjDHi +j = −V iηjκFj − 1 +V 2V iκV kηjGkj − V iηkηjGkj − κηjGi +j ++ 1 +V 2κηlGj +lHi +j − 2 +V 2κV lηjRi +k,ljqk − ηlηjRi +k,ljqk , +which is equivalent to +(4.13) +DHi +j = κ(−FjV i + Gi +j + 1 +V 2V iV kGkj − 1 +V 2Gk +jHi +k + 2 +V 2V lRi +k,ljqk) ++ ηkGkjV i + ηlRi +k,ljqk + ηkAi +jk + κVjBi + VjηkCi +k, +where the last three terms with arbitrary Bi, Cik and symmetric Aijk = Aikj parameterize +the general solution of the homogeneous equation ηjDHij = 0. Just as for Hij itself, the +transversality condition can be imposed on Aijk and Cik, +Ai +jkV k = 0 , +Ci +kV k = 0 . +(4.14) +Analogously for equation (4.5), with the only difference that we now impose the unfolded +equations on the field F i, +DF i = κJi + ηjKi +j +(4.15) +9 + +again demanding KijV j = 0. Then the compatibility condition for V i yields +DDV i = (Dκ)F i(x) − κ(DF i) + (Dηj)Gi +j − ηj(DGi +j) = += −ejκFjF i − elηjGljF i − κηjKi +j + 1 +V 2κηkGj +kGi +j − ηjDGi +j = elejRi +k,ljV k , +(4.16) +or, equivalently, +(4.17) +ηjDGi +j = ηjκ(−FjF i + Ki +j + 1 +V 2F iV kGkj − 1 +V 2Gk +jGi +k + 2 +V 2V lRi +k,ljV k) ++ ηjηk(GkjF i + Ri +l,kjV l). +The solution again consists of the inhomogeneous part and the terms parameterizing a +general solution of the homogeneous equation, +(4.18) +DGi +j = κ(−FjF i + Ki +j + 1 +V 2F iV kGkj − 1 +V 2Gk +jGi +k + 2 +V 2V lRi +k,ljV k) ++ ηk(GkjF i + Ri +l,kjV l) + ηlMi +jl + κVjNi + VjηkLi +k +with Mijl = Milj. Once again, Mijl and Lik obey the transversality conditions +Mi +jkV k = 0 , +Li +kV k = 0 . +(4.19) +In its turn, consistency of equations (4.13) and (4.18) imposes differential constraints +on the yet unconstrained coefficients A, B, C and M, N, L in terms of new unconstrained +variables. This process continues indefinitely leading eventually to a totally consistent infinite +set of equations on the infinite set of variables. Since the analysis of all these conditions in +terms of component fields like H, G, A, B, C, M, N, L quickly gets complicated we now revisit +them in a more compact form of generating functions. +4.2 +Covariant constraints +Though the description of a point particle considered in Section 4.1 is clear in principle +it is algebraically involved and not instructive. It can be simplified at least in Minkowski +background by imposing appropriate constraints in terms of generating functions of Section +3. To this end, we introduce auxiliary variables yi as in (3.7), rewriting the system (4.4), +(4.5) as +Dqi(x, y) = ej d +dyj qi(x, y), +(4.20) +DV i(x, y) = ej d +dyj V i(x, y). +(4.21) +10 + +These equations are clearly consistent, as derivatives commute while the vielbein one-forms +anticommute. They do not describe any dynamics, imposing no conditions on qi(x, 0) and +V i(x, 0). The results of Section 4.1 can be reproduced by imposing the following conditions: +V i(x, y) d +dyiqj(x, y) = V 2(x, y)V j(x, y) . +(4.22) +One has to check that this constraint is compatible with (4.20), (4.21), i.e. its differenti- +ation does not produce new constraints, giving zero by virtue of (4.22). Indeed, +D +� +V i(x, y) d +dyiqj(x, y)−V 2(x, y)V j(x, y) +� += ek d +dyk +� +V i(x, y) d +dyiqj(x, y)−V 2(x, y)V j(x, y) +� += 0, +(4.23) +(In the sequel, the arguments of the generating functions qi(x, y), V i(x, y) are implicit.) +Let us now show that supplemented with constraint (4.22) equations (4.20), (4.21) repro- +duce the equations from the previous section. By virtue of (4.22), and since ei = κV i +V 2 + ηi, +Eq. (4.20) yields +Dqi = κV i + ηj d +dyj qi . +(4.24) +Equation (4.4) is reproduced with ηj d +dyj qi|y=0= ηjHij. To fix an obvious freedom up to a +function φiVj in a way preserving trasversality one can set +Hi +j = +� d +dyj qi − 1 +V 2VjV k d +dykqi���� +y=0 . +Analogously, (4.21) yields equation (4.5) with +V j d +dyj V i��� +y=0 = F i , +(4.25) +� d +dyj V i − 1 +V 2VjV k d +dyk V i���� +y=0 = Gi +j . +(4.26) +Let us note that the second derivative of the generating function has d2(d+1) +2 +independent +components, of which, keeping in mind the transversality conditions, d2(d+1) +2 +−d2 are encoded +in Aijk, d2 − d in Cik and d more in Bi. That means that the system (4.20)-(4.22) indeed +concisely reproduces the off-shell formulation of Section 4.1 in all orders. +5 +Examples of on-shell systems +To put the system on-shell one has to set the field F i, that determines the evolution of +V i along itself, to some function F i(q, V ). Restriction of some combination of derivatives +parameterized by F i then would impose some partial differential equations on qi giving rise +to the equations of motion. Generally, a non-zero force F i(q, V ) would demand some higher +11 + +components of additional fields associated with the higher components in yj of qi(x, y) and +V i(x, y) (descendants) to be nonzero. There are two somewhat opposite options. +One is that all these descendants are kept non-zero and arbitrary in the sense that they +parameterize a general solution to the compatibility conditions. Another one is that these +descendants give as simple as possible specific solution to the compatibility conditions. In the +former case the system turns out to be off-shell in all directions transversal to the trajectory. +In the latter, the evolution along transversal directions has a specific form compatible with +F i(q, V ) ̸= 0 in the full unfolded system. Postponing a general analysis of this issue for the +future publication here we consider a few simple examples of the second kind. +5.1 +Lorentz force in a constant field +A particular choice of F i(q, V ) linear in V , F i(q, V ) = F ij(q)V j, Fij = −Fji, replicates +the Lorentz force. +As a toy example consider a particular solution to the compatibility +conditions of (4.4), (4.5) in flat space, that easily puts the system on-shell. Namely, let +F ij(q) be a constant field, i.e. dF ij = 0. Antisymmetry of Fij allows us to impose condition +(3.11) and write down the following on-shell system: +dqi(x) = κV i + ηi = ei, +(5.1) +dV i(x) = κV jF i +j + ηjF i +j = ejF i +j, +(5.2) +which is obviously consistent without introducing higher components in yj of qi(x, y) and +V i(x, y). +Note that the free particle case considered in Section 3 is reproduced at F i = 0 and +also corresponds to the specific (trivial) choice of the descendants associated with higher +components in yj of qi(x, y) and V i(x, y). +5.2 +Gravitational interaction +Within the exterior algebra formalism underlying the unfolded dynamics approach, the +gravitational background is naturally taken into account by using appropriate covariant +derivatives of the Cartan formulation of gravity. To introduce it in the metric formalism, i.e. +with Christoffel symbols, one has to distinguish between laboratory Lorentz indices denoted +by Latin letters and the underlined world sheet indices. For instance, V i = eiiV i, where the +vielbein eii relates laboratory and world indices. Let us start with the Cartan formulation. +The on-shell covariant condition (4.5) for V i with zero force reads as +∂kV i = −ωk +i +jV j + ηk +jGi +j . +(5.3) +On-shell, it is possible to impose the condition (3.11), compatibility with which then implies +the anticipated antisymmetry of ω, +ωk +i +j = −ωk +j +i . +(5.4) +12 + +Here the higher-order compatibility with Gij = 0 is easily achieved for the case of flat +(zero-curvature) gravitational fields while the general case demands some Gij ̸= 0. +It is not difficult to see that the condition (5.3), after applying the frame postulate +∂kel +i − Γi +klei +i + ωk +i +jel +j = 0 , +(5.5) +can be equivalently rewritten, leaving out the derivatives of the vielbein, i.e only in terms of +V i and the Christoffel symbols. +∂kV i = (∂kei +i)V i + ei +i∂kV i = −ωk +i +jei +jV i + ηk +jGi +j =⇒ ∂kV i = Γi +klV l + ηk +jGi +j . +(5.6) +From here it is possible to use the metric formalism in the equations for V i +dV i = κV k∂kV i + ηk∂kV i = κΓi +klV kV l + ηjGi +j , +(5.7) +where +∂k = ek +k∂k . +(5.8) +After the vielbein reduction on V : P(ei) = κV i, one gets the expected geodesic equation. +The example above of the link between spin-connection and Christoffel symbols realizes +the transition between worldsheet and fiber indices. This highlights the difference between +the usual dynamics formulated in terms of xi, ˙xk and our unfolded system formulated in +terms of qi, V i. +The relation between the two formalisms can be uplifted to the action level. As noted in +Section 3, after an appropriate reduction of the vielbein, Dqi becomes dqi +dτ ((3.15), (3.16)), +where τ is the natural evolution parameter. Using that qi(x) are embedding functions for +the coordinates xi, let us write an action, quadratic in dqi +dτ using the metric gij in the target +space (not necessarily corresponding to Minkowski’s space) and adding the gauge parameters +α for reparametrization invariance by α′dτ ′ = αdτ, +S = 1 +2 +� +dτα +� 1 +α2gij +dqi +dτ +dqj +dτ + m2� +. +(5.9) +In terms of xi, we obtain the regular action +S = 1 +2 +� +dτα +� 1 +α2gij +dqi +dxk +dqj +dxl +dxk +dτ +dxl +dτ + m2� += 1 +2 +� +dτα +� 1 +α2 ˜gkl +dxk +dτ +dxl +dτ + m2� +. +(5.10) +Here ˜gkl is the induced metric from the target space. As usual, Euler-Lagrange equations +for α are algebraic +α = 1 +m +� +˜gkl +dxk +dτ +dxl +dτ , +(5.11) +which allows us to substitute them back into the action to arrive at the conventional result +S = m +� +dτ +� +˜gkl +dxk +dτ +dxl +dτ . +(5.12) +13 + +5.3 +Interaction with higher spins +Note that the condition (5.3) allows a direct generalization onto higher-spin interactions +via introducing an appropriate higher-spin connection ωn1,...ns−1,m = dxkωkn1,...ns−1,m [20], +dV i = −ωn1,...ns−1, +iV n1...V ns−1 + ηjGi +j . +(5.13) +In this case, the compatibility with the constraint (3.11) demands ω(n1,...ns−1,m) = 0, which +means that, in agreement with the general higher-spin theory [20], ωn1,...ns−1,m is described +by the Young diagram n1 ... ns−1 +m +. +Note that the force associated with the field of an arbitrary spin can also be written in +terms of generalized Christoffel symbols [21] (see also [22, 23]). +6 +Conclusion +In this paper, we suggest an approach to the description of a relativistic classical point +particle as a field on which relativistic symmetries act geometrically. This is achieved by +rewriting equations in the unfolded formalism that supports manifest invariance under dif- +feomorphisms and the Lorentz group. The point particle is represented as a field obeying +unfolded equations. +A mechanism of projectors specifying the evolution parameter in a +covariant way is introduced. +The proposed approach can be useful for different types of theories including the double +field theory, [24, 25] where, as we hope, it can be used to provide an alternative way of +enforcing the section constraint (see, e.g., [26].) More generally, the description of lower- +dimensional objects within a proper extension of the proposed approach is of great interest, +in particular, for the description of branes in superstring theory as well as string theory +itself: it would be interesting to reformulate the string theory as a 2d theory described from +the start in terms of fields in the target space. It would also be interesting to analyze the +relation of the suggested mechanism with the models with non-linearly realized symmetries +[27, 28]. +It should be stressed that in the presence of extra dimensions the unfolded dynamics +solely along the parameters associated with the particle trajectory does not allow for com- +patible unfolded equations demanding an evolution along the transverse directions respecting +appropriate compatibility conditions. This raises a number of questions for the future study +such as, for instance, whether any solutions of the compatibility conditions can be associated +with an evolution along a one-dimensional trajectory and, if not, what are the sufficient con- +ditions for this to be true? A related interesting problem is to obtain the on-shell conditions +from the variational principle along the lines of [29]. +14 + +Acknowledgement +We are grateful to Ruslan Metsaev for the correspondence. 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B 825 (2022), 136882 [arXiv:2111.12691 +[hep-th]]. +16 + diff --git a/7tE1T4oBgHgl3EQf7QXS/content/tmp_files/load_file.txt b/7tE1T4oBgHgl3EQf7QXS/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..f9eaeaf523679a969c0b46afd8618e90c37cc8b2 --- /dev/null +++ b/7tE1T4oBgHgl3EQf7QXS/content/tmp_files/load_file.txt @@ -0,0 +1,678 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf,len=677 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='03533v1 [hep-th] 9 Jan 2023 FIAN/TD/16/2022 Unfolded Point Particle as a Field in Minkowski Space A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' Tarusov1,2 and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' Vasiliev1,2 1 I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' Tamm Department of Theoretical Physics, Lebedev Physical Institute, Leninsky prospect 53, 119991, Moscow, Russia 2 Moscow Institute of Physics and Technology, Institutsky pereulok 9, 141701, Dolgoprudny, Moscow region, Russia Abstract Point-particle dynamics is reformulated as a field theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' This is achieved by using the unfolded dynamics approach that makes it possible to give dynamical interpretation to the concept of physical dimension which is 1 for a point particle in the d-dimensional space-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' The main idea for the description of a k-dimensional on-shell system in the d-dimensional space is to keep the evolution along d − k dimensions off-shell or, alternatively, restrict it in a specific way respecting the compatibility conditions of the resulting unfolded system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' The developed approach gives some hints how a non-linear realization of the symmetry G of a larger-dimensional space in a lower-dimensional system can emerge from a geometrical realization on the fields in an appropriate G- invariant space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' For the example of a relativistic point particle considered in this paper, G is the Poincar´e group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' The proposed general scheme is illustrated by simple examples that reproduce conventional results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' 1 Table of contents 1 Introduction 3 2 Unfolding 3 3 Free particle 6 4 Off-shell system 8 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='1 General setup .' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' 10 5 Examples of on-shell systems 11 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='1 Lorentz force in a constant field .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' 12 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='3 Interaction with higher spins .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' 14 6 Conclusion 14 2 1 Introduction Among different approaches to relativistic theories, one can distinguish between the world-line particle approach and the field-theoretic one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' In this article, we unify these ap- proaches within the unfolded formulation of dynamical systems [1, 2], which will allow us to ascribe a dynamical sense to physical dimensions of a system [3, 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' An example of this phenomenon was given in [5], where, following Fronsdal’s prescription [6], an infinite system of massless fields of all spins in four-dimensional space has been described by a single field in the ten-dimensional space (Analogous results were achieved in the particle approach somewhat earlier in [7]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' In this approach space-time geometry in which the dynamical equations are formulated is determined by the symmetry acting on the space, while the physical dimension is associated with the set of initial data, that determine the evolution of the system [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' In the papers [5, 3, 7] (see also [8]) this idea was realized for the symmetry acting linearly on the fields in both the four- and ten-dimensional space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' The description of the dynamics of point particles elaborated in this article assumes a non-linear realization of the Lorentz symmetry on the dynamic variables of the point particle as a consequence of the Einstein constraint on the velocity four-vector unun = 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' (By selecting any un, that obeys this constraint, the Lorentz symmetry gets spontaneously broken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=') The application of the unfolded formalism in this case differs significantly from the linear case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' It requires the introduction of additional fields which encode unconstrained or specific evolution along ”extra” dimensions of space-time as is explained in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' The paper is organised as follows: Section 2 recalls the unfolded formalism used in the paper, illustrated by the well-known scalar field example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' Section 3 details the application of that formalism to the case of a free point particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' In Section 4 an off-shell formulation of the unfolded particle dynamics is presented both in terms of component fields and in terms of generating functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' In Section 5 it is explained how an external force to the equations of motion can be introduced and a number of simple examples of the on-shell systems is considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' Section 6 contains brief conclusions with some emphasize on the further applications and open problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' 2 Unfolding The possibility of decreasing the order of a differential equation by introducing new variables and transitioning to an equivalent system of differential equations is well known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' Extension of this approach to partial differential equations is based on the jet formalism [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' Also it was elaborated in the framework of BV-BRST formalism e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' in [10, 11, 12, 13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' Unfolded dynamics approach [1, 2] (see also [14]), that is most appropriate for the gauge theories in the framework of gravity, is a generalization of the first-order formulation of a system via replacing a partial derivative by de Rham derivative d := ξn ∂ ∂xn and dynamical 3 variables by space-time differential forms W(x), which allows one to rewrite the system of equations in the form dW Ω(ξn, x) = GΩ(W(ξn, x)) , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='1) where ξn is the anticommuting differential used as a placeholder for dxn, and GΩ(W(ξ, x)) is some function of W containing only exterior products of the differential forms W(ξ, x) at the same x (no space-time derivatives in GΩ(W(ξ, x));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' wedge products are implicit).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' The functions GΩ(W(ξ, x)) cannot be arbitrary, as the de Rham derivative is nilpotent and thus the compatibility condition dGΩ = 0 must hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' This demands GΛ(W)∂GΩ(W) ∂W Λ = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='2) This constraint allows one to show that system (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='1) is manifestly invariant under the fol- lowing gauge transformations: δgaugeW Ω(ξn, x) = dǫΩ(ξn, x) + ǫΛ(ξn, x)∂GΩ(W(ξn, x)) ∂W Λ(ξn, x) , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='3) where deg ǫΛ(ξn, x) = deg W Λ(ξn, x) − 1 , with deg ω being a differential form degree of ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' Generally speaking, gauge invariance only takes place in so-called universal systems [15, 16], in which the compatibility conditions hold as a consequence of the system itself without taking into account the number of space-time dimensions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' the fact that d+1-forms vanish in d-dimensional space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' Indeed, for non-universal systems partial derivative ∂GΩ ∂W Λ might have no sense, leading to a non-zero derivative of zero represented by a d + 1-form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' In other terms, the fact that GΛ(ξn, x) is a function of W Ω(ξn, x) can be written as GΛ = ∞ � n=1 f Λ Ω1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='ΩnW Ω1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='W Ωn , f Λ Ω1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='Ωk,Ωk+1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='Ωn = (−1)degΩkdegΩk+1f Λ Ω1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='Ωk+1,Ωk,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='Ωn .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='4) (From now on we omit arguments of GΛ and W Ω if it does not lead to misunderstandings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=') Compatibility condition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='2) then yields generalized Jacobi identities on the structure con- stants f m � n=0 (n + 1)f Λ [Ω1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='Ωm−nf Φ Λ,Ωm−n+1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='Ωm} = 0 , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='5) where [} indicates an appropriate (anti)symmetrisation of indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' From this point of view, the universality of a system means that the generalized Jacobi identities hold true regardless of the dimension d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' The underlying mathematical structure is called the strong homotopy L∞ algebra [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' For universal systems it is also possible to introduce a Q-differential (homological vector field) of the form [16] Q = GΩ ∂ ∂W Ω , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='6) 4 which turns out to be nilpotent, Q2 = 0 , as a consequence of compatibility conditions (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' In these terms, any universal unfolded system can be rewritten as dF(W) = QF(W) , where F(W) is an arbitrary function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' This way of describing the system as a so-called Q-manifold relates the de Rham derivation on the world-sheet with coordinates xn to the derivation Q on the target space with coordinates W Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' Unfolded formalism allows for a natural way of description of background geometry via Maurer–Cartan equations which have the unfolded form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' Indeed, let g be a Lie algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' Setting W = w and G = 1 2[w , w] for a one-form w ∈ g, one observes that equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='1) yields the zero-curvature condition dw + 1 2[w , w] = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='7) For g being Poincar´e algebra with the one-form gauge fields (connection) en and wnm, the unfolded system (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='7) yields the coordinate-independent description of Minkowski space in the form Rn = 0 , Rnm = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='8) Consider a system of equations DCA(x) = 0 , where the fields CA(x) in Minkowski space are valued in a Poincar´e-module V while D is the exteriour covariant derivative in V with the flat connection w = enPn +ωnmLnm obeying (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='7): D = d + w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' (For instance, Lorentz transformations act on the tensor indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=') This system is invariant under the following gauge transformations: δCA = −εA BCB, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='9) δw(x) = Dε(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='10) From here it follows that for a fixed w = w0 the gauge symmetry parameters are restricted by the condition D0ε(x) = 0 , D0 = d + w0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='11) Since D2 0 = 0, in the topologically trivial case the zero-form parameter εBA, can be recon- structed from any point hence generating global symmetries of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' Minkowski space in Cartesian coordinates is described by the connection en = dxn , ωnm = 0 , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='12) in which case (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='11) yields ∂nεn − εn men m = 0 , ∂nεnm = 0 , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='13) which can be solved as εnm = −εmn = const, εn = εnmxm + ǫn, where ǫn is x-independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' This obviously forms the Poincar´e transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' 5 3 Free particle The classical point particle is conventionally described by generalized coordinates qi(s) depending on the evolution parameter s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' In this paper, we also use the generalized coor- dinates qi = qi(x) which, however, will be treated as space-time fields, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=', functions of all space-time coordinates xn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' Let us, for simplicity, work with Cartesian coordinates xn of a flat Minkowski space, which will allow us to omit the Lorentz connection dependent terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' We start with a first-step equation Dqi(x) = ejqi j(x) , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='1) where qij(x) is an arbitrary (for now) matrix and D is the Lorentz covariant derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' In Cartesian coordinates this yields dqi(x) = ejqi j(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='2) At j = 0 one arrives at a differential equation with respect to time t = x0 with an arbitrary right hand side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' However, in contrast with classical dynamics, other values of j also produce non-trivial equations, which means that the equations of motion involve all space-time variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' This unusual modification makes sense when treating generalized coordinates as embedding functions from our laboratory system to some other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' In this case the space sector of the right hand side is just a Jacobian of that transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' It is convenient to assume that the dimensions of the original and target spaces are the same, thus the space sector of our matrix has to be non-degenerate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' Apart from the dependence on ”extra” variables, this is similar to the description of a classical particle in terms of the transformation from a chosen reference frame to the one in which the particle is at rest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' The analysis does not stop here though, since qij must satisfy the compatibility condition ejDqi j(x) = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='3) The general solution to this condition has the form of a one-form with tensor coefficients that are symmetric in lower indices, which solves (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='3) due to anticommutativity of the one-forms en, eiej = −ejei, Dqi j(x) = ekqi jk(x) , qi jk(x) = qi kj(x) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='4) Equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='4) also produces the compatibility condition which has the analogous form Dqi jk(x) = elqi (jkl)(x) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='5) The process can be continued resulting in the infinite set of equations Dqi (j1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='jn)(x) = ekqi (j1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='jnk)(x) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='6) This system provides an example of an off-shell unfolded system that does not describe any non-trivial equations of motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' It is fully analogous to that described in [18] for the scalar field case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' (Supersymmetric extensions of the off-shell unfolded systems were recently 6 considered in [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=') Every equation expresses the compatibility of the previous one, but involves a new object that requires its own compatibility condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' To get nontrivial dy- namics, however, one has to introduce additional conditions on the coefficients qi(j1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='jnk) describing higher derivatives of the field qi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' Imposing constraints on the fields qi(j1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='jnk) is equivalent to imposing some differential equations on the fields qi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' The equations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='6) admit a more compact form using auxiliary variables yi, so that the right hand side of the equations results from differentiation of the generating functions with respect to yi, Dqi (j1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='jn)(x, y) = ek d dykqi (j1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='jn)(x, y) , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='7) where qi(x, y) is the generating function qi(x, y) = ∞ � n=0 1 n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='qi j1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='jn(x)yj1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='yjn , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='8) with the original field qi(x) recovered at y = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' To describe a free relativistic particle this way we introduce a time-like “velocity 4-vector” V i(x) as a new variable, imposing the equations Dqi(x) = ejVj(x)V i(x), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='9) DV i(x) = 0, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='10) V i(x)Vi(x) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='11) Let us show that this system indeed describes a free relativistic particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' In Cartesian coordinates the system takes the form ej ∂ ∂xj qi(x) = ejVj(x)V i(x), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='12) ej ∂ ∂xj V i(x) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='13) Here the second equation implies that V i is a constant while the first one contains a one-form κ = eiVi , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='14) which, in a sense, serves as a projector on the world line of the particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' There is some freedom in the parametrization of the world line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' For example, to take time x0 as the evolution parameter, one has to reduce dxn to dx0 (equivalently, en → dxnδ0 n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' This reproduces the familiar equations of motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' Indeed, after such a reduction, the velocity vector produces a factor of V0, which is just a relativistic gamma-factor γ = (1 − v2 c2 )−1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' This is not surprising, since the differentiation on the left hand side is over laboratory time, ˙qi(x) = γV i(x), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='15) ˙V i(x) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='16) 7 Thus, equations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='9)-(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='11) indeed describe propagation of a free point particle with the 4-velocity V i(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' Since our unfolded system can be easily extended to include the Lorentz connection by appending (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='7), it inherits the full Poincar´e symmetries as outlined at the end of Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' The unfolded formalism allows us to straightforwardly derive the symmetries of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' In this case (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='3) generates the background Poincar´e transformation as well as a transformation of qi δqi = 0ǫj ∂ekVkV i ∂ej = 0ǫjVjV i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='17) Note that in this formalism nontrivial particle dynamics is only along the direction asso- ciated with κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' In other (transversal) directions the dynamics is trivial with no dependence on the other coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' 4 Off-shell system 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='1 General setup The world-line one-form κ (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='14) makes it possible to formulate the off-shell unfolded system of a specific form distinguishing between the directions along κ and transversal ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' The evolution of the system in the transversal directions is necessary for consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' Indeed, the naive system Dqi(x) = κV i(x) , DV i(x) = κF i(x) , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='1) is inconsistent for arbitrary F i because now κ is not closed Dκ = −eiDVi = −eiκFi .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='2) While the classical behavior of the system is defined by the terms aligned with V i, to achieve compatibility in all directions one has to adjust the evolution along the transversal directions appropriately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' To achieve this it is convenient to introduce the transversal one-forms ηi := ei − κV i V 2 , V iηi = 0 , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='3) where an additional normalization is introduced, since the condition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='11) is relaxed, as it is not necessarily true off-shell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' (Still we assume that V 2(x) := V i(x)Vi(x) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=') The system then takes the form Dqi = κV i(x) + ηjHi j(x), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='4) DV i = κF i(x) + ηjGi j(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='5) Since ηi is V i-transversal, this system is invariant under the “gauge” transformations H ′i j(x) = Hi j(x) + φi(x)Vj(x), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='6) G ′i j(x) = Gi j(x) + ψi(x)Vj(x) , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='7) 8 with arbitrary functions φi(x), ψi(x), that can be gauge fixed by demanding Hi jV j = 0 , Gi jV j = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='8) Firstly, let us note that the system is indeed off-shell as long as the condition V iVi = 1 is not enforced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' Indeed, the left hand sides contain d2 components of first derivatives of qi(x) (or V i(x) for the second equation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' On the right hand side, the Hij (Gij) contain d(d−1) components due to the transversality condition while the V i (F i) span the d leftover components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' To check the compatibility conditions of this system, one has to act by D on the both sides of the equations then solving them with respect to Gij and Hij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' We analyze the system in an arbitrary torsion free geometry with Dei = 0, which yields Dκ = −eiκFi − eiηjGij, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='9) Dηi = 1 V 4 � (ekκFkV i + ekηjGkjV i + κηjGi j)V 2 − 2κV iVkηjGk j � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='10) The compatibility of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='4) yields using DDAi = RikAk = elejRik,ljAk .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' DDqi = (Dκ)V i(x) − κDV i(x) + D(ηj)Hi j(x) − ηjDHi j(x) = = −V iejκFj − V iekηjGkj − κηjGi j + 1 V 2κηlGj lHi j − ηjDHi j = elejRi k,ljqk .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='11) Expanding the last equation in the basis two-forms κηi and ηiηj, we obtain (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='12) ηjDHi j = −V iηjκFj − 1 V 2V iκV kηjGkj − V iηkηjGkj − κηjGi j + 1 V 2κηlGj lHi j − 2 V 2κV lηjRi k,ljqk − ηlηjRi k,ljqk , which is equivalent to (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='13) DHi j = κ(−FjV i + Gi j + 1 V 2V iV kGkj − 1 V 2Gk jHi k + 2 V 2V lRi k,ljqk) + ηkGkjV i + ηlRi k,ljqk + ηkAi jk + κVjBi + VjηkCi k, where the last three terms with arbitrary Bi, Cik and symmetric Aijk = Aikj parameterize the general solution of the homogeneous equation ηjDHij = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' Just as for Hij itself, the transversality condition can be imposed on Aijk and Cik, Ai jkV k = 0 , Ci kV k = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='14) Analogously for equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='5), with the only difference that we now impose the unfolded equations on the field F i, DF i = κJi + ηjKi j (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='15) 9 again demanding KijV j = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' Then the compatibility condition for V i yields DDV i = (Dκ)F i(x) − κ(DF i) + (Dηj)Gi j − ηj(DGi j) = = −ejκFjF i − elηjGljF i − κηjKi j + 1 V 2κηkGj kGi j − ηjDGi j = elejRi k,ljV k , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='16) or, equivalently, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='17) ηjDGi j = ηjκ(−FjF i + Ki j + 1 V 2F iV kGkj − 1 V 2Gk jGi k + 2 V 2V lRi k,ljV k) + ηjηk(GkjF i + Ri l,kjV l).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' The solution again consists of the inhomogeneous part and the terms parameterizing a general solution of the homogeneous equation, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='18) DGi j = κ(−FjF i + Ki j + 1 V 2F iV kGkj − 1 V 2Gk jGi k + 2 V 2V lRi k,ljV k) + ηk(GkjF i + Ri l,kjV l) + ηlMi jl + κVjNi + VjηkLi k with Mijl = Milj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' Once again, Mijl and Lik obey the transversality conditions Mi jkV k = 0 , Li kV k = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='19) In its turn, consistency of equations (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='13) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='18) imposes differential constraints on the yet unconstrained coefficients A, B, C and M, N, L in terms of new unconstrained variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' This process continues indefinitely leading eventually to a totally consistent infinite set of equations on the infinite set of variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' Since the analysis of all these conditions in terms of component fields like H, G, A, B, C, M, N, L quickly gets complicated we now revisit them in a more compact form of generating functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='2 Covariant constraints Though the description of a point particle considered in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='1 is clear in principle it is algebraically involved and not instructive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' It can be simplified at least in Minkowski background by imposing appropriate constraints in terms of generating functions of Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' To this end, we introduce auxiliary variables yi as in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='7), rewriting the system (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='4), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='5) as Dqi(x, y) = ej d dyj qi(x, y), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='20) DV i(x, y) = ej d dyj V i(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='21) 10 These equations are clearly consistent, as derivatives commute while the vielbein one-forms anticommute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' They do not describe any dynamics, imposing no conditions on qi(x, 0) and V i(x, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' The results of Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='1 can be reproduced by imposing the following conditions: V i(x, y) d dyiqj(x, y) = V 2(x, y)V j(x, y) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='22) One has to check that this constraint is compatible with (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='20), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='21), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' its differenti- ation does not produce new constraints, giving zero by virtue of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' Indeed, D � V i(x, y) d dyiqj(x, y)−V 2(x, y)V j(x, y) � = ek d dyk � V i(x, y) d dyiqj(x, y)−V 2(x, y)V j(x, y) � = 0, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='23) (In the sequel, the arguments of the generating functions qi(x, y), V i(x, y) are implicit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=') Let us now show that supplemented with constraint (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='22) equations (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='20), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='21) repro- duce the equations from the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' By virtue of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='22), and since ei = κV i V 2 + ηi, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='20) yields Dqi = κV i + ηj d dyj qi .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='24) Equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='4) is reproduced with ηj d dyj qi|y=0= ηjHij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' To fix an obvious freedom up to a function φiVj in a way preserving trasversality one can set Hi j = � d dyj qi − 1 V 2VjV k d dykqi���� y=0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' Analogously, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='21) yields equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='5) with V j d dyj V i��� y=0 = F i , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='25) � d dyj V i − 1 V 2VjV k d dyk V i���� y=0 = Gi j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='26) Let us note that the second derivative of the generating function has d2(d+1) 2 independent components, of which, keeping in mind the transversality conditions, d2(d+1) 2 −d2 are encoded in Aijk, d2 − d in Cik and d more in Bi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' That means that the system (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='20)-(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='22) indeed concisely reproduces the off-shell formulation of Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='1 in all orders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' 5 Examples of on-shell systems To put the system on-shell one has to set the field F i, that determines the evolution of V i along itself, to some function F i(q, V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' Restriction of some combination of derivatives parameterized by F i then would impose some partial differential equations on qi giving rise to the equations of motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' Generally, a non-zero force F i(q, V ) would demand some higher 11 components of additional fields associated with the higher components in yj of qi(x, y) and V i(x, y) (descendants) to be nonzero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' There are two somewhat opposite options.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' One is that all these descendants are kept non-zero and arbitrary in the sense that they parameterize a general solution to the compatibility conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' Another one is that these descendants give as simple as possible specific solution to the compatibility conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' In the former case the system turns out to be off-shell in all directions transversal to the trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' In the latter, the evolution along transversal directions has a specific form compatible with F i(q, V ) ̸= 0 in the full unfolded system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' Postponing a general analysis of this issue for the future publication here we consider a few simple examples of the second kind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='1 Lorentz force in a constant field A particular choice of F i(q, V ) linear in V , F i(q, V ) = F ij(q)V j, Fij = −Fji, replicates the Lorentz force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' As a toy example consider a particular solution to the compatibility conditions of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='4), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='5) in flat space, that easily puts the system on-shell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' Namely, let F ij(q) be a constant field, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' dF ij = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' Antisymmetry of Fij allows us to impose condition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='11) and write down the following on-shell system: dqi(x) = κV i + ηi = ei, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='1) dV i(x) = κV jF i j + ηjF i j = ejF i j, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='2) which is obviously consistent without introducing higher components in yj of qi(x, y) and V i(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' Note that the free particle case considered in Section 3 is reproduced at F i = 0 and also corresponds to the specific (trivial) choice of the descendants associated with higher components in yj of qi(x, y) and V i(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='2 Gravitational interaction Within the exterior algebra formalism underlying the unfolded dynamics approach, the gravitational background is naturally taken into account by using appropriate covariant derivatives of the Cartan formulation of gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' To introduce it in the metric formalism, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' with Christoffel symbols, one has to distinguish between laboratory Lorentz indices denoted by Latin letters and the underlined world sheet indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' For instance, V i = eiiV i, where the vielbein eii relates laboratory and world indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' Let us start with the Cartan formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' The on-shell covariant condition (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='5) for V i with zero force reads as ∂kV i = −ωk i jV j + ηk jGi j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='3) On-shell, it is possible to impose the condition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='11), compatibility with which then implies the anticipated antisymmetry of ω, ωk i j = −ωk j i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='4) 12 Here the higher-order compatibility with Gij = 0 is easily achieved for the case of flat (zero-curvature) gravitational fields while the general case demands some Gij ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' It is not difficult to see that the condition (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='3), after applying the frame postulate ∂kel i − Γi klei i + ωk i jel j = 0 , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='5) can be equivalently rewritten, leaving out the derivatives of the vielbein, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='e only in terms of V i and the Christoffel symbols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' ∂kV i = (∂kei i)V i + ei i∂kV i = −ωk i jei jV i + ηk jGi j =⇒ ∂kV i = Γi klV l + ηk jGi j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='6) From here it is possible to use the metric formalism in the equations for V i dV i = κV k∂kV i + ηk∂kV i = κΓi klV kV l + ηjGi j , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='7) where ∂k = ek k∂k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='8) After the vielbein reduction on V : P(ei) = κV i, one gets the expected geodesic equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' The example above of the link between spin-connection and Christoffel symbols realizes the transition between worldsheet and fiber indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' This highlights the difference between the usual dynamics formulated in terms of xi, ˙xk and our unfolded system formulated in terms of qi, V i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' The relation between the two formalisms can be uplifted to the action level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' As noted in Section 3, after an appropriate reduction of the vielbein, Dqi becomes dqi dτ ((3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='15), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='16)), where τ is the natural evolution parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' Using that qi(x) are embedding functions for the coordinates xi, let us write an action, quadratic in dqi dτ using the metric gij in the target space (not necessarily corresponding to Minkowski’s space) and adding the gauge parameters α for reparametrization invariance by α′dτ ′ = αdτ, S = 1 2 � dτα � 1 α2gij dqi dτ dqj dτ + m2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='9) In terms of xi, we obtain the regular action S = 1 2 � dτα � 1 α2gij dqi dxk dqj dxl dxk dτ dxl dτ + m2� = 1 2 � dτα � 1 α2 ˜gkl dxk dτ dxl dτ + m2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='10) Here ˜gkl is the induced metric from the target space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' As usual, Euler-Lagrange equations for α are algebraic α = 1 m � ˜gkl dxk dτ dxl dτ , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='11) which allows us to substitute them back into the action to arrive at the conventional result S = m � dτ � ˜gkl dxk dτ dxl dτ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='12) 13 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='3 Interaction with higher spins Note that the condition (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='3) allows a direct generalization onto higher-spin interactions via introducing an appropriate higher-spin connection ωn1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='ns−1,m = dxkωkn1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='ns−1,m [20], dV i = −ωn1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='ns−1, iV n1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='V ns−1 + ηjGi j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='13) In this case, the compatibility with the constraint (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='11) demands ω(n1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='ns−1,m) = 0, which means that, in agreement with the general higher-spin theory [20], ωn1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='ns−1,m is described by the Young diagram n1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' ns−1 m .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' Note that the force associated with the field of an arbitrary spin can also be written in terms of generalized Christoffel symbols [21] (see also [22, 23]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' 6 Conclusion In this paper, we suggest an approach to the description of a relativistic classical point particle as a field on which relativistic symmetries act geometrically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' This is achieved by rewriting equations in the unfolded formalism that supports manifest invariance under dif- feomorphisms and the Lorentz group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' The point particle is represented as a field obeying unfolded equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' A mechanism of projectors specifying the evolution parameter in a covariant way is introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' The proposed approach can be useful for different types of theories including the double field theory, [24, 25] where, as we hope, it can be used to provide an alternative way of enforcing the section constraint (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=', [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=') More generally, the description of lower- dimensional objects within a proper extension of the proposed approach is of great interest, in particular, for the description of branes in superstring theory as well as string theory itself: it would be interesting to reformulate the string theory as a 2d theory described from the start in terms of fields in the target space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' It would also be interesting to analyze the relation of the suggested mechanism with the models with non-linearly realized symmetries [27, 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' It should be stressed that in the presence of extra dimensions the unfolded dynamics solely along the parameters associated with the particle trajectory does not allow for com- patible unfolded equations demanding an evolution along the transverse directions respecting appropriate compatibility conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' This raises a number of questions for the future study such as, for instance, whether any solutions of the compatibility conditions can be associated with an evolution along a one-dimensional trajectory and, if not, what are the sufficient con- ditions for this to be true?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' A related interesting problem is to obtain the on-shell conditions from the variational principle along the lines of [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' 14 Acknowledgement We are grateful to Ruslan Metsaev for the correspondence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' This work was supported by the Russian Basic Research Foundation Grant No 20-02-00208.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQf7QXS/content/2301.03533v1.pdf'} +page_content=' References [1] M.' metadata={'source': 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b/89E5T4oBgHgl3EQfQw7M/content/tmp_files/2301.05516v1.pdf.txt @@ -0,0 +1,4257 @@ +arXiv:2301.05516v1 [math.PR] 13 Jan 2023 +CLT for real β-Ensembles at High Temperature∗ +Charlie Dworaczek Guera†, Ronan Memin‡ +Abstract +We establish a central limit theorem for the fluctuations of the empirical measure in the beta- +ensemble of dimension N at a temperature proportional to N and with convex, smooth potential. +The space of test functions for which the CLT holds includes C1, vanishing functions at infinity. It is +obtained by the inversion of an operator which is a pertubation of a Sturm-Liouville operator. The +method that we use is based on a change of variables introduced in [BFG15] and in [Shc14]. +Contents +1 +Introduction and main result +1 +2 +Regularity of the equilibrium measure and Hilbert transform +6 +3 +Concentration inequality, proof of Theorem 1.5 +9 +4 +Localization of the edge of a configuration +14 +5 +Laplace transform for smooth test functions, proof of Theorem 1.3 +17 +6 +Inversion of L +22 +7 +Regularity of the inverse of L and completion of the proof of Theorem 1.3 +26 +A Appendix: proof of Theorem 6.2 +29 +1 +Introduction and main result +The beta-ensemble of dimension N ⩾ 1 with parameter β > 0 and potential V is the probability measure +on RN given by +dPβ,V +N (x1, . . . , xN) = +1 +ZN(V, β) +� +i max(1, β), +lim inf +|x|→∞ +V (x) +Nβ′ ln |x| > 1 , +(2) +see [AGZ10, equation (2.6.2)]. The parameter β, which is allowed to depend on N, is the so-called inverse +temperature. +Under the special choice V (x) = x2 +2 , the measure (1) can be seen as the joint law of the (unordered) +eigenvalues of certain matrix models: +• For β = 1 (resp. β = 2), it is the law of the eigenvalues of the Gaussian Orthogonal Ensemble +(resp. Gaussian Unitary Ensemble), [AGZ10][Theorem 2.5.2]. +• For general β > 0, potentially depending on N, it is the law of the spectrum of certain tri-diagonal +random matrices as shown by Dumitriu and Edelman in [DE02]. +We consider here the high temperature regime where β scales as 1/N, and write β = 2P +N for some +P > 0. The corresponding measure is therefore +dPV,P +N (x1, . . . , xN) = +1 +ZV,P +N +� +i 0 has drawn a lot of attention from the random matrix +and statistical physics communities lately. The limiting density was first described in the case of the +quadratic potential in [ABG12], as a crossover between the Wigner semicircle law (fixed β > 0 case) and +the Gaussian density (case β = 0). The fluctuations of the eigenvalues in the bulk and at the edge of a +configuration were studied for example in [BGP15],[NT18],[NT20],[Pak18], [Lam21]. These fluctuations +were shown to be described by Poisson statistics in this regime. Recently, Spohn uncovered in [Spo20] +a link between the study of the Classical Toda chain and the β-ensemble in this regime, the result was +later extended to more general potentials in [GM22]. It is explained in [Spo21][Section 6] how the central +2 + +limit theorem for the empirical measure in the high temperature β ensemble is linked with the currents +of the Toda chain. +The Central Limit Theorem for the fluctuations of the linear statistics of beta-ensembles was first +established by [Joh98] for β = 2 polynomial potential, then generalized and further developed in the +regime where β is fixed in [Shc13], [BG13a], [BG13b], [BLS18]. Also an optimal local law was found +in this regime in [BMP22]. The CLT was obtained in the high-temperature regime βN → 2P > 0 by +Nakano and Trinh in [NT18, Theorem 4.9] for quadratic V , relying on the tridiagonal representation for +the beta-ensemble with quadratic potential in [DE02]. In [HL21], the authors prove the CLT in the case +of the circular beta-ensemble at high temperature with general potential, using a normal approximation +method involving the spectral analysis of an operator associated to the limiting covariance structure. +Their method allowed them to derive a Berry-Esseen bound, i.e. a speed of convergence of the fluctuations +towards a Gaussian variable. +In this paper, we adapt part of the arguments of [HL21] to our setup. More precisely, we show that +for a class of regular, convex potentials V satisfying a growth condition of the type +lim +|x|→∞ +V ′′(x) +V ′(x)2 = 0 , +denoting νN = ˆµN − µV +P and considering test functions f belonging to the range of a certain integro- +differential operator, the scaled fluctuations of ˆµN, defined by +√ +NνN(f) := +√ +N +�ˆ +R +fdµN − +ˆ +R +fdµV +P +� +, +converge in law towards centered Gaussian law with variance depending on f. +When considering the fixed temperature regime, i.e. β fixed, one has to renormalize the xi’s by +√ +N. +It is shown in [AGZ10][Theorem 2.6.1] that the measure +1 +N +N +� +i=1 +δxi/ +√ +N +satisfies a large deviation principle, and the limiting measure is characterized in [AGZ10][Lemma 2.6.2] +by an equation similar to (5). In fact, the term ln ρV +P in the left-hand side of (5) is the only difference in +the equation characterizing the limiting measure in the fixed β case. We point out the very similar char- +acterization of the equilibrium measure corresponding to the minimization problem arising in [BGK16]. +There again, the limiting measure is compactly supported. The term ln ρV +P is of prime importance because +its presence implies that the support of ρV +P is the whole real line. It leads to technicalities to deal with +the behavior at infinity of most of the associated objects, namely dealing with weighted Lebesgue spaces +L2(ρV +P ) and the corresponding Sobolev spaces Hk(ρV +P ). +Our strategy is based on a change of variables in the partition function ZV,P +N +(4), used for the beta- +ensemble at fixed temperature introduced in [BFG15] and [Shc14], and used in [Gui19] and in [BGK16] +to derive the loop equations and in [BLS18] to derive a CLT in the β-ensemble with β fixed. The outline +of the argument goes as follows: Take φ : R → R smooth, vanishing fast enough at infinity, and do the +change of variables in ZV,P +N +, xi = yi + +t +√ +N φ(yi), 1 ⩽ i ⩽ N, to get +ZP,V +N += +ˆ +RN +� +i 0, Q +� +V ′(x) +� +e−V (x) = +o +|x|→∞(x−α) . +iii) |V ′(x)| +−→ +|x|→+∞ +∞ and x �→ +1 +V ′(x)2 is integrable at infinity. Furthermore, for any sequence xN +such that |xN| goes to infinity, and for all real a < b, we have, as N goes to infinity, +1 +V ′(xN)2 +sup +a⩽x⩽b +|V ′′(xN + x)| −→ +N→∞ 0 . +iv) V ′′(x) +V ′(x) = +O +|x|→∞(1) and V (3)(x) +V ′(x) += +O +|x|→∞(1). +The convexity assumption in i) will guarantee the Poincaré inequality, stated in Proposition 2.4 for the +equilibrium measure µV +P . Because i) implies that V goes to infinity faster than linearly, we will see that +it also ensures exponential decay at infinity of ρV +P . Recalling the sufficient condition for PV,P +N +of equation +(2) to be defined, this first assumption implies that there exists α > 0 such that lim inf|x|→∞ +V (x) +|x| +> α. +This guarantees in particular that the beta-ensemble (3) is well-defined for all N ⩾ 1 and P ⩾ 0. +The second assumption ensures that any power of V ′ and V ′′ is in L2(ρV +P ) and ρV +P , which behaves +like e−V up to a sub-exponential factor, belongs to the Sobolev space H2(R) ⊂ C1(R). Indeed, for k ⩽ 2, +using iv), ρV +P +(k) behaves at infinity like (V ′)kρV +P as shown in (2.2) which is in L2(R) by assumption ii). +Assumption iii) will be used to localize the minimum/maximum point of a typical configuration +(x1, . . . , xN) following the law PV,P +N : this will be done in Corollary 4.2, which comes as a consequence of +4 + +[Lam21][Theorem 3.4]. More precisely, Corollary 4.2 establishes that for sequences (α+ +N)N, (α− +N)N going +to infinity, the random variables +α+ +N +� +max +1⩽j⩽N xj − E+ +N +� +and +α− +N +� +max +1⩽j⩽N xj − E− +N +� +converge in distribution. For large N, the scalars E+ +N and E− +N can thus be seen as the edges of a typical +configuration. Furthermore, +V (E± +N) ∼ ln N . +(9) +We refer to Section 4 for detailed statements. We will need another technical assumption to ensure that +Taylor remainders arising in the proof of Theorem 5.2 are negligible. This final step in the proof of +Theorem 1.3 consists in lifting the result of Proposition 5.1 from compactly supported functions to more +general functions. We use Assumption iv) to ensure that L−1 is regular enough ie that for sufficiently +smooth functions f, +� +L−1f +�′ +∈ H2(R). +Assumption 1.2. With the notations of Theorem 4.1, we have +sup +d(x,IN)⩽1 +���V (3)(x) +��� = o(N 1/2) , +where IN = +� +E− +N − 2; E+ +N + 2 +� +. +In view of the asymptotics (9), Assumption 1.2 is reasonable, at least in the cases where V is of the +form |x|a + R(x) for a = 2 or a > 3 (we choose a so that V is three times differentiable), and with +R ∈ C3(R), convex, small enough (see the comment following Theorem 1.3) or V = cosh, for example. +On the other hand a scaled potential like ex2 doesn’t satisfy assumptions iii) ans iv).. +We are now able to state the main result, ie the central limit theorem for functions belonging to the +image of the operator L introduced in (8). +Theorem 1.3. Assume that V satisfies Assumptions 1.1 and Assumption 1.2. Then for φ verifying the +following conditions: +• φ ∈ C1(R) +• φ(x) = O(1/x) and φ′(x) = O(1/x2) at infinity +• +ˆ +R +φ(x)dµV +P (x) = 0 +we have the convergence in law +√ +NνN(φ) → N +� +0, (σV +P )2(φ) +� +(10) +where the limiting variance (σV +P )2(φ) is given by +(σV +P )2(φ) = +ˆ +R +� +� +L−1φ +�′′(x)2 + V ′′(x) +� +L−1φ +�′(x)2 +� +dµV +P (x) ++ P +¨ +R2 +�� +L−1φ +�′(x) − +� +L−1φ +�′(y) +x − y +�2 +dµV +P (x)dµV +P (y) . +(11) +5 + +Remark 1.4. Since νN(φ + c) = νN(φ) for all constant c ∈ R, the assumption +ˆ +R +φ(x)dµV +P = 0 can be +dropped by replacing φ by φ − +ˆ +R +φ(x)dµV +P in the expression of the limiting variance. +As a tool to deal with the error term of equation (6), we establish a concentration inequality for the +empirical measure. This inequality is stated in terms of the following distance over the set of probability +distributions P(R). +For µ, µ′ ∈ P(R) we define the distance +d(µ, µ′) = +sup +∥f∥Lip⩽1 +∥f∥1/2⩽1 +����� +ˆ +fdµ − +ˆ +fdµ′ +���� +� +, +(12) +where ∥f∥Lip denotes the Lipschitz constant of f, and ∥f∥2 +1/2 = +ˆ +R +|t| |F[f](t)|2 dt, where F denotes the +Fourier transform on L2(R) which takes the following expression F[f](t) = +ˆ +R +f(x)e−itxdx for +f ∈ L1(R) ∩ L2(R). +We then have +Theorem 1.5. There exists K ∈ R (depending on P and on V ), such that for any N ⩾ 1 and r > 0, +PV,P +N +� +d(ˆµN, µV +P ) > r +� +⩽ e−Nr2 P π2 +2 ++5P ln N+K . +(13) +This result is the analog of [HL21, Theorem 1.4]. +The paper is organized as follows. In Section 2 we discuss the regularity of the equilibrium +density ρV +P under Assumption 1.1. In Section 3 we prove Theorem 1.5. Section 4 is dedicated to the +localization of the edge of a typical configuration, mentioned in the discussion preceding the statement of +Assumption 1.2. We next prove in Section 5 the convergence of the Laplace transform of +√ +NνN(Lφ) for +general functions φ which establishes Theorem 1.3 for functions of the form Lφ. Section 6 is dedicated to +the diagonalization and inversion of L given by (8). In Section 7, we show regularity properties of L−1 to +establish Theorem 1.3. We detail in Appendix A elements of proof for the spectral theory of Schrödinger +operators, used in Section 6. +Acknowledgments The authors wish to thank Alice Guionnet and Karol Kozlowski for their helpful +suggestions. We also thank Arnaud Debussche for pointing out the link with Schrödinger operators theory +and Gautier Lambert for pointing out [Lam21]. We would also like to thank Jeanne Boursier, Corentin +Le Bihan and Jules Pitcho for their intuition about the regularity of the inverse operator. +2 +Regularity of the equilibrium measure and Hilbert transform +In this section, we discuss the regularity properties of the equilibrium density ρV +P , namely its decay at +infinity and its smoothness, and give formulas for its two first derivatives. +The Hilbert transform, whose definition we recall, plays a central role in the analysis of the equilibrium +measure. It is first defined on the Schwartz class through ∀φ ∈ S(R), ∀x ∈ R, +H[φ](x) := + +R +φ(t) +t − xdt = lim +ε↓0 +ˆ +|t−x|>ε +φ(t) +t − xdt = +ˆ +∞ +0 +φ(x + t) − φ(x − t) +t +dt, +(14) +where + +denotes the Cauchy principal value integral, and then extended to L2(R) thanks to property ii) +of Lemma 2.1: ∥f∥L2(dx) = 1 +π ∥H[f]∥L2(dx). The last expression in (14) is a definition where the integral +converges in the classical sense. +6 + +We also recall the definition of the logarithmic potential U f of a density of probability f : R → R, +given for x ∈ R by +U f(x) = − +ˆ +R +ln |x − y|f(y)dy . +(15) +Because we assume f ∈ L1(R) to be nonnegative, U f takes values in [−∞, +∞). If f integrates the +function ln, i.e +´ +R ln |x|f(x)dx < +∞, then U f takes real values. +Additionally, one can check that the logarithmic potential and the Hilbert transform of f are linked +through the distributional identity +� +U f�′ = H[f]. +We recall in the next lemma some properties of the Hilbert transform that we will use in the rest of +the paper. +Lemma 2.1 (Properties of the Hilbert transform). +i) Fourier transform: For all φ ∈ L2(R), F +� +H[φ] +� +(ω) = iπsgn(ω)F[φ](ω) for all ω ∈ R. +ii) As a consequence, 1 +π H is an isometry of L2(R), and H satisfies on L2(R) the identity H2 = −π2I. +iii) Derivative: For any f ∈ H1(R), H[f] is also H1(R) and H[f]′ = H[f ′]. +iv) For all p > 1, the Hilbert transform can be extended as a bounded operator H : Lp(R) → Lp(R). +v) Skew-self adjointness: For any f, g ∈ L2(R), ⟨H[f], g⟩L2(R) = −⟨f, H[g]⟩L2(R). +Proof. We refer to [Kin09] for the proofs of these properties. +As a consequence of [GZ19], ˆµN converges almost surely under PV,P +N +towards the unique minimizer of +the energy-functional EV +P , defined for µ ∈ P(R) by +EV +P (µ) = + + + +ˆ +R +� +V + ln +�dµ +dx +�� +dµ − P +¨ +R2 ln +��x − y +��dµ(x)dµ(y) if µ ≪ dx ++∞ otherwise +. +(16) +(Here we wrote µ ≪ dx for "µ is absolutely continuous with respect to Lebesgue measure") +Consequently, following [GM22, Lemma 3.2], the density ρV +P of µV +P satisfies equation (5), which we +rewrite here for convenience. +V (x) − 2P +ˆ +R +ln |x − y|ρV +P (y)dy + ln ρV +P (x) = λV +P , +(17) +where λV +P is a constant (depending on V and P). Using this equation, we show in the next lemma that +ρV +P decays exponentially and is twice continuously differentiable. +We now drop the superscript of ρV +P and µV +P and denote it ρP and µP for convenience. +Lemma 2.2. Under Assumption 1.1, +• The support of µP is R and there exists a constant CV +P such that for all x ∈ R, +ρP (x) ⩽ CV +P (1 + |x|)2P e−V (x) . +• The density ρP is in C2(R) and we have +ρ′ +P = − +� +V ′ + 2PH[ρP ] +� +ρP +(18) +and +ρ′′ +P = +� +− 2PH[ρP ]′ − V ′′ + V ′2 + 4P 2H[ρP ]2 + 4PV ′H[ρP ] +� +ρP . +(19) +7 + +Proof. For the first point, [GM22, Lemma 3.2] establishes that the support of µP is the whole real axis, +and that under the first condition of 1.1, we have the bound, valid for all x ∈ R +ρP (x) ⩽ +KV +P +(1 + |x|)2 , +(20) +with KV +P a positive constant. Using (17) and the fact that +ln |x − y| ⩽ ln +� +1 + |x| +� ++ ln +� +1 + |y| +� +, +we see that for all x ∈ R, +ρP (x) ⩽ CV +P exp +� +− V (x) + 2P ln(1 + |x|) +� +, +(21) +with +CV +P = exp +� +2P +ˆ +R +ln(1 + |y|)ρP (y)dy + λV +P +� +which is indeed finite by (20). +For the second point, we use that +� +U ρP �′ = H[ρP ] weakly and equation (17) to conclude on the +distributional identity +ρ′ +P = +� +− V ′ − 2PH[ρP ] +� +ρP . +By the second point of Assumption 1.1, V ′(x)e−V (x)+2P ln(1+|x|) = o(x−1) as |x| → ∞, thus by (21), +V ′ρP ∈ L2(R). Also since ρP is L2(R) and bounded, we deduce, by using that H +� +L2(R) +� += L2(R), that +H[ρP ]ρP ∈ L2(R). Adding up these terms we get ρP ∈ H1(R). Because H[ρP ]′ = H[ρ′ +P ] in a weak sense +by Lemma 2.1, H[ρP ] ∈ H1(R). By the classical fact that H1(R) is contained in the set of 1/2-Hölder +functions C1/2(R), we have H[ρP ] ∈ C1/2(R) and so U ρP ∈ C1,1/2(R), the set of functions in C1(R) with +derivative of class 1/2-Hölder. +Using the fact that V is continuously differentiable, the previous equation for the weak derivative of ρP +then ensures that ρP ∈ C1(R) and equation (18) holds in the strong sense. +Differentiating (in a weak sense) equation (18) we obtain +ρ′′ +P = +� +− 2PH[ρP ]′ − V ′′ + V ′2 + 4P 2H[ρP ]2 + 4PV ′H[ρP ] +� +ρP . +The three first terms belong to L2(R) for the same reasons as before. By equation (21), ρP ∈ L4(R) and +by lemma 1.1, so is H[ρP ], thus using the boundedness of ρP we see that ρP H[ρP ]2 ∈ L2(R). For the last +term, we use that V ′ρP and H[ρP ] belong to L4(R) to ensure by Cauchy-Schwarz inequality that it is in +L2(R). Finally, we can conclude that ρP ∈ H2(R) and so that H[ρP ] ∈ H2(R) with H[ρP ]′′ = H[ρ′′ +P ] (in a +weak sense). As before, we conclude that ρP ∈ C2(R) and that equation (19) holds in a strong sense. +We next show that the Hilbert transform of ρP is continuous and decays at infinity. +Lemma 2.3. Let u ∈ L2(R) such that +´ +R u(t)dt exists and f : t �→ tu(t) ∈ H1(R) then +H[u](x) +∼ +|x|→∞ +− +´ +R u(t)dt +x +. +Moreover if +ˆ +R +u(t)dt = 0, +´ +R f(t)dt exists and g : t �→ t2u(t) ∈ H1(R), then +H[u](x) +∼ +|x|→∞ +− +´ +R tu(t)dt +x2 +. +As a consequence, we obtain that H[ρP ](x) +∼ +|x|→∞ x−1 and the logarithmic potential U ρP is Lipschitz +bounded, with bounded derivative H[ρP ]. +8 + +Proof. Let u ∈ L2(R), such that +´ +R u(t)dt exists and f : t �→ tu(t) ∈ H1(R). Then +xH[u](x) + +ˆ +R +u(t)dt = +ˆ +R +�xu(x + t) − xu(x − t) +2t ++ u(x + t) +2 ++ u(x − t) +2 +� +dt = H[f](x). +Since f ∈ H1(R), so is H[f], proving that it goes to zero at infinity. Hence +H[u](x) +∼ +|x|→∞ +− +´ +R u(t)dt +x +Moreover if +ˆ +R +u(t)dt = 0, +´ +R f(t)dt exists and g : t �→ t2u(t) ∈ H1(R), then by the same argument: +x2H[u](x) = xH[f](x) = H[g](x) − +ˆ +R +f(t)dt +where g(t) = t2u(t). We deduce that H[u](x) +∼ +|x|→∞ +− +´ +R tu(t)dt +x2 +since H[g] goes to zero at infinity. +We conclude this section by stating the Poincaré inequality for the measure µP under the assumption +that V is convex. +Proposition 2.4. The measure µP satisfies the following Poincaré inequality: There exists a constant +C such that for all f ∈ C∞ +c (R), +VarµP (f) ⩽ C +ˆ +R +|f ′|2dµP . +(22) +This fact comes as a direct consequence of [BBCG08][Corollary 1.9], which states that if the probability +measure µ has a log-concave density on R, then it satisfies (22) for f smooth enough (actually the result +is true in Rd, replacing f ′ by ∇f). Indeed, by convexity of V and concavity of x �→ ´ +R ln |x − y|ρP (y)dy, +equation (17) implies that ln ρP is concave. +Remark 2.5. We will apply later inequality (22) to more general functions than C∞ +c (R), namely functions +of the weighted Sobolev space H1(ρP ), defined in Section 6; which can be seen as the completion of C∞ +c (R) +with respect to the norm ∥u∥L2(ρP ) + ∥u′∥L2(ρP ). +3 +Concentration inequality, proof of Theorem 1.5 +We prove in this section the concentration Theorem 1.5. Its proof is a direct adaptation of Theorem 1.4 +of [HL21], which shows the analogous estimate in the circular setup. It is inspired by [MMS14] and based +on a comparison between a configuration xN = (x1, . . . , xN) sampled with PV,P +N +and a regularized version +yN = (y1, . . . , yN), which we describe here. +Definition 3.1. Let xN = (x1, . . . , xN) ∈ RN, and suppose (up to reordering) that x1 ⩽ x2 . . . ⩽ xN. +We define yN ∈ RN by: +y1 := x1, and for 0 ⩽ k ⩽ N − 1, yk+1 := yk + max{xk+1 − xk, N −3}. +Note that the configuration yN given by the previous definition satisfies yk+1 − yk ⩾ N −3, and yN is +close to xN in the sense that +N +� +k=1 +|xk − yk| ⩽ +1 +2N . +(23) +Indeed, by construction we have |xk − yk| = yk − xk ⩽ (k − 1)N −3, and we get the bound by summing +these inequalities. +9 + +The key point of the proof of Theorem 1.5 is comparing the empirical measure ˆµN = 1 +N +�N +i=1 δxi, where +xN follows PV,P +N , to the regularized measure +�µN := λN −5 ∗ 1 +N +N +� +i=1 +δyi, +(24) +ie the convolution of λN −5 and the empirical measure, where λN −5 is the uniform measure on [0, N −5]. +The interest of introducing the measure �µN is that it is close to ˆµN, while having a finite energy EV +P (�µN), +given by (16). Finally, notice that the empirical measure doesn’t change when reordering x1, . . . , xN, and +thus we do not lose in generality for our purposes in assuming that x1 ⩽ . . . ⩽ xN in definition 3.1. +We now introduce a distance on P(R) which is well-suited to our context. +Definition 3.2. For µ, µ′ ∈ P(R) we define the distance (possibly infinite) D(µ, µ′) by +D(µ, µ′) := +� +− +ˆ +ln |x − y|d(µ − µ′)(x)d(µ − µ′)(y) +�1/2 +(25) += +�ˆ +∞ +0 +1 +t +��F[µ − µ′] +��2dt +�1/2 +. +(26) +where the Fourier transform of a signed measure ν is defined by F[ν](x) := +´ +e−itxd(µ − µ′)(x) +Let f : R → R with finite 1/2 norm ∥f∥1/2 := +�´ +R |t| |F[f](t)|2 dt +�1/2 +. By Plancherel theorem and +Hölder inequality, for any µ, µ′ ∈ P(R), setting ν = µ − µ′, +���� +ˆ +R +fdµ − +ˆ +R +fdµ′ +���� +2 += +����� +1 +2π +ˆ +R +|t|1/2F[f](t)F[ν](t) +|t|1/2 dt +����� +2 +⩽ +1 +2π2 ∥f∥2 +1/2D2(µ, µ′). +Therefore the metric d defined in (12) is dominated by D: +d(µ, µ′) ⩽ +1 +√ +2π D(µ, µ′). +(27) +The following lemma shows how the distance D is related to the energy-functional EV +P defined in (16), +we will write EP for simplicity. +Lemma 3.3. We have for any absolutely continuous µ ∈ P(R) with finite energy EV +P (µ), +EP (µ) − EP (µP ) = PD2(µ, µP ) + +ˆ +ln +� dµ +dµP +� +dµ . +(28) +Proof of Lemma 3.3. Subtracting EP (µ) − EP (µP ) we find +EP (µ) − EP (µP ) = +ˆ +V d(µ − µP ) + +ˆ +ln dµ +dx dµ − +ˆ +ln ρP dµP − P +¨ +ln |x − y|dµ(x)dµ(y) ++ P +¨ +ln |x − y|dµP (x)dµP (y) . +(29) +Now, if ν is a signed measure of mass zero, integrating (17) we get +ˆ +V (x)dν(x) − 2P +¨ +ln |x − y|dν(x)dµP (y) + +ˆ +ln(ρP )(x)dν(x) = 0 . +10 + +We take ν = µ − µP , and get +ˆ +V (x)d(µ − µP )(x) = 2P +¨ +ln |x − y|dµ(x)dµP (y) − 2P +¨ +ln |x − y|dµP (x)dµP (y) +− +ˆ +ln(ρP )(x)dµ(x) + +ˆ +ln(ρP )(x)dµP (x) . +Plugging this last identity in (29), we find +EP (µ) − EP (µP ) = −P +¨ +ln |x − y|dν(x)dν(y) + +ˆ +ln +� dµ +dµP +� +(x)dµ(x) +which establishes the result. +Proof of Theorem 1.5. We first give a lower bound for the partition function ZV,P +N +(4) of PV,P +N . We rewrite +it as +ZV,P +N += +ˆ +RN exp +� +2P +N +� +i 0 +PV,P +N +� +D2(�µN, µP ) > r +� +⩽ e−NP r+cN+5P ln N +ˆ +RN exp +� +−2PN +ˆ +U ρP d(�µN − ˆµN) +� N +� +i=1 +ρP (xi)dxi . (32) +Next, we show that −N ´ U ρP d(�µN − ˆµN) is bounded. +By Lemma 2.3, U ρP is differentiable with bounded derivative H[ρP ] on R. As a consequence, +����N +ˆ +U ρP d(�µN − ˆµN) +���� ⩽ +N +� +i=1 +ˆ +|U ρP (yi + u) − U ρP (xi)| dλN −5(u) +⩽ ∥H[ρP ]∥∞ +� N +� +i=1 +|yi − xi| + N +ˆ +udλN −5(u) +� +⩽ ∥H[ρP ]∥∞ +� 1 +2N + N −4/2 +� +, +where we used (23) in the last inequality. Therefore, we deduce from (32) +PV,P +N +� +D2(�µN, µP ) > r +� +⩽ e−NP r+cN+5P ln N+2P ∥H[ρP ]∥∞ = e−NP r+5P ln N+KN +(33) +with KN := cN + 2P∥H [ρP ] ∥∞. Since cN is bounded, so is KN. +Finally, let f be a Lipschitz bounded function with ∥f∥Lip ⩽ 1, then, we have (as we did for U ρP ) +���� +ˆ +fdˆµN − +ˆ +fd�µN +���� ⩽ N −2 . +Thus, +d(ˆµN, µP ) ⩽ d(ˆµN, �µN) + d(�µN, µP ) ⩽ N −2 + +1 +√ +2π D(�µN, µP ) , +and for any N such that r − N −2 ⩾ r/2 (in particular r − N −2 > 0) we get +PV,P +N +(d(ˆµN, µP ) > r) ⩽ PV,P +N +� 1 +2π2 D2(�µN, µP ) > (r − N −2)2 +� +⩽ PV,P +N +� 1 +2π2 D2(�µN, µP ) > r2/4 +� +, +and the last term is bounded by e−Nr2 P π2 +2 ++5P ln N+K for some K large enough, which concludes the +proof. +12 + +As a consequence of Theorem 1.5, we are able to control the quantities +ζN(φ) := +¨ +R2 +φ(x) − φ(y) +x − y +d(ˆµN − µP )(x)d(ˆµN − µP )(y) +(34) +for a certain class of test functions φ. +Corollary 3.4. There exists C, K > 0 such that for all φ ∈ C2(R)∩H2(R) with bounded second derivative, +we have for ε > 0 and N large enough, +PV,P +N +�√ +N|ζN(φ)| ⩽ N −1/2+ε� +⩾ 1 − exp +� +− +PN ε +2CN2(φ) + 5P ln N + K +� +with N2(φ) = ∥φ′∥L2(dx) + ∥φ′′∥L2(dx). +Proof. We follow the proof given in [Gui19][Cor. 4.16] and adapt it to our setting. Let us denote by +� +ζN(φ) the quantity +¨ +R2 +φ(x) − φ(y) +x − y +d(�µN − µP )(x)d(�µN − µP )(y) . +We have the almost sure inequality, by a Taylor estimate +|ζN(φ) − � +ζN(φ)| ⩽ 2N −2∥φ′′∥∞ . +(35) +Thus, for any δ > 0, +PV,P +N +(|ζN(φ)| > δ) ⩽ PV,P +N +� +|ζN(φ) − � +ζN(φ)| > δ/2 +� ++ PV,P +N +� +|� +ζN(φ)| > δ/2 +� +⩽ PV,P +N +� +2N −2∥φ′′∥∞ > δ/2 +� ++ PV,P +N +� +|� +ζN(φ)| > δ/2 +� +, +where the first term of the right-hand side is either 0 or 1. With δ = N −1+ε, ε > 0, it is zero for N large +enough. For such a choice of δ, and for N large enough, +PV,P +N +� +|ζN(φ)| > N −1+ε� +⩽ PV,P +N +� +|� +ζN(φ)| > 1 +2N −1+ε +� +. +We next show that, for some C > 0 independent of φ, we have +|� +ζN(φ)| ⩽ CD2(�µN, µP )N2(φ) . +(36) +We begin by showing this inequality for ψ ∈ S(R). By using the inverse Fourier transform we have +�ζN(ψ) = 1 +2π +¨ ´ +dtF[ψ](t)e−itx − +´ +dtF[ψ](t)e−ity +x − y +d +� +�µN − µP +� +(x)d +� +�µN − µP +� +(y) += −1 +2π +ˆ +dtitF[ψ](t) +¨ +e−ity e−it(x−y) − 1 +−it(x − y) d +� +�µN − µP +� +(x)d +� +�µN − µP +� +(y) += −1 +2π +ˆ +dtitF[ψ](t) +¨ +e−ity +ˆ 1 +0 +dαe−iαt(x−y)d +� +�µN − µP +� +(x)d +� +�µN − µP +� +(y) += −1 +2π +ˆ +dtitF[ψ](t) +ˆ 1 +0 +dα +ˆ +e−iαtxd +� +�µN − µP +� +(x) +ˆ +e−i(1−α)tyd +� +�µN − µP +� +(y) +13 + +We then apply in order the triangular inequality, Cauchy-Schwarz inequality, a change of variable and +the fact that |F [�µN − µP ]|2 is an even function. +|�ζN(ψ)| ⩽ 1 +2π +ˆ +R +dt |tF[ψ](t)| +ˆ 1 +0 +dα |F [�µN − µP ] (αt)| . +��F [�µN − µP ] +� +(1 − α)t +��� +⩽ 1 +2π +ˆ +R +dt |tF[ψ](t)| +� ˆ 1 +0 +dα |F [�µN − µP ] (αt)|2 � 1 +2 � ˆ 1 +0 +dα +��F [�µN − µP ] +� +(1 − α)t +���2 � 1 +2 +⩽ 1 +2π +ˆ +R +dt |tF[ψ](t)| +ˆ 1 +0 +dα |F [�µN − µP ] (αt)|2 +⩽ 1 +2π +ˆ +∞ +0 +dt |tF[ψ](t)| +ˆ 1 +0 +tdα +tα |F [�µN − µP ] (αt)|2 + 1 +2π +ˆ 0 +−∞ +dt |tF[φ](t)| +ˆ 1 +0 +−tdα +−tα |F [�µN − µP ] (αt)|2 +⩽ 1 +2π +ˆ +R +dt |tF[ψ](t)| D2(�µN, µP ) +⩽ 1 +2π +� ˆ +R +dt |tF[ψ](t)|2 (1 + t2) +� 1 +2 � ˆ +R +dt +1 + t2 +� 1 +2 D2(�µN, µP ) +⩽ +1 +2√π D2(�µN, µP )N2(ψ) +By density of S(R) in L2(R), and since �ζN : +� +H2(R), N2 +� +→ R is continuous, the inequality still holds +for φ. Thus, using equation (33), +PV,P +N +� +|� +ζN(φ)| > 1 +2N −1+ε +� +⩽ PV,P +N +� +D2(�µN, µP ) > +N −1+ε +2CN2(φ) +� +⩽ exp +� +−P +N ε +2CN2(φ) + 5P ln N + K +� +, +which concludes the proof. +4 +Localization of the edge of a configuration +In [Lam21][Theorem 1.8, Theorem 3.4], Lambert was able to control the edge (i.e the minimum and the +maximum) of a typical configuration (x1, . . . , xN) distributed according to PV,P +N , by showing that the +random measure +ΞN := +N +� +j=1 +δϕ−1 +N (xj) +converges in distribution towards a Poisson point process for a function ϕN which takes the form +ϕN(x) := EN + α−1 +N x . +Before being more precise on the construction of (EN)N and (αN)N, we explain, following [Lam21], how +one can use this convergence to localize the edge of a typical configuration (x1, . . . , xN). Let us assume for +a moment that ΞN converges towards a Poisson point process with intensity θ(x) = e−x, with EN → +∞. +In particular, the random variable +ΞN(t, +∞) +converges in distribution towards a Poisson random variable with mean +´ +∞ +t +e−xdx. Combined with the +equalities +PV,P +N +� +ΞN(t, +∞) = 0 +� += PV,P +N +� +∀ 1 ⩽ j ⩽ N, ϕ−1 +N (xj) = αN(xj − EN) ⩽ t +� += PV,P +N +� +αN +� +max +1⩽j⩽N xj − EN +� +⩽ t +� +, +14 + +we deduce that for all t ∈ R +PV,P +N +� +αN +� +max +1⩽j⩽N xj − EN +� +⩽ t +� +−→ +N→∞ exp(−e−t) . +Therefore, the random variable +αN +� +max +1⩽j⩽N xj − EN +� +converges in distribution to the Gumbel law, showing that the maximum of a configuration is of order +EN. Furthermore, as will be clear from the construction of αN and EN, αN is positive, and goes to +infinity as N goes to infinity. +Replacing in the previous analysis θ(x) = ex and EN → −∞, we would have deduced in the same +fashion that +αN +� +min +1⩽j⩽N xj − EN +� +converges in law. +With the above notations, we can apply [Lam21][Theorem 3.4] to our context. +Theorem 4.1. Let v = ±. There exists (Ev +N)N, (αv +N)N sequences of real numbers with |Ev +N| → +∞, +αv +N > 0 for large enough N, satisfying V ′(Ev +N) = αv +Nv, such that: +a) Ne−V (Ev +N)+2P ln |Ev +N|+λP +V +αv +N +−→ +N→∞ 1 (recall λV +P is defined through equation (5)), +b) +ln(αv +N) +N +−→ +N→∞ 0 and αv +N|Ev +N| −→ +N→∞ +∞ , +c) For all compact K ⊂ R, +(αv +N)−2 sup +x∈K +|V ′′(ϕN(x))| −→ +N→∞ 0 . +As a consequence, the random measure ΞN converges in distribution as N → ∞ to a Poisson point process +with intensity θ(x) = e−vx. +Proof. We prove it in the case v = +, the case where v = − being similar. We show that there exists a +sequence (E+ +N)N going to +∞ satisfying f(E+ +N) = − ln N, where we defined the function f by +f(x) = −V (x) + 2P ln |x| + λV +P − ln |V ′(x)| . +(we will then have α+ +N = V ′(E+ +N) > 0. In the case v = −1 we would have looked for a sequence E− +N going +to −∞ and α− +N = −V ′(E− +N)) +As a consequence of Assumptions 1.1,ii), one shows that ln |V ′| is negligible with respect to V at infinity. +Therefore, because ln |x| +V (x) +−→ +|x|→∞ 0, +f(x) = −V (x) + o(V (x)) +at infinity (in particular, f(x) +−→ +x→+∞ −∞). We deduce that for 0 < ε < 1 fixed, there exists A > 0 such +that for all x > A, +−(1 + ε)V (x) < f(x) < −(1 − ε)V (x) , +(37) +and because f(x) +−→ +x→+∞ −∞ there exists (E+ +N)N going to infinity such that for all N ⩾ 1, f(E+ +N) = − ln N. +Setting x = E+ +N in (37), we obtain that −V (E+ +N) ∼ f(E+ +N) = − ln N. By convexity of V and the fact that +it goes to infinity at infinity, V is increasing on some [M, +∞[, where M ⩾ 0. Thus −(1±ε)V (x) = − ln N +15 + +iff x = V −1 +� ln N +1 ± ε +� +, where V −1 denotes +� +V|[M,+∞[ +�−1. We conclude by (37) that such an E+ +N must +satisfy +V −1 +� ln N +1 + ε +� +⩽ E+ +N ⩽ V −1 +� ln N +1 − ε +� +. +(38) +By convexity of V and the fact that it goes to infinity at infinity, (α+ +N)N is non-decreasing and goes to +infinity. It is thus positive for N large enough, ensuring that αN|E+ +N| −→ +N→∞ +∞. Property c) of the +theorem follows from Assumptions 1.1, point ii). It remains to show that ln(α+ +N) +N += ln |V ′(E+ +N)| +N +−→ +N→∞ 0. +By construction, we have +ln |V ′(E+ +N)| +N += +ln +� +Ne−V (E+ +N)+2P ln N+λP +� +N += −V (E+ +N) +N ++ o(1) . +Using that V (E+ +N) ∼ ln N, we can conclude that ln |V ′(E+ +N)| = o(N) which concludes the proof. +By the discussion preceding Theorem 4.1, we deduce +Corollary 4.2 (Edge of a configuration). Let E± +N, α± +N := |V ′(E± +N)| be the sequences of Theorem 4.1 +associated with v = ±1. Then, both random variables +α+ +N +� +max +1⩽j⩽N xj − E+ +N +� +and +α− +N +� +min +1⩽j⩽N xj − E− +N +� +converge to a Gumbel law, whose distribution function is given for t ⩾ 0 by G([0, t]) = exp(e−t). Further- +more, V (E± +N) ∼ ln N and α± +N −→ +N→∞ ±∞. +Remark 4.3. Note that[Lam21][Theorem 3.4] applies for V of class C2 outside of a compact set, allowing +to take V (x) = |x|a for a ⩾ 1. Furthermore, if V (x) = |x|a + R(x) for a ⩾ 1, R ∈ C2(R) and convex +and where R(x), R′(x) and R′′(x) are negligible respectively with respect to xa, xa−1 and xa−2, we find +E± +N ∼ ±(ln N)1/a. +If V (x) = cosh(x), we find E+ +N ∼ −E− +N ∼ arg cosh(ln N) ∼ ln ln N. +The next lemma will be convenient in the proof of Theorem 5.2 when dealing with error terms. +Lemma 4.4. With the notations of Corollary 4.2, we have +µP ([E− +N, E+ +N]c) = o(N −1/2) . +Proof. Let 0 < δ < 1, to be specified later. We have +ˆ +∞ +E+ +N +ρP dx = +ˆ +∞ +E+ +N +(ρP )δ(ρP )1−δdx ⩽ +ˆ +R +(ρP )δdx +sup +[E+ +N,+∞[ +(ρP )1−δ . +By the first inequality of Lemma 2.2, the integral is finite. Also from the same inequality, we have for +some constant C′ and x big enough ρP (x) ⩽ C′e− 3 +4 V (x). Because V is increasing in a neighborhood of ++∞, we get for N large enough +sup +[E+ +N,+∞[ +(ρP )1−δ ⩽ C′1−δe−(1−δ) 3 +4 V (E+ +N) . +16 + +Taking δ > 0 such that 1 +2 − (1 − δ) 3 +4 =: −γ < 0 and using that V (E+ +N) = ln N + o(ln N) (established in +the proof of Theorem 4.1), +√ +N +ˆ +∞ +E+ +N +ρP dx ⩽ Ke−γ ln N+(1−δ) 3 +4 o(ln N) , +and the right-hand side goes to zero as N goes to infinity. We deal with the integral +´ E− +N +−∞ ρP dx in the +same way. +Remark 4.5. We could improve the proof to show that µP ([E− +N, E+ +N]c) ∼ 1 +N but showing that it is o(N +1 +2 ) +is sufficient for what we need and requires less carefulness. +5 +Laplace transform for smooth test functions, proof of Theo- +rem 1.3 +Section 3 allows us to justify in Proposition 5.1 the heuristics we gave in equation (6) for φ having +compact support. We will then extend in Theorem 5.2 this result to a more general set of functions, by +an approximation by compactly supported functions, using Corollary 4.2. +Proposition 5.1. For φ ∈ C1(R, R) with compact support, we have for any real t, as N goes to infinity, +EV,P +N +� +et +√ +NνN (Ξφ)� +→ exp +�t2 +2 qP (φ) +� +, +(39) +where Ξφ is given by equation (7), and qP (φ) is given by +qP (φ) := +ˆ +R +� +φ′(x)2 + V ′′(x)φ(x)2 +� +dµP (x) + P +¨ +R2 +�φ(x) − φ(y) +x − y +�2 +dµP (x)dµP (y) . +(40) +Proof. Let φ ∈ C1 +c(R, R), and let t ∈ R. We perform in equation (4) the change of variables +xi = yi + +t +√ +N φ(yi), 1 ⩽ i ⩽ N, which is a diffeomorphism for N big enough. We thus have +ZV,P +N += +ˆ +� +1⩽i 0 if N is big enough, and where +|RN,1(i, j)| ⩽ +|t|3 +3N 3/2 ∥φ′∥3 +∞. +17 + +Again by Taylor-Lagrange theorem, the second term in (41) equals +exp +� +− +N +� +i=1 +� +V (yi) + +t +√ +N +V ′(yi)φ(yi) + t2 +2N V ′′(yi)φ(yi)2 + RN,2(i) +� � +where RN,2(i) = +t3 +6N 3/2 V (3) � +yi + tθi +√ +N φ(yi) +� +φ(yi)3 for some θi ∈ [0, 1], thus for N large enough +|RN,2(i)| ⩽ +|t|3 +6N 3/2 ∥φ∥3 +∞ +sup +d(x,supp φ)⩽1 +|V (3)(x)|. +The last term reads +N +� +i=1 +� +1 + +t +√ +N +φ′(yi) +� += exp +� N +� +i=1 +� +t +√ +N +φ′(yi) − t2 +2N φ′(yi)2 + RN,3(i) +� � +, +with |RN,3(i)| ⩽ +t3 +3N 3/2 ∥φ′∥3 +∞. Dividing both sides of equation (41) by ZV,P +N +we get +EV,P +N +� +exp +� +t +√ +N +� +P +¨ +R2 +φ(x) − φ(y) +x − y +dˆµN(x)dˆµN(y) + +ˆ +R +(φ′ − V ′φ)dˆµN +�� +× exp {KN(t, φ)} +× exp +� +t2 +2 +� +−P +¨ +R2 +�φ(x) − φ(y) +x − y +�2 +dˆµN(x)dˆµN(y) − +ˆ +R +(V ′′φ2 + φ′2)dˆµN +�� � += 1, +with |KN(t, φ)| ⩽ c(t,φ) +√ +N +where c(t, φ) ⩾ 0 is independent of N. This bound shows that taking the limit +N → ∞ we can get rid of KN: +lim +N→∞ EV,P +N +� +exp +� +t +√ +N +� +P +¨ +R2 +φ(x) − φ(y) +x − y +dˆµN(x)dˆµN(y) + +ˆ +R +(φ′ − V ′φ)dˆµN +�� +× exp +� +t2 +2 +� +−P +¨ +R2 +�φ(x) − φ(y) +x − y +�2 +dˆµN(x)dˆµN(y) − +ˆ +R +(V ′′φ2 + φ′2)dˆµN +�� � += 1. +Using Fubini’s theorem (the function (x, y) �→ φ(x)−φ(y) +x−y +being bounded continuous on R2), the first line +in the expectation value can be rewritten as et +√ +NΛN with +ΛN := 2P +¨ +R2 +φ(x) − φ(y) +x − y +dµP (x)d(ˆµN − µP )(y) + +ˆ +R +(φ′ − V ′φ)d(ˆµN − µP ) + PζN(φ) +(42) +where we used equation (5) and ζN(φ) is given by (34). Let F : P(R) → R be defined by +F(µ) = −P +¨ +R2 +�φ(x) − φ(y) +x − y +�2 +dµ(x)dµ(y) − +ˆ +R +(V ′′φ2 + φ′2)dµ . +(43) +It is continuous for the topology of weak convergence since all the functions in the integrals are bounded +continuous. So far we have established that +lim +N→∞ EV,P +N +� +et +√ +NΛN+ t2 +2 F (ˆµN )� += 1, +with ΛN given by (42). We now replace in the latter equation the term F(ˆµN) by its limiting expression, +F(µP ). Fix a metric that is compatible with the weak convergence of probability measures on R. For +example, +dLip(µ, ν) = sup +���� +ˆ +fdµ − +ˆ +fdν +���� , +(44) +18 + +where the supremum runs over f : R → R bounded and Lipschitz with ∥f∥∞ ⩽ 1 and Lipschitz constant +|f|Lip ⩽ 1. By the large deviations principle for (ˆµN)N under the probability (3) established by [GZ19, +Theorem 1.1], for all δ > 0 the event {dLip(ˆµN, µP ) > δ} has (for N big enough) probability smaller than +e−Ncδ where cδ > 0. Hence, +lim +N→∞ EV,P +N +� +et +√ +NΛN + t2 +2 F (ˆµN)� += lim +N→∞ EV,P +N +� +1{dLip(ˆµN ,µP )⩽δ}et +√ +NΛN + t2 +2 F (ˆµN)� +. +By continuity of F there is some ε(δ) which goes to 0 as δ → 0 such that, for dLip(ν, µP ) ⩽ δ, we have +|F(ν) − F(µP )| ⩽ ε(δ). Taking the (decreasing) limit as δ goes to zero we deduce +lim +N→∞ EV,P +N +� +et +√ +NΛN + t2 +2 F (ˆµN )� += lim +δ→0 lim +N→∞ EV,P +N +� +1{dLip(ˆµN ,µP )⩽δ}et +√ +NΛN � +e +t2 +2 F (µP ). +But the same large deviations argument shows that +lim +δ→0 lim +N→∞ EV,P +N +� +1{dLip(ˆµN ,µP )⩽δ}et +√ +NΛN � += lim +N→∞ EV,P +N +� +et +√ +NΛN � +. +Thus, we have shown that +lim +N→∞ EV,P +N +� +et +√ +N� +2P +˜ +R2 +φ(x)−φ(y) +x−y +dµP (x)d(ˆµN−µP )(y)+ +´ +R(φ′−V ′φ)d(ˆµN−µP )+P ζN (φ)�� += e− t2 +2 F (µP ) , +(45) +Which establishes that +√ +NΛN = +√ +N +� +νN(Ξφ) + PζN(φ) +� +converges in law towards a centered Gaussian +random variable with announced variance. We finally get rid of the remaining term ζN(φ), using Corollary +3.4: taking ε = 1/4 for example, we see in particular that +√ +NζN(φ) converges in probability towards +zero. The conclusion follows from Slutsky’s lemma. +We now extend the result of Proposition 5.1 to a more general set of functions. With the notations +of Proposition 5.1, we have +Theorem 5.2. Let φ ∈ H2(R) ∩ C2(R) such that φ′′ is bounded. Additionally, suppose that V (3)φ2, +V ′′φφ′, V ′′φ2 and V ′φ are bounded. Then, recalling (40) we have the convergence in distribution as N +goes to infinity +√ +NνN(Ξφ) → N(0, qP (φ)) . +Proof. For N ⩾ 1, let E− +N, E+ +N be given by Corollary 4.2. Let χN : R → [0, 1] be C2 with compact support +such that +χN(x) = 1 for x ∈ [E− +N − 1, E+ +N + 1] and χN(x) = 0 for x ∈ [E− +N − 2, E+ +N + 2]c +and such that, denoting φN = φχN, supN ∥φ′ +N∥∞ + ∥φ′ +N∥L2(R), supN ∥φ′′ +N∥∞ + ∥φ′′ +N∥L2(R) < +∞ (we +assumed φ ∈ H2(R), in particular φ′ is bounded and such a χN exists). The point of cutting φ outside +the set [E− +N −1, E+ +N +1] is that with high probability, the empirical measure ˆµN doesn’t see the difference +between φ and φN. +The support of φN is then contained in [E− +N −2, E+ +N+2], and we now argue that the proof of Proposition +5.1 can be adapted so that +√ +NνN(ΞφN) → N(0, qP (φ)) . +(46) +Similarly as in Proposition 5.1, we perform in ZV,P +N +the change of variables xi = yi + +t +√ +N φN(yi), +1 ⩽ i ⩽ N, which is the same as before, but with φ replaced by φN. First, with IN := [E− +N − 2, E+ +N + 2], +the error term +KN(t, φN) ⩽ 2 +t3 +3N 1/2 ∥φ′ +N∥3 +∞ + +t3 +6N 1/2 ∥φN∥∞ +sup +d(x,IN)⩽1 +|V (3)(x)| +19 + +of the proof of Proposition 5.1 is still going to zero, because of our choice of χN and Assumption 1.2. As +previously, we then have +lim +N→∞ EV,P +N +� +et +√ +NΛN (φN)+ t2 +2 FN(ˆµN )� += 1 +(47) +with +ΛN(φN) := 2P +¨ +R2 +φN(x) − φN(y) +x − y +dµP (x)d(ˆµN − µP )(y) + +ˆ +R +(φ′ +N − V ′φN)d(ˆµN − µP ) + PζN(φN) , +where ζN is given by (34), and +FN(ˆµN) = −P +¨ +R2 +�φN(x) − φN(y) +x − y +�2 +dˆµN(x)dˆµN(y) − +ˆ +R +(V ′′φ2 +N + φ′2 +N)dˆµN . +Taking again the distance dLip defined in (44), one can check that for µ, ν probability measures over R, +|FN(µ) − FN(ν)| ⩽ CNdLip(µ, ν) , +where CN is a term depending on the norms ∥φ′ +N∥∞, ∥φ′′ +N∥∞, ∥V ′′φ2 +N∥∞ and ∥(V ′′φ2 +N)′∥∞. The choice +of χN and the fact that φ is chosen so that V (3)φ2 and V ′′φφ′ are bounded guarantee that ∥(V ′′φ2 +N)′∥∞ +is bounded in N. The other norms are easily bounded by hypothesis. Therefore CN can be seen to be +uniformly bounded in N, and we find some C ⩾ 0 independent of N such that +|FN(µ) − FN(ν)| ⩽ CdLip(µ, ν) . +As in proposition 5.1, we use the large deviation principle for (ˆµN) to deduce +lim +N→+∞ EV,P +N +� +et +√ +NΛN (φN)+ t2 +2 FN(ˆµN )� += +lim +N→+∞ EV,P +N +� +et +√ +NΛN (φN)� +e +t2 +2 FN (µP ) . +By dominated convergence, FN(µP ) converges to F(µP ), the function F being given by (43). This shows +the convergence as N goes to infinity +lim +N→+∞ EV,P +N +� +et +√ +NΛN (φN)� += e− t2 +2 F (µP ) , +and +√ +N +� +νN(ΞφN)+PζN(φN) +� +converges towards a centered Gaussian variable with variance −F(µP ) = +qP (φ). Because supN ∥φ′ +N∥L2(dx) + ∥φ′′ +N∥L2(dx) is finite, we can apply again Corollary 3.4 to deduce the +convergence in law (46). +We now have the ingredients to conclude, by showing that the characteristic function +EV,P +N +� +eit +√ +NνN (Ξφ)� += EV,P +N +� +eit +√ +N +´ +ΞφdˆµN� +e−it +√ +N +´ +ΞφdµP +converges to the characteristic of a Gaussian variable with appropriate variance. By Corollary 4.2, the +probability under PV,P +N +of the event EN = +� +x1, . . . , xN ∈ [E− +N − 1, E+ +N + 1] +� +converges to 1. Along with +the convergence (46), we deduce +e− t2 +2 qP (φ) = lim +N EV,P +N +� +eit +√ +N +´ +ΞφNdˆµN � +e−it +√ +N +´ +ΞφNdµP = lim +N EV,P +N +� +1ENeit +√ +N +´ +ΞφNdˆµN� +e−it +√ +N +´ +ΞφNdµP , +Where we used +���EV,P +N +� +1Ec +Neit +√ +N +´ +ΞφNdˆµN � +e−it +√ +N +´ +ΞφNdµP +��� ⩽ PV,P +N (Ec +N) −−−−−→ +N→+∞ 0 . +20 + +Using that φN = φ on JN = [E− +N − 1, E+ +N + 1], +ˆ +ΞφNdµP = 2P +¨ φN(x) − φN(y) +x − y +dµP (x)dµP (y) + +ˆ +(φ′ +N − V ′φN)dµP += 2P +¨ +J2 +N +φ(x) − φ(y) +x − y +dµP (x)dµP (y) + 2P +¨ +(J2 +N)c +φN(x) − φN(y) +x − y +dµP (x)dµP (y) ++ +ˆ +JN +(φ′ − V ′φ)dµP + +ˆ +Jc +N +(φχ′ +N + φ′χN − V ′φχN)dµP . +By boundedness of (∥φ′ +N∥∞)N, the second term is bounded by +CP +¨ +(J2 +N)c dµP dµP ⩽ 2CP µP (Jc +N) = o(N −1/2) , +where we used the union bound and Lemma 4.4. By the same estimate and the fact that χN can be +chosen so that (∥χ′ +N∥∞)N is bounded, and because φ′, V ′φ are bounded, the last term is also o(N −1/2). +By the previous arguments, we also conclude that +2P +¨ +(J2 +N)c +φ(x) − φ(y) +x − y +dµP (x)dµP (y) + +ˆ +Jc +N +(φ′ − V ′φ)dµP = o(N −1/2) , +thus +ˆ +ΞφNdµP = +ˆ +ΞφdµP + o(N −1/2) , +and so far we have +e− t2 +2 qP (φ) = lim +N EV,P +N +� +1ENeit +√ +N +´ +ΞφNdˆµN� +e−it +√ +N +´ +ΞφdµP . +Finally, on EN, using φN = φ and that ˆµN is supported in JN, +ˆ +ΞφNdˆµN = 2P +¨ +J2 +N +φ(x) − φ(y) +x − y +dµP (x)dˆµN(y) + 2P +¨ +(J2 +N)c +φN(x) − φN(y) +x − y +dµP (x)dˆµN(y) + +ˆ +JN +(φ′ − V ′φ)dˆµN += 2P +¨ φ(x) − φ(y) +x − y +dµP (x)dˆµN(y) + +ˆ +(φ′ − V ′φ)dˆµN + o(N −1/2) , +Where in the second line we used, using Lemma 4.4 again, that +¨ +(J2 +N)c +φN(x) − φN(y) +x − y +dµP (x)dˆµN(y) = +¨ +JN×Jc +N +φN(x) − φN(y) +x − y +dµP (x)dˆµN(y) = o(N −1/2) , +and the same estimate holds for φN replaced by φ. Therefore, +e− t2 +2 qP (φ) = lim +N EV,P +N +� +1ENeit +√ +N +´ +ΞφdˆµN� +e−it +√ +N +´ +ΞφdµP . +This establishes that +lim +N EV,P +N +� +eit +√ +N +´ +ΞφdˆνN� += e− t2 +2 qP (φ) , +which concludes the proof. +Remark 5.3. Taking φ such that φ′ satisfies the conditions of Theorem 5.2, we then have +(48) +EV,P +N +� +et +√ +NνN (Lφ)� +−→ +N→∞ exp +�t2 +2 qP (φ′) +� +, +21 + +where the operator L is defined as Lφ := Ξφ′, ie +Lφ = 2P +ˆ +R +φ′(x) − φ′(y) +x − y +dµP (y) + φ′′(x) − V ′(x)φ′(x) . +(49) +Note that qV +P (φ′) = +� +σV +P +�2(Lφ) where σV +P is defined in (11). By Theorem 7.1, the class of functions in +L−1(T ) where +T := +� +f ∈ C1(R), f = O(1/x), f ′ = O(1/x2), +ˆ +R +fρP = 0 +� +satisfies (48). This proves Theorem 1.3. +6 +Inversion of L +This section is dedicated to the definition of L given by (8) and its domain and then we focus on its +inversion. We rely heavily on results of Appendix A: the diagonalization of the operator A by the use of +the theory of Schrödinger operators. +Let P > 0 be fixed. We introduce the operators A and W, acting on sufficiently smooth functions of +L2(ρP ), by +Aφ = − +� +φ′ρP +�′ +ρP += − +� +φ′′ + ρ′ +P +ρP +φ′ +� +and +Wφ = −H +� +φ′ρP +� +. +(50) +We first show the following decomposition of L. +Lemma 6.1. For φ twice differentiable we have the following pointwise identity +−Lφ = Aφ + 2PWφ . +(51) +Proof. We write for x ∈ R +2P +ˆ +R +φ′(x) − φ′(y) +x − y +ρP (y)dy = −2Pφ′(x)H[ρP ](x) + 2PH[φ′ρP ](x) . +(52) +Then, +Lφ = φ′′ − V ′φ′ − 2Pφ′H[ρP ] + 2PH +� +φ′ρP +� +. +By (18) we have −V ′ − 2PH[ρP ] = ρ′ +P +ρP , which concludes the proof. +In order to state the next theorem, whose proof we detail in the Appendix, we introduce the following +Sobolev-type spaces. Let +H2 +V ′(R) := +� +u ∈ H2(R), uV ′ ∈ L2(R) +� +. +We now define +H2 +V ′(ρP ) := ρ−1/2 +P +H2 +V ′(R) +and its homogeneous counterpart +H2 +V ′,0(ρP ) := +� +u ∈ H2 +V ′(ρP ), +ˆ +R +uρP dx = 0 +� +. +Finally, we let L2 +0(ρP ) be the subset of L2(ρP ) of zero mean functions with respect to ρP . +We detail the proof of the following theorem in Appendix A which is based on Schrödinger operators +theory. +22 + +Theorem 6.2 (Diagonalization of A in L2 +0(ρP )). There exists a sequence 0 < λ1 < λ2 < . . . going to +infinity, and a complete orthonormal set (φn)n⩾1 of L2 +0(ρP ) of associated eigenfunctions for A, meaning +that +• span{φn, n ⩾ 1} is dense in L2 +0(ρP ), +• For all i, j, ⟨φi, φj⟩L2(ρP ) = δi,j, +• For all n ⩾ 1, Aφn = λnφn. +Furthermore, each φn is in H2 +V ′,0(ρP ), A : H2 +V ′,0(ρP ) → L2 +0(ρP ) is bijective, and we have the writing, for +u ∈ L2 +0(ρP ) +A−1u = +� +n⩾1 +λ−1 +n ⟨u, φn⟩L2(ρP ) φn . +We see the operators A and W as unbounded operators on the space +H = +� +u ∈ H1(ρP ) | +ˆ +R +uρP dx = 0 +� +endowed with the inner product ⟨u, v⟩H = ⟨u′, v′⟩L2(ρP ). This defines an inner product on H and makes +it a complete space: it can be seen that H1(ρP ) is the completion of C∞ +c (R) with respect to the inner +product ⟨u, v⟩L2(ρP ) +⟨u′, v′⟩L2(ρP ). The space H is then the kernel of the bounded (with respect to ∥·∥H) +linear form, ⟨�1, ·⟩L2(ρP ) on H1(ρP ), and both inner products are equivalent on H because of the Poincaré +inequality, Proposition 2.4. The use of H is motivated by the fact that both A and W are self-adjoint +positive on this space as we show in Lemma 6.4. +In the next proposition, we deduce from Theorem 6.2 the diagonalization of A in H. +Proposition 6.3 (Diagonalization of A in H). With the same eigenvalues 0 < λ1 < λ2 < . . . as in +Theorem 6.2, there exists a complete orthonormal set (ψn)n⩾1 of H formed by eigenfunctions of A. +Proof. With (φn)n⩾1 of Theorem 6.2, +δi,j = ⟨φi, φj⟩L2(ρP ) = 1 +λj +⟨φi, Aφj⟩L2(ρP ) += 1 +λj +⟨φ′ +i, φ′ +j⟩L2(ρP ) += 1 +λj +⟨φi, φj⟩H. +With ψn = +1 +√λn φn, (ψn)n⩾1 is then orthonormal with respect to the inner product of H. To show that +span{ψn, n ⩾ 1} is dense in H, let u ∈ H be such that for all j ⩾ 1, ⟨u, φj⟩H = 0. In the last series of +equalities, replace φi by u: we see that u is orthogonal to each φj in L2(ρP ), thus u is a constant as +shown in the proof of Lemma A.10, and because u ∈ H it has zero mean against ρP . This shows that +u = 0. +We set for what follows D(A) = +� +u ∈ H2 +V ′,0(ρP ) | Au ∈ H +� +and D(W) = {u ∈ H | Wu ∈ H}. +Lemma 6.4. The following properties hold: +• The operator W : D(W) → H is positive: for all φ ∈ D(W), +⟨Wφ, φ⟩H = 1 +2∥φ′ρP ∥2 +1/2 ⩾ 0 , +with equality only for φ = 0, where the 1/2-norm of u is given by +∥u∥2 +1/2 = +ˆ +R +|x|. |F[u](x)|2 dx . +23 + +• Both A and W are self-adjoint for the inner product of H. +Proof. To prove the first point, let φ ∈ D(W). Then, +2π ⟨Wφ, φ⟩H = −2π ⟨H[φ′ρP ]′, φ′ρP ⟩L2(dx) = − +� +ixF +� +H[φ′ρP ] +� +, F[φ′ρP ] +� +L2(dx) += π ⟨ | x | F[φ′ρP ], F[φ′ρP ]⟩L2(dx) = π∥φ′ρP ∥2 +1/2 ⩾ 0 , +and because φ is in H, this last quantity is zero if and only if φ vanishes. +For the second point, let u, v ∈ D(W). Using Plancherel’s isometry and i) of Lemma 2.1, +⟨Wu, v⟩H = ⟨(Wu)′, v′ρP ⟩L2(dx) = 1 +2 ⟨ | x | F[u′ρP ], F[v′ρP ]⟩L2(dx) , +and this last expression is symmetric in (u, v). +The proof of the self-adjointness of A follows from +integration by partss. +Definition 6.5 (Quadratic form associated to −L). We define for all u, v ∈ H ∩ C∞ +c (R) the quadratic +form associated to −L by +q−L(u, v) = ⟨Au, Av⟩L2(ρP ) + 2P ⟨F[u′ρP ], F[v′ρP ]⟩L2(|x|dx) +Note that for all u, v ∈ H ∩ C∞ +c (R), q−L(u, v) = ⟨−Lu, v⟩H and that whenever u ∈ D(A) ∩ D(W), +q−L(u, u) = ⟨Au, u⟩H + 2P ⟨Wu, u⟩H ⩾ λ1(A)∥u∥2 +H +(53) +by Proposition 6.3 and Lemma 6.4. After extending the q−L to its form domain Q(L) which is equal +to +� +u ∈ H, Au ∈ L2(ρP ), F[u′ρP ] ∈ L2(|x|dx) +� += H2 +V ′,0(ρP ). The equality comes from the fact that +A−1� +L2 +0(ρP ) +� += H2 +V ′,0(ρP ), that H ⊂ H2 +V ′,0(ρP ) and that F[u′ρP ] ∈ L2(x2dx) whenever u ∈ H2 +V ′,0(ρP ), +indeed u′ρP ∈ H1(R) because (u′ρP )′ = −ρP Au ∈ L2(R). We now define D(L) the domain of definition +of −L by: +D(L) := +� +u ∈ Q(L), v �→ q−L(u, v) can be extended to a continuous linear form on H +� +Proposition 6.6. D(L) = D(A) ∩ D(W). +Proof. Let u ∈ D(L), by Riesz’s theorem there exists fu ∈ H, such that q−L(u, v) = ⟨fu, v⟩H for all v ∈ H, +we set −Lu := fu, it is called the Friedrichs extension of −L. Then for all v ∈ H ∩ C∞ +c (R), by integration +by part we get: +⟨−Lu, v⟩H = q−L(u, v) = ⟨u, Av⟩H + 2P ⟨u, Wv⟩H , +hence we deduce the distributionnal identity −Lu = Au + 2PWu. Since u ∈ H2 +V ′,0(ρP ), Wu ∈ H1(ρP ) +implying that Au ∈ H and then that Wu ∈ H. +We are now ready to state the main theorem of this section, that is the inversion of L on D(L). +Theorem 6.7 (Inversion of L). −L : D(L) −→ H is bijective. Furthermore, (−L)−1 is continuous from +(H, ∥.∥H) to (D(L), q−L). +Proof. Let f ∈ H, since ⟨f, .⟩H is a linear form on Q(L) = H2 +V ′,0(ρP ) which is, by (53), continuous with +respect to q−L, one can applies Riesz’s theorem so there exists a unique uf ∈ H2 +V ′,0(R), such that for all +v ∈ H, ⟨f, v⟩H = q−L(uf, v). Since, uf is clearly in D(L) by definition of the Friedrichs extension of −L, +we have −Lu = f. +Remark 6.8. We can diagonalize L by the same argument we used in Appendix A to diagonalize A in +L2 +0(ρP ). +24 + +We now state a result that could allow one to extract more regularity for L−1 by the help of an +explicit form that uses Fredholm determinant theory for Hilbert-Schmidt operators, the reader can refer +to [GGK12]. +Definition 6.9 (Fredholm determinant). Let U be a self-adjoint Hilbert-Schmidt operator, we denote the +Fredholm determinant by det(I + U). +Theorem 6.10 (Determinant formula for L−1). For all u ∈ H, such that x �→ +1 +ρP (x) +ˆ +∞ +x +u(t)ρP (t)dt +is integrable at +∞, we have: +L−1u = A−1u − ρ−1/2 +P +R +� +ρ1/2 +P A−1u +� +(54) +where R is the kernel operator defined for all v ∈ L2(R) by: +R[v](x) = +ˆ +R +R(x, y)v(y)dy +where +R(x, y) = +1 +det(I + K) +� +n⩾0 +1 +n! +ˆ +Rn det +n+1 +� K(x, y) +K(x, λb) +K(λa, y) +K(λa, λb) +� +a,b=1...n +dλ1 . . . dλn +where K is the kernel operator defined for all w ∈ L2(ρP ) by: +K[v](x) = +ˆ +R +K(x, y)w(y)dy +(55) +with +K(x, y) = −2PρP(x)ρP (y) ln +���1 − y +x +���. +(56) +Proof. Let f ∈ H, there exists a unique u ∈ D(A) such that Au = f. Since (u′ρP )′ = ρP Au ∈ L2(R), +hence u′ρP ∈ H1(R) so u′(x)ρP (x) +−→ +|x|→+∞ 0 −(u′ρP )′ +ρP += f hence +(A−1f)′(x)ρP (x) = u′(x)ρP (x) = +ˆ +∞ +x +f(t)ρP (t)dt. +(57) +Using the fact that +´ +R u(x)ρP (x)dx = 0, integrating again we get: +u(x) = − +ˆ +∞ +x +ds +ρP (s) +ˆ +∞ +s +f(t)ρP (t)dt + C +where C = +ˆ +R +ρP (x)dx +ˆ +∞ +x +ds +ρP (s) +ˆ +∞ +s +f(t)ρP (t)dt. Now let g ∈ H, there exists a unique v ∈ D(L), +such that −Lv = Av + 2PWv = g and then v + 2PWA−1v = A−1g. using (57), we get: +WA−1v(x) = + +R +ds +s − x +ˆ +∞ +s +dtv(t)ρP (t) +By Sokhotski-Plejmel formula, we have: + +R +ds +s − x +ˆ +∞ +s +dtv(t)ρP (t) = +lim +M→+∞ lim +ε→0 +ˆ M +−M +ds +2 +� +1 +x − s + iε + +1 +x − s − iε +� ˆ +∞ +s +dtv(t)ρP (t) +25 + +We then proceed to an integration by part: + +R +ds +s − x +ˆ +∞ +s +dtv(t)ρP (t) = +lim +M→+∞ lim +ε→0 +� +− ln +� +(x − s)2 + ε2� +2 +ˆ +∞ +s +dtv(t)ρP (t) +�M +−M +− +ˆ +R +ds ln |x − s|v(s)ρP (s) +To conclude that WA−1v(x) = − +´ +R ds ln |x − s|v(s)ρP (s), we just need to show that +ln(x) +ˆ +∞ +x +dtv(t)ρP (t) −→ +|x|→∞ 0 +which can be seen by Cauchy-Schwarz inequality: +��� ln(x) +ˆ +∞ +x +dtv(t)ρP (t) +��� ⩽ | ln(x)|∥v∥L2(ρP ).ρP (x)1/4� ˆ +R +ρP (t)1/2dt +�1/2 +. +In this inequality, we used that ρP is decreasing in a neighborhood of +∞, hence +ln |x| +ˆ +∞ +s +dtv(t)ρP (t) +−→ +x→+∞ 0 +the exact same argument allows us to conclude when x goes to −∞. Using the fact that +´ +R v(t)ρP (t)dt = 0, +we obtain the following equality: +v − 2P +ˆ +R +ds ln |x − s|v(s)ρP (s) = A−1g := h. +Now setting ˜v(t) = ρ1/2 +P (t)v(t) and ˜h = ρ1/2 +P (t)h(t), we obtain ˜v + K[˜v] = ˜h where K is defined in +(55). Since its kernel (defined in (56)) belongs to L2(R2), K is Hilbert-Schmidt. Hence by Fredholm +determinant theory: +˜v = ˜h − R[˜h] +or L−1g = A−1g − ρ−1/2 +P +R +� +ρ1/2 +P A−1g +� +as expected. +7 +Regularity of the inverse of L and completion of the proof of +Theorem 1.3 +Since we have proven the central limit theorem for functions of the type Lφ with φ regular enough and +satisfying vanishing asymptotic conditions at infinity, we exhibit a class of functions f for which L−1f is +regular enough to satisfy conditions of Theorem 5.2. We define T the subset of H defined by +T := +� +f ∈ C1(R), f(x) = +O +|x|→∞(x−1), f ′(x) = +O +|x|→∞(x−2), +ˆ +R +fρP = 0 +� +Theorem 7.1. For all f ∈ T , there exists a unique u ∈ C3(R) such that u′ ∈ H2(R) with u(3) bounded +wich verifies: +• u′(x) = +O +|x|→∞ +� +1 +xV ′(x) +� +• u′′(x) = +O +|x|→∞ +� +1 +xV ′(x) +� +26 + +• u(3)(x) = +O +|x|→∞ +� 1 +x +� +such that f = Lu. +Proof. Let f ∈ T ⊂ H, then since −L is bijective from D(L) → H, there exists a unique u ∈ D(L) such +that −Lu = f ie: +−u′′ − ρ′ +P +ρP +u′ − 2PH[u′ρP ] = f +(58) +Hence we have +−(u′ρP )′ = ρP +� +f + 2PH[u′ρP ] +� +. +(59) +Since u ∈ D(L) ⊂ {u ∈ H2 +V ′,0(R), Au ∈ H}, the functions u′ρP and its derivatives (u′ρP )′ = −ρP Au and +(u′ρP )′′ = −ρ′ +P +ρP +ρ1/2 +P . +� +ρ1/2 +P Au +� +− ρP +� +Au +�′ +are in L2(dx). In particular u′ρP goes to zero at infinity, and +H[u′ρP ] ∈ H2(R) ⊂ C1(R). So we can integrate (59) on [x, +∞[ , since by Lemma 2.3, the right-hand +side behaves like a +O +|x|→∞ +�ρP (x) +x +� +, to get the following expression +u′(x)ρP (x) = +ˆ +∞ +x +ρP (t) +ρ′ +P (t)(f + 2PH[u′ρP ]).ρ′ +P (t)dt +(60) +From this expression, we can see that u′ ∈ C2(R) so we just have to check the integrability condition at +infinity and the fact that u(3) is bounded. After proceeding to an integration by parts, which is permitted +by the previous argument, we obtain: +u′(x) = −ρP (x) +ρ′ +P (x) +� +f(x) + 2PH[u′ρP ](x) +� +− +1 +ρP (x) +ˆ +∞ +x +� +ρP (t) +ρ′ +P (t)(f + 2PH[u′ρP ]) +�′ +ρP (t)dt +(61) +and we define R1(x) := +1 +ρP (x) +ˆ +∞ +x +� +ρP (t) +ρ′ +P (t)(f + 2PH[u′ρP ]) +�′ +ρP (t)dt, we will need to show that R1 is +a remainder of order O +� +1 +xV ′(x)2 +� +at infinity. In this case we will have u′(x) = O +� +1 +xV ′(x) +� +which will +be useful for the following. If we reinject (61) in (58), we find: +u′′ = −(f + 2PH[u′ρP ]) − ρ′ +P +ρP +� +− ρP +ρ′ +P +� +f + 2PH[u′ρP ] +� +− R1 +� += ρ′ +P +ρP +R1 +(62) +Hence +u′′(x) = ρ′ +P +ρ2 +P +(x) +ˆ +∞ +x +ρP (t)dt +� +�ρP +ρ′ +P +�′ +(t) +� +�� +� += +O +t→+∞ +� +V ′′(t) +V ′(t)2 +� +� +f + 2PH[u′ρP ] +� +(t) +� +�� +� += +O +t→+∞ +� +1 +t +� ++ +ρP +ρ′ +P +(t) +� �� � += +O +t→+∞ +� +1 +V ′(t) +� +� +f ′ − 2PH[ρP Au] +� +(t) +� +�� +� += +O +t→+∞ +� +1 +t2 +� +� +. +The fact that H[ρP Au](t) = +O +t→+∞(t−2) comes again from lemma 2.3. Finally we have that, +u(3)(x) = +�ρ′ +P +ρ2 +P +�′ +(x)ρP (x)R1(x) − +�ρ′ +P +ρ2 +P +� +(x) +� +ρP +ρ′ +P +(f + 2PH[u′ρP ]) +�′ +(x)ρP (x) += +� ρ′′ +P +ρP +− 2ρ′2 +P +ρ2 +P +� +(x) +� +�� +� += +O +x→+∞ +� +V ′(x)2 +� +R1(x) − +�ρ′ +P +ρP +� +(x) +� +ρP +ρ′ +P +(f + 2PH[u′ρP ]) +�′ +(x) +� +�� +� += +O +x→+∞ +� +V ′′(x) +xV ′(x) +x−2 +� +27 + +The second term is +O +x→+∞ +�1 +x +� +by the assumption that V ′′ +V ′ (x) = +O +|x|→∞(1). Hence, we just have to check +that R1(x) = +O +x→+∞ +� +1 +xV ′(x)2 +� +to establish that u′, u′′, u(3) are in L2(R). By a comparison argument, +we control R1 by controlling +I1(x) := +ˆ +∞ +x +ρP (t) +tV ′(t)dt +By integration by parts: +I1(x) := − +ρP (x) +xV ′(x) +ρP +ρ′ +P +(x) +� +�� +� += +O +x→+∞ +� +ρP (x) +xV ′(x)2 +� +− +ˆ +∞ +x +ρP (t) +� 1 +tV ′ +ρP +ρ′ +P +�′ +(t)dt += +O +x→+∞ +� ρP (x) +xV ′(x)2 +� +− +ˆ +∞ +x +ρP (t)dt +� +− +1 +t2V ′(t) +ρP +ρ′ +P +(t) − V ′′(t) +tV ′(t)2 +ρP +ρ′ +P ++ +1 +tV ′(t) +�ρP +ρ′ +P +�′ +(t) +� +(63) +By the same argument as before, the last integral is of the form +ˆ +∞ +x +O +t→+∞ +� ρP (t) +tV ′(t)2 +� +dt so if +I2(x) := +ˆ +∞ +x +ρP (t) +tV ′(t)2 dt = +O +x→+∞ +� ρP (x) +xV ′(x)2 +� +then so is I1. By integration by parts, we obtain: +I2(x) = ρP (x) +1 +xV ′(x)2 +ρP +ρ′ +P +(x) − +ˆ +∞ +x +ρP (t)dt +��ρP +ρ′ +P +�′ +(t) +1 +tV ′(t)2 − ρP +ρ′ +P +(t) +� +1 +t2V ′(t)2 + 2V ′′(t) +tV ′(t)3 +�� += +O +x→+∞ +� ρP (x) +xV ′(x)2 +� ++ +ˆ +∞ +x +O +t→+∞ +� ρP (t) +tV ′(t)3 +� +dt +The last integral is a +O +x→+∞ +� ρP (x) +xV ′(x)2 +� +because, again, by integration by part: +ˆ +∞ +x +ρP (t) +tV ′(t)3 dt = ρP (x) +1 +xV ′(x)3 +ρP +ρ′ +P +(x) − +ˆ +∞ +x +O +t→+∞ +� ρP (t) +tV ′(t)4 +� +and finally +ˆ +∞ +x +ρP (t) +tV ′(t)4 dt ⩽ +ρP (x) +xV ′(x)2 +ˆ +∞ +x +dt +V ′(t)2 = +O +x→+∞ +� ρP (x) +xV ′(x)2 +� +In the final step, we used the fact that x �→ +ρP (x) +xV ′(x) is decreasing in a neighborhood of +∞ (which +can be checked by differentiating) and that x �→ +1 +V ′(x)2 is integrable at ∞ by assumption iii). Hence +R1(x) = +O +x→+∞ +� +1 +xV ′(x)2 +� +(the exact same result can be shown at −∞), which leads to the fact +u′(x) = +O +|x|→+∞ +� +1 +xV ′(x) +� +, +u′′(x) = +O +|x|→+∞ +� +1 +xV ′(x) +� +and +u(3)(x) = +O +|x|→+∞ +�1 +x +� +(64) +which establishes that these functions are in L2 in a neighborhood of ∞. Since we already showed that +u ∈ C3(R) ⊂ H3 +loc(R), it establishes that u ∈ H3(R) ∩ C3(R) with u(3) bounded. To complete the proof +we just have to show that (u′)2V (3), u′u′′V ′′, (u′)2V ′′ and u′V ′ are bounded which is easily checked by +(64) and Assumption 1.1 iv). +28 + +A +Appendix: proof of Theorem 6.2 +In order to analyze A, we let, for u ∈ L2(dx), +Su := ρ1/2 +P Aρ−1/2 +P +u . +Note that u ∈ +� +L2(dx), ∥.∥L2(dx) +� +�→ ρ−1/2 +P +u ∈ +� +L2(ρP ), ∥.∥L2(ρP ) +� +is an isometry. It turns out that it will +be easier to study first the operator S in order to get the spectal properties of A. +Proposition A.1. The operator S is a Schrödinger operator: it admits the following expression for all +u ∈ C2 +c (R): Su = −u′′ + wP u with +wP = 1 +2 +�1 +2V ′2 − V ′′ + 2PV ′H[ρP ] − 2PH[ρ′ +P ] + 2P 2H[ρP ]2 +� += 1 +2 +� +(ln ρP )′′ + 1 +2(ln ρP )′2� +. +Furthermore, wP is continuous and we have wP (x) ∼ +∞ +V ′(x)2 +4 +−→ +|x|→∞ +∞. +Proof. We compute directly +� +ρP +� +ρ−1/2 +P +u +�′�′ +ρP += +� +ρ−1/2 +P +u +�′′ + ρ′ +P +ρP +� +ρ−1/2 +P +u +�′ += +� +ρ−1/2 +P +u′ − 1 +2ρ−3/2 +P +ρ′ +P u +�′ + ρ′ +P ρ−3/2 +P +u′ − 1 +2ρ−5/2 +P +� +ρ′ +P +�2u += ρ−1/2 +P +u′′ + 1 +4ρ−5/2 +P +� +ρ′ +P +�2u − 1 +2ρ−3/2 +P +ρ′′ +P u += ρ−1/2 +P +� +u′′ + 1 +4ρ−2 +P +� +ρ′ +P +�2u − 1 +2ρ−1 +P ρ′′ +P u +� += ρ−1/2 +P +� +u′′ − 1 +2 +��ρ′′ +P +ρP +� +− 1 +2 +�ρ′ +P +ρP +�2� +u +� += ρ−1/2 +P +� +u′′ − 1 +2 +� +(ln ρP )′′ + 1 +2(ln ρP )′2� +u +� += ρ−1/2 +P +� +u′′ − wP u +� +. +Now, using Lemma 2.2, we have +wP = 1 +2 +�1 +2V ′2 − V ′′ + 2PV ′H[ρP ] − 2PH[ρ′ +P ] + 2P 2H[ρP ]2 +� +. +Notice that H[ρ′ +P ] and H[ρP ] are bounded since they belong to H1(R), as we showed in Lemma 2.2 that +ρP is H2(R). Along with Assumption 1.1iii) and Lemma 2.3, we deduce wP (x) ∼ +∞ +1 +4V ′2(x). +Remark A.2. Note that the function wP need not be positive on R. In fact, neglecting the terms involving +the Hilbert transforms of ρP and ρ′ +P , wP would only be positive outside of a compact set. However, using +the positivity of A, which will be shown further in the article, we can show that the operator −u′′ + wP u +is itself positive on L2(dx). It can also be checked that, by integration by partss, S is symmetric on C2 +c(R) +with the inner product of L2(dx). +We now introduce an extension of S by defining its associated bilinear form. +Definition A.3 (Quadratic form associated to S). +Let α ⩾ 0 such that wP + α ⩾ 1. We define the quadratic form associated to S + αI, defined for all +u ∈ C2 +c(R) by +qα(u, u) := +ˆ +R +u′2dx + +ˆ +R +u2(wP + α)dx +29 + +This quadratic form can be extended to a larger domain denoted by Q(S+αI), called the form domain +of the operator S + αI. By the theory of Schrödinger operators, it is well-known (see [Dav96][Theorem +8.2.1]) that such a domain is given by +Q(S + αI) = +� +u ∈ H1(R), u(wP + α)1/2 ∈ L2(R) +� += +� +u ∈ H1(R), uV ′ ∈ L2(R) +� +=: H1 +V ′(R) . +The space H1 +V ′(R) can be seen to be the completion under the norm qα of C∞ +c . Now that the quadratic +form associated to S + αI has been extended to its form domain, it is possible to go back to the operator +and extend it by its Friedrichs extension. +Theorem A.4 (Friedrichs extension of S + αI). +There exists an extension (S + αI)F of the operator S + αI, called the Friedrichs extension of S + αI +defined on H2 +V ′(R) := +� +u ∈ H2(R), uV ′ ∈ L2(R) +� +. +Proof. We denote +D +� +(S + αI)F +� += +� +v ∈ H1 +V ′(R), u ∈ H1 +V ′(R) �−→ qα(u, v) can be extended to a continuous linear form on L2(R) +� +If v ∈ D +� +(S + αI)F +� +, by Riesz’s theorem there exists a unique fv ∈ L2(R) such that qα(u, v) = +⟨u, fv⟩L2(dx) holds for all u ∈ L2(R) and we can set (S + αI)F v := fv. Note that it is indeed a way +of extending S + αI since for all u, v ∈ C2 +c(R), qα(u, v) = ⟨u, (S + αI)v⟩L2(dx). +We want to show that D +� +(S+αI)F +� += H2 +V ′(R). Let f ∈ D +� +(S+αI)F +� +and g := (S+αI)F f ∈ L2(R). +By definition of qα, for all u ∈ C2 +c(R): +ˆ +R +gudx = +ˆ +R +f ′u′dx + +ˆ +R +(wP + α)fudx = − +ˆ +R +fu′′dx + +ˆ +R +(wP + α)fudx +Therefore in the sense of distributions, we get −f ′′ = g − (wP + α)f which is a function in L2(R), hence +f ∈ H2 +V ′(R). Conversely, if f ∈ H2 +V ′(R), it is possible to extend u �→ qα(f, u) to a continuous linear form +on L2(R) by +u �→ − +ˆ +R +uf ′′dx + +ˆ +R +uf(wP + α)dx +which concludes the fact that D +� +(S + αI)F +� += H2 +V ′(R). +In the following, we will deal only with (S + αI)F : H2 +V ′(R) −→ L2(R) and denote it (S + αI). +Remark A.5. Note that in the previous proof, the application of Riesz’s theorem doesn’t allow to say that +(S + αI) : v ∈ +� +H2 +V ′(R), ∥.∥qα +� +�→ fv ∈ +� +L2(R), ∥.∥L2(dx) +� +, where ∥.∥qα stands for the norm associated +to the bilinear positive definite form qα, is continuous. It can be seen by the fact that +v ∈ +� +D(S + αI), ∥.∥qα +� +�→ q(., v) ∈ +� +L2(R)′, ∥.∥L2(dx)′ +� +, where L2(R)′ stands for the topological dual of +L2(R) equipped with its usual norm, is not continuous. Indeed the ∥.∥qα norm doesn’t control the second +derivative of v and hence doesn’t provide any module of continuity for the L2(R)-extended linear form +q(., v). +Theorem A.6 (Inversion of S + αI). +For every f ∈ L2(R), there exists a unique u ∈ H2 +V ′(R) such that (S + αI)u = f. Furthermore, the map +(S + αI)−1 is continuous from +� +L2(R), ∥.∥L2(dx) +� +to +� +H2 +V ′(R), ∥.∥qα +� +. +30 + +Proof. Let f ∈ L2(R), the map u �−→ ⟨u, f⟩L2(dx) is continuous on +� +H1 +V ′(R), ∥.∥qα +� +which is a Hilbert +space. Therefore by Riesz’s theorem, there exists a unique vf ∈ H1 +V ′(R) such that for all u ∈ H1 +V ′(R), +⟨f, u⟩L2(dx) = qα(vf, u) from which we deduce that, in the sense of distributions, f = −v′′ +f + (wP + α)vf +which implies that vf ∈ H2 +V ′(R). Since vf ∈ D +� +S + αI +� +, we have then for all u ∈ L2(R), ⟨f, u⟩L2(dx) = +qα(vf, u) = ⟨(S + α)vf, u⟩L2(dx), hence (S + αI)vf = f. Finally, by Riesz’s theorem, f ∈ L2(R) �→ vf ∈ +H1 +V ′(R) is continuous hence so is (S + αI)−1. +Remark A.7. It would be tempting to use Banach’s isomorphism theorem to say that since (S + αI)−1 +is bijective and continuous, so must be S + αI. But since +� +H2 +V ′(R), ∥.∥qα +� +is not a Banach space (it’s not +closed in H1 +V ′(R)) we can’t apply it. +We are now able to diagonalize the resolvent of S. +Theorem A.8 (Diagonalization of (S + αI)−1). +There exists a complete orthonormal set (ψn)n⩾0 of L2(dx) (meaning that +span{ψn, n ⩾ 0} +∥.∥L2(dx) = L2(dx) +and ⟨ψi, ψj⟩L2(dx) = δi,j), where each ψn ∈ H2 +V ′ and +� +µn(α) +� +n⩾0 ∈ [0, 1]N with µn(α) −→ +N→∞ 0 such that +(S + αI)−1ψn = µn(α)ψn for all n ⩾ 0. We also have +��� +��� +��� +� +S + αI +�−1��� +��� +��� +L +� +L2(dx) +� ⩽ 1. +Proof. By Proposition A.1, wP + α is continuous and goes to infinity at infinity. By Rellich criterion +[RS78][see Theorem XIII.65], the unit ball of H2 +V ′(R), ie the set +� +u ∈ H2 +V ′(R), +ˆ +R +u′2 + +ˆ +R +(wP + α)u2 ⩽ 1 +� +considered as a subset of L2(R) is relatively compact in +� +L2(dx), ∥.∥L2(dx) +� +. Hence, we can conclude +that the injection ι : +� +H2 +V ′(R), ∥.∥qα +� +−→ +� +L2(dx), ∥.∥L2(dx) +� +is a compact operator. Since (S + αI)−1 : +� +L2(dx), ∥.∥L2(dx) +� +−→ +� +H2 +V ′(R), ∥.∥qα +� +is continuous then (S+αI)−1 is compact from +� +L2(dx), ∥.∥L2(dx) +� +to itself. The fact that (S + αI)−1 is self-adjoint and positive allows us to apply the spectral theorem +to obtain +� +µn(α) +� +n⩾0 positive eigenvalues verifying µn(α) −→ +N→∞ 0 by compactness and a Hilbertian basis +(ψn)n⩾0 ∈ L2(R)N, such that for all n ⩾ 0, (S + αI)−1ψn = µn(α)ψn. It is then easy to see that for +all n, ψn ∈ H2 +V ′(R) since they belong to the range of (S + αI)−1. Finally, since for all φ ∈ L2(R), +⟨(S + αI)φ, φ⟩L2(dx) ⩾ ∥φ∥2 +L2(dx), the spectrum of (S + αI)−1 is contained in [0, 1]. +It allows us to +conclude that +������(S + αI)−1������ +L2(dx) ⩽ 1. +It is now straightforward to see how to extend A = ρ−1/2 +P +Sρ1/2 +P +on H2 +V ′(ρP ) := ρ−1/2 +P +H2 +V ′(R) equipped +with the norm ∥.∥qα,ρP to +� +L2(ρP ), ∥.∥L2(ρP ) +� +. The norm ∥.∥qα,ρP is defined for all u ∈ H2(ρP ) by +∥u∥qα,ρP = +ˆ +R +u′2ρP dx + +ˆ +R +u2(wP + α)ρP dx . +It is easy to see that (A + αI)−1 is continuous. We stress the fact that H2 +V ′(ρP ) ̸= +� +u ∈ H2(ρP ), uV ′ ∈ +L2(ρP ) +� +. Indeed if u ∈ H2 +V ′(R), even though uρ−1/2 +P +and its derivative belong to L2(ρP ), (ρ−1/2 +P +u)′′ /∈ +L2(ρP ). The reader can check that for such a function to be in L2(ρP ), it would be necessary to have +that u2V ′4 ∈ L2(dx) which is not the case. +Remark A.9. The kernel of A is generated by the function �1. Indeed if φ ∈ H2 +V ′(ρP ) is in the kernel of +A then +31 + +0 = − +� +φ′ρP +�′ +ρP +⇒ ∃c ∈ R, φ′ = c +ρP +But since φ′ is in L2(ρP ) then c = 0 which implies that φ is constant. We must restrict A to the orthogonal +of KerA with respect to the inner product of L2(ρP ), ie +H2 +V ′,0(ρP ) := +� +u ∈ H2 +V ′(ρP ) | +ˆ +R +uρP = 0 +� +. +Doing so makes A injective. +Before inverting A, we need the following lemma: +Lemma A.10. The following equality holds +(A + αI) +� +H2 +V ′,0(ρP ) +� += L2 +0(ρP ) := +� +u ∈ L2(ρP ), +ˆ +R +uρPdx = 0 +� +Proof. Let φ = �c for c ∈ R, (A + αI)φ = � +αc then (A + αI)(R.�1) = R�1. Hence since A + αI is self-adjoint +with respect to the inner product of L2(ρP ) and that R�1 is stable by A + αI, then (A + αI) +� +(R.�1)⊥ ∩ +H2 +V ′(ρP ) +� +⊂ (R.�1)⊥. For the converse, let u ∈ (R.�1)⊥, since A + αI is bijective, there exists v ∈ H2 +V ′(ρP ) +such that u = (A + αI)v. For all w ∈ R.�1, +0 = ⟨u, w⟩L2(ρP ) = ⟨(A + αI)v, w⟩L2(ρP ) = ⟨v, (A + αI)w⟩L2(ρP ) +Hence v ∈ +� +(A + αI)(R�1) +�⊥ = R�1⊥ and so (R.�1)⊥ ⊂ (A + αI) +� +(R.�1)⊥� +. +It is easy to see that L2 +0(ρP ) is a closed subset of L2(ρP ) as it is the kernel of the linear form +φ ∈ L2(ρP ) �→ +� +φ,�1 +� +L2(ρP ), making it a Hilbert space. +Proposition A.11 (Diagonalization and invertibility of A). There exists a complete orthonormal set of +� +L2 +0(ρP ), ⟨., .⟩L2(ρP ) +� +, (φn)n∈N ∈ H2 +V ′,0(ρP )N such that Aφn = λnφn (meaning that +span{φn, n ⩾ 0} +∥.∥L2(ρP ) = L2 +0(ρP ) +and ⟨φi, φj⟩L2(ρP ) = δi,j). Furthermore, A : H2 +V ′,0(ρP ) −→ L2 +0(ρP ) := +� +u ∈ L2(ρP ), +´ +R uρPdx = 0 +� +is +bijective, A−1 is continuous when considered as an operator of L2 +0(ρP ). +Proof. Since (S + αI)−1 considered as an operator of L2(dx), is compact so is (A + αI)−1 on L2(ρP ) +and since A is self-adjoint, by the spectral theorem, (A + αI)−1 is diagonalizable. With the notations of +Theorem A.8, (A+αI)−1 has eigenvalues +� +µn(α) +� +n⩾0 and corresponding eigenfunctions φn = ρ−1/2 +P +ψn ∈ +H2 +V ′(ρP ). Hence for all n ∈ N, Aφn = λnφn with λn := +� +1 +µn(α) − α +� +. Now, +λn∥φn∥L2(ρP ) = +ˆ +R +(Aφn)φnρP dx = − +ˆ +R +(ρP φ′ +n)′φn = +ˆ +R +φ′2 +n ρP ⩾ 0 . +Furthermore, the kernel of A is R.�1, thus the spectrum of A restricted to H2 +V ′,0(ρP ) is positive. But +since (A + αI)−1 is a compact operator of L2(ρP ) and that (A + αI) maps R.�1⊥ to R.�1⊥ with respect +to the inner product of L2(ρP ) (see lemma A.10), then +� +A + αI +�−1 is compact as an operator from +L2 +0(ρP ) to itself. By Fredholm alternative, for every λ ∈ R λ ̸= 0, either (A + αI)−1 − λI is bijective +32 + +either λ ∈ Sp +� +(A + αI)−1� +. These conditions are equivalent to: either A + (α − 1 +λ)I is bijective as an +operator from H2 +V ′,0(ρP ) to L2 +0(ρP ), either −α + 1 +λ ∈ Sp +� +A +� +. If we set λ = 1 +α then either A is bijective +either 0 ∈ Sp(A), since the latter is wrong then A : H2 +V ′,0(ρP ) → L2 +0(ρP ) is bijective. The spectrum of +A is +� +1 +µn(α) − α +� +n⩾0 +⊂ (λ1, +∞) ⊂ (0, +∞), where λ1 is the smallest eigenvalue. Hence, we deduce +������A−1������ +L(L2(ρP )) ⩽ λ−1 +1 . +References +[ABG12] +R. Allez, J.P. Bouchaud, and A. Guionnet. Invariant beta ensembles and the gauss-wigner +crossover. Physical Review Letters, Aug. 2012. +[AGZ10] +G. W. Anderson, A. Guionnet, and O. Zeitouni. An introduction to random matrices, volume +118 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, +2010. +[BBCG08] D. Bakry, F. Barthe, P. Cattiaux, and A. Guillin. A simple proof of the Poincaré inequality +for a large class of probability measures including the log-concave case. Electron. Commun. +Probab., 13:60–66, 2008. +[BFG15] +F. Bekerman, A. Figalli, and A. Guionnet. Transport maps for β-matrix models and univer- +sality. 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ArXiv 2101.06528, 2021. +34 + diff --git a/89E5T4oBgHgl3EQfQw7M/content/tmp_files/load_file.txt b/89E5T4oBgHgl3EQfQw7M/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..03fc5cb2ad7dda4faecc64c0c796569013427574 --- /dev/null +++ b/89E5T4oBgHgl3EQfQw7M/content/tmp_files/load_file.txt @@ -0,0 +1,1202 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E5T4oBgHgl3EQfQw7M/content/2301.05516v1.pdf,len=1201 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E5T4oBgHgl3EQfQw7M/content/2301.05516v1.pdf'} +page_content='05516v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E5T4oBgHgl3EQfQw7M/content/2301.05516v1.pdf'} +page_content='PR] 13 Jan 2023 CLT for real β-Ensembles at High Temperature∗ Charlie Dworaczek Guera†, Ronan Memin‡ Abstract We establish a central limit theorem for the fluctuations of the empirical measure in the beta- ensemble of dimension N at a temperature proportional to N and with convex, smooth potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E5T4oBgHgl3EQfQw7M/content/2301.05516v1.pdf'} +page_content=' The space of test functions for which the CLT holds includes C1, vanishing functions at infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E5T4oBgHgl3EQfQw7M/content/2301.05516v1.pdf'} +page_content=' It is obtained by the inversion of an operator which is a pertubation of a Sturm-Liouville operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E5T4oBgHgl3EQfQw7M/content/2301.05516v1.pdf'} +page_content=' The method that we use is based on a change of variables introduced in [BFG15] and in [Shc14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E5T4oBgHgl3EQfQw7M/content/2301.05516v1.pdf'} +page_content=' Contents 1 Introduction and main result 1 2 Regularity of the equilibrium measure and Hilbert transform 6 3 Concentration inequality, proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E5T4oBgHgl3EQfQw7M/content/2301.05516v1.pdf'} +page_content='5 9 4 Localization of the edge of a configuration 14 5 Laplace transform for smooth test functions, proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E5T4oBgHgl3EQfQw7M/content/2301.05516v1.pdf'} +page_content='3 17 6 Inversion of L 22 7 Regularity of the inverse of L and completion of the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E5T4oBgHgl3EQfQw7M/content/2301.05516v1.pdf'} +page_content='3 26 A Appendix: proof of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E5T4oBgHgl3EQfQw7M/content/2301.05516v1.pdf'} +page_content='2 29 1 Introduction and main result The beta-ensemble of dimension N ⩾ 1 with parameter β > 0 and potential V is the probability measure on RN given by dPβ,V N (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E5T4oBgHgl3EQfQw7M/content/2301.05516v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E5T4oBgHgl3EQfQw7M/content/2301.05516v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E5T4oBgHgl3EQfQw7M/content/2301.05516v1.pdf'} +page_content=' , xN) = 1 ZN(V, β) � i 0 there exists a sequence X with a subsequence (not necessarily +consecutive) X′ ⊆ X such that cost(GF, X′) ≥ (2 − ϵ) · cost(GF, X). +There exists a sequence Y with a subsequence (not necessarily consecutive) Y ′ ⊆ Y such +that cost(GF, Y ′) − cost(GF, Y ) = Ω(m · lg lg n). +STACS 2023 + +39:6 +Dynamic BSTs: Improved Lower Bounds for Greedy-Future +▶ Theorem 7. Let S be a sequence, we define rev(S) to be the sequence S in reverse. For +any ϵ > 0 there exists a sequence X such that cost(GF, rev(X)) ≥ (2 − ϵ) · cost(GF, X). +There exists a sequence Y such that cost(GF, rev(Y )) − cost(GF, Y ) = Ω(m · lg lg n). +The motivation for studying subsequences (Theorem 6) is the fact that OPT always saves +costs when queries are removed from its sequence. Formally, if X′ ⊆ X, then cost(OPT, X′) ≤ +cost(OPT, X). Indeed, OPT can serve X′ by simulating a run on X. More generally, this +relation of costs when comparing a sequence to a subsequence of it, is an important property +which even has a name: +▶ Definition 8 (Approximate-monotonicity [12, 15]). An algorithm A is approximately- +monotone with a constant c if for any sequence X, subsequence X′ ⊆ X, and initial tree T, +it holds that cost(A, X′, T) ≤ c · cost(A, X, T). +▶ Corollary 9. If GF is approximately-monotone with a constant c, then c ≥ 2. +As noted, OPT is approximately-monotone with c = 1 (strictly monotone). The reason +that approximate-monotonicity is of interest, in particular for GF, is because it is one of two +properties that together are necessary and sufficient for any dynamically-optimal algorithm. +The complementing property, which GF is known to satisfy, is simulation-embedding: +▶ Definition 10 (Simulation-Embedding [15]). An algorithm A has the simulation-embedding +property with a constant c if for any algorithm B and any sequence X, there exists a +supersequence Y ⊇ X such that cost(A, Y ) ≤ c · cost(B, X). (X is a subsequence of Y , not +necessarily of consecutive queries.) +An algorithm A which is approximately-monotone with a constant c1 and has the +simulation-embedding property with a constant c2 is dynamically-optimal with a constant +c1 · c2. Indeed, for any sequence X, there is some supersequence Y (X) ⊇ X such that +cost(A, X) ≤ c1 · cost(A, Y (X)) ≤ c1 · c2 · cost(OPT, X). Harmon [12] proved that GG, and +hence GF, has the simulation-embedding property, hence GF is dynamically-optimal if and +only if it is approximately-monotone. An alternative indirect proof was given by [6], proving +that GG is O(1)-competitive versus the move-to-root algorithm, therefore inheriting the +property from move-to-root. +The motivation for studying reversal (Theorem 7) is that OPT is oblivious to reversing +the sequence of queries, up to an additive difference of n. Indeed, to serve a sequence X in +reverse, we can pay n to restructure the initial tree T0 to the final tree Tm, and then “reverse +the arrow of time”: when serving query xt, also modify the tree from Tt to Tt−1 where Ti is +the tree that OPT would get by the end of processing the i-th query of X, in order. This +means that any dynamically-optimal algorithm must be able to serve a sequence of requests +and its reverse with the same cost up to a constant factor. Theorem 7 does not disprove +dynamic-optimality for GF, but gives some insight of how reversal affects GF. +3.1 +Maintaining a Static Tree for GF +In this section we describe the basic “tool” which we use to fix a tree structure for GF despite +its dynamic nature. That is, we describe a class of sequences which we call mixed-stable +sequences such that GF never restructures its tree when serving a sequence in this class. +For the sake of simplicity, we assume that the initial tree is structured as we need it to be. +Appendix A.1 explains how to enforce a specific “initial” tree given an arbitrary initial tree, +and also argues why this minor issue does not affect the competitive ratio of GF. + +Y. Sadeh and H. Kaplan +39:7 +As noted, our objective is to produce a sequence that “tricks” GF into having unnecessary +nodes in the core of the tree, such that the requested values are only at the leaves. As +an example, consider the classic sequence of queries X = [1, 3, 1, 3, . . .] with an initial tree +containing 2 at the root, 1 as a left child of the root and 3 as a right child of the root. +Because of the alternating pattern, GF never re-structures the tree, and the cost per query +is 2 rather than 1.5 on average (e.g. when 1 is in the root, and 3 is its right child). +▶ Definition 11 (Stable Nodes and Sequences). Let T be a full binary search tree, and let X +be a sequence of queries over the items in the leaves of T. We define the stability of nodes as +follows, see also Figure 1. +We say that an inner node v in T is strongly-stable if it has two children, and the +subsequence of X consisting only of the items in the subtree of v, alternates between accesses +to the left and right subtrees of v. +We say that an inner node v with a left child u in T is weakly-stable with a left-bias if +both v and u have two children, and the subsequence of X consisting only of the items in the +subtree of v, repeats the following 3-cycle. First it accesses the left-subtree of u, then the +right subtree of u, and finally right subtree of v. (It is left-biased because 2 +3 of the accesses +are to the left of v). Symmetrically, we say that v is weakly-stable with a right-bias if v has +two children, its right child u has two children, and the restriction of X to accesses in the +subtree of v repeats a 3-cycle consisting of an access to the right subtree of u, the left subtree +of u, and the left subtree of v. Notice that u is a strongly-stable node by definition, and we +refer to it as the favored-child of v. +We regard the sequence X as being induced by the tree T with stability “attached” to its +inner nodes. We assume that every node is stable, and refer to X as a mixed-stable sequence +and to T as a mixed-stable tree. We distinguish two special cases: If all inner nodes are +strongly-stable then we refer to X and T as strongly-stable, and if exactly half of the inner +nodes of T are weakly-stable then we refer to X and T as weakly-stable (recall that each +weakly-stable node has a strongly-stable favored-child). +Figure 1 Node and sequence stability (Definition 11). First, consider the repeated sequence +421, i.e. X = 421421421 . . .. Then v is a weakly-stable right-biased node because its visits pattern +is a repetition of right(u), left(u), left(v). u is a strongly-stable node because its visits pattern +is right(u), left(u). w is not stable at all, because its visits pattern is always left(w). Second, +consider the repetition of the access pattern 12141314. One can verify that all three inner nodes +are strongly-stable. Hence, this is a strongly-stable sequence. Third, note that no weakly-stable +sequence corresponds to the figure, because it requires an even number of inner nodes, but if we +make w a leaf (removing 2, 3), then the repeated access pattern of 4w1 is a weakly-stable sequence. +To motivate Definition 11 a little, note that the sequence X = [1, 3, 1, 3, . . .] is a strongly- +stable sequence that corresponds to a tree over the items {1, 2, 3} where 2 is in the root. +X yields a lower-bound of 4 +3 on the competitive ratio of GF. Similarly, the sequence X′ = +[5, 3, 1, 5, 3, 1, . . .] is a weakly-stable sequence that corresponds to the tree over {1, 2, 3, 4, 5} +with 2 at the root and 4 its right-child. X′ yields a lower-bound of 8 +5 on the competitive ratio +of GF, which is already an improvement over the best known lower bound, see also Figure 2. +The distinction between strongly-stable and weakly-stable nodes is that GF may modify +STACS 2023 + +39:8 +Dynamic BSTs: Improved Lower Bounds for Greedy-Future +(a) X = [1, 3, 1, 3, . . .] +(b) X′ = [5, 3, 1, 5, 3, 1, . . .] +Figure 2 Examples of the simplest strongly-stable (a) and weakly-stable (b) sequences. Their +corresponding trees are the left tree in each pair while the right tree in each pair is an optimized static +tree to serve the same sequence. Queried nodes are colored in blue. One can verify that ˆc(X, GF) = 2 +and ˆc(X′, GF) = 8 +3 while based on the optimized tree, ˆc(X, OPT) ≤ 3 +2 and ˆc(X′, OPT) ≤ 5 +3. +the structure of the tree when a weakly-stable node is considered, but only temporarily and +without affecting the cost. In our example with X′, after querying 5, GF may put 4 in the +root instead of 2, but following the query of 3 this change will be reverted. +Motivated by the power of stable sequences over small trees, we proceed to a more general +analysis of stable sequences. +▶ Definition 12 (Atomic Sequence). A tree T, along with stability type (weak/strong) for +each node, and a subtree of each node to be accessed initially, induce a stable sequence. This +sequence is unique up to its length, which can be extended indefinitely. We define the “atomic +unit” of this sequence as the shortest sequence X such that any repetition of X is also a +stable sequence that corresponds to T. +Throughout the paper we work with whole multiples of the atomic sequence. Moreover, +unless stated otherwise, we work with the atomic sequence itself (a single repetition). +▶ Lemma 13. Let X be a mixed-stable sequence with respect to a tree T. Then every leaf u +is visited once every 2a(u) · 3b(u) queries where a(u) and b(u) are non-negative integers. In +particular, the atomic length of X is 2maxleaf u a(u) · 3maxleaf u b(u) (the lcm). Moreover, if X +is strongly-stable then ∀u : b(u) = 0, and if X is weakly-stable then ∀u : a(u) = 0. +Proof. Consider a leaf u. Define the frequency of visiting an ancestor w of u to be the +frequency of accessing a leaf in the subtree of w. If w is a strongly-stable ancestor then the +frequency of visiting a child of w is 1 +2 of the frequency of visiting w. If w is weakly-stable, v +is its favored-child, and x is a child of v then the frequency of visiting x is 1 +3 of the frequency +of visiting w. Similarly if w is weakly-stable, v is its non-favored-child then the frequency +of visiting v is 1 +3 of the frequency of visiting w. It follows that u is visited exactly once +every 2a(u) · 3b(u) queries where a(u) is the number of strongly-stable nodes that are not +favored-children (there are no such nodes if X is weakly-stable), and b(u) is the number of +weakly-stable nodes (no such nodes if X is strongly-stable), on the path to u. Finally, since +every leaf u is visited with a specific period, the whole sequence has a period which is the +lcm of all periods. +◀ +▶ Lemma 14. Let X be a mixed-stable sequence with respect to a tree T. If GF serves +X with T as initial tree, and breaks ties in favor of nodes of smaller-depth, then it never +restructures T. +Proof. The proof is by induction on the size of the tree. If T has a single node, then it is +trivial. Otherwise, the root r is an inner-node, and we prove that it always remains the root. + +Y. Sadeh and H. Kaplan +39:9 +It then follows, by restricting the access sequence to values within each subtree, that the rest +of the tree remains fixed as well. We use the notations of τ(v) and vi as in Algorithm 1. +First, consider the case that r is a strongly-stable node (Definition 11). Given an access +to some value x in the left subtree of r, by definition, the next access would be to a value +in the right subtree of r, hence τ(r) < τ(vi) for any vi ̸= r on the path from r to x, and +therefore GF will keep r in the root. The same argument holds if x is in the right subtree of +r, and the next access is in the left subtree. +Next, consider the case that r is a weakly-stable node. Without loss of generality, assume +that it is left-biased, and denote its favored-child (left child) by u. Denote the left and right +subtrees of u by A and B respectively, and the right subtree of r by C. The access pattern +of subtrees is ABC(ABC . . .). +If the current access was to some x ∈ A, both r and u have been touched. The next +access queries in B, so τ(u) = τ(r) < τ(vi) for any vi ̸= u, r on the access path to x. +Since GF tie-breaks in favor of smaller-depth, it will keep r in the root.4 +If the current access was to some x ∈ B, then both r and u have been touched. The next +access touches C, so τ(r) < τ(vi) for any vi ̸= r on the access path to x, including u, +thus r must remain the root. +If the current access was to some x ∈ C, since the next access touches A, τ(r) < τ(vi) for +any vi ̸= r on the access path to x, thus r must remain the root. In this case u was not +touched, but nonetheless it remains the left child of r. +◀ +▶ Lemma 15. If X is a mixed-stable sequence, the frequency of accessing x ∈ X is in the +range of [ +1 +3d(x) , +1 +3d(x)/2 ]. In particular, if X is strongly-stable then the frequency equals +1 +2d(x) . +Proof. The frequency of visiting a node depends on the path to it. +The frequency is +multiplied by 1 +2 when passing through a strongly-stable node, and multiplied by either 1 +3 or +2 +3 when passing through a weakly-stable node. Every factor of 2 +3 is followed by 1 +2, due to +the strongly-stable favored-child of the weakly-stable node. Thus the frequency is bounded +between +1 +3d(x) and +1 +2d(x)/2 · +� 2 +3 +�d(x)/2 = +1 +3d(x)/2 . +◀ +▶ Corollary 16. Let X be a strongly-stable sequence, then: ˆc(GF, X) = � +x∈X +d(x)+1 +2d(x) . +3.2 +Promotions and Recursive Trees +The way in which we show our lower bounds relies on the fact that serving the leaves of a +static tree is sub-optimal, since a trivial static optimization is to move the leaves closer to +the root. We refer to this operation as a promotion of the leaf that we move. We emphasize +that for the purpose of our result, we analyze the improvement one gets from promotions, +but the actual OPT, which is dynamic, may be able to reduce the cost further. +▶ Definition 17 (Promotion). Consider trees T and T ′. We say that a node x was promoted +in T ′ by h (with respect to T), if dT (x) − dT ′(x) = h. Given a mixed-stable sequence X, the +average promotion of T to T ′ is the weighted average promotion in T ′ of the nodes of T, +weighted by the query frequencies of the nodes. +4 This is the reason we defined this kind of access pattern as weakly-stable, because the stability can be +chosen, but is not forced. We emphasize that putting u as a parent of r will not make the next access +cheaper as both u and r will be touched anyway, and then r will be reinstated as the root. +STACS 2023 + +39:10 +Dynamic BSTs: Improved Lower Bounds for Greedy-Future +By definition, static optimization of a tree T to T ′ for a mixed-stable sequence X, implies +a cost improvement for OPT which is at least the average promotion of T to T ′, per query. +Intuitively, promoting leaves that are closer to the root contributes more to the average +promotion than promoting deeper leaves since the access frequencies decrease exponentially +with depth. That being said, our promotion scheme will be relatively uniform, promoting +most leaves by roughly the same amount, as in the following example. +▶ Example 18. To clarify promotions, consider Figure 3. There, we can safely promote every +node by one, except for one of the deepest nodes. Therefore, we immediately conclude that for +the corresponding strongly-stable sequence X, we have: ˆc(GF, X) ≥ ˆc(OPT, X) + (1 − +1 +2n ). +(a) Before promotions. +(b) After promotions. +Figure 3 (a) A tree which induces a strongly-stable sequence X, only blue nodes are queried. +The frequency of querying an odd number v = 2i − 1 in this tree is +1 +2i except for v = 2n + 1 which +has the same frequency as v = 2n − 1. (b) An improved static tree, in which each node except for +one has been promoted one step closer to the root. The cost of serving X over this tree is cheaper +by almost 1 per query. +We define our trees using recursive structures. +▶ Definition 19. A recursive tree, Tr, of depth r is defined by a specific full binary tree T +(independent of r) such that at least one of its leaves is an actual leaf, and some of its leaves +are roots of recursive trees, Tr−1, of depth r − 1. We refer to the inner nodes of T as the +trunk of Tr, and define T0 to be a single node. See Figure 4 for two examples.5 +Figure 4 Two recursive trees of depth r. Each of the trees T and F is a full binary tree with +at least one actual leaf (in blue), and some hanging subtrees. At the bottom of the recursion (for +r = 0), the subtrees are nodes. Note that: (a) Expanding T for r = n results in the tree in Figure 3; +(b) The pattern F is important for Theorem 4. +5 The name of the pattern F in Figure 4, stands for Fibonacci: One can verify that for r ≥ 2, the number +of leaves at depth 1 ≤ d ≤ r − 1 is the (d − 1)th Fibonnaci number Fd−1 (we define F0 = 0). Moreover, +this can be used to prove the nice equation: �∞ +d=0 +Fd +2d = 2. + +2n- 1 +2n + 12n - 1 +2n : +2n + 1Y. Sadeh and H. Kaplan +39:11 +3.3 +Multiplicative Lower Bound for GF +In this section we prove Theorem 4. We do it by describing a concrete weakly-stable sequence, +whose average cost per query is 6 while an average promotion of 3 is possible, resulting in an +optimal cost of at most 3. We start by stating a purely mathematical lemma that will be +used in the analysis. +▶ Lemma 20. Let br be a sequence defined by an initial value b0 and the relation br = +α · br−1 + β + γ · r +2r for some constants α, β, γ where α ̸= 1 +2, 1. Then br = +β +1−α(1 − αr) + αr · +b0 + +2αγ +(2α−1)2 ·(αr − 1 +2r )− +γ +(2α−1) · r +2r . In particular, when γ = 0 then br = +β +1−α(1−αr)+αr ·b0. +Proof Sketch. Either use induction, or “guess” that a geometric sequence yr with a multiplier +of α satisfies yr = p · r +2r + q · 1 +2r + s + br, and determine the fixed coefficients p, q, s. +◀ +▶ Lemma 21. Let X be a weakly-stable sequence implied by the recursive tree Fr in Figure 4, +where the root is a weakly-stable node with a right-bias. Then for any ϵ > 0, there is a +sufficiently large recursive depth r such that (1) ˆc(GF, X) > 6 − ϵ, (2) a static optimization +of the tree saves an average cost of at least 3 − ϵ, and (3) regardless of r, ˆc(OPT, X) < 3. +Proof. Let cr denote the average cost of serving X with Fr. +Then c0 = 1 and cr = +1 +3(cr−1 + 1) + 1 +3 · 3 + 1 +3(cr−1 + 2) = +2 +3cr−1 + 2, which yields by Lemma 20 that cr = +2 +1−2/3(1 − (2/3)r) + (2/3)r · 1 = 6 · (1 − (2/3)r) + (2/3)r. To analyze the average promotion, +we re-structure Fr to a new static structure F ′ +r as follows, see Figure 5. The leaf is moved +to the root, whose children are the recursive subtrees, optimized themselves by the same +logic. The old root is moved to be a right child of the maximal value in the new left subtree, +and the old right-child (of the old-root) is moved to be a left child of the minimal value in +the new right subtree. F ′ +r maintains the order of values as was in Fr. The demotions of the +old root and its right child do not affect the cost, because X does not query these values. +Denote by pr the average promotion of Fr to F ′ +r. Then p0 = 0 since nothing is promoted for +a singleton, and pr = 1 +3pr−1 + 1 +3 · 2 + 1 +3(pr−1 + 1) = 2 +3pr−1 + 1. Again by Lemma 20 we get +that pr = +1 +1−2/3(1 − (2/3)r) + (2/3)r · 0 = 3 · (1 − (2/3)r). Observe that for r → ∞ we get +that cr → 6 and pr → 3, thus parts (1) and (2) of the claim follow. For part (3), observe +that cr − pr = 6 · (1 − (2/3)r) + (2/3)r − 3 · (1 − (2/3)r) = 3 − 2 · (2/3)r < 3. +◀ +(a) F-tree pattern. +(b) Promotion scheme. +Figure 5 The F-tree pattern and its promotion scheme in Lemma 21. +Only the top-level +promotions are presented in (b), but more promotions are done recursively within each subtree. +▶ Theorem 4. If GF is (c, d)-competitive where the additive term d is sublinear in the length +of the sequence, i.e. d = o(m), then c ≥ 2. +Proof. Assume by contradiction that GF is (2 − δ, f(m))-competitive for some δ > 0 +and a function f(m) = o(m). Let X′ be a sequence that consists of s repetitions of the +atomic weakly-stable sequence that corresponds to the recursive tree Fr. It follows that +ˆc(GF, X′) ≤ (2 − δ) · ˆc(OPT, X′) + f(|X′|) +|X′| . By Lemma 21, we can choose r large enough such +STACS 2023 + +Fr: +Fr-139:12 +Dynamic BSTs: Improved Lower Bounds for Greedy-Future +that ˆc(GF, X′) > 6 − δ, and regardless of r, ˆc(OPT, X′) < 3. Then, since f is sub-linear, we +can choose the number of repetitions s to be large enough such that f(|X′|) +|X′| +< 2δ. But then +we also get that ˆc(GF, X′) < (2 − δ) · 3 + 2δ = 6 − δ, which is a contradiction. +◀ +By Analyzing mixed-stable sequences we proved a lower bound of 2 on the competitve +ratio of GF. Theorem 22 gives an upper bound. +▶ Theorem 22. Let X be a mixed-stable sequence and let T be the tree that corresponds to +it. Then cost(GF, X, T) < c · cost(OPT, X, T) for c = 5 +2. If X is strongly-stable, then c = 2. +We defer the proof of Theorem 22 to Appendix A.2. The upper-bound in Theorem 22 is +clearly not tight, since in the proof of Theorem 22 we neglected a term using the inequality +ˆc(GF, X) ≤ +2 +α · ˆc(OPT, X) − 1 +α +� +1 − n−1 +2m +� +< +2 +α · ˆc(OPT, X), for a constant α. The lack +of tightness is more prominent when ˆc(OPT, X) is small, like in the sequence studied in +Lemma 21 (for Theorem 4). We suspect that the lower bound in Theorem 4 is tight, and +more strongly, that the F-tree pattern is the best pattern to use. This is based on studying +several other recursive patterns, including those in Figure 4 and Figure 6: None was stronger, +and it also seems that patterns with large costs do not “compensate” with large enough +promotions. +As a closing remark to the multiplicative results, we note that by the static optimality +theorem for GG [9], competitive analysis against a static algorithm (i.e. an algorithm that +does not change its initial tree) cannot show a super-constant lower bound. Concretely, +the theorem states that cost(GF, X) ≡ cost(GG, X) = O(m + �n +i=1 ni lg m +ni ) and one can +verify that the actual constants are 5m + 6 �n +i=1 ni lg m +ni . This bound can be re-written as +5m + 6m · H2(X) where H2(X) = �n +i=1 +ni +m lg m +ni is the base-2 entropy of the frequencies of +the values in X. By [17], cost(OPT s, X) ≥ m · H2(X) +lg 3 +where OPT s is the static optimum, +and therefore cost(GF, X) ≤ (5 + 6 lg 3) · cost(OPT s, X). Thus, no static argument can show +a lower bound larger than ≈ 11.59. +3.4 +Additive Lower Bounds for GF +In this section we move on to analyze the additive gap between GF and OPT. For this, +we construct and analyze more elaborate patterns of recursively-defined trees, in order to +get a large average promotion when optimizing the structure of the trees. The analysis is +more involved since we cannot simply assume that the depth of the recurrence, r, approaches +infinity. Here n is a function of r and the difference of cost can be meaningful in terms of n +only if n is finite. +▶ Definition 23. For k ≥ 2, and r ≥ 0 we define a (k, r)-tree Tr as follows. The tree is +recursive of depth r (as in Definition 19), such that its trunk is composed of a root and a +left-chain of length k − 1 that starts in the right-child of the root. The left child of the deepest +node of the trunk is an actual leaf, and the rest of the leaves are Tr−1 subtrees. T0 is a single +node. See Figure 6. When k is clear from the context, we also refer to the tree as Tr. +Observe that the tree Fr that was used to prove Theorem 4 is in fact a (k, r)-tree with +k = 2. When we conclude the analysis, we will get the two ends of a “tradeoff” such that +on the one end we have a relatively high cost ratio, and on the other a relatively high cost +difference. Moreover, we will show that the higher the difference of costs on a sequence +induced by (k, r)-tree, the closer the cost ratio is to 1 (comparing GF to OPT). +▶ Lemma 24. The depth of a (k, r)-tree is k · r, and its left-most node is at depth r. + +Y. Sadeh and H. Kaplan +39:13 +(a) (k, r)-tree pattern. +(b) Promotion scheme. The main gain is due to the first step. +Figure 6 (a) The recursive pattern of a (k, r)-tree, Tr. The trunk of the tree has k nodes: the +root, and a chain of k − 1 nodes leading to an actual leaf. The rest of the leaves are (k, r − 1)-trees. +(b) The promotion scheme used later in Lemma 26, exemplified for k = 4 (see also Figure 5 for the +degenerate case of k = 2). The main gain is from the first step of promoting the actual leaf to the +root, and its sibling subtree (marked with +) one step upwards. Additional gain is achieved by +promoting the left-most node of each hanging right subtree to the trunk at the expense of demoting +trunk nodes. More promotions are done recursively within each subtree. The nodes marked 0, 1, 2 +are indeed consecutive, and also: 2 < x and x + 1 = w < y = z − 1. +Proof. Trivial by induction: For r = 0, the deepest node is the root, at depth 0. For r ≥ 1, +observe that the deepest node belongs to the deepest subtree Tr−1, which is rooted at depth +k since the path to it includes k trunk nodes. Similarly, the depth of the left-most node is +increased by 1 per recursive level of the tree. +◀ +▶ Lemma 25. Let Tr be a (k, r)-tree. Then |Tr| = (2 + +2 +k−1)kr − (1 + +2 +k−1) where |Tr| is the +number of nodes in Tr. In rougher terms, |Tr| = Θ(kr). +Proof. Denote nr = |Tr|. By definition, n0 = 1 and nr = (k + 1) + k · nr−1. Hence, by +Lemma 20 (with γ = 0): nr = k+1 +1−k(1 − kr) + kr = (2 + +2 +k−1)kr − (1 + +2 +k−1). +◀ +▶ Lemma 26. Let X be any mixed-stable sequence corresponding to a (k, r)-tree Tr. Denote +the average weighted promotion possible in Tr by pr, where weighting is according to the +frequency of querying each leaf. Then pr > k · (1 − αr) for α = 1 − 1 +3k . In particular, if X is +a strongly-stable sequence, then pr = (k + 1) · (1 − αr) + δ for α = 1 − +1 +2k and 0 ≤ δ < αr. +Proof. We can promote by k every explicit leaf in every Tr′ for all recursive levels 1 ≤ r′ ≤ r, +from its location to the root of Tr′. Only nodes that are T0 leaves do not contribute an explicit +promotion of at least k, therefore pr > k · (1 − f) where f is the sum of query-frequencies of +all T0 leaves (the inequality is strict due to unaccounted subtree promotions). To conclude, +we argue that f ≤ (1 − +1 +3k )r. The frequency of accessing the explicit leaf of Tr is at least +1 +3k by Lemma 15, hence with frequency of at most 1 − +1 +3k we query a value in some Tr−1 +subtree. Similarly, within the chosen subtree there is again a relative frequency of at most +1 − +1 +3k to query within some Tr−2 subtree. Overall, since there are r levels of recursion, we +conclude that f ≤ (1 − +1 +3k )r. +Proving the second part of the claim required a more careful analysis. We define the +following method of promotion, depicted in Figure 6. In the (k, r)-tree we promote the +(only) explicit leaf to the root, and promote its sibling subtree by 1. Then we apply similar +promotions recursively within every (k, r − 1)-subtree. Finally, we promote the left-most +node within each (k, r − 1)-subtree that hangs as a right-subtree from the trunk to the parent +of this subtree. Denote the total average (weighted) promotion by pr. Note that it does not +matter if we promote the left-most nodes of the right subtrees before or after the recursive +promotions, because the total order on the items guarantees that there is only one value +STACS 2023 + +nodes +Tr-039:14 +Dynamic BSTs: Improved Lower Bounds for Greedy-Future +that can be put instead of every demoted trunk node, and the recursive promotions within a +specific subtree do not change the depth of its leftmost leaf. +The promotion of the explicit leaf of Tr saves a cost of k weighted by a factor (query +frequency) of +1 +2k . The promotion of the sibling subtree saves 1 weighted by a factor of +1 +2k . +The recursive promotions are pr−1 weighted by �k +i=1 +1 +2i (for all the k subtrees), and finally +the last promotions are technically negligible (as seen in the analysis below), but for the +sake of completeness we consider them in the analysis as well: promoting the left-most node +from each subtree saves (r − 1) + 1 = r since the leaf that we promote last is at depth r − 1 +within the recursive subtree, and this promotion is weighted by +1 +2r · �k−1 +i=2 +1 +2i (factor of +1 +2r +follows from Lemma 24). We get that: pr = k+1 +2k + pr−1 · +� +1 − +1 +2k +� ++ r +2r · 1 +2 +� +1 − +1 +2k−2 +� +. Then +by Lemma 20, with α = 1 − +1 +2k and γ = 1 +2(1 − +1 +2k−2 ), we get: +pr = (k + 1) · (1 − αr) + δ +, +δ ≡ αr · p0 + +2αγ +(2α − 1)2 · +� +αr − 1 +2r +� +− +γ +(2α − 1) · r +2r +It remains to show that 0 ≤ δ < αr. It is simple to see that δ = 0 for k = 2, because then +γ = 0 and p0 = 0 is the average weighted promotion in a tree with a single node. For k ≥ 3, +by the definition of α and γ we have that +2αγ +(2α−1)2 = (2k−1)(2k−4) +(2k−2)2 += 1 − +1 +2k−4+ 4 +2k ∈ ( 3 +4, 1) and +γ +2α−1 = 1 +2 − +1 +2k−2 ∈ [ 1 +3, 1 +2). Substituting these bounds and p0 = 0 into the formula for δ gives +δ < αr. Moreover, δ is positive since δ > 3 +4(αr − 1 +2r )− 1 +2 · r +2r = 3 +4(α− 1 +2)·�r−1 +i=0 αi · +� 1 +2 +�r−1−i− +r +2r+1 = 3(α− 1 +2 ) +2r+1 +· �r−1 +i=0 (2α)i − +r +2r+1 > 3(α− 1 +2 ) +2r+1 +· r − +r +2r+1 = (3( 1 +2 − 1 +2k ) − 1) · +r +2r+1 > 0 for k ≥ 3. +Note that indeed the gain from promoting the left-most node of each subtree is negligible, +since the effect is merely having γ ̸= 0, which only contributes 0 ≤ δ < αr < 1. +◀ +▶ Corollary 27. The average cost-per-query of GF on a strongly-stable sequence induced +by a (k, r)-tree is larger than the optimal cost by at least (k + 1) · +� +1 − +� +1 − +1 +2k +�r� +. On any +mixed-stable sequence, the difference is at least k · +� +1 − +� +1 − +1 +3k +�r� +. +We are ready to prove Theorem 5. +▶ Theorem 5. For every n ≥ 2 there exist sequences X ∈ [n]m such that cost(GF, X) = +cost(OPT, X) + Ω(m · lg lg n). Among these sequences, there exists a sequence whose length +is m = nΘ( +lg lg n +lg lg lg n ). (There exist other longer sequences too.) +Proof. Let X be the strongly-stable sequence induced by a (k, r)-tree Tr, and for simplicity +assume that the initial tree is Tr.6 By Lemma 25, n = (2+ +2 +k−1)kr−(1+ +2 +k−1) therefore lg lg n = +lg r + lg lg k + O(1).7 By Corollary 27, ˆc(GF, X) − ˆc(OPT, X) ≥ ∆ ≡ (k + 1) · (1 − (1 − 1 +2k )r). +By choosing r = 2k we get that ∆ = (k + 1) · (1 − (1 − +1 +2k )2k) ≈ (1 − 1 +e) · (k + 1).8 We +also get that lg lg n = k + lg lg k + O(1), therefore ∆ ≈ (1 − 1 +e) lg lg n and we conclude that +ˆc(GF, X) − ˆc(OPT, X) ≥ Ω(lg lg n). +By Lemma 13 the length of the atomic strongly-stable sequence of Tr is m = 2d(Tr), hence +m = 2rk by Lemma 24. By Lemma 25, n+(1+2/(k−1)) +2+2/(k−1) += kr = 2r lg k. Together we get that +m = 2rk = 2(r lg k)·(k/ lg k) = +� +n+(1+2/(k−1)) +2+2/(k−1) +�(k/ lg k) += nΘ( +lg lg n +lg lg lg n ). +◀ +6 We remove this assumption in Remark 35. +7 k ≥ 2 ⇒ kr ≤ n < 4kr ⇒ lg n = r lg k + c for c ∈ [0, 2), and so lg lg n = lg r + lg lg k + O(1). +8 The approximation is off by less than 10% for k ≥ 2. (60% and 20% for k = 0, 1 respectively.) + +Y. Sadeh and H. Kaplan +39:15 +▶ Remark 28. In the proof of Theorem 5, the sequence X does not have to be strongly- +stable, and any mixed-stable sequence X induced by a (k, r)-tree Tr works as well. Indeed, +Corollary 27 guarantees that ˆc(GF, X)−ˆc(OPT, X) ≥ ∆ for ∆ = k· +� +1− +� +1− 1 +3k +�r� +, and then +by choosing r = 3k we get that ∆ = Θ(k), and k = Θ(lg lg n), and m = 2Θ(rk) = nΘ( +lg lg n +lg lg lg n ). +▶ Remark 29. The choice of r = 2k in the proof of Theorem 5 maximizes our lower +bound on the additive gap cost(GF, X) − cost(OPT, X) (up to constants) for our (k, r)- +trees. Indeed, revisiting the proof, we have that ∆ and n are both functions of k and r, +and we need to choose r and k to maximize ∆ as a function of n. Note that ∆ = O(k) +regardless of r, and lg lg(n) = lg r + lg lg k + O(1). To simplify and eliminate a parameter +we define r = 2k · f(k) for some monotone function f. Now we get simplified relations: +∆ = (k+1)·(1−(1− 1 +2k )2kf(k)) ≈ (k+1)·(1−e−f(k)) and lg lg n = k+lg f(k)+lg lg k+O(1). +Consider the following two cases. +If f(k) = Ω(1): then ∃c ∈ R such that ∀k ≥ 1 : lg f(k) ≥ c, and therefore lg lg n = Ω(k), +written differently k = O(lg lg n), which yields ∆ = O(k) = O(lg lg n). +If f(k) = o(1): Being o(1) means that limk→∞ f(k) = 0, so for sufficiently large values +of k we can use the approximation ex ≈ 1 + x (that holds for small x) to get: ∆ ≈ +(k + 1) · f(k) = +k+1 +1/f(k). If +1 +f(k) grows faster than (k + 1), we get ∆ = O(1) which does +not even grow with n. Therefore +1 +f(k) is increasing, but at a sub-linear rate. Recall that +lg lg n = k − lg +1 +f(k) + lg k + O(1). Since +1 +f(k) is sub-linear, we get that k = Θ(lg lg n), +which yields ∆ = O(k) = O(lg lg n). +▶ Corollary 30. GF is not (1, O(m))-competitive. If the multiplicative term is 1, then the +additive term is at least Ω(m · lg lg n). +We have yet to analyze the cost of OPT on a strongly-stable sequence X corresponding to +a (k, r)-tree that produces the gap of Ω(m·lg lg n) in Theorem 5. Allegedly, if the cost is cheap, +say linear, we would get a large competitive ratio as well. However, by Theorem 22 we expect +a competitive ratio of at most 2, and therefore we can conclude without further analysis, +that cost(OPT, X) = Ω(m · lg lg n). In fact, we prove that cost(OPT, X) = Θ(m · +lg n +lg lg lg n). +It follows that the competitive ratio deteriorates when the additive gap increases. +▶ Lemma 31. Define the constants α ≡ 1 − +1 +2k and β ≡ �k +j=1 +j +2j + k+1 +2k . Let X be a +strongly-stable sequence induced by a (k, r)-tree. Then ˆc(GF, X) = 2k · β · (1 − αr) + αr. In +asymptotic terms: ˆc(GF, X) = Θ(2k · (1 − αr)). +Proof. We write a recurrence for the average cost, cr, of GF on the strongly-stable sequence +induced by Tr. We have c0 = 1, and +cr+1 = 1 + k +2k ++ +k +� +j=1 +j + cr +2j += +� +1 − 1 +2k +� +cr + +k +� +j=1 +j +2j + 1 + k +2k +≡ α · cr + β +( 1+k +2k +is due to the actual leaf, and the summation is the contribution of all the Tr subtrees.) +By Lemma 20 (with γ = 0), cr = +β +1−α(1 − αr) + αr · c0 = 2k · β(1 − αr) + αr. Since +α = 1 − +1 +2k ∈ [ 3 +4, 1) clearly αr < 1. Furthermore, β = Θ(1). To see this note that β only +depends on k. Denote β = β(k) and observe that: β(k+1)−β(k) = +� k+1 +2k+1 + k+2 +2k+1 +� +− k+1 +2k = +1 +2k+1 . +Therefore, β(k) = β(2) + �k +i=3 +� +β(i) − β(i − 1) +� += +� +1 +2 + 2 +4 + 3 +4 +� ++ �k +i=3 +1 +2i = 2 − +1 +2k , and +β(k) ∈ [ 7 +4, 2) ⇒ β = Θ(1). Because 2k · (1 − αr) ≥ 2k · (1 − α) = 1 > αr, we conclude that +cr = Θ(2k · (1 − αr)). +◀ +STACS 2023 + +39:16 +Dynamic BSTs: Improved Lower Bounds for Greedy-Future +▶ Lemma 32. Let X be a sequence from the family of sequences in Theorem 5, then +cost(OPT, X) = Θ(m · +lg n +lg lg lg n). +Proof. Let X be a strongly-stable sequence induced by querying a (k, r)-tree. We know that +1 +2ˆc(GF, X) < ˆc(OPT, X) ≤ ˆc(GF, X) where the lower-bound is by Theorem 22. Therefore, +ˆc(OPT, X) = Θ(2k · (1 − αr)) by Lemma 31. By Lemma 25, lg n = r · lg k + O(1), or r = +lg n−O(1) +lg k +. When we substitute r = 2k as in the proof of Theorem 5, we get that (1−αr) = Θ(1) +and 2k = r = lg n−O(1) +lg k += Θ( +lg n +lg lg lg n). Therefore, cost(OPT, X) = Θ(m · +lg n +lg lg lg n). +◀ +As a concluding remark, we recall that the Fr-tree is a (k, r)-tree for k = 2. If we +substitute k = 2 in the formula of Lemma 31 we get that α = 3 +4, β = 7 +4, and ˆc(GF, X) = +7 · (1 − (3/4)r) + (3/4)r. By Lemma 26, the average promotion is 3 · (1 − (3/4)r) (for k = 2, +we have δ = 0). These values are the strongly-stable analogues of Lemma 21, and can be +used to show a weaker lower bound of 7 +4, on the competitive ratio of GF. +4 +Conclusions and Open Questions +In this paper we gave improved lower bounds on the competitiveness of the Greedy Future +(GF) algorithm for serving a sequence of queries by a dynamic binary search tree (BST). In +contrast to many of the previous results on GF that are obtained using the geometric-view by +studying the equivalent Geometric Greedy (GG) algorithm, we used the standard “tree-view” +and the treap-based definition of GF. We showed that the competitive ratio of GF is at +least 2, and that there are sequences X ∈ [n]m for which the cost difference (additive gap) +between GF and OPT is Ω(m · lg lg n). These lower bounds enabled us to show that if GF +is approximately-monotone (Definition 8) with some constant c then c ≥ 2. Also, the lower +bounds show that the cost of GF on a sequence compared to its cost on its reverse, may +differ by a factor as close as we like to 2, or by a difference of Ω(m · lg lg n). In contrast, the +cost of OPT on a sequence compared to its reverse may differ by at most n. +Our results give new insights on the “tradeoff” between the additive term and the +multiplicative term in the competitiveness of GF, showing that the multiplicative term is +typically larger when the total cost of the algorithm on the sequence is smaller. Indeed, our +best multiplicative term is achieved for a sequence whose average cost per query is 6. This +tradeoff is not surprising since a fixed difference implies a larger ratio when the quantities are +small. It may be interesting to figure out if this tradeoff hints of some underlying property +of GF, or is just an artifact of our technique that requires high costs on average per query in +order to increase the additive gap between GF and OPT. +Clearly, these improved lower bounds still don’t settle the deeper question of whether GF +(and GG) is dynamically-optimal. Our techniques focused on a smaller family of sequences +which we named mixed-stable sequences, whereas “most” sequences are not stable. While +it is possible that an improved lower bound (larger than 2) can be found by a more clever +pattern of mixed-stable sequences, it seems more likely to be found by analyzing sequences +for which the tree maintained by GF is not static. In addition, we note that GF was not +investigated too deeply directly, as most of the work has been done in the geometric view +with respect to its counterpart GG. Therefore, studying other problems in tree-view may give +complementing insights. One such problem is the deque conjecture, which has been partially +settled for GG, in the case when deletions are only allowed on the minimum item [2]. + +Y. Sadeh and H. Kaplan +39:17 +References +1 +Parinya Chalermsook, Julia Chuzhoy, and Thatchaphol Saranurak. Pinning down the Strong +Wilber 1 Bound for Binary Search Trees. In Approximation, Randomization, and Combinatorial +Optimization. Algorithms and Techniques (APPROX/RANDOM), pages 33:1–33:21, 2020. +2 +Parinya Chalermsook, Mayank Goswami, László Kozma, Kurt Mehlhorn, and Thatchaphol +Saranurak. Greedy is an almost optimal deque. In 14th International Conference on Algorithms +and Data Structures (WADS), pages 152–165, 2015. +3 +Parinya Chalermsook, Mayank Goswami, László Kozma, Kurt Mehlhorn, and Thatchaphol +Saranurak. Pattern-avoiding access in binary search trees. 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In 17th Annual ACM-SIAM Symposium on Discrete +Algorithm (SODA), page 374–383, 2006. +22 +Robert Wilber. Lower bounds for accessing binary search trees with rotations. SIAM Journal +on Computing, 18(1):56–67, 1989. +STACS 2023 + +39:18 +Dynamic BSTs: Improved Lower Bounds for Greedy-Future +A +Appendix: Deferred Proofs and Discussions +A.1 +Enforcing a Stable Tree for GF +We describe how to restructure any initial tree, to a desired tree, when GF is considered. +The initial tree cannot simply be re-organized since GF updates the tree in a specific way +following each query. Moreover, when we are given a sequence X and add a prefix P to it, +denote the concatenation by P ◦ X, even if P enforces the desired tree when served alone, +serving P ◦X may give a different tree following P when X starts. The reason for this is that +GF restructures the tree while serving P according to future queries, therefore the existence +of X may affect its decisions while serving P. Nevertheless, the idea is to restructure the +tree top-down from the root, such that we “propagate” stability, as in Definition 11, over the +nodes that have already been fixed in their correct places. See Figure 7 for an example. +Figure 7 Example of enforcing a tree as detailed in Theorem 33. Consider the mixed-stable +sequence X = [11, 7, 1, 13, 9, 3, 11, 7, 5, 15, 9, 1, 11, 7, 3, 17, 9, 5]. Its desired tree T is the right-most +tree in the figure. Nodes 6, 4, 16 are weakly-stable biased towards their starred-edge (leading to +10, 2, 14 respectively), all other internal nodes are strongly-stable. Left-to-right: Initially, there are +no stabilized nodes (left). The first step queries only Y1 = [A] as if it was a leaf. Duplicated six times +and substituted for the pattern of a weakly-stable node, we get Z1 = [10, 10, 6, 6, 10, 6], stabilizing +{6, 10}. In the second step, the next nodes that are stabilized are {2, 4, 8, 12}. The weakly-stable +sequence of the subtrees is Y2 = [C, B, A]. Duplicated six times, and substituted 2, 2, 4, 4, 2, 4 for +A, 8, . . . , 8 for B and 12, . . . , 12 for C, we get: Z2 = [12, 8, 2, 12, 8, 2, 12, 8, 4, 12, 8, 4, 12, 8, 2, 12, 8, 4]. +Next Y3 = [F, D, A, G, E, B, F, D, C, G, E, A, F, D, B, G, E, C], and Z3 is the result of duplication +and substitution. Since A−F are leaves, only the substitution of G requires the non-trivial pattern +(14, 14, 16, 16, 14, 16). This stabilizes the nodes {14, 16} and is the last step of this example. In total, +the enforcing sequence is Z1 ◦ Z2 ◦ Z3 (◦ for concatenation). In general, there may be more steps, +each stabilizes at least one more node until all inner nodes are stable. +▶ Theorem 33. Let X be a mixed-stable sequence that corresponds to a tree T.9 There is a +sequence S(X) such that when GF serves the concatenation S(X) followed by X, then when +it finishes serving S(X) its current tree is T regardless of the initial tree T0. We refer to S(X) +as the enforcing sequence of X. Additionally, if X is the atomic sequence corresponding to T +(Definition 12), then |S(X)| < 3n · |X|. If X is strongly-stable or weakly-stable (and atomic), +then |S(X)| < 2|X| and |S(X)| < 3|X| respectively. +Proof. We construct S(X) in steps. Initially, the tree of GF (which is the initial tree T0) and +the desired mixed-stable tree T may be completely different. Each step of the construction +adds a sequence of queries to S(X) that extends a rooted and connected subtree that belongs +9 The type of stability of each node of T can be deduced from X. + +Y. Sadeh and H. Kaplan +39:19 +both to T and to the current tree of GF and will not change subsequently. We regard nodes +that already joined this subtree as stabilized. Once all inner nodes have been stabilized then +S(X) is complete and the tree of GF is exactly T. Every stabilized node remains stable +with respect to the continuation of the sequence as in Definition 11. See Figure 7 for an +illustration. We first describe how to stabilize the desired root (of T), r, which is the base +of the construction. We emphasize that stabilizing a weakly-stable node also stabilizes its +favored-child (Definition 11). We split into three cases. +1. If there is only a single node, then it is r which is also already the root. We do nothing. +2. If r is strongly-stable in T: We query [r, r] to make it the root. The second query +guarantees that r is placed at the root after the first query. +3. If r is weakly-stable in T: Let z be r’s favored-child. We add the queries [z, z, r, r, z, r] +to S(X). This stabilizes r and z, and by the end of these six queries, r is the root and +z is its child. To see why this happens, consider how GF works: The second query to +z guarantees that it is placed at the root after the first query. The third query touches +both z (the root) and r (queried), and due to the fourth and fifth queries places r at +the root and z as its child. The purpose of the sixth query is to ensure that z does not +become the root due to future queries when the fifth query is processed. +Assume that we already have a tree with a connected subtree of stabilized internal nodes. +The subtrees that hang off the stable nodes are not empty since stable nodes are internal +nodes. If each of these subtrees is a leaf then we are done and S(X) is complete. Otherwise, +we stabilize the root of every subtree that is not a leaf as we describe next. Recall that if the +root of a subtree is weakly-stable, we also stabilize its favored-child. +For convenience, denote the index of the current step by ℓ. We regard each unstabilized +subtree as a leaf, and generate an atomic mixed-stable sequence over the current stabilized +connected subtree, according to the stability types of the inner nodes. Denote this sequence +by Yℓ. Note that Yℓ is a sequence of the leaves that correspond to the unstabilized subtrees. +We derive from Yℓ the stabilizing sequence, Zℓ, for the new nodes, by repeating Yℓ six times, +and replacing each leaf in Yℓ +6 by an appropriate node from the subtree that it represents. +This replacement is done as follows. If the root r of the subtree is a leaf or a strongly-stable +node then we simply access r. If r is weakly-stable and its favored-child is z then we +replace each query of the leaf (subtree) by the next query of the sequence z, z, r, r, z, r, while +cyclically repeating it. Since Yℓ was repeated six times, the sequence z, z, r, r, z, r repeats +an integral number of times in Zℓ. Because Zℓ is based on the mixed-stable sequence Yℓ, +the already stabilized nodes remain stable. Only the subtrees hanging off them are affected. +The restriction of Zℓ to each hanging subtree is either z, z, r, r, z, r or r, r, r, r, r, r so by an +argument analogous to the one in the base case the root of each hanging subtree is stabilized, +as well as each favored-child of a weakly-stable root. +We repeat stabilizing steps until all the inner nodes of T have been stabilized. If the last +step is d, then S(X) = Z1 ◦ Z2 ◦ . . . ◦ Zd where ◦ denotes concatenation. We emphasize that +when GF serves S(X) ◦ X, by the end of serving S(X) its tree is indeed T, because we can +think of X as just another step of the stabilization process, in which all the subtrees are leaves. +It remains to analyze the length of S(X). The length of any Yi is at most that of X. +Every step stabilizes at least one inner node, hence there are at most n−1 +2 +steps (the rest +n+1 +2 +nodes are leaves), so we get that |S(X)| = 6 �d +i=1 |Yi| ≤ 6 · n−1 +2 +· |X| < 3n · |X|. +For the refined analysis of the length of S(X) in case X is strongly-stable, or weakly-stable, +we investigate the relation between |Yi| and |Yi+1|. For convenience we define Yd+1 = X. +By Lemma 13, |Yi| = 2ai · 3bi, where ai = maxleaf u a(u) and bi = maxleaf u b(u). Observe +that when we extend the stability from a node v which is a leaf of the stabilized subtree at +STACS 2023 + +39:20 +Dynamic BSTs: Improved Lower Bounds for Greedy-Future +step i, one or two levels deeper, then we have for every new stabilized leaf descendant u of +v that either a(u) = a(v) + 1 and b(u) = b(v), or a(u) = a(v) and b(u) = b(v) + 1. However, +the lcm of the preiods of all leaves of Yi+1 may be affected by two different branches, thus +allowing any of the combinations of ai ≤ ai+1 ≤ ai + 1 and bi ≤ bi+1 ≤ bi + 1. So we get +that |Yi| divides |Yi+1| and |Yi| ≤ |Yi+1| ≤ 6|Yi|. In the general case of mixed-stability it +could be that the length grows initially six-fold (starting from the second step, two differ- +ent branches might each increase a and b, respectively) while the second half of the steps +satisfies |Yd/2+1| = |Yd/2+2| = . . . = |X| (another branch that “mixes” the increase in a and +b “catches up” with the lcm), so |S(X)| = Θ(n · |X|) is possible. As an example, revisit +Figure 7: there we have |Y1| = 1, |Y2| = 3, |Y3| = 18 and |Y4| = |X| = 18. However, when +X is strongly-stable, we have |Yi+1| = 2|Yi| for every i because bi = 0 (Lemma 13), and +ai+1 = ai + 1 for every i. Therefore, |S(X)| = 6 �d +i=1 |Yi| = 6 �d +i=1 +|X| +2i < 6|X|. We can +optimize further by noting that we can define Zi to be only twice longer than Yi (no need for +six repetitions). So in fact we can define S(X) for strongly-stable X such that |S(X)| < 2|X|. +If X is weakly-stable, similarly |Yi+1| = 3|Yi| because for every i we have ai = 0 (Lemma 13) +and bi+1 = bi + 1, therefore |S(X)| = 6 �d +i=1 |Yi| = 6 �d +i=1 +|X| +3i < 3|X|. +◀ +▶ Theorem 34. For any tree T there is a sequence S(T) such that for any suffix of queries +Y , when GF serves S(T) ◦ Y , its tree when is it done with the last query of S(T) is T. +Proof. We rely on the ideas from the proof of Theorem 33. To generate an oblivious enforcing +prefix, we concatenate several enforcing sequences, each enforcing higher nodes in the tree. +Define T [1] ≡ T and T [i+1] is the tree T [i] stripped of all of its leaves, until the final tree T [h] +contains only the root. For each tree T [i], denote by Xi its corresponding strongly-stable +sequence Xi, and by S1 the enforcing sequence of X1. Note that if T [i] is not full, we relax +the definition of the corresponding strongly-stable sequence and instead of querying a missing +leaf we query its unary parent. This modification works as-well because we can imagine that +the query proceeds to the missing leaf, which would anyway remain below the parent. We +claim that S(T) ≡ S1 ◦ X2 ◦ . . . ◦ Xh satisfies the theorem. Indeed, S1 enforces the desired +tree, and in particular the position of the leaves. Following S1, all the nodes except for the +leaves are touched again, so these leaves can never become parents, regardless of the suffix Y . +The argument holds similarly for the following steps, and since the structure of T is already +in place, it suffices to use Xi instead of their enforcing sequences. +◀ +Adding a prefix to our sequence may affect the competitive ratio. However, once we fixed +the stable tree, we can repeat the corresponding stable sequence to “amplify” the original +competitive ratio making the effect of the prefix negligible. One difficulty raised by repetitions +is when we care about the length of the sequence in our claim. This is the case in Theorem 5 +where we claim the existence of a sequence of length nΘ( +lg lg n +lg lg lg n ). In the proof of this theorem +we assumed for simplicity that we can choose the initial tree. The following remark shows +that indeed we can start with an arbitrary initial tree without weakening the theorem. +▶ Remark 35. Let X be an atomic mixed-stable sequence used to prove Theorem 5. Consider +the sequence Z = S(X) ◦ Xn, where S(X) is the prefix (guaranteed by Theorem 33) that is +enforcing the desired “initial” tree T, Xn are n repetitions of X, and ◦ represents concaten- +ation. By Theorem 33, n|X| ≤ |Z| < 4n|X| therefore we have: |Z| = Θ(n|X|) = nΘ( +lg lg n +lg lg lg n ) +(the second equality is by Theorem 5). Since after processing S(X) the tree of GF is fixed: +cost(GF, Z, T0)−cost(OPT, Z, T0) ≥ n·(cost(GF, X, T)−cost(OPT, X, T)). By the proof of +Theorem 5 and Remark 28: cost(GF, X, T) − cost(OPT, X, T) = Ω(|X| · lg lg n), and putting +everything together we get that: cost(GF, Z, T0) − cost(OPT, Z, T0) = Ω(|Z| · lg lg n). Note + +Y. Sadeh and H. Kaplan +39:21 +that if X is strongly-stable, or weakly-stable, then it suffices to define Z = S(X) ◦ X without +repetitions and we get that |Z| = Θ(|X|), and the rest of the arguments remain the same. +A.2 +Omitted Proofs +In this subsection we restate and prove Lemmas and Theorems that were omitted from the +main text. For convenience, we restate the claims in their original numbering. +The proof of Theorem 22 makes use of Wilber’s first bound [22]. We use the original +presentation of this bound which is a bit tighter than later simplified versions such as [14]. +▶ Definition 36 (Wilber’s First Bound [22]). Let X be a sequence of queries, and let T be a +static reference tree such that every query of X is in a leaf of T. An alternation at an inner +node u of T is defined to be two queries closest in time such that one accesses either the left +or right subtree of u and the other accesses the other subtree of u. Define ALT(u) to be the +number of alternations at node u. Then: cost(OPT, X) ≥ m + 1 +2 +� +inner u∈T ALT(u). +▶ Theorem 22. Let X be a mixed-stable sequence and let T be the tree that corresponds to +it. Then cost(GF, X, T) < c · cost(OPT, X, T) for c = 5 +2. If X is strongly-stable, then c = 2. +Proof. We use the tree that corresponds to the mixed-stable sequence as the reference tree +for Wilber’s first bound. Arithmetic manipulations will yield an expression that we can tie +to the cost of GF, according to the claim. +Let X be a mixed-stable sequence, with a corresponding tree T. Let S be the set of +values that are in the leaves of T, and let U be the set of inner nodes, |U| = n−1 +2 . We also +denote by A(i) the set of proper ancestors of i. By the definition of the cost of a static tree, +we know that ˆc(GF, X) = � +i∈S (d(i) + 1) · f(i) where d(i) is the depth of i and f(i) is the +frequency of accessing i. We extend f(u) to refer to the frequency of visiting any node u. +Note that f(u) = � +i∈S∧u∈A(i) f(i) and that � +i∈S f(i) = 1. +Now consider Wilber’s bound for X, with T as the reference tree. We can use T as the +reference tree since X only accesses leaves of T, by definition. We also denote αu ≡ ALT (u)+1 +f(u)·m +(ALT(u) is defined in Defintion 36, and note that 0 ≤ ALT(u) ≤ f(u) · m − 1). We have +αu ∈ (0, 1], where αu = 1 corresponds to fully alternating accesses to the subtree rooted +at u. The lower bound is cost(OPT, X) ≥ m + 1 +2 +� +u∈U (ALT(u) + 1) − |U| +2 = m +2 + m +2 (1 + +� +u∈U αu · f(u)) − n−1 +4 += +� m +2 − n−1 +4 +� ++ m +2 +� +i∈S (1 + � +u∈A(i) αu)f(i). Let α ≤ minu∈U αu, +we get that ˆc(OPT, X) ≥ +� 1 +2 − n−1 +4m +� ++ α +2 +� +i∈S (d(i) + 1) · f(i) = α +2 ˆc(GF, X) + +� 1 +2 − n−1 +4m +� +where the equality holds since GF maintains a static tree. Thus ˆc(GF, X) ≤ 2 +α · ˆc(OPT, X)− +1 +α +� +1 − n−1 +2m +� +< 2 +α · ˆc(OPT, X). +In order to choose a suitable α, recall that a strongly-stable node u has a coefficient of +αu = 1, which means that for strongly-stable sequences, in which all inner nodes are stable, +we can pick α = 1 and conclude that ˆc(GF, X) < 2 · ˆc(OPT, X). If u is a weakly-stable node, +then its coefficient is αu = 2 +3. So for a mixed-stable sequence we can naively pick α = 2 +3, +resulting in ˆc(GF, X) < 3 · ˆc(OPT, X). +In order to improve from 3 to 5 +2, we observe that by definition, every weakly-stable node has +a strongly-stable child. Let u be a weakly-stable node and let w be its (strongly-stable) favored- +child (recall Definition 11). Since ALT(u) = ALT(w) (by definition of the access pattern in u), +we can present Wilber’s bound differently, summing (ALT(u)+1)·(1+β)+(ALT(w)+1)·(1−β) +instead of (ALT(u)+1)+(ALT(w)+1). We get modified coefficients α′ +u = (ALT (u)+1)·(1+β) +m·f(u) += +αu · (1 + β) = 2(1+β) +3 +and similarly α′ +w = αw(1 − β) = (1 − β). Choosing β = 1 +5 balances the +coefficients: α′ +u = α′ +w = 4 +5. Now we can choose α = 4 +5, and get ˆc(GF, X) < 5 +2 · ˆc(OPT, X) +for mixed-stable sequences. +◀ +STACS 2023 + +39:22 +Dynamic BSTs: Improved Lower Bounds for Greedy-Future +▶ Theorem 6. For any ϵ > 0 there exists a sequence X with a subsequence (not necessarily +consecutive) X′ ⊆ X such that cost(GF, X′) ≥ (2 − ϵ) · cost(GF, X). +There exists a sequence Y with a subsequence (not necessarily consecutive) Y ′ ⊆ Y such +that cost(GF, Y ′) − cost(GF, Y ) = Ω(m · lg lg n). +Proof. Denote the initial tree by T0. Let Z be the weakly-stable sequence used for proving +Theorem 4. Let TP be the tree that corresponds to Z and TQ the optimized tree, in which +the leaves are promoted as in Lemma 21. Let P and Q be the sequences that enforce TP +and TQ by Theorem 34, respectively. Note that ϵ determines Z, P and Q since it tells us +how close to a ratio of 2 we need to get. +Revisit Figure 5 to see the (recursive) structures of TP (on the left) and TQ (on the right, +post-promotions). Observe that TP remains static when GF serves Z with it, by definition. +Moreover, TQ remains static when GF serves Z with it. Indeed, let r be the root of TQ. Z +queries the item in r every third access and the other accesses are alternating between its left +and right subtrees, hence r remains the root of TQ. The rest of TQ remains static recursively. +Define X = P ◦ Q ◦ Zk for a large k, and X′ = P ◦ Zk ⊂ X (◦ for concatena- +tion). Since GF does not change TP and TQ while serving Z we get that cost(GF,X′) +cost(GF,X) = +cost(GF,P,T0)+k·cost(GF,Z,TP ) +cost(GF,P ◦Q,T0)+k·cost(GF,Z,TQ). This ratio approaches cost(GF,Z,TP ) +cost(GF,Z,TQ) for large enough k, and +since Tp and TQ are exactly the trees used in the proof of Lemma 21, we conclude that we can +make the resulting ratio as close to 2 as we like (choosing Z, P, Q according to the desired ϵ). +The proof for Y and Y ′ is similar. We define Z to be the strongly-stable sequence used +in Theorem 5, and define the appropriate tree-enforcing sequences P and Q by Theorem 34. +Revisit Figure 6 to see the (recursive) structures of TP (on the left) and TQ (on the right, +post-promotions). We set Y = P ◦ Q ◦ Zk and Y ′ = P ◦ Zk. The main difference is in the +argument of why GF does not change TQ when serving Z (TP is static by definition). For this, +observe that the root of TQ, denote it r, is the left-most leaf in the right subtree of TP . This +means that the access pattern at r is alternating between its left subtree and its right subtree +including itself, thus again we conclude that r remains at the root of TQ, and the rest of TQ +remains static recursively. Therefore, cost(GF, Y ′) − cost(GF, Y ) ≥ k · +� +cost(GF, Z, TP ) − +cost(GF, Z, TQ)) − cost(GF, P ◦ Q, T0) = Ω(m · lg lg n) for large enough k. +◀ +▶ Theorem 7. Let S be a sequence, we define rev(S) to be the sequence S in reverse. For +any ϵ > 0 there exists a sequence X such that cost(GF, rev(X)) ≥ (2 − ϵ) · cost(GF, X). +There exists a sequence Y such that cost(GF, rev(Y )) − cost(GF, Y ) = Ω(m · lg lg n). +Proof. The proof is similar to that of Theorem 6, and we define Z, T0, P, TP , Q and TQ the +same way. Here we define X = Q ◦ (rev(Z))k+1 ◦ rev(P) and Y = Q ◦ (rev(Z))k+1 ◦ rev(P), +for a large k. Recall that Z is different between X (by Theorem 4) and Y (by Theorem 5). +We claim that TQ remains static when GF serves rev(Z) over it, rather than Z, by the +same argument as in the proof of Theorem 6, because the interleaving pattern in the root +is preserved under reversal. Moreover, cost(GF, rev(Z), TQ) = cost(GF, Z, TQ) because the +cost on a static tree depends only on the access frequencies. Putting everything together, we +get: +cost(GF,rev(X)) +cost(GF,X) += +cost(GF,P,T0)+k·cost(GF,Z,TP )+cost(GF,Z◦rev(Q),TP ) +cost(GF,Q,T0)+k·cost(GF,rev(Z),TQ)+cost(GF,rev(Z)◦rev(P ),TQ). Note that +the suffix contains one repetition of Z so that the rest of it (rev(P) or rev(Q)) does not +affect the restructuring decisions of GF during the earlier repetitions of Z. The limit of this +ratio for large k is cost(GF,Z,TP ) +cost(GF,Z,TQ). We finish the argument as in the proof of Theorem 6. +In the case of Y , we get that cost(GF, Y ′) − cost(GF, Y ) ≥ k · +� +cost(GF, Z, TP ) − +cost(GF, Z, TQ)) − +� +cost(GF, Q, T0) + cost(GF, rev(Z) ◦ rev(P), TQ) +� += Ω(m · lg lg n) for +large enough k. +◀ + diff --git a/8dE1T4oBgHgl3EQfUAND/content/tmp_files/load_file.txt b/8dE1T4oBgHgl3EQfUAND/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..88126609f0f040d07debf40495b48faa7f17ca09 --- /dev/null +++ b/8dE1T4oBgHgl3EQfUAND/content/tmp_files/load_file.txt @@ -0,0 +1,915 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf,len=914 +page_content='Dynamic Binary Search Trees: Improved Lower Bounds for the Greedy-Future Algorithm Yaniv Sadeh � � Tel Aviv University, Israel Haim Kaplan � � Tel Aviv University, Israel Abstract Binary search trees (BSTs) are one of the most basic and widely used data structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' The best static tree for serving a sequence of queries (searches) can be computed by dynamic programming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' In contrast, when the BSTs are allowed to be dynamic (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' change by rotations between searches), we still do not know how to compute the optimal algorithm (OPT) for a given sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' One of the candidate algorithms whose serving cost is suspected to be optimal up-to a (multiplicative) constant factor is known by the name Greedy Future (GF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' In an equivalent geometric way of representing queries on BSTs, GF is in fact equivalent to another algorithm called Geometric Greedy (GG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Most of the results on GF are obtained using the geometric model and the study of GG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Despite this intensive recent fruitful research, the best lower bound we have on the competitive ratio of GF is 4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Furthermore, it has been conjectured that the additive gap between the cost of GF and OPT is only linear in the number of queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' In this paper we prove a lower bound of 2 on the competitive ratio of GF, and we prove that the additive gap between the cost of GF and OPT can be Ω(m · log log n) where n is the number of items in the tree and m is the number of queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' 2012 ACM Subject Classification Theory of computation → Online algorithms;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Theory of compu- tation → Sorting and searching Keywords and phrases Binary Search Trees, Greedy Future, Geometric Greedy, Lower Bounds, Dynamic Optimality Conjecture Digital Object Identifier 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content='4230/LIPIcs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content='STACS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content='2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content='39 Funding The work of the authors is partially supported by Israel Science Foundation (ISF) grant number 1595-19, German Science Foundation (GIF) grant number 1367 and the Blavatnik research fund at Tel Aviv University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' © Yaniv Sadeh and Haim Kaplan;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' licensed under Creative Commons License CC-BY 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content='0 40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Editors: Petra Berenbrink, Mamadou Moustapha Kanté, Patricia Bouyer, and Anuj Dawar;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Article No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' 39;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' 39:1–39:22 Leibniz International Proceedings in Informatics Schloss Dagstuhl – Leibniz-Zentrum für Informatik, Dagstuhl Publishing, Germany arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content='03084v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content='DS] 8 Jan 2023 39:2 Dynamic BSTs: Improved Lower Bounds for Greedy-Future 1 Introduction Binary search trees (BSTs) are one of the most basic and widely used data-structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' They are used to store a sorted set of keys from a totally ordered universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Traversing BSTs is usually done by using a single pointer, initially pointing to the root, and moving to the left or right child according to the order of the searched key and the key of the item at the current node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Therefore, we typically define the cost1 of a search to be the length of the search path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' The data structure itself may be static, or change dynamically throughout time, in response to insertions and deletions of items, and possibly even restructured during queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Static BSTs are well understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' One can guarantee that the longest path from the root to a leaf is of length O(log n) if the number of keys is n, by using a balanced tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' If the access sequence is known in advance (in fact only the frequency of accesses of each key matters) then an O(n2) time algorithm computing the optimal static tree for the particular set of frequencies was given by Knuth [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' It is also notable that the lower bound on the cost when the known frequencies are ⃗f = [f1, f2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' , fn] and the number of queries is m, is Ω(m · H(⃗f)) where H(⃗f) = �n i=1 fi log 1 fi is the entropy function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' A simple way with O(n log n) running time to construct a near-optimal static (centroid) tree whose cost is O(m · H(⃗f)), has been described by Mehlhorn [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' The running time has been improved to O(n) by Fredman [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' In contrast to the static case, the dynamic case is less understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' One can, of course, serve the sequence with a static tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' But, for many sequences we must change the structure of the tree as we make the searches in order to be efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' For example, the requested items may be different in different parts of the sequence so a different set of items has to be placed near the root during different parts of the sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Restructuring is done by rotations that maintain the symmetric order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' When rotations are allowed, the cost is defined to be the size of the subtree that contains the search path and all edges which we rotate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Here, we assume that the set of values stored in the tree does not change (no insertions or deletions), yet restructuring the tree is allowed to speed up future searches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' One famous dynamic algorithm for doing this is the Splay algorithm of Sleator and Tarjan [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' After each query, the splay algorithm moves the queried item to the root of the tree, according to three simple rules called zig-zag, zig-zig and zig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' The splay algorithm is efficient in the sense that it is able to exploit the structure of many families of sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' In particular splay is proven to be as good as the static optimum (up to a constant factor), which also implies that the cost of splay on any given sequence is at most O(log n) times the (dynamic) optimum cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Sleator and Tarjan conjectured that splay is in fact dynamically-optimal, meaning that its cost is like the cost of an optimal algorithm that knows the whole sequence of queries in advance, up to some constant factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' However, this dynamic-optimality conjecture of splay is still open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' In fact, it is open whether there is any dynamically-optimal online binary search tree algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' The best competitive ratio achievable to date is O(log log n), and it is obtained by Tango [8], Multi-splay [21] and Chain-splay [11] trees, and a geometric divide-and-conquer approach of [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' While seeking for (better) guaranteed competitiveness, other dynamic algorithms were considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' A promising candidate was independently proposed by Lucas [16] and Munro [18], which is now commonly referred to as Greedy Future, henceforth: GF in short.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' As its name suggests, GF is a greedy algorithm that rearranges the nodes on the path from the root to the current queried item as a treap whose priorities are according to the future accesses2 1 Our cost model is formally defined in Definition 1, in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' 2 Each item in a treap has two keys: value and priority.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' The treap is a binary search tree with respect to Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Sadeh and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Kaplan 39:3 (as this paper deals with analyzing GF, we detail it formally in Algorithm 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Note that unlike splay, GF, by definition, is required to know the future in order to restructure the tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Surprisingly however, Demaine et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' [7] showed that one can simulate GF without knowing the future by a hierarchy of split-trees while losing only a constant factor in performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Additionally, [7] presented a geometric view of an algorithm serving queries by a dynamic binary search tree using a two dimensional grid on which we mark the sequence as well as the items accessed by the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' In this presentation there is yet another natural promising candidate for dynamic optimality, which is commonly known as Geometric Greedy and sometimes simply Greedy, which we shall refer to as GG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' [7] showed that GG is in fact the same algorithm as GF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' The geometric view proved useful to obtain new results regarding GG and hence GF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Fox [9] proved that an access-lemma that is analogues to the so called access-lemma of splay trees holds for GG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' From this follows that most of the nice properties that hold for splay also hold for GF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' In particular, it follows that GF is O(log n) competitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Chalermsook et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' [3] analyzed upper bounds on the cost of GG for access patterns which are permutations, and in particular found that for highly structured permutations, which they called k-decomposable, the cost is n · 2α(n)O(k) where α(n) is the inverse-Ackermann function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Chalermsook et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' [5] study special access patterns that belong to a broader family of pattern-avoiding permutations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' See [4] for a survey of currently known properties of greedy and splay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Our Contributions: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' It is known that GF is not exactly optimal, but it is conjectured, like splay, to be optimal up to a constant factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' In fact, it has been even more strongly conjectured by Demaine et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' [7] to be optimal up to an additive O(m) term, and possibly even exactly m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Kozma [14] refuted the second part and gave a specific sequence for which this additive gap is m + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' In this paper we refute the linear gap conjecture and show a family of sequences for which the additive gap is at least Ω(m log log n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' The largest lower bound on the competitive ratio of GF is 4 3 by Demaine et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' They show a family of sequences on which after an initial query, the optimum pays 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content='5 on average per query while GF pays 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content='3 We describe a technique that allows us to improve this lower bound to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' We note that the best known lower bound on the competitive ratio of splay is 2 (see [15, Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content='5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' In both cases, the construction requires a rather large number of items (large n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Based on the multiplicative lower bound described above we show the following two interesting properties of GF: (1) There are sequences X such that the cost of GF on the reverse sequence is twice larger than the cost of GF on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' (2) There are sequences X such that we can remove some queries from them and get a subsequence X′, such that the cost of GF on X′ is twice larger than the cost of GF on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Based on the additive lower bound described above, the two properties of GF in the previous bullet also hold if we replace the (multiplicative) relation of “costs twice more” by the (additive) relation “costs Ω(m log log n) more”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' We study subsequences and reversal (contributions 3-4) since any dynamically-optimal the values of the items and a heap with respect to their priorities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' That is, the priority of an item is no larger than the priorities of its children.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' In our case, the priorities are deterministically defined by future requests in a way that we define precisely in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' 3 Reddmann [19] found an example in which the cost ratio between GF and the optimum is 26 17 ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content='53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' But this is for one particular sequence of a fixed length so it does not rule out any competitive ratio if we allow an additive constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' STACS 2023 39:4 Dynamic BSTs: Improved Lower Bounds for Greedy-Future algorithm A must have a “nice” behavior in these cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Concretely, A must satisfy the approximately-monotone property (Definition 8) which states that there is a fixed constant c such that the cost of A on any subsequence of any sequence is never more than c times the cost on the whole sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' As for reversal, the optimum can process a sequence and its reversal with similar costs up to a difference of n, thus any dynamically-optimal algorithm must be able to do so with costs that differ by at most a constant factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' We discuss this motivation in more detail in Section 3 (right after stating Theorem 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Our contributions are all based on the same technique, which is quite simple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' We enforce GF to maintain a static tree and only query the leaves of this tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Although being dynamic in general, there are some access-patterns that cause GF not to change the tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' By studying these patterns, we can study GF on a static tree, and the analysis of its cost simplifies to the weighted-average of the depth of the queries (weighted by frequency).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' To lower-bound the gap between GF and OPT, we analyze the average cost that can be saved by promoting the items in the leaves to locations closer to the root.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Note that any other item can be placed further away from the root since it is never queried by the sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' 2 Model In this section we describe the model which we use, and define our notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' First, we note that throughout the paper lg x is used to denote the base two logarithm of x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' We consider a totally ordered universe of (fixed size) n items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' For simplicity, one may think of the values {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' , n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' The items are organized in some initial BST which we denote by T0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Then, a sequence of queries, denoted by X = [x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' , xm], is given, one query at a time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' We reserve m to denote the length of the sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' The tree before serving xt is denoted by Tt−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' An algorithm has to find the queried value xt, by traversing Tt−1 from its root.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' After finding xt, the algorithm is allowed to re-structure Tt−1 to get Tt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' We define the cost of the algorithm at time t to be the total number of nodes that were touched at time t, both on the path to xt and for restructuring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' The cost of an algorithm for the whole sequence is simply the sum of its costs over all times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' We define it formally below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' ▶ Definition 1 (Cost).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Let X be a sequence of queries, and let T0 be an initial tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Let A be an algorithm that serves X and let Tt be the tree that A has after serving xt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Let Pt be the set of nodes on the path from the root to xt in Tt−1 and let Ut be the set of nodes of the minimal subtree that contains all the edges that were rotated by A to transform Tt−1 to Tt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Then the cost of A for serving X at time t is |Pt ∪ Ut|, and the cost of A for serving X is the sum of costs over t = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' , m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' We denote the cost of A to serve X starting with T0 by cost(A, X, T0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' We denote the average cost per query by ˆc(A, X, T0) = cost(A,X,T0) m .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' When T0 is clear from the context, or immaterial, we write cost(A, X) and ˆc(A, X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' ▶ Definition 2 (Depth).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Let T be a tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' The depth of a node v ∈ T, denoted by d(v), is the number of edges in the path from the root to v (in particular d(root) = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Note that the cost of querying v (without restructuring) is d(v) + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' We also define the depth of the tree, denoted by d(T), as the maximum depth of a node in T, that is d(T) = maxv∈T d(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' ▶ Definition 3 (Competitiveness).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' We say that an algorithm A is (α, β)-competitive for initial tree T0 if for any sequence of queries X, it holds that cost(A, X, T0) ≤ α·cost(OPT, X, T0)+β where OPT is a best algorithm to serve X given T0 (with full knowledge of X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' When we do not specify T0 we mean that the relation holds for all initial trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' We refer to α as the multiplicative term and to β as the additive term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' For ease of language, we regard the multiplicative term as the competitive ratio, and also write “the competitive ratio of” instead Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Sadeh and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Kaplan 39:5 of “the multiplicative term of the competitiveness of”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' In such cases, we assume that the additive term is o(m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' It is easiest to think of β = O(n) while assuming that m = ω(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' To conclude this section, we give a precise description of the GF algorithm, in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' We emphasize that its implementation is complex and probably would not be good in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' However, its main benefit is its theoretical value, as a candidate for dynamic optimality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Should it be proven to be dynamically-optimal, then we would get a better understanding of the problem and also a stepping-stone to analyze simpler algorithms, such as splay, in comparison to GF rather than against some “vague” optimum that depends on the sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Algorithm 1 GreedyFuture (GF) Algorithm Input: A sequence of queries X ∈ [n]m and an initial BST T0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' We restructure Tt−1 to Tt after serving the request xt with Tt−1 for t = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' , m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Function Restructure(query value v, current tree Tt−1, future accesses X′): Let v1 < v2 < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' < vk be the nodes on the path from the root of Tt−1 to the queried value v (including v and the root).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' We also define v0 = −∞ and vk+1 = +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Denote the subtrees hanging off this path by R0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' , Rk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' For each i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' , k, let τ(vi) be the index of the first appearance of a query of a value x ∈ (vi−1, vi+1) in X′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Restructure the nodes v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' , vk as a treap: maintain a BST ordering, while the heap’s priorities are set to be the τ values, where the root’s τ is smallest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Tie-break arbitrarily, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' in favor of smaller values, or smaller depth prior to restructuring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Then, hang the subtrees R0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' , Rk unchanged at their appropriate locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' The resulting tree is Tt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' 3 Stable Sequences and Lower Bounds In this section we properly define the family of stable sequences (Definition 11) for which the tree maintained by GF is never changed (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' the access path of the current query is a treap with respect to the suffix of the sequence).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' To prove our lower bounds we use such sequences in which only the items at the leaves of GF are requested, and the internal nodes cause some extra cost that OPT avoids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' We use a natural way to represent such sequences as trees, and use this representation to prove the following lower bounds, which are the main results of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' ▶ Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' If GF is (c, d)-competitive where the additive term d is sublinear in the length of the sequence, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' d = o(m), then c ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' ▶ Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' For every n ≥ 2 there exist sequences X ∈ [n]m such that cost(GF, X) = cost(OPT, X) + Ω(m · lg lg n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Among these sequences, there exists a sequence whose length is m = nΘ( lg lg n lg lg lg n ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' (There exist other longer sequences too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=') Theorems 4 and 5 enable us to prove the following two theorems, proven in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' ▶ Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' For any ϵ > 0 there exists a sequence X with a subsequence (not necessarily consecutive) X′ ⊆ X such that cost(GF, X′) ≥ (2 − ϵ) · cost(GF, X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' There exists a sequence Y with a subsequence (not necessarily consecutive) Y ′ ⊆ Y such that cost(GF, Y ′) − cost(GF, Y ) = Ω(m · lg lg n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' STACS 2023 39:6 Dynamic BSTs: Improved Lower Bounds for Greedy-Future ▶ Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Let S be a sequence, we define rev(S) to be the sequence S in reverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' For any ϵ > 0 there exists a sequence X such that cost(GF, rev(X)) ≥ (2 − ϵ) · cost(GF, X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' There exists a sequence Y such that cost(GF, rev(Y )) − cost(GF, Y ) = Ω(m · lg lg n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' The motivation for studying subsequences (Theorem 6) is the fact that OPT always saves costs when queries are removed from its sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Formally, if X′ ⊆ X, then cost(OPT, X′) ≤ cost(OPT, X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Indeed, OPT can serve X′ by simulating a run on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' More generally, this relation of costs when comparing a sequence to a subsequence of it, is an important property which even has a name: ▶ Definition 8 (Approximate-monotonicity [12, 15]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' An algorithm A is approximately- monotone with a constant c if for any sequence X, subsequence X′ ⊆ X, and initial tree T, it holds that cost(A, X′, T) ≤ c · cost(A, X, T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' ▶ Corollary 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' If GF is approximately-monotone with a constant c, then c ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' As noted, OPT is approximately-monotone with c = 1 (strictly monotone).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' The reason that approximate-monotonicity is of interest, in particular for GF, is because it is one of two properties that together are necessary and sufficient for any dynamically-optimal algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' The complementing property, which GF is known to satisfy, is simulation-embedding: ▶ Definition 10 (Simulation-Embedding [15]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' An algorithm A has the simulation-embedding property with a constant c if for any algorithm B and any sequence X, there exists a supersequence Y ⊇ X such that cost(A, Y ) ≤ c · cost(B, X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' (X is a subsequence of Y , not necessarily of consecutive queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=') An algorithm A which is approximately-monotone with a constant c1 and has the simulation-embedding property with a constant c2 is dynamically-optimal with a constant c1 · c2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Indeed, for any sequence X, there is some supersequence Y (X) ⊇ X such that cost(A, X) ≤ c1 · cost(A, Y (X)) ≤ c1 · c2 · cost(OPT, X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Harmon [12] proved that GG, and hence GF, has the simulation-embedding property, hence GF is dynamically-optimal if and only if it is approximately-monotone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' An alternative indirect proof was given by [6], proving that GG is O(1)-competitive versus the move-to-root algorithm, therefore inheriting the property from move-to-root.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' The motivation for studying reversal (Theorem 7) is that OPT is oblivious to reversing the sequence of queries, up to an additive difference of n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Indeed, to serve a sequence X in reverse, we can pay n to restructure the initial tree T0 to the final tree Tm, and then “reverse the arrow of time”: when serving query xt, also modify the tree from Tt to Tt−1 where Ti is the tree that OPT would get by the end of processing the i-th query of X, in order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' This means that any dynamically-optimal algorithm must be able to serve a sequence of requests and its reverse with the same cost up to a constant factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Theorem 7 does not disprove dynamic-optimality for GF, but gives some insight of how reversal affects GF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content='1 Maintaining a Static Tree for GF In this section we describe the basic “tool” which we use to fix a tree structure for GF despite its dynamic nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' That is, we describe a class of sequences which we call mixed-stable sequences such that GF never restructures its tree when serving a sequence in this class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' For the sake of simplicity, we assume that the initial tree is structured as we need it to be.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content='1 explains how to enforce a specific “initial” tree given an arbitrary initial tree, and also argues why this minor issue does not affect the competitive ratio of GF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Sadeh and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Kaplan 39:7 As noted, our objective is to produce a sequence that “tricks” GF into having unnecessary nodes in the core of the tree, such that the requested values are only at the leaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' As an example, consider the classic sequence of queries X = [1, 3, 1, 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content='] with an initial tree containing 2 at the root, 1 as a left child of the root and 3 as a right child of the root.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Because of the alternating pattern, GF never re-structures the tree, and the cost per query is 2 rather than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content='5 on average (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' when 1 is in the root, and 3 is its right child).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' ▶ Definition 11 (Stable Nodes and Sequences).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Let T be a full binary search tree, and let X be a sequence of queries over the items in the leaves of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' We define the stability of nodes as follows, see also Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' We say that an inner node v in T is strongly-stable if it has two children, and the subsequence of X consisting only of the items in the subtree of v, alternates between accesses to the left and right subtrees of v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' We say that an inner node v with a left child u in T is weakly-stable with a left-bias if both v and u have two children, and the subsequence of X consisting only of the items in the subtree of v, repeats the following 3-cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' First it accesses the left-subtree of u, then the right subtree of u, and finally right subtree of v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' (It is left-biased because 2 3 of the accesses are to the left of v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Symmetrically, we say that v is weakly-stable with a right-bias if v has two children, its right child u has two children, and the restriction of X to accesses in the subtree of v repeats a 3-cycle consisting of an access to the right subtree of u, the left subtree of u, and the left subtree of v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Notice that u is a strongly-stable node by definition, and we refer to it as the favored-child of v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' We regard the sequence X as being induced by the tree T with stability “attached” to its inner nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' We assume that every node is stable, and refer to X as a mixed-stable sequence and to T as a mixed-stable tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' We distinguish two special cases: If all inner nodes are strongly-stable then we refer to X and T as strongly-stable, and if exactly half of the inner nodes of T are weakly-stable then we refer to X and T as weakly-stable (recall that each weakly-stable node has a strongly-stable favored-child).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Figure 1 Node and sequence stability (Definition 11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' First, consider the repeated sequence 421, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' X = 421421421 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content='. Then v is a weakly-stable right-biased node because its visits pattern is a repetition of right(u), left(u), left(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' u is a strongly-stable node because its visits pattern is right(u), left(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' w is not stable at all, because its visits pattern is always left(w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Second, consider the repetition of the access pattern 12141314.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' One can verify that all three inner nodes are strongly-stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Hence, this is a strongly-stable sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Third, note that no weakly-stable sequence corresponds to the figure, because it requires an even number of inner nodes, but if we make w a leaf (removing 2, 3), then the repeated access pattern of 4w1 is a weakly-stable sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' To motivate Definition 11 a little, note that the sequence X = [1, 3, 1, 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content='] is a strongly- stable sequence that corresponds to a tree over the items {1, 2, 3} where 2 is in the root.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' X yields a lower-bound of 4 3 on the competitive ratio of GF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Similarly, the sequence X′ = [5, 3, 1, 5, 3, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content='] is a weakly-stable sequence that corresponds to the tree over {1, 2, 3, 4, 5} with 2 at the root and 4 its right-child.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' X′ yields a lower-bound of 8 5 on the competitive ratio of GF, which is already an improvement over the best known lower bound, see also Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' The distinction between strongly-stable and weakly-stable nodes is that GF may modify STACS 2023 39:8 Dynamic BSTs: Improved Lower Bounds for Greedy-Future (a) X = [1, 3, 1, 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content='] (b) X′ = [5, 3, 1, 5, 3, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content='] Figure 2 Examples of the simplest strongly-stable (a) and weakly-stable (b) sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Their corresponding trees are the left tree in each pair while the right tree in each pair is an optimized static tree to serve the same sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Queried nodes are colored in blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' One can verify that ˆc(X, GF) = 2 and ˆc(X′, GF) = 8 3 while based on the optimized tree, ˆc(X, OPT) ≤ 3 2 and ˆc(X′, OPT) ≤ 5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' the structure of the tree when a weakly-stable node is considered, but only temporarily and without affecting the cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' In our example with X′, after querying 5, GF may put 4 in the root instead of 2, but following the query of 3 this change will be reverted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Motivated by the power of stable sequences over small trees, we proceed to a more general analysis of stable sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' ▶ Definition 12 (Atomic Sequence).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' A tree T, along with stability type (weak/strong) for each node, and a subtree of each node to be accessed initially, induce a stable sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' This sequence is unique up to its length, which can be extended indefinitely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' We define the “atomic unit” of this sequence as the shortest sequence X such that any repetition of X is also a stable sequence that corresponds to T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Throughout the paper we work with whole multiples of the atomic sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Moreover, unless stated otherwise, we work with the atomic sequence itself (a single repetition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' ▶ Lemma 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Let X be a mixed-stable sequence with respect to a tree T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Then every leaf u is visited once every 2a(u) · 3b(u) queries where a(u) and b(u) are non-negative integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' In particular, the atomic length of X is 2maxleaf u a(u) · 3maxleaf u b(u) (the lcm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Moreover, if X is strongly-stable then ∀u : b(u) = 0, and if X is weakly-stable then ∀u : a(u) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Consider a leaf u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Define the frequency of visiting an ancestor w of u to be the frequency of accessing a leaf in the subtree of w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' If w is a strongly-stable ancestor then the frequency of visiting a child of w is 1 2 of the frequency of visiting w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' If w is weakly-stable, v is its favored-child, and x is a child of v then the frequency of visiting x is 1 3 of the frequency of visiting w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Similarly if w is weakly-stable, v is its non-favored-child then the frequency of visiting v is 1 3 of the frequency of visiting w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' It follows that u is visited exactly once every 2a(u) · 3b(u) queries where a(u) is the number of strongly-stable nodes that are not favored-children (there are no such nodes if X is weakly-stable), and b(u) is the number of weakly-stable nodes (no such nodes if X is strongly-stable), on the path to u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Finally, since every leaf u is visited with a specific period, the whole sequence has a period which is the lcm of all periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' ◀ ▶ Lemma 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Let X be a mixed-stable sequence with respect to a tree T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' If GF serves X with T as initial tree, and breaks ties in favor of nodes of smaller-depth, then it never restructures T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' The proof is by induction on the size of the tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' If T has a single node, then it is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Otherwise, the root r is an inner-node, and we prove that it always remains the root.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Sadeh and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Kaplan 39:9 It then follows, by restricting the access sequence to values within each subtree, that the rest of the tree remains fixed as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' We use the notations of τ(v) and vi as in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' First, consider the case that r is a strongly-stable node (Definition 11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Given an access to some value x in the left subtree of r, by definition, the next access would be to a value in the right subtree of r, hence τ(r) < τ(vi) for any vi ̸= r on the path from r to x, and therefore GF will keep r in the root.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' The same argument holds if x is in the right subtree of r, and the next access is in the left subtree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Next, consider the case that r is a weakly-stable node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Without loss of generality, assume that it is left-biased, and denote its favored-child (left child) by u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Denote the left and right subtrees of u by A and B respectively, and the right subtree of r by C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' The access pattern of subtrees is ABC(ABC .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' If the current access was to some x ∈ A, both r and u have been touched.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' The next access queries in B, so τ(u) = τ(r) < τ(vi) for any vi ̸= u, r on the access path to x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Since GF tie-breaks in favor of smaller-depth, it will keep r in the root.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content='4 If the current access was to some x ∈ B, then both r and u have been touched.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' The next access touches C, so τ(r) < τ(vi) for any vi ̸= r on the access path to x, including u, thus r must remain the root.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' If the current access was to some x ∈ C, since the next access touches A, τ(r) < τ(vi) for any vi ̸= r on the access path to x, thus r must remain the root.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' In this case u was not touched, but nonetheless it remains the left child of r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' ◀ ▶ Lemma 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' If X is a mixed-stable sequence, the frequency of accessing x ∈ X is in the range of [ 1 3d(x) , 1 3d(x)/2 ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' In particular, if X is strongly-stable then the frequency equals 1 2d(x) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' The frequency of visiting a node depends on the path to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' The frequency is multiplied by 1 2 when passing through a strongly-stable node, and multiplied by either 1 3 or 2 3 when passing through a weakly-stable node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Every factor of 2 3 is followed by 1 2, due to the strongly-stable favored-child of the weakly-stable node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Thus the frequency is bounded between 1 3d(x) and 1 2d(x)/2 · � 2 3 �d(x)/2 = 1 3d(x)/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' ◀ ▶ Corollary 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Let X be a strongly-stable sequence, then: ˆc(GF, X) = � x∈X d(x)+1 2d(x) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content='2 Promotions and Recursive Trees The way in which we show our lower bounds relies on the fact that serving the leaves of a static tree is sub-optimal, since a trivial static optimization is to move the leaves closer to the root.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' We refer to this operation as a promotion of the leaf that we move.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' We emphasize that for the purpose of our result, we analyze the improvement one gets from promotions, but the actual OPT, which is dynamic, may be able to reduce the cost further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' ▶ Definition 17 (Promotion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Consider trees T and T ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' We say that a node x was promoted in T ′ by h (with respect to T), if dT (x) − dT ′(x) = h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Given a mixed-stable sequence X, the average promotion of T to T ′ is the weighted average promotion in T ′ of the nodes of T, weighted by the query frequencies of the nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' 4 This is the reason we defined this kind of access pattern as weakly-stable, because the stability can be chosen, but is not forced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' We emphasize that putting u as a parent of r will not make the next access cheaper as both u and r will be touched anyway, and then r will be reinstated as the root.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' STACS 2023 39:10 Dynamic BSTs: Improved Lower Bounds for Greedy-Future By definition, static optimization of a tree T to T ′ for a mixed-stable sequence X, implies a cost improvement for OPT which is at least the average promotion of T to T ′, per query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Intuitively, promoting leaves that are closer to the root contributes more to the average promotion than promoting deeper leaves since the access frequencies decrease exponentially with depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' That being said, our promotion scheme will be relatively uniform, promoting most leaves by roughly the same amount, as in the following example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' ▶ Example 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' To clarify promotions, consider Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' There, we can safely promote every node by one, except for one of the deepest nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Therefore, we immediately conclude that for the corresponding strongly-stable sequence X, we have: ˆc(GF, X) ≥ ˆc(OPT, X) + (1 − 1 2n ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' (a) Before promotions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' (b) After promotions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Figure 3 (a) A tree which induces a strongly-stable sequence X, only blue nodes are queried.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' The frequency of querying an odd number v = 2i − 1 in this tree is 1 2i except for v = 2n + 1 which has the same frequency as v = 2n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' (b) An improved static tree, in which each node except for one has been promoted one step closer to the root.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' The cost of serving X over this tree is cheaper by almost 1 per query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' We define our trees using recursive structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' ▶ Definition 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' A recursive tree, Tr, of depth r is defined by a specific full binary tree T (independent of r) such that at least one of its leaves is an actual leaf, and some of its leaves are roots of recursive trees, Tr−1, of depth r − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' We refer to the inner nodes of T as the trunk of Tr, and define T0 to be a single node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' See Figure 4 for two examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content='5 Figure 4 Two recursive trees of depth r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Each of the trees T and F is a full binary tree with at least one actual leaf (in blue), and some hanging subtrees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' At the bottom of the recursion (for r = 0), the subtrees are nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Note that: (a) Expanding T for r = n results in the tree in Figure 3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' (b) The pattern F is important for Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' 5 The name of the pattern F in Figure 4, stands for Fibonacci: One can verify that for r ≥ 2, the number of leaves at depth 1 ≤ d ≤ r − 1 is the (d − 1)th Fibonnaci number Fd−1 (we define F0 = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Moreover, this can be used to prove the nice equation: �∞ d=0 Fd 2d = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' 2n- 1 2n + 12n - 1 2n : 2n + 1Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Sadeh and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Kaplan 39:11 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content='3 Multiplicative Lower Bound for GF In this section we prove Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' We do it by describing a concrete weakly-stable sequence, whose average cost per query is 6 while an average promotion of 3 is possible, resulting in an optimal cost of at most 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' We start by stating a purely mathematical lemma that will be used in the analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' ▶ Lemma 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Let br be a sequence defined by an initial value b0 and the relation br = α · br−1 + β + γ · r 2r for some constants α, β, γ where α ̸= 1 2, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Then br = β 1−α(1 − αr) + αr · b0 + 2αγ (2α−1)2 ·(αr − 1 2r )− γ (2α−1) · r 2r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' In particular, when γ = 0 then br = β 1−α(1−αr)+αr ·b0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Proof Sketch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Either use induction, or “guess” that a geometric sequence yr with a multiplier of α satisfies yr = p · r 2r + q · 1 2r + s + br, and determine the fixed coefficients p, q, s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' ◀ ▶ Lemma 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Let X be a weakly-stable sequence implied by the recursive tree Fr in Figure 4, where the root is a weakly-stable node with a right-bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Then for any ϵ > 0, there is a sufficiently large recursive depth r such that (1) ˆc(GF, X) > 6 − ϵ, (2) a static optimization of the tree saves an average cost of at least 3 − ϵ, and (3) regardless of r, ˆc(OPT, X) < 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Let cr denote the average cost of serving X with Fr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Then c0 = 1 and cr = 1 3(cr−1 + 1) + 1 3 · 3 + 1 3(cr−1 + 2) = 2 3cr−1 + 2, which yields by Lemma 20 that cr = 2 1−2/3(1 − (2/3)r) + (2/3)r · 1 = 6 · (1 − (2/3)r) + (2/3)r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' To analyze the average promotion, we re-structure Fr to a new static structure F ′ r as follows, see Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' The leaf is moved to the root, whose children are the recursive subtrees, optimized themselves by the same logic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' The old root is moved to be a right child of the maximal value in the new left subtree, and the old right-child (of the old-root) is moved to be a left child of the minimal value in the new right subtree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' F ′ r maintains the order of values as was in Fr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' The demotions of the old root and its right child do not affect the cost, because X does not query these values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Denote by pr the average promotion of Fr to F ′ r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Then p0 = 0 since nothing is promoted for a singleton, and pr = 1 3pr−1 + 1 3 · 2 + 1 3(pr−1 + 1) = 2 3pr−1 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Again by Lemma 20 we get that pr = 1 1−2/3(1 − (2/3)r) + (2/3)r · 0 = 3 · (1 − (2/3)r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Observe that for r → ∞ we get that cr → 6 and pr → 3, thus parts (1) and (2) of the claim follow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' For part (3), observe that cr − pr = 6 · (1 − (2/3)r) + (2/3)r − 3 · (1 − (2/3)r) = 3 − 2 · (2/3)r < 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' ◀ (a) F-tree pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' (b) Promotion scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Figure 5 The F-tree pattern and its promotion scheme in Lemma 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Only the top-level promotions are presented in (b), but more promotions are done recursively within each subtree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' ▶ Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' If GF is (c, d)-competitive where the additive term d is sublinear in the length of the sequence, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' d = o(m), then c ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Assume by contradiction that GF is (2 − δ, f(m))-competitive for some δ > 0 and a function f(m) = o(m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Let X′ be a sequence that consists of s repetitions of the atomic weakly-stable sequence that corresponds to the recursive tree Fr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' It follows that ˆc(GF, X′) ≤ (2 − δ) · ˆc(OPT, X′) + f(|X′|) |X′| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' By Lemma 21, we can choose r large enough such STACS 2023 Fr: Fr-139:12 Dynamic BSTs: Improved Lower Bounds for Greedy-Future that ˆc(GF, X′) > 6 − δ, and regardless of r, ˆc(OPT, X′) < 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Then, since f is sub-linear, we can choose the number of repetitions s to be large enough such that f(|X′|) |X′| < 2δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' But then we also get that ˆc(GF, X′) < (2 − δ) · 3 + 2δ = 6 − δ, which is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' ◀ By Analyzing mixed-stable sequences we proved a lower bound of 2 on the competitve ratio of GF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Theorem 22 gives an upper bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' ▶ Theorem 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Let X be a mixed-stable sequence and let T be the tree that corresponds to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Then cost(GF, X, T) < c · cost(OPT, X, T) for c = 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' If X is strongly-stable, then c = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' We defer the proof of Theorem 22 to Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' The upper-bound in Theorem 22 is clearly not tight, since in the proof of Theorem 22 we neglected a term using the inequality ˆc(GF, X) ≤ 2 α · ˆc(OPT, X) − 1 α � 1 − n−1 2m � < 2 α · ˆc(OPT, X), for a constant α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' The lack of tightness is more prominent when ˆc(OPT, X) is small, like in the sequence studied in Lemma 21 (for Theorem 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' We suspect that the lower bound in Theorem 4 is tight, and more strongly, that the F-tree pattern is the best pattern to use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' This is based on studying several other recursive patterns, including those in Figure 4 and Figure 6: None was stronger, and it also seems that patterns with large costs do not “compensate” with large enough promotions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' As a closing remark to the multiplicative results, we note that by the static optimality theorem for GG [9], competitive analysis against a static algorithm (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' an algorithm that does not change its initial tree) cannot show a super-constant lower bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Concretely, the theorem states that cost(GF, X) ≡ cost(GG, X) = O(m + �n i=1 ni lg m ni ) and one can verify that the actual constants are 5m + 6 �n i=1 ni lg m ni .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' This bound can be re-written as 5m + 6m · H2(X) where H2(X) = �n i=1 ni m lg m ni is the base-2 entropy of the frequencies of the values in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' By [17], cost(OPT s, X) ≥ m · H2(X) lg 3 where OPT s is the static optimum, and therefore cost(GF, X) ≤ (5 + 6 lg 3) · cost(OPT s, X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Thus, no static argument can show a lower bound larger than ≈ 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content='59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content='4 Additive Lower Bounds for GF In this section we move on to analyze the additive gap between GF and OPT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' For this, we construct and analyze more elaborate patterns of recursively-defined trees, in order to get a large average promotion when optimizing the structure of the trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' The analysis is more involved since we cannot simply assume that the depth of the recurrence, r, approaches infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Here n is a function of r and the difference of cost can be meaningful in terms of n only if n is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' ▶ Definition 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' For k ≥ 2, and r ≥ 0 we define a (k, r)-tree Tr as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' The tree is recursive of depth r (as in Definition 19), such that its trunk is composed of a root and a left-chain of length k − 1 that starts in the right-child of the root.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' The left child of the deepest node of the trunk is an actual leaf, and the rest of the leaves are Tr−1 subtrees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' T0 is a single node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' See Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' When k is clear from the context, we also refer to the tree as Tr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Observe that the tree Fr that was used to prove Theorem 4 is in fact a (k, r)-tree with k = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' When we conclude the analysis, we will get the two ends of a “tradeoff” such that on the one end we have a relatively high cost ratio, and on the other a relatively high cost difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Moreover, we will show that the higher the difference of costs on a sequence induced by (k, r)-tree, the closer the cost ratio is to 1 (comparing GF to OPT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' ▶ Lemma 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' The depth of a (k, r)-tree is k · r, and its left-most node is at depth r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Sadeh and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Kaplan 39:13 (a) (k, r)-tree pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' (b) Promotion scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' The main gain is due to the first step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Figure 6 (a) The recursive pattern of a (k, r)-tree, Tr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' The trunk of the tree has k nodes: the root, and a chain of k − 1 nodes leading to an actual leaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' The rest of the leaves are (k, r − 1)-trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' (b) The promotion scheme used later in Lemma 26, exemplified for k = 4 (see also Figure 5 for the degenerate case of k = 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' The main gain is from the first step of promoting the actual leaf to the root, and its sibling subtree (marked with +) one step upwards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Additional gain is achieved by promoting the left-most node of each hanging right subtree to the trunk at the expense of demoting trunk nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' More promotions are done recursively within each subtree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' The nodes marked 0, 1, 2 are indeed consecutive, and also: 2 < x and x + 1 = w < y = z − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Trivial by induction: For r = 0, the deepest node is the root, at depth 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' For r ≥ 1, observe that the deepest node belongs to the deepest subtree Tr−1, which is rooted at depth k since the path to it includes k trunk nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Similarly, the depth of the left-most node is increased by 1 per recursive level of the tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' ◀ ▶ Lemma 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Let Tr be a (k, r)-tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Then |Tr| = (2 + 2 k−1)kr − (1 + 2 k−1) where |Tr| is the number of nodes in Tr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' In rougher terms, |Tr| = Θ(kr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Denote nr = |Tr|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' By definition, n0 = 1 and nr = (k + 1) + k · nr−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Hence, by Lemma 20 (with γ = 0): nr = k+1 1−k(1 − kr) + kr = (2 + 2 k−1)kr − (1 + 2 k−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' ◀ ▶ Lemma 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Let X be any mixed-stable sequence corresponding to a (k, r)-tree Tr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Denote the average weighted promotion possible in Tr by pr, where weighting is according to the frequency of querying each leaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Then pr > k · (1 − αr) for α = 1 − 1 3k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' In particular, if X is a strongly-stable sequence, then pr = (k + 1) · (1 − αr) + δ for α = 1 − 1 2k and 0 ≤ δ < αr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' We can promote by k every explicit leaf in every Tr′ for all recursive levels 1 ≤ r′ ≤ r, from its location to the root of Tr′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Only nodes that are T0 leaves do not contribute an explicit promotion of at least k, therefore pr > k · (1 − f) where f is the sum of query-frequencies of all T0 leaves (the inequality is strict due to unaccounted subtree promotions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' To conclude, we argue that f ≤ (1 − 1 3k )r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' The frequency of accessing the explicit leaf of Tr is at least 1 3k by Lemma 15, hence with frequency of at most 1 − 1 3k we query a value in some Tr−1 subtree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Similarly, within the chosen subtree there is again a relative frequency of at most 1 − 1 3k to query within some Tr−2 subtree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Overall, since there are r levels of recursion, we conclude that f ≤ (1 − 1 3k )r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Proving the second part of the claim required a more careful analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' We define the following method of promotion, depicted in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' In the (k, r)-tree we promote the (only) explicit leaf to the root, and promote its sibling subtree by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Then we apply similar promotions recursively within every (k, r − 1)-subtree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Finally, we promote the left-most node within each (k, r − 1)-subtree that hangs as a right-subtree from the trunk to the parent of this subtree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Denote the total average (weighted) promotion by pr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Note that it does not matter if we promote the left-most nodes of the right subtrees before or after the recursive promotions, because the total order on the items guarantees that there is only one value STACS 2023 nodes Tr-039:14 Dynamic BSTs: Improved Lower Bounds for Greedy-Future that can be put instead of every demoted trunk node, and the recursive promotions within a specific subtree do not change the depth of its leftmost leaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' The promotion of the explicit leaf of Tr saves a cost of k weighted by a factor (query frequency) of 1 2k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' The promotion of the sibling subtree saves 1 weighted by a factor of 1 2k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' The recursive promotions are pr−1 weighted by �k i=1 1 2i (for all the k subtrees),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' and finally the last promotions are technically negligible (as seen in the analysis below),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' but for the sake of completeness we consider them in the analysis as well: promoting the left-most node from each subtree saves (r − 1) + 1 = r since the leaf that we promote last is at depth r − 1 within the recursive subtree,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' and this promotion is weighted by 1 2r · �k−1 i=2 1 2i (factor of 1 2r follows from Lemma 24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' We get that: pr = k+1 2k + pr−1 · � 1 − 1 2k � + r 2r · 1 2 � 1 − 1 2k−2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Then by Lemma 20, with α = 1 − 1 2k and γ = 1 2(1 − 1 2k−2 ), we get: pr = (k + 1) · (1 − αr) + δ , δ ≡ αr · p0 + 2αγ (2α − 1)2 · � αr − 1 2r � − γ (2α − 1) · r 2r It remains to show that 0 ≤ δ < αr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' It is simple to see that δ = 0 for k = 2, because then γ = 0 and p0 = 0 is the average weighted promotion in a tree with a single node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' For k ≥ 3, by the definition of α and γ we have that 2αγ (2α−1)2 = (2k−1)(2k−4) (2k−2)2 = 1 − 1 2k−4+ 4 2k ∈ ( 3 4, 1) and γ 2α−1 = 1 2 − 1 2k−2 ∈ [ 1 3, 1 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Substituting these bounds and p0 = 0 into the formula for δ gives δ < αr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Moreover, δ is positive since δ > 3 4(αr − 1 2r )− 1 2 · r 2r = 3 4(α− 1 2)·�r−1 i=0 αi · � 1 2 �r−1−i− r 2r+1 = 3(α− 1 2 ) 2r+1 �r−1 i=0 (2α)i − r 2r+1 > 3(α− 1 2 ) 2r+1 r − r 2r+1 = (3( 1 2 − 1 2k ) − 1) · r 2r+1 > 0 for k ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Note that indeed the gain from promoting the left-most node of each subtree is negligible, since the effect is merely having γ ̸= 0, which only contributes 0 ≤ δ < αr < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' ◀ ▶ Corollary 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' The average cost-per-query of GF on a strongly-stable sequence induced by a (k, r)-tree is larger than the optimal cost by at least (k + 1) · � 1 − � 1 − 1 2k �r� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' On any mixed-stable sequence, the difference is at least k · � 1 − � 1 − 1 3k �r� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' We are ready to prove Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' ▶ Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' For every n ≥ 2 there exist sequences X ∈ [n]m such that cost(GF, X) = cost(OPT, X) + Ω(m · lg lg n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Among these sequences, there exists a sequence whose length is m = nΘ( lg lg n lg lg lg n ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' (There exist other longer sequences too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=') Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Let X be the strongly-stable sequence induced by a (k, r)-tree Tr, and for simplicity assume that the initial tree is Tr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content='6 By Lemma 25, n = (2+ 2 k−1)kr−(1+ 2 k−1) therefore lg lg n = lg r + lg lg k + O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content='7 By Corollary 27, ˆc(GF, X) − ˆc(OPT, X) ≥ ∆ ≡ (k + 1) · (1 − (1 − 1 2k )r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' By choosing r = 2k we get that ∆ = (k + 1) · (1 − (1 − 1 2k )2k) ≈ (1 − 1 e) · (k + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content='8 We also get that lg lg n = k + lg lg k + O(1), therefore ∆ ≈ (1 − 1 e) lg lg n and we conclude that ˆc(GF, X) − ˆc(OPT, X) ≥ Ω(lg lg n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' By Lemma 13 the length of the atomic strongly-stable sequence of Tr is m = 2d(Tr), hence m = 2rk by Lemma 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' By Lemma 25, n+(1+2/(k−1)) 2+2/(k−1) = kr = 2r lg k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Together we get that m = 2rk = 2(r lg k)·(k/ lg k) = � n+(1+2/(k−1)) 2+2/(k−1) �(k/ lg k) = nΘ( lg lg n lg lg lg n ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' ◀ 6 We remove this assumption in Remark 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' 7 k ≥ 2 ⇒ kr ≤ n < 4kr ⇒ lg n = r lg k + c for c ∈ [0, 2), and so lg lg n = lg r + lg lg k + O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' 8 The approximation is off by less than 10% for k ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' (60% and 20% for k = 0, 1 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=') Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Sadeh and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Kaplan 39:15 ▶ Remark 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' In the proof of Theorem 5, the sequence X does not have to be strongly- stable, and any mixed-stable sequence X induced by a (k, r)-tree Tr works as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Indeed, Corollary 27 guarantees that ˆc(GF, X)−ˆc(OPT, X) ≥ ∆ for ∆ = k· � 1− � 1− 1 3k �r� , and then by choosing r = 3k we get that ∆ = Θ(k), and k = Θ(lg lg n), and m = 2Θ(rk) = nΘ( lg lg n lg lg lg n ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' ▶ Remark 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' The choice of r = 2k in the proof of Theorem 5 maximizes our lower bound on the additive gap cost(GF, X) − cost(OPT, X) (up to constants) for our (k, r)- trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Indeed, revisiting the proof, we have that ∆ and n are both functions of k and r, and we need to choose r and k to maximize ∆ as a function of n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Note that ∆ = O(k) regardless of r, and lg lg(n) = lg r + lg lg k + O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' To simplify and eliminate a parameter we define r = 2k · f(k) for some monotone function f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Now we get simplified relations: ∆ = (k+1)·(1−(1− 1 2k )2kf(k)) ≈ (k+1)·(1−e−f(k)) and lg lg n = k+lg f(k)+lg lg k+O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Consider the following two cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' If f(k) = Ω(1): then ∃c ∈ R such that ∀k ≥ 1 : lg f(k) ≥ c, and therefore lg lg n = Ω(k), written differently k = O(lg lg n), which yields ∆ = O(k) = O(lg lg n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' If f(k) = o(1): Being o(1) means that limk→∞ f(k) = 0, so for sufficiently large values of k we can use the approximation ex ≈ 1 + x (that holds for small x) to get: ∆ ≈ (k + 1) · f(k) = k+1 1/f(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' If 1 f(k) grows faster than (k + 1), we get ∆ = O(1) which does not even grow with n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Therefore 1 f(k) is increasing, but at a sub-linear rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Recall that lg lg n = k − lg 1 f(k) + lg k + O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Since 1 f(k) is sub-linear, we get that k = Θ(lg lg n), which yields ∆ = O(k) = O(lg lg n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' ▶ Corollary 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' GF is not (1, O(m))-competitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' If the multiplicative term is 1, then the additive term is at least Ω(m · lg lg n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' We have yet to analyze the cost of OPT on a strongly-stable sequence X corresponding to a (k, r)-tree that produces the gap of Ω(m·lg lg n) in Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Allegedly, if the cost is cheap, say linear, we would get a large competitive ratio as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' However, by Theorem 22 we expect a competitive ratio of at most 2, and therefore we can conclude without further analysis, that cost(OPT, X) = Ω(m · lg lg n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' In fact, we prove that cost(OPT, X) = Θ(m · lg n lg lg lg n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' It follows that the competitive ratio deteriorates when the additive gap increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' ▶ Lemma 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Define the constants α ≡ 1 − 1 2k and β ≡ �k j=1 j 2j + k+1 2k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Let X be a strongly-stable sequence induced by a (k, r)-tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Then ˆc(GF, X) = 2k · β · (1 − αr) + αr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' In asymptotic terms: ˆc(GF, X) = Θ(2k · (1 − αr)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' We write a recurrence for the average cost, cr, of GF on the strongly-stable sequence induced by Tr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' We have c0 = 1, and cr+1 = 1 + k 2k + k � j=1 j + cr 2j = � 1 − 1 2k � cr + k � j=1 j 2j + 1 + k 2k ≡ α · cr + β ( 1+k 2k is due to the actual leaf, and the summation is the contribution of all the Tr subtrees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=') By Lemma 20 (with γ = 0), cr = β 1−α(1 − αr) + αr · c0 = 2k · β(1 − αr) + αr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Since α = 1 − 1 2k ∈ [ 3 4, 1) clearly αr < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Furthermore, β = Θ(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' To see this note that β only depends on k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Denote β = β(k) and observe that: β(k+1)−β(k) = � k+1 2k+1 + k+2 2k+1 � − k+1 2k = 1 2k+1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Therefore, β(k) = β(2) + �k i=3 � β(i) − β(i − 1) � = � 1 2 + 2 4 + 3 4 � + �k i=3 1 2i = 2 − 1 2k , and β(k) ∈ [ 7 4, 2) ⇒ β = Θ(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Because 2k · (1 − αr) ≥ 2k · (1 − α) = 1 > αr, we conclude that cr = Θ(2k · (1 − αr)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' ◀ STACS 2023 39:16 Dynamic BSTs: Improved Lower Bounds for Greedy-Future ▶ Lemma 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Let X be a sequence from the family of sequences in Theorem 5, then cost(OPT, X) = Θ(m · lg n lg lg lg n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Let X be a strongly-stable sequence induced by querying a (k, r)-tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' We know that 1 2ˆc(GF, X) < ˆc(OPT, X) ≤ ˆc(GF, X) where the lower-bound is by Theorem 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Therefore, ˆc(OPT, X) = Θ(2k · (1 − αr)) by Lemma 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' By Lemma 25, lg n = r · lg k + O(1), or r = lg n−O(1) lg k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' When we substitute r = 2k as in the proof of Theorem 5, we get that (1−αr) = Θ(1) and 2k = r = lg n−O(1) lg k = Θ( lg n lg lg lg n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Therefore, cost(OPT, X) = Θ(m · lg n lg lg lg n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' ◀ As a concluding remark, we recall that the Fr-tree is a (k, r)-tree for k = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' If we substitute k = 2 in the formula of Lemma 31 we get that α = 3 4, β = 7 4, and ˆc(GF, X) = 7 · (1 − (3/4)r) + (3/4)r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' By Lemma 26, the average promotion is 3 · (1 − (3/4)r) (for k = 2, we have δ = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' These values are the strongly-stable analogues of Lemma 21, and can be used to show a weaker lower bound of 7 4, on the competitive ratio of GF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' 4 Conclusions and Open Questions In this paper we gave improved lower bounds on the competitiveness of the Greedy Future (GF) algorithm for serving a sequence of queries by a dynamic binary search tree (BST).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' In contrast to many of the previous results on GF that are obtained using the geometric-view by studying the equivalent Geometric Greedy (GG) algorithm, we used the standard “tree-view” and the treap-based definition of GF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' We showed that the competitive ratio of GF is at least 2, and that there are sequences X ∈ [n]m for which the cost difference (additive gap) between GF and OPT is Ω(m · lg lg n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' These lower bounds enabled us to show that if GF is approximately-monotone (Definition 8) with some constant c then c ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Also, the lower bounds show that the cost of GF on a sequence compared to its cost on its reverse, may differ by a factor as close as we like to 2, or by a difference of Ω(m · lg lg n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' In contrast, the cost of OPT on a sequence compared to its reverse may differ by at most n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Our results give new insights on the “tradeoff” between the additive term and the multiplicative term in the competitiveness of GF, showing that the multiplicative term is typically larger when the total cost of the algorithm on the sequence is smaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Indeed, our best multiplicative term is achieved for a sequence whose average cost per query is 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' This tradeoff is not surprising since a fixed difference implies a larger ratio when the quantities are small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' It may be interesting to figure out if this tradeoff hints of some underlying property of GF, or is just an artifact of our technique that requires high costs on average per query in order to increase the additive gap between GF and OPT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Clearly, these improved lower bounds still don’t settle the deeper question of whether GF (and GG) is dynamically-optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Our techniques focused on a smaller family of sequences which we named mixed-stable sequences, whereas “most” sequences are not stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' While it is possible that an improved lower bound (larger than 2) can be found by a more clever pattern of mixed-stable sequences, it seems more likely to be found by analyzing sequences for which the tree maintained by GF is not static.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' In addition, we note that GF was not investigated too deeply directly, as most of the work has been done in the geometric view with respect to its counterpart GG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Therefore, studying other problems in tree-view may give complementing insights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' One such problem is the deque conjecture, which has been partially settled for GG, in the case when deletions are only allowed on the minimum item [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Sadeh and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Kaplan 39:17 References 1 Parinya Chalermsook, Julia Chuzhoy, and Thatchaphol Saranurak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Pinning down the Strong Wilber 1 Bound for Binary 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' The landscape of bounds for binary search trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' arXiv, abs/1603.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content='04892, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' 5 Parinya Chalermsook, Manoj Gupta, Wanchote Jiamjitrak, Nidia Obscura Acosta, Akash Pareek, and Sorrachai Yingchareonthawornchai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Improved pattern-avoidance bounds for greedy BSTs via matrix decomposition.' metadata={'source': 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Georgakopoulos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Chain-splay trees, or, how to achieve and prove loglogN- competitiveness by splaying.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Information Processing Letters, 106(1):37–43, 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' 12 Dion Harmon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' New Bounds on Optimal Binary Search Trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' PhD thesis, Massachusetts Institute of Technology, 2006.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Springer-Verlag, 2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' 19 Hauke Reddmann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' On the geometric equivalent of instance optimal binary search tree algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Master’s thesis, Universität Hamburg, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' 20 Daniel Dominic Sleator and Robert Endre Tarjan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Self-adjusting binary search trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Journal of ACM, 32(3):652–686, 1985.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' 21 Chengwen Chris Wang, Jonathan Derryberry, and Daniel Dominic Sleator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' O(log log n)- competitive dynamic binary search trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' In 17th Annual ACM-SIAM Symposium on Discrete Algorithm (SODA), page 374–383, 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' 22 Robert Wilber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Lower bounds for accessing binary search trees with rotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' SIAM Journal on Computing, 18(1):56–67, 1989.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' STACS 2023 39:18 Dynamic BSTs: Improved Lower Bounds for Greedy-Future A Appendix: Deferred Proofs and Discussions A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content='1 Enforcing a Stable Tree for GF We describe how to restructure any initial tree, to a desired tree, when GF is considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' The initial tree cannot simply be re-organized since GF updates the tree in a specific way following each query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Moreover, when we are given a sequence X and add a prefix P to it, denote the concatenation by P ◦ X, even if P enforces the desired tree when served alone, serving P ◦X may give a different tree following P when X starts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' The reason for this is that GF restructures the tree while serving P according to future queries, therefore the existence of X may affect its decisions while serving P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Nevertheless, the idea is to restructure the tree top-down from the root, such that we “propagate” stability, as in Definition 11, over the nodes that have already been fixed in their correct places.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' See Figure 7 for an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Figure 7 Example of enforcing a tree as detailed in Theorem 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Consider the mixed-stable sequence X = [11, 7, 1, 13, 9, 3, 11, 7, 5, 15, 9, 1, 11, 7, 3, 17, 9, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Its desired tree T is the right-most tree in the figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Nodes 6, 4, 16 are weakly-stable biased towards their starred-edge (leading to 10, 2, 14 respectively), all other internal nodes are strongly-stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Left-to-right: Initially, there are no stabilized nodes (left).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' The first step queries only Y1 = [A] as if it was a leaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Duplicated six times and substituted for the pattern of a weakly-stable node, we get Z1 = [10, 10, 6, 6, 10, 6], stabilizing {6, 10}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' In the second step, the next nodes that are stabilized are {2, 4, 8, 12}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' The weakly-stable sequence of the subtrees is Y2 = [C, B, A].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Duplicated six times, and substituted 2, 2, 4, 4, 2, 4 for A, 8, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' , 8 for B and 12, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' , 12 for C, we get: Z2 = [12, 8, 2, 12, 8, 2, 12, 8, 4, 12, 8, 4, 12, 8, 2, 12, 8, 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Next Y3 = [F, D, A, G, E, B, F, D, C, G, E, A, F, D, B, G, E, C], and Z3 is the result of duplication and substitution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Since A−F are leaves, only the substitution of G requires the non-trivial pattern (14, 14, 16, 16, 14, 16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' This stabilizes the nodes {14, 16} and is the last step of this example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' In total, the enforcing sequence is Z1 ◦ Z2 ◦ Z3 (◦ for concatenation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' In general, there may be more steps, each stabilizes at least one more node until all inner nodes are stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' ▶ Theorem 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Let X be a mixed-stable sequence that corresponds to a tree T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content='9 There is a sequence S(X) such that when GF serves the concatenation S(X) followed by X, then when it finishes serving S(X) its current tree is T regardless of the initial tree T0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' We refer to S(X) as the enforcing sequence of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Additionally, if X is the atomic sequence corresponding to T (Definition 12), then |S(X)| < 3n · |X|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' If X is strongly-stable or weakly-stable (and atomic), then |S(X)| < 2|X| and |S(X)| < 3|X| respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' We construct S(X) in steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Initially, the tree of GF (which is the initial tree T0) and the desired mixed-stable tree T may be completely different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Each step of the construction adds a sequence of queries to S(X) that extends a rooted and connected subtree that belongs 9 The type of stability of each node of T can be deduced from X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Sadeh and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Kaplan 39:19 both to T and to the current tree of GF and will not change subsequently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' We regard nodes that already joined this subtree as stabilized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Once all inner nodes have been stabilized then S(X) is complete and the tree of GF is exactly T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Every stabilized node remains stable with respect to the continuation of the sequence as in Definition 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' See Figure 7 for an illustration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' We first describe how to stabilize the desired root (of T), r, which is the base of the construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' We emphasize that stabilizing a weakly-stable node also stabilizes its favored-child (Definition 11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' We split into three cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' If there is only a single node, then it is r which is also already the root.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' We do nothing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' If r is strongly-stable in T: We query [r, r] to make it the root.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' The second query guarantees that r is placed at the root after the first query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' If r is weakly-stable in T: Let z be r’s favored-child.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' We add the queries [z, z, r, r, z, r] to S(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' This stabilizes r and z, and by the end of these six queries, r is the root and z is its child.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' To see why this happens, consider how GF works: The second query to z guarantees that it is placed at the root after the first query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' The third query touches both z (the root) and r (queried), and due to the fourth and fifth queries places r at the root and z as its child.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' The purpose of the sixth query is to ensure that z does not become the root due to future queries when the fifth query is processed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Assume that we already have a tree with a connected subtree of stabilized internal nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' The subtrees that hang off the stable nodes are not empty since stable nodes are internal nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' If each of these subtrees is a leaf then we are done and S(X) is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Otherwise, we stabilize the root of every subtree that is not a leaf as we describe next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Recall that if the root of a subtree is weakly-stable, we also stabilize its favored-child.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' For convenience, denote the index of the current step by ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' We regard each unstabilized subtree as a leaf, and generate an atomic mixed-stable sequence over the current stabilized connected subtree, according to the stability types of the inner nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Denote this sequence by Yℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Note that Yℓ is a sequence of the leaves that correspond to the unstabilized subtrees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' We derive from Yℓ the stabilizing sequence, Zℓ, for the new nodes, by repeating Yℓ six times, and replacing each leaf in Yℓ 6 by an appropriate node from the subtree that it represents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' This replacement is done as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' If the root r of the subtree is a leaf or a strongly-stable node then we simply access r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' If r is weakly-stable and its favored-child is z then we replace each query of the leaf (subtree) by the next query of the sequence z, z, r, r, z, r, while cyclically repeating it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Since Yℓ was repeated six times, the sequence z, z, r, r, z, r repeats an integral number of times in Zℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Because Zℓ is based on the mixed-stable sequence Yℓ, the already stabilized nodes remain stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Only the subtrees hanging off them are affected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' The restriction of Zℓ to each hanging subtree is either z, z, r, r, z, r or r, r, r, r, r, r so by an argument analogous to the one in the base case the root of each hanging subtree is stabilized, as well as each favored-child of a weakly-stable root.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' We repeat stabilizing steps until all the inner nodes of T have been stabilized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' If the last step is d, then S(X) = Z1 ◦ Z2 ◦ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' ◦ Zd where ◦ denotes concatenation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' We emphasize that when GF serves S(X) ◦ X, by the end of serving S(X) its tree is indeed T, because we can think of X as just another step of the stabilization process, in which all the subtrees are leaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' It remains to analyze the length of S(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' The length of any Yi is at most that of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Every step stabilizes at least one inner node, hence there are at most n−1 2 steps (the rest n+1 2 nodes are leaves), so we get that |S(X)| = 6 �d i=1 |Yi| ≤ 6 · n−1 2 |X| < 3n · |X|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' For the refined analysis of the length of S(X) in case X is strongly-stable, or weakly-stable, we investigate the relation between |Yi| and |Yi+1|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' For convenience we define Yd+1 = X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' By Lemma 13, |Yi| = 2ai · 3bi, where ai = maxleaf u a(u) and bi = maxleaf u b(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Observe that when we extend the stability from a node v which is a leaf of the stabilized subtree at STACS 2023 39:20 Dynamic BSTs: Improved Lower Bounds for Greedy-Future step i, one or two levels deeper, then we have for every new stabilized leaf descendant u of v that either a(u) = a(v) + 1 and b(u) = b(v), or a(u) = a(v) and b(u) = b(v) + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' However, the lcm of the preiods of all leaves of Yi+1 may be affected by two different branches, thus allowing any of the combinations of ai ≤ ai+1 ≤ ai + 1 and bi ≤ bi+1 ≤ bi + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' So we get that |Yi| divides |Yi+1| and |Yi| ≤ |Yi+1| ≤ 6|Yi|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' In the general case of mixed-stability it could be that the length grows initially six-fold (starting from the second step, two differ- ent branches might each increase a and b, respectively) while the second half of the steps satisfies |Yd/2+1| = |Yd/2+2| = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' = |X| (another branch that “mixes” the increase in a and b “catches up” with the lcm), so |S(X)| = Θ(n · |X|) is possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' As an example, revisit Figure 7: there we have |Y1| = 1, |Y2| = 3, |Y3| = 18 and |Y4| = |X| = 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' However, when X is strongly-stable, we have |Yi+1| = 2|Yi| for every i because bi = 0 (Lemma 13), and ai+1 = ai + 1 for every i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Therefore, |S(X)| = 6 �d i=1 |Yi| = 6 �d i=1 |X| 2i < 6|X|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' We can optimize further by noting that we can define Zi to be only twice longer than Yi (no need for six repetitions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' So in fact we can define S(X) for strongly-stable X such that |S(X)| < 2|X|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' If X is weakly-stable, similarly |Yi+1| = 3|Yi| because for every i we have ai = 0 (Lemma 13) and bi+1 = bi + 1, therefore |S(X)| = 6 �d i=1 |Yi| = 6 �d i=1 |X| 3i < 3|X|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' ◀ ▶ Theorem 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' For any tree T there is a sequence S(T) such that for any suffix of queries Y , when GF serves S(T) ◦ Y , its tree when is it done with the last query of S(T) is T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' We rely on the ideas from the proof of Theorem 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' To generate an oblivious enforcing prefix, we concatenate several enforcing sequences, each enforcing higher nodes in the tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Define T [1] ≡ T and T [i+1] is the tree T [i] stripped of all of its leaves, until the final tree T [h] contains only the root.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' For each tree T [i], denote by Xi its corresponding strongly-stable sequence Xi, and by S1 the enforcing sequence of X1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Note that if T [i] is not full, we relax the definition of the corresponding strongly-stable sequence and instead of querying a missing leaf we query its unary parent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' This modification works as-well because we can imagine that the query proceeds to the missing leaf, which would anyway remain below the parent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' We claim that S(T) ≡ S1 ◦ X2 ◦ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' ◦ Xh satisfies the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Indeed, S1 enforces the desired tree, and in particular the position of the leaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Following S1, all the nodes except for the leaves are touched again, so these leaves can never become parents, regardless of the suffix Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' The argument holds similarly for the following steps, and since the structure of T is already in place, it suffices to use Xi instead of their enforcing sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' ◀ Adding a prefix to our sequence may affect the competitive ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' However, once we fixed the stable tree, we can repeat the corresponding stable sequence to “amplify” the original competitive ratio making the effect of the prefix negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' One difficulty raised by repetitions is when we care about the length of the sequence in our claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' This is the case in Theorem 5 where we claim the existence of a sequence of length nΘ( lg lg n lg lg lg n ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' In the proof of this theorem we assumed for simplicity that we can choose the initial tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' The following remark shows that indeed we can start with an arbitrary initial tree without weakening the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' ▶ Remark 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Let X be an atomic mixed-stable sequence used to prove Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Consider the sequence Z = S(X) ◦ Xn, where S(X) is the prefix (guaranteed by Theorem 33) that is enforcing the desired “initial” tree T, Xn are n repetitions of X, and ◦ represents concaten- ation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' By Theorem 33, n|X| ≤ |Z| < 4n|X| therefore we have: |Z| = Θ(n|X|) = nΘ( lg lg n lg lg lg n ) (the second equality is by Theorem 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Since after processing S(X) the tree of GF is fixed: cost(GF, Z, T0)−cost(OPT, Z, T0) ≥ n·(cost(GF, X, T)−cost(OPT, X, T)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' By the proof of Theorem 5 and Remark 28: cost(GF, X, T) − cost(OPT, X, T) = Ω(|X| · lg lg n), and putting everything together we get that: cost(GF, Z, T0) − cost(OPT, Z, T0) = Ω(|Z| · lg lg n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Note Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Sadeh and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Kaplan 39:21 that if X is strongly-stable, or weakly-stable, then it suffices to define Z = S(X) ◦ X without repetitions and we get that |Z| = Θ(|X|), and the rest of the arguments remain the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content='2 Omitted Proofs In this subsection we restate and prove Lemmas and Theorems that were omitted from the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' For convenience, we restate the claims in their original numbering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' The proof of Theorem 22 makes use of Wilber’s first bound [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' We use the original presentation of this bound which is a bit tighter than later simplified versions such as [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' ▶ Definition 36 (Wilber’s First Bound [22]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Let X be a sequence of queries, and let T be a static reference tree such that every query of X is in a leaf of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' An alternation at an inner node u of T is defined to be two queries closest in time such that one accesses either the left or right subtree of u and the other accesses the other subtree of u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Define ALT(u) to be the number of alternations at node u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Then: cost(OPT, X) ≥ m + 1 2 � inner u∈T ALT(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' ▶ Theorem 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Let X be a mixed-stable sequence and let T be the tree that corresponds to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Then cost(GF, X, T) < c · cost(OPT, X, T) for c = 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' If X is strongly-stable, then c = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' We use the tree that corresponds to the mixed-stable sequence as the reference tree for Wilber’s first bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Arithmetic manipulations will yield an expression that we can tie to the cost of GF, according to the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Let X be a mixed-stable sequence, with a corresponding tree T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Let S be the set of values that are in the leaves of T, and let U be the set of inner nodes, |U| = n−1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' We also denote by A(i) the set of proper ancestors of i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' By the definition of the cost of a static tree, we know that ˆc(GF, X) = � i∈S (d(i) + 1) · f(i) where d(i) is the depth of i and f(i) is the frequency of accessing i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' We extend f(u) to refer to the frequency of visiting any node u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Note that f(u) = � i∈S∧u∈A(i) f(i) and that � i∈S f(i) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Now consider Wilber’s bound for X, with T as the reference tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' We can use T as the reference tree since X only accesses leaves of T, by definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' We also denote αu ≡ ALT (u)+1 f(u)·m (ALT(u) is defined in Defintion 36, and note that 0 ≤ ALT(u) ≤ f(u) · m − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' We have αu ∈ (0, 1], where αu = 1 corresponds to fully alternating accesses to the subtree rooted at u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' The lower bound is cost(OPT, X) ≥ m + 1 2 � u∈U (ALT(u) + 1) − |U| 2 = m 2 + m 2 (1 + � u∈U αu · f(u)) − n−1 4 = � m 2 − n−1 4 � + m 2 � i∈S (1 + � u∈A(i) αu)f(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Let α ≤ minu∈U αu, we get that ˆc(OPT, X) ≥ � 1 2 − n−1 4m � + α 2 � i∈S (d(i) + 1) · f(i) = α 2 ˆc(GF, X) + � 1 2 − n−1 4m � where the equality holds since GF maintains a static tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Thus ˆc(GF, X) ≤ 2 α · ˆc(OPT, X)− 1 α � 1 − n−1 2m � < 2 α · ˆc(OPT, X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' In order to choose a suitable α, recall that a strongly-stable node u has a coefficient of αu = 1, which means that for strongly-stable sequences, in which all inner nodes are stable, we can pick α = 1 and conclude that ˆc(GF, X) < 2 · ˆc(OPT, X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' If u is a weakly-stable node, then its coefficient is αu = 2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' So for a mixed-stable sequence we can naively pick α = 2 3, resulting in ˆc(GF, X) < 3 · ˆc(OPT, X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' In order to improve from 3 to 5 2, we observe that by definition, every weakly-stable node has a strongly-stable child.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Let u be a weakly-stable node and let w be its (strongly-stable) favored- child (recall Definition 11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Since ALT(u) = ALT(w) (by definition of the access pattern in u), we can present Wilber’s bound differently, summing (ALT(u)+1)·(1+β)+(ALT(w)+1)·(1−β) instead of (ALT(u)+1)+(ALT(w)+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' We get modified coefficients α′ u = (ALT (u)+1)·(1+β) m·f(u) = αu · (1 + β) = 2(1+β) 3 and similarly α′ w = αw(1 − β) = (1 − β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Choosing β = 1 5 balances the coefficients: α′ u = α′ w = 4 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Now we can choose α = 4 5, and get ˆc(GF, X) < 5 2 · ˆc(OPT, X) for mixed-stable sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' ◀ STACS 2023 39:22 Dynamic BSTs: Improved Lower Bounds for Greedy-Future ▶ Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' For any ϵ > 0 there exists a sequence X with a subsequence (not necessarily consecutive) X′ ⊆ X such that cost(GF, X′) ≥ (2 − ϵ) · cost(GF, X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' There exists a sequence Y with a subsequence (not necessarily consecutive) Y ′ ⊆ Y such that cost(GF, Y ′) − cost(GF, Y ) = Ω(m · lg lg n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Denote the initial tree by T0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Let Z be the weakly-stable sequence used for proving Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Let TP be the tree that corresponds to Z and TQ the optimized tree, in which the leaves are promoted as in Lemma 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Let P and Q be the sequences that enforce TP and TQ by Theorem 34, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Note that ϵ determines Z, P and Q since it tells us how close to a ratio of 2 we need to get.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Revisit Figure 5 to see the (recursive) structures of TP (on the left) and TQ (on the right, post-promotions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Observe that TP remains static when GF serves Z with it, by definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Moreover, TQ remains static when GF serves Z with it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Indeed, let r be the root of TQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Z queries the item in r every third access and the other accesses are alternating between its left and right subtrees, hence r remains the root of TQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' The rest of TQ remains static recursively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Define X = P ◦ Q ◦ Zk for a large k, and X′ = P ◦ Zk ⊂ X (◦ for concatena- tion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Since GF does not change TP and TQ while serving Z we get that cost(GF,X′) cost(GF,X) = cost(GF,P,T0)+k·cost(GF,Z,TP ) cost(GF,P ◦Q,T0)+k·cost(GF,Z,TQ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' This ratio approaches cost(GF,Z,TP ) cost(GF,Z,TQ) for large enough k, and since Tp and TQ are exactly the trees used in the proof of Lemma 21, we conclude that we can make the resulting ratio as close to 2 as we like (choosing Z, P, Q according to the desired ϵ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' The proof for Y and Y ′ is similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' We define Z to be the strongly-stable sequence used in Theorem 5, and define the appropriate tree-enforcing sequences P and Q by Theorem 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Revisit Figure 6 to see the (recursive) structures of TP (on the left) and TQ (on the right, post-promotions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' We set Y = P ◦ Q ◦ Zk and Y ′ = P ◦ Zk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' The main difference is in the argument of why GF does not change TQ when serving Z (TP is static by definition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' For this, observe that the root of TQ, denote it r, is the left-most leaf in the right subtree of TP .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' This means that the access pattern at r is alternating between its left subtree and its right subtree including itself, thus again we conclude that r remains at the root of TQ, and the rest of TQ remains static recursively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Therefore, cost(GF, Y ′) − cost(GF, Y ) ≥ k · � cost(GF, Z, TP ) − cost(GF, Z, TQ)) − cost(GF, P ◦ Q, T0) = Ω(m · lg lg n) for large enough k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' ◀ ▶ Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Let S be a sequence, we define rev(S) to be the sequence S in reverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' For any ϵ > 0 there exists a sequence X such that cost(GF, rev(X)) ≥ (2 − ϵ) · cost(GF, X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' There exists a sequence Y such that cost(GF, rev(Y )) − cost(GF, Y ) = Ω(m · lg lg n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' The proof is similar to that of Theorem 6, and we define Z, T0, P, TP , Q and TQ the same way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Here we define X = Q ◦ (rev(Z))k+1 ◦ rev(P) and Y = Q ◦ (rev(Z))k+1 ◦ rev(P), for a large k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Recall that Z is different between X (by Theorem 4) and Y (by Theorem 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' We claim that TQ remains static when GF serves rev(Z) over it, rather than Z, by the same argument as in the proof of Theorem 6, because the interleaving pattern in the root is preserved under reversal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Moreover, cost(GF, rev(Z), TQ) = cost(GF, Z, TQ) because the cost on a static tree depends only on the access frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Putting everything together, we get: cost(GF,rev(X)) cost(GF,X) = cost(GF,P,T0)+k·cost(GF,Z,TP )+cost(GF,Z◦rev(Q),TP ) cost(GF,Q,T0)+k·cost(GF,rev(Z),TQ)+cost(GF,rev(Z)◦rev(P ),TQ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' Note that the suffix contains one repetition of Z so that the rest of it (rev(P) or rev(Q)) does not affect the restructuring decisions of GF during the earlier repetitions of Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' The limit of this ratio for large k is cost(GF,Z,TP ) cost(GF,Z,TQ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' We finish the argument as in the proof of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' In the case of Y , we get that cost(GF, Y ′) − cost(GF, Y ) ≥ k · � cost(GF, Z, TP ) − cost(GF, Z, TQ)) − � cost(GF, Q, T0) + cost(GF, rev(Z) ◦ rev(P), TQ) � = Ω(m · lg lg n) for large enough k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} +page_content=' ◀' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE1T4oBgHgl3EQfUAND/content/2301.03084v1.pdf'} diff --git a/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf b/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..36095185cbbf7db8ae5af862a49cd21eea31a7cc --- /dev/null +++ b/9dE3T4oBgHgl3EQfrApq/content/2301.04656v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:c76c92754fd97ce725448660803db600374581abb016b3d309d03918ea16cdbf +size 1517122 diff --git a/9dE3T4oBgHgl3EQfrApq/vector_store/index.pkl b/9dE3T4oBgHgl3EQfrApq/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..8c814f8cb55f2092cb2bdefa4acf48e7778cbef2 --- /dev/null +++ b/9dE3T4oBgHgl3EQfrApq/vector_store/index.pkl @@ -0,0 +1,3 @@ +version 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b/DtE4T4oBgHgl3EQf6Q5X/content/tmp_files/2301.05330v1.pdf.txt @@ -0,0 +1,932 @@ +Strain-induced Landau levels of Majorana fermions in an anisotropically interacting +Kitaev model on a honeycomb lattice +Takuto Yamada1 and Sei-ichiro Suga1 +1Graduate School of Engineering, University of Hyogo, Himeji 671-2280, Japan +(Dated: January 16, 2023) +The low-energy states of an anisotropically interacting Kitaev model on a honeycomb lattice +under triaxial strain are investigated. A numerical calculation shows that quantized states appear +in the low-energy region and their energy is proportional to the square root of the quantum number. +Furthermore, the quantized state at zero energy appears only on one sublattice. The obtained results +are characteristic of the Landau levels of Dirac fermions with time-reversal symmetry, indicating +the emergence of the strain-induced Landau levels of Majorana fermions, which is also determined +in the anisotropic Kitaev model by an analytical calculation. +I. +INTRODUCTION +The Kitaev model is an S = 1/2 quantum spin model +that has bond-dependent Ising-type interactions on a +honeycomb lattice1, called Kitaev interactions. A Ma- +jorana representation of the spin operators has shown +that this model is described by noninteracting itiner- +ant Majorana fermions coupled with Z2 gauge fluxes1. +In the ground state, the model is equivalent to the +noninteracting itinerant Majorana fermions on a hon- +eycomb lattice. +Therefore, the low-lying dispersion is +described by the type of Dirac fermions. +Fascinating +properties related with Majorana fermions have been +revealed by intensive theoretical studies. +Since Majo- +rana fermions are charge-neutral particles acting as their +own antiparticles, they are difficult to interact directly +to electromagnetic fields. Materials exhibiting Kitaev in- +teractions, called Kitaev candidate materials have been +found, including A2IrO3 (A = Na, Ir)2–10, α-RuCl310–18, +and H3LiIr2O619. +The behavior caused by Majorana +fermions in these materials has been studied using various +methods20–24. In their results, half-integer thermal quan- +tum Hall effect can be a conclusive evidence for the emer- +gent Majorana fermions. This phenomenon has been first +pointed out theoretically1 and then observed experimen- +tally in α-RuCl325,26. +Strain fields can induce an artificial vector potential, +which has opposite signs at two Dirac points due to time- +reversal symmetry27. Experiments on artificial graphene +have revealed a strong pseudomagnetic field in the range +of 10 T–100 T and the presence of Landau levels28–32. +The strain-induced pseudomagnetic field can interact di- +rectly with itinerant Majorana fermions. Indeed, numer- +ical calculations have shown the emergence of Landau +levels of itinerant Majorana fermions in the isotropically +interacting Kitaev model under triaxial strain33. Thus, +the phenomena related to the strain-induced Landau lev- +els in the Kitaev candidate materials can be a hallmark +of itinerant Majorana fermions. +According to the ab-initio calculations for the Kitaev +candidate materials, the Kitaev interactions include a +spatial anisotropy6,7,15. +So far, the Landau levels of +itinerant Majorana fermions and the related phenom- +ena have been investigated for the strained Kitaev model +with isotropic interactions33,34, while whether these Lan- +dau levels could emerge in the anisotropically interact- +ing strained Kitaev model is still unclear. Here, in the +present study, we explore the low-energy properties of +the anisotropically interacting Kitaev model on a hon- +eycomb lattice under triaxial strain. +We focus on the +parameter region where the itinerant Majorana fermions +exhibit a gapless dispersion relation in the absence of a +strain field. Through a numerical calculation, we demon- +strate that the strain-induced Landau levels of Majorana +fermions emerge also in the anisotropically interacting +Kitaev system, which is confirmed also by an analytical +calculation. +The rest of the paper is organized as follows. +Sec- +tion II outlines the deformation of the Kitaev model for +the numerical calculation using a singular-value decom- +position method. We then determine the Z2 gauge-flux +sector of the ground state. Section III presents the nu- +merical results for the local density of states (LDOS) of +the itinerant Majorana fermions; we show the presence +of the Landau levels typical of massless Dirac fermions +with time-reversal symmetry in the low-energy region of +the considered model, indicating the emergence of the +strain-induced Landau levels of Majorana fermions in the +anisotropically interacting Kitaev model. Section IV dis- +cusses the low-energy states of the system based on the +analytical calculation, illustrating results consistent with +the numerical outcomes. Finally, the study is summa- +rized in Sec. V. +II. +MODEL AND METHOD +A. +Formulation for numerical calculations +The Hamiltonian is described by +H = − +� +⟨jk⟩x +Jx +jkσx +j σx +k − +� +⟨jk⟩y +Jy +jkσy +j σy +k − +� +⟨jk⟩z +Jz +jkσz +j σz +k, (1) +where σα +j (α = x, y, z) is an α component of the Pauli +matrix at the j site and Jα +jk is the coupling constant be- +tween the nearest-neighbor atoms on the α bond in the +arXiv:2301.05330v1 [cond-mat.str-el] 12 Jan 2023 + +2 +C +C +C +rz +ry +rx +Jz +Jy +Jx +A +B +R = 3 +R = 2 +R = 1 +Jz +Jy +Jx +(a) +(b) +FIG. 1: (Color online) (a) Unstrained honeycomb flakes ex- +pressed by R: R = 1 is a central hexagon (a cross denotes its +center.), R = 2 consists of a central hexagon and six surround- +ing hexagons, R = 3 consists of the R = 2 and twelve sur- +rounding hexagons, and so on. Thus, R honeycomb flake in- +cludes 2N = 6R2 spins. The A and B sublattices are shown in +black and white, respectively. (b) Central hexagon of the un- +strained honeycomb lattice. The coupling constants, Jx, Jy, +and Jz, on the X, Y , and Z bonds are represented in blue, red, +and green, respectively. The vectors connect correspondingly +the nearest-neighbor sites along these bonds. Triaxial strain +C is represented schematically using three brown arrows. +honeycomb lattice. We use a zigzag-terminated honey- +comb lattice with an open boundary condition. The size +of the honeycomb flakes is expressed by R [Fig. 1(a)]33, +and it includes 2N = 6R2 spins, where N is the number of +the unit cells. The triaxial strain originates at the center +of the central hexagon marked by an cross in Fig. 1(a). +In the unstrained honeycomb lattice, the coupling con- +stants are independent of the site: Jα +jk = Jα(> 0). When +a weak triaxial strain is applied as schematically shown +in Fig. 1(b), the coupling constant Jα +jk becomes33,35–37 +Jα +jk ≈ Jα [1 − β (1 − |rj − rk|/a0)], where β is the mag- +netoelastic coupling and a0 is the unstrained bond length. +The position vector of an atom is given by rj = r0 +j + uj, +where r0 +j = (x0 +j, y0 +j ) is the position vector in the un- +strained lattice and uj is the displacement vector; they +are expressed respectively as r0 +j = |r0 +j |(cos θ0 +j, sin θ0 +j) and +uj = (C/a0) |r0 +j |2(cos 3θ0 +j, sin 3θ0 +j) using the polar coordi- +nate, where C is the triaxial strain strength. Jα +jk must be +positive on the whole nearest-neighbor bonds. According +to our numerical calculation, this condition is satisfied for +CR ⪅ 0.3. We thus set CR = 0.2 in the following nu- +merical calculation. In the honeycomb flakes possessing +the same constant CR, a scaling holds concerning the +honeycomb flake shapes for different R values38. +To +diagonalize +the +Hamiltonian, +four +Majorana +fermions, cj and bα +j , are set at each site1, satisfying +{cj, ck} = 2δjk, {cj, bα +k} = 0, and {bα +j , bβ +k} = 2δαβδjk. +To project the enlarged Hilbert space into the physi- +cal Hilbert space, the constraint cjbx +j by +j bz +j = 1 is im- +posed. +In this procedure, the spin operator is repre- +sented as σα +j += icjbα +j and the Hamiltonian reads as +Hu = i � +α∈{x,y,z} +� +⟨jk⟩α Jα +jkuα +jkcjck, where uα +jk = ibα +j bα +k +is a bond operator with an eigenvalue of ±1 and satisfies +[Hu, uα +jk] = 0. Thus, uα +jk is identified with a static Z2 +gauge field between the nearest-neighbor j and k sites on +the α bond. We then introduce a relevant gauge-flux op- +erator defined as a product of the six Z2 gauge fields sur- +rounding a hexagon1. The gauge-flux operator commutes +with Hu and its eigenvalue becomes ±1. Therefor, the +system can be mapped to itinerant Majorana fermions +coupled with the Z2 gauge fluxes on the hexagonal pla- +quettes. For every configurations of the Z2 gauge fluxes, +the Hamiltonian Hu can be expressed as33 +Hu = i +2 +� +¯cT +A ¯cT +B +� � +0 +M +−M T +0 +� � +¯cA +¯cB +� +, +(2) +where Mjk = Jα +jkuα +jk and ¯cA(B) is an N-component vec- +tor representing the itinerant Majorana fermions on the +A(B) sublattice. We call the Z2 gauge-flux having −1 +‘flux’. When at least two of the three coupling constants +are equal in the unstrained system, the Lieb’s theorem39 +states that the exact ground state is in the sector where +all the Z2 gauge fluxes take unity (the flux-free sector)1. +The sector where the n gauge fluxes become −1 is called +the n-flux sector. +By using a singular-value decomposition method, we +calculate the eigenvalues ϵm,n (m = 1, 2, · · · , N) and the +eigenvectors for a given n-flux configuration n; then we +obtain the LDOS, ρj,A(B)(E), of the itinerant Majorana +fermions on the A(B) sublattice in the j-th unit cell. The +magnetoelastic coupling is set as β = 1 for simplicity. +The coupling constants in the unstrained lattice satisfy +Jx + Jy + Jz = 1, forming the triangle in the parameter +space expressed by Jx, Jy, and Jz [left panel of Fig. +3(a)]1, while the central downward triangle enlarged in +the right panel represents the gapless phase. +B. +One-flux gap and ground-state sector +In the strained honeycomb lattice, the translational +invariance is broken, and hence the Lieb’s theorem can- +not be adopted. +Thus, we must confirm whether the +ground state is in the flux-free sector for the given R +with CR = 0.2. The ground-state energy for a n-flux +configuration, n, is given by EGS,n = − � +m ϵm,n. In the +open boundary system, the one-flux state is possible and +can be a candidate competing with the flux-free state38. +We calculate the one-flux gap ∆1 = EGS,1 − EGS,0 for +all the one-flux configurations at various R up to 90 for +the given Jx, Jy, and Jz. Figure 2 depicts the typical +behavior of the minimum one-flux gap ∆min +1 +for a given +R. For R ≥ 35, ∆min +1 +is well described by the follow- +ing polynomial: ∆min +1 += aR−4 + bR−2 + c, where a, b, +and c are the constants. The extrapolated c values for +R → ∞ are 1.61 × 10−3, 4.44 × 10−4, and 3.73 × 10−4 in +Figs. 2(a)-2(c), respectively. We perform the same cal- +culations for the given coupling constants used in Figs. +3(b)-3(g), finding that all the extrapolated c values for +R → ∞ positive. Thus, we can deduce that the ground +state of the anisotropically interacting Kitaev model for +CR = 0.2 is in the flux-free sector. In the following nu- + +3 +0.000 +0.005 +0.010 +∆min +1 +(a) Jx = 1/3, Jy = 1/3, Jz = 1/3 +∆min +1 +|R→∞ = 1.61 × 10−3 +R ≥ 35 +R < 35 +0.0000 +0.0005 +0.0010 +0.0015 +0.0020 +∆min +1 +(b) Jx = 7/24, Jy = 7/24, Jz = 5/12 +∆min +1 +|R→∞ = 4.44 × 10−4 +R ≥ 35 +R < 35 +0 +1 +202 +1 +252 +1 +302 +1 +352 +1 +502 +1 +802 +1/R2 +0.00000 +0.00025 +0.00050 +0.00075 +0.00100 +∆min +1 +(c) Jx = 11/48, Jy = 17/48, Jz = 5/12 +∆min +1 +|R→∞ = 3.73 × 10−4 +R ≥ 35 +R < 35 +FIG. 2: (Color online) (a) Minimum one-flux gap (∆min +1 +) for +a given R. The dashed lines reprsent the polynomial, ∆min +1 += +aR−4 + bR−2 + c, that well describes ∆min +1 +for R ≥ 35 for +the constants a, b, and c having the following values: (a) +−2.95 × 103, 1.04 × 101, 1.61 × 10−3; (b) 6.56 × 101, 2.81 × +10−1, 4.44 × 10−4; (c) −6.77 × 101, 2.42 × 10−1, 3.73 × 10−4. +merical calculations, we set R = 60 (2N = 21600) and +C = 1/300. +III. +NUMERICAL RESULTS +Figures 3(b)-3(g) illustrate the LDOS, ρj,A(E) and +ρj,B(E), at the site in the central hexagon of the sys- +tem. The left and right panels show the results for the +A and B sublattices, respectively. We also evaluate the +integral value: Ij,A(B)(E) = +� E +0 ρj,A(B)(E′)dE′. Figure +3(b) displays the results for the isotropic interactions that +are plotted using the small open circle in the right panel +of Fig. 3(a). The coupling constants of the top, middle, +and bottom panels in Fig. 3(c)-3(g) correspond to the +black dots from close to the center toward the edge along +the lines A-E [right panel of Fig. 3(a)], respectively. +We find that Ij,A(B)(E) forms plateaus. +We mea- +sure them in units of the lowest-energy plateau in the +A sublattice, finding that the pronounced plateaus are +described as 2n + 1 (n = 0, 1, 2, · · · ) on the A sublattice +and 2n (n = 0, 1, 2, · · · ) on the B sublattice from the +low energy in turn. At or in the vicinity of the bound- +ary between pronounced neighboring plateaus, ρj,A/B(E) +reaches a peak, as indicated by the vertical dashed lines +in Figs. +3(b)-3(g). +The peak at E = 0 generally ap- +pears only on the A sublattice, and it is called the sub- +lattice polarization40. We then plot the peak energies, +En (n = 0, 1, 2, · · · ), on the A sublattice (Fig. 4), whose +coupling constants correspond to the black dots along the +lines A-E [right panel of Fig. 3(a)]. As shown in Fig. 4, +En satisfies the relation En ∝ √n. These results are char- +acteristic of the Landau levels of massless Dirac fermions +with time-reversal symmetry29,36,37,41; thus, the itinerant +Majorana fermions under triaxial strain are quantized to +the Landau levels. +Figure 3(c)-3(g) indicate that as the system leaves the +isotropically interacting point, the Landau levels of Ma- +jorana fermions are smeared at the higher energies and +their number is reduces at the lower energies. Within the +shaded areas on the lines A-E in the right panel in Fig. +3(a), at least three Landau levels of Majorana fermions +from n = 0 appear on the A sublattice, confirming the +relation En ∝ √n. From the permutation of Jx, Jy, and +Jz, there are six equivalent regions in the phase diagram +shown in Fig. 3(a). We apply our results to the five ad- +ditional regions and summarize the results in the right +panel of Fig. 3(a). In the shaded area, the Landau lev- +els of Majorana fermions emerge. In the outer unshaded +area, instead, one or two peaks appear at and next to +E = 0 in ρj,A(E), and the sublattice polarization is sat- +isfied. +We perform the same calculations for R = 40, +45, and 50 while keeping CR = 0.2. As R increases, the +region where the Landau levels of Majorana fermions ap- +pear expands toward the boundary between gapless and +gapped phases. We, therefore, expect the Landau levels +of Majorana fermions to emerge in the whole unstrained +gapless phase, when the system becomes large enough. +IV. +EFFECTIVE LOW-ENERGY THEORY +A. +Formulation, eigenenergy, and sublattice +polarization +We now discuss the low-energy states of the itiner- +ant Majorana fermions on the triaxially strained honey- +comb lattice through an analytical calculation. Following +Refs.40,42, we adopt the effects of weak triaxial strain as +Jα(r) = Jα � +1 + τ +� +r · rα/3a02�� +, where rz = a0(0, 1), +rx = a0(− +√ +3/2, −1/2), and ry = a0( +√ +3/2, −1/2) are +vectors that connect the unstrained nearest-neighbor +sites, as shown in Fig. 1(b), and τ controls the strain +strength. Also in the effective low-energy theory, the cou- +pling constants in the unstrained lattice satisfy Jx+Jy + +Jz = 1. The strain effect is considered around the Dirac +points, K and K′, of the isotropically interacting system +in the unstrained honeycomb lattice. Therefore, this for- +mulation is effective for a system with weakly anisotropic + +4 +A B C D E +(a) Phase diagram +(c) A +(d) B +(e) C +(f) D +(g) E +(b) Isotropic point +FIG. 3: (Color online) (a) Phase diagram of the unstrained Kitaev model on the plane Jx +Jy +Jz = 1, where Jα (α = x, y, z) +are the coupling constants of the unstrained system with Jα ≥ 0. In the left panel, the inner triangle represents the gapless +phase and the three outer triangles are gapped phases. +A gapless phase is enlarged in the right panel, where over three +Landau levels of Majorana fermions appear in the inner shaded region and the sublattice polarization is satisfied for R = 60 +and C = 1/300. In the outer unshaded area, one or two peaks appear at and next to E = 0 in ρj,A(E), and the sublattice +polarization is satisfied. (b)-(g) Local density of states, ρj,A/B(E), at the central hexagon of the system and the integral values, +Ij,A/B(E) = +� E +0 ρj,A/B(E′)dE′, for R = 60 and C = 1/300; (b) ρj,A/B(E) and Ij,A/B(E) for the isotropic interactions plotted +using the small open circle in the right panel of (a); (c)-(g) the coupling constants of the top, middle, and bottom panels +correspond to the black dots from close to the center toward the edge along the lines A-E [right panel of (a)], respectively. +Kitaev interactions. +The pseudovector potential, Aξ = (ξAξ +x, ξAξ +y), induced +by triaxial strain is given as +Aξ +x = vξ +x +−1 �� +Jz − 1 +2Jx − 1 +2Jy +� ++ (Jx − Jy) +xτ +4 +√ +3a0 ++ +� +Jz + 1 +4Jx + 1 +4Jy +� yτ +3a0 +� +, +(3) + +5 +(a) A +(b) B +(c) C +(d) D +(e) E +FIG. 4: (Color online) Peak energies, En (n = 0, 1, 2, · · · ), of ρj,A(E) as functions of √n. The coupling constants in (a)-(e) +correspond to those in Figs. 3(c)-3(g), respectively. +Aξ +y = vξ +y +−1 +√ +3 +2 +� +(Jx − Jy) − (Jx + Jy) +xτ +2 +√ +3a0 +− (Jx − Jy) yτ +6a0 +� +, +(4) +where ξ = ±1 for K/K′ and +vξ +x = +√ +3a0 +4ℏ +�√ +3 (Jx + Jy) + iξ (Jx − Jy) +� +≡ |vx|eiξφx, +vξ +y = +√ +3a0 +4ℏ +� 1 +√ +3 (3Jz + 1) − iξ (Jx − Jy) +� +≡ |vy|eiξφy. +The pseudomagnetic field, Bξ = (0, 0, ξBξ +z), is given as +Bξ +z = cos φx∂xAξ +y−cos φy∂yAξ +x. The Hamiltonian around +K and K′ reads +Hξ = ξ +� +|vxvy| +� +0 +Πξ +x +∗ − iΠξ +y +∗ +Πξ +x + iΠξ +y +0 +� +, +(5) +where Πξ +x += +� +|vx/vy| +� +eiξφxpx + ξAξ +x +� +and Πξ +y += +� +|vy/vx| +� +eiξφypy + ξAξ +y +� +. By defining the annihilation +operators as +aK = +lB +√ +2ℏ +� +ΠK +x +∗ − iΠK +y +∗� +, aK′ = +lB +√ +2ℏ +� +ΠK′ +x ++ iΠK′ +y +� +(6) +with lB +2 += ℏ/|Bξ +z|, the eigenenergy is obtained as +En = +� +2 +√ +2ℏ/lB +� � +|vxvy|√n, (n = 0, 1, 2, · · · ). +The +n = 0 eigenenstates are |Ψ+ +0 ⟩ = (0, |ψ0⟩)T for K and +|Ψ− +0 ⟩ = (|ψ0⟩, 0)T for K′. Since the components of the +two-dimensional spinor in the sublattice basis are ex- +changed in K and K′, the eigenstates at n = 0 are +nonzero only on the A sublattice (sublattice polariza- +tion), while they are nonzero on both the A and B sub- +lattices at n ̸= 0. These eigenenergy and eigenstate fea- +tures agree with the numerical results, providing the evi- +dence of the emergence of the Landau levels of Majorana +fermions. +B. +Numerical results in terms of the effective +low-energy theory +FIG. 5: (Color online) Peak energy, E1, of ρj,A(E) as a func- +tion of +√ +C at the isotropically interacting system obtained in +the numerical calculation. R =40, 50, and 60 are set. Corre- +sponding C is obtained for the fixed CR =1/25, 2/25, 3/25, +4/25, 5/25, 6/25, and 7/25, respectively. +Let us discuss the numerical results in terms of the +effective low-energy theory. +As the system leaves the +isotropically interacting point in the numerical calcula- +tion, the Landau levels of Majorana fermions are smeared +at the higher energies and their number is reduced at the +lower energies, as shown in Figs. 3(c)-3(g). When the +anisotropy of the interactions becomes strong, the Dirac +points deviate considerably from those of the isotropi- +cally interacting unstrained system. This situation is op- + +6 +TABLE I: Parameters a and b to fit the coefficient (E1) of En ∝ √n using a linear regression, E1 = a +√ +8C +b. The coefficients +(E1) are obtained by the numerical calculation for R =40, 50, and 60 at the fixed CR =1/25, 2/25, 3/25, 4/25, 5/25, 6/25, +and 7/25, respectively. The coupling constants are for the isotropically interacting system and for the system closest to the +isotropically interacting point on each line A-E in Fig. 3(a). The third line (below the results for a) denotes the ratios of a to +0.99951 at the isotropic point are denoted. +(Jx, Jy, Jz) +( 1 +3, 1 +3, 1 +3) +A ( 31 +96, 31 +96, 17 +48) B ( 121 +384, 127 +384, 17 +48) C ( 59 +192, 65 +192, 17 +48) +D ( 115 +384, 133 +384, 17 +48) +E ( 7 +24, 17 +48, 17 +48) +a +0.99951 +0.99966 +1.00109 +1.00420 +1.00720 +1.00989 +Ration of a to 0.99951 +1 +1.00015 +1.00016 +1.00047 +1.00769 +1.01039 +b +9.03667 × 10−5 1.48732 × 10−4 2.51854 × 10−5 −2.13822 × 10−4 −3.83096 × 10−4 −4.69701 × 10−4 +posed to the condition for the effective low-energy the- +ory to hold. Thus, as the system leaves the isotropically +interacting point, the higher-order terms of Aξ become +relevant, reducing the number of the Landau levels of +Majorana fermions. +We next investigate the relation between the control +parameters of the strain strength, C and τ, in the numer- +ical and analytical calculations. To this end, we evaluate +the coefficient of En ∝ √n obtained in the numerical cal- +culation for R =40, 50, and 60 at the seven values of the +fixed CR within CR < 0.3. Figure 5 illustrates the coef- +ficient (E1) as a function of +√ +C in the isotropically inter- +acting system. The data follows the line E1 = a +√ +8C + b +with a ≈ 0.99951 and b ≈ 9.03667×10−5, indicating that +the relation, En = +√ +8Cn, is deduced within our numer- +ical accuracy. The coefficient E1 in the system closest +to the isotropically interacting point on each line A-E is +evaluated in the same way. The evaluated a and b are +summarized in Table I, indicating that a ≈ 1.0 and b ≈ 0, +and thus En ≈ +√ +8Cn. +The coefficient of En ∝ √n in the effective low-energy +theory, +� +2 +√ +2ℏ/lB +� � +|vxvy|, is given by using τ and one +of the three coupling constants, leading to the expression +for En as +En = +� +(Jz − 1)2 + +� +Jz + 1 +3 +�2� 1 +2 � +3τn +2 . +(7) +The eigenenergy in the isotropically interacting system +is obtained as En = 2 +� +τn/3. +Comparing En in the +numerical and analytical calculations for the isotropi- +cally interacting system, the relation, τ = 6C, is de- +rived. This relation is consistent with that derived by +comparing the pseudomagnetic field in the numerical43 +and analytical42 calculations for the isotropically inter- +acting system, where |Bξ +z| = 4Cℏ/a02 and 2ℏτ/(3a02), +respectively. Since Jz’s in the anisotropically interacting +system in Table I are the same, their coefficients of √τ +take the same value, +� +1025/768, according to Eq. (7). +The ratio of this coefficient to that in the isotropically +interacting system is 1.00049. This ratio is close to that +denoted in Table I. Therefore, the relation τ = 6C is +approximately satisfied in the anisotropically interacting +systems denoted in Table I. +V. +SUMMARY +We have investigated the low-energy states of an +anisotropically interacting Kitaev model under triaxial +strain. Based on the numerical results, we argue that the +Landau levels of Majorana fermions emerge in the wide +gapless region around the isotropically interacting point +in the phase diagram of the unstraind system. The emer- +gence of the strain-induced Landau levels of Majorana +fermions in the anisotropically interacting Kitaev system +is confirmed also by an analytical calculation. When uni- +axial strain is applied, the same features (i.e., En ∝ √n +and sublattice polarization) are expected to appear. To +observe the features of the strain-induced Landau lev- +els of Majorana fermions, scanning tunneling microscopy +(STM) is promising. According to the STM theory for +the Kitaev model, the differential conductance through +a single site provides direct information on the LDOS +of Majorana fermions at low temperatures44. +In fact, +the two features of the strain-induced Landau levels (i.e., +En ∝ √n and sublattice polarization) have been observed +in artificial graphene under simulated triaxial strain eval- +uated pseudomagnetic fields up to 60 T29. The fabrica- +tion of the α-RuCl3 thin films has been studied actively +in recent years. This situation favors experiments for in- +vestigating whether the strain-induced Landau levels of +Majorana fermions emerge in Kitaev candidate materials +such as α-RuCl3. We hope that our results contribute to +such studies. +Acknowledgments +We would like to thank T. Suzuki and R. Taniguchi for +valuable discussions. 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Jauho, Phys. Rev. B +93, 035456 (2016). +44 M. Udagawa, S. Takayoshi, and T. Oka, Phys. Rev. Lett. +126, 127201 (2021). + diff --git a/DtE4T4oBgHgl3EQf6Q5X/content/tmp_files/load_file.txt b/DtE4T4oBgHgl3EQf6Q5X/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..7baaa827846cb4d7e43053faee73c4a8aa36a1ec --- /dev/null +++ b/DtE4T4oBgHgl3EQf6Q5X/content/tmp_files/load_file.txt @@ -0,0 +1,751 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf,len=750 +page_content='Strain-induced Landau levels of Majorana fermions in an anisotropically interacting Kitaev model on a honeycomb lattice Takuto Yamada1 and Sei-ichiro Suga1 1Graduate School of Engineering, University of Hyogo, Himeji 671-2280, Japan (Dated: January 16, 2023) The low-energy states of an anisotropically interacting Kitaev model on a honeycomb lattice under triaxial strain are investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' A numerical calculation shows that quantized states appear in the low-energy region and their energy is proportional to the square root of the quantum number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Furthermore, the quantized state at zero energy appears only on one sublattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' The obtained results are characteristic of the Landau levels of Dirac fermions with time-reversal symmetry, indicating the emergence of the strain-induced Landau levels of Majorana fermions, which is also determined in the anisotropic Kitaev model by an analytical calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' INTRODUCTION The Kitaev model is an S = 1/2 quantum spin model that has bond-dependent Ising-type interactions on a honeycomb lattice1, called Kitaev interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' A Ma- jorana representation of the spin operators has shown that this model is described by noninteracting itiner- ant Majorana fermions coupled with Z2 gauge fluxes1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' In the ground state, the model is equivalent to the noninteracting itinerant Majorana fermions on a hon- eycomb lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Therefore, the low-lying dispersion is described by the type of Dirac fermions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Fascinating properties related with Majorana fermions have been revealed by intensive theoretical studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Since Majo- rana fermions are charge-neutral particles acting as their own antiparticles, they are difficult to interact directly to electromagnetic fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Materials exhibiting Kitaev in- teractions, called Kitaev candidate materials have been found, including A2IrO3 (A = Na, Ir)2–10, α-RuCl310–18, and H3LiIr2O619.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' The behavior caused by Majorana fermions in these materials has been studied using various methods20–24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' In their results, half-integer thermal quan- tum Hall effect can be a conclusive evidence for the emer- gent Majorana fermions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' This phenomenon has been first pointed out theoretically1 and then observed experimen- tally in α-RuCl325,26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Strain fields can induce an artificial vector potential, which has opposite signs at two Dirac points due to time- reversal symmetry27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Experiments on artificial graphene have revealed a strong pseudomagnetic field in the range of 10 T–100 T and the presence of Landau levels28–32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' The strain-induced pseudomagnetic field can interact di- rectly with itinerant Majorana fermions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Indeed, numer- ical calculations have shown the emergence of Landau levels of itinerant Majorana fermions in the isotropically interacting Kitaev model under triaxial strain33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Thus, the phenomena related to the strain-induced Landau lev- els in the Kitaev candidate materials can be a hallmark of itinerant Majorana fermions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' According to the ab-initio calculations for the Kitaev candidate materials, the Kitaev interactions include a spatial anisotropy6,7,15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' So far, the Landau levels of itinerant Majorana fermions and the related phenom- ena have been investigated for the strained Kitaev model with isotropic interactions33,34, while whether these Lan- dau levels could emerge in the anisotropically interact- ing strained Kitaev model is still unclear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Here, in the present study, we explore the low-energy properties of the anisotropically interacting Kitaev model on a hon- eycomb lattice under triaxial strain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' We focus on the parameter region where the itinerant Majorana fermions exhibit a gapless dispersion relation in the absence of a strain field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Through a numerical calculation, we demon- strate that the strain-induced Landau levels of Majorana fermions emerge also in the anisotropically interacting Kitaev system, which is confirmed also by an analytical calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' The rest of the paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Sec- tion II outlines the deformation of the Kitaev model for the numerical calculation using a singular-value decom- position method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' We then determine the Z2 gauge-flux sector of the ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Section III presents the nu- merical results for the local density of states (LDOS) of the itinerant Majorana fermions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' we show the presence of the Landau levels typical of massless Dirac fermions with time-reversal symmetry in the low-energy region of the considered model, indicating the emergence of the strain-induced Landau levels of Majorana fermions in the anisotropically interacting Kitaev model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Section IV dis- cusses the low-energy states of the system based on the analytical calculation, illustrating results consistent with the numerical outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Finally, the study is summa- rized in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' MODEL AND METHOD A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Formulation for numerical calculations The Hamiltonian is described by H = − � ⟨jk⟩x Jx jkσx j σx k − � ⟨jk⟩y Jy jkσy j σy k − � ⟨jk⟩z Jz jkσz j σz k, (1) where σα j (α = x, y, z) is an α component of the Pauli matrix at the j site and Jα jk is the coupling constant be- tween the nearest-neighbor atoms on the α bond in the arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content='05330v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content='str-el] 12 Jan 2023 2 C C C rz ry rx Jz Jy Jx A B R = 3 R = 2 R = 1 Jz Jy Jx (a) (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' 1: (Color online) (a) Unstrained honeycomb flakes ex- pressed by R: R = 1 is a central hexagon (a cross denotes its center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' ), R = 2 consists of a central hexagon and six surround- ing hexagons, R = 3 consists of the R = 2 and twelve sur- rounding hexagons, and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Thus, R honeycomb flake in- cludes 2N = 6R2 spins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' The A and B sublattices are shown in black and white, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' (b) Central hexagon of the un- strained honeycomb lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' The coupling constants, Jx, Jy, and Jz, on the X, Y , and Z bonds are represented in blue, red, and green, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' The vectors connect correspondingly the nearest-neighbor sites along these bonds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Triaxial strain C is represented schematically using three brown arrows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' honeycomb lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' We use a zigzag-terminated honey- comb lattice with an open boundary condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' The size of the honeycomb flakes is expressed by R [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' 1(a)]33, and it includes 2N = 6R2 spins, where N is the number of the unit cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' The triaxial strain originates at the center of the central hexagon marked by an cross in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' 1(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' In the unstrained honeycomb lattice, the coupling con- stants are independent of the site: Jα jk = Jα(> 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' When a weak triaxial strain is applied as schematically shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' 1(b), the coupling constant Jα jk becomes33,35–37 Jα jk ≈ Jα [1 − β (1 − |rj − rk|/a0)], where β is the mag- netoelastic coupling and a0 is the unstrained bond length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' The position vector of an atom is given by rj = r0 j + uj, where r0 j = (x0 j, y0 j ) is the position vector in the un- strained lattice and uj is the displacement vector;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' they are expressed respectively as r0 j = |r0 j |(cos θ0 j, sin θ0 j) and uj = (C/a0) |r0 j |2(cos 3θ0 j, sin 3θ0 j) using the polar coordi- nate, where C is the triaxial strain strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Jα jk must be positive on the whole nearest-neighbor bonds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' According to our numerical calculation, this condition is satisfied for CR ⪅ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' We thus set CR = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content='2 in the following nu- merical calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' In the honeycomb flakes possessing the same constant CR, a scaling holds concerning the honeycomb flake shapes for different R values38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' To diagonalize the Hamiltonian, four Majorana fermions, cj and bα j , are set at each site1, satisfying {cj, ck} = 2δjk, {cj, bα k} = 0, and {bα j , bβ k} = 2δαβδjk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' To project the enlarged Hilbert space into the physi- cal Hilbert space, the constraint cjbx j by j bz j = 1 is im- posed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' In this procedure, the spin operator is repre- sented as σα j = icjbα j and the Hamiltonian reads as Hu = i � α∈{x,y,z} � ⟨jk⟩α Jα jkuα jkcjck, where uα jk = ibα j bα k is a bond operator with an eigenvalue of ±1 and satisfies [Hu, uα jk] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Thus, uα jk is identified with a static Z2 gauge field between the nearest-neighbor j and k sites on the α bond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' We then introduce a relevant gauge-flux op- erator defined as a product of the six Z2 gauge fields sur- rounding a hexagon1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' The gauge-flux operator commutes with Hu and its eigenvalue becomes ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Therefor, the system can be mapped to itinerant Majorana fermions coupled with the Z2 gauge fluxes on the hexagonal pla- quettes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' For every configurations of the Z2 gauge fluxes, the Hamiltonian Hu can be expressed as33 Hu = i 2 � ¯cT A ¯cT B � � 0 M −M T 0 � � ¯cA ¯cB � , (2) where Mjk = Jα jkuα jk and ¯cA(B) is an N-component vec- tor representing the itinerant Majorana fermions on the A(B) sublattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' We call the Z2 gauge-flux having −1 ‘flux’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' When at least two of the three coupling constants are equal in the unstrained system, the Lieb’s theorem39 states that the exact ground state is in the sector where all the Z2 gauge fluxes take unity (the flux-free sector)1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' The sector where the n gauge fluxes become −1 is called the n-flux sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' By using a singular-value decomposition method, we calculate the eigenvalues ϵm,n (m = 1, 2, · · · , N) and the eigenvectors for a given n-flux configuration n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' then we obtain the LDOS, ρj,A(B)(E), of the itinerant Majorana fermions on the A(B) sublattice in the j-th unit cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' The magnetoelastic coupling is set as β = 1 for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' The coupling constants in the unstrained lattice satisfy Jx + Jy + Jz = 1, forming the triangle in the parameter space expressed by Jx, Jy, and Jz [left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' 3(a)]1, while the central downward triangle enlarged in the right panel represents the gapless phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' One-flux gap and ground-state sector In the strained honeycomb lattice, the translational invariance is broken, and hence the Lieb’s theorem can- not be adopted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Thus, we must confirm whether the ground state is in the flux-free sector for the given R with CR = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' The ground-state energy for a n-flux configuration, n, is given by EGS,n = − � m ϵm,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' In the open boundary system, the one-flux state is possible and can be a candidate competing with the flux-free state38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' We calculate the one-flux gap ∆1 = EGS,1 − EGS,0 for all the one-flux configurations at various R up to 90 for the given Jx, Jy, and Jz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Figure 2 depicts the typical behavior of the minimum one-flux gap ∆min 1 for a given R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' For R ≥ 35, ∆min 1 is well described by the follow- ing polynomial: ∆min 1 = aR−4 + bR−2 + c, where a, b, and c are the constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' The extrapolated c values for R → ∞ are 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content='61 × 10−3, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content='44 × 10−4, and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content='73 × 10−4 in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' 2(a)-2(c), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' We perform the same cal- culations for the given coupling constants used in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' 3(b)-3(g), finding that all the extrapolated c values for R → ∞ positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Thus, we can deduce that the ground state of the anisotropically interacting Kitaev model for CR = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content='2 is in the flux-free sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' In the following nu- 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content='010 ∆min 1 (a) Jx = 1/3, Jy = 1/3, Jz = 1/3 ∆min 1 |R→∞ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content='61 × 10−3 R ≥ 35 R < 35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content='0000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content='0005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content='0010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content='0015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content='0020 ∆min 1 (b) Jx = 7/24, Jy = 7/24, Jz = 5/12 ∆min 1 |R→∞ = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content='44 × 10−4 R ≥ 35 R < 35 0 1 202 1 252 1 302 1 352 1 502 1 802 1/R2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content='00000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content='00025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content='00050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content='00075 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content='00100 ∆min 1 (c) Jx = 11/48, Jy = 17/48, Jz = 5/12 ∆min 1 |R→∞ = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content='73 × 10−4 R ≥ 35 R < 35 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' 2: (Color online) (a) Minimum one-flux gap (∆min 1 ) for a given R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' The dashed lines reprsent the polynomial, ∆min 1 = aR−4 + bR−2 + c, that well describes ∆min 1 for R ≥ 35 for the constants a, b, and c having the following values: (a) −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content='95 × 103, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content='04 × 101, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content='61 × 10−3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' (b) 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content='56 × 101, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content='81 × 10−1, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content='44 × 10−4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' (c) −6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content='77 × 101, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content='42 × 10−1, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content='73 × 10−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' merical calculations, we set R = 60 (2N = 21600) and C = 1/300.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' NUMERICAL RESULTS Figures 3(b)-3(g) illustrate the LDOS, ρj,A(E) and ρj,B(E), at the site in the central hexagon of the sys- tem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' The left and right panels show the results for the A and B sublattices, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' We also evaluate the integral value: Ij,A(B)(E) = � E 0 ρj,A(B)(E′)dE′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Figure 3(b) displays the results for the isotropic interactions that are plotted using the small open circle in the right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' 3(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' The coupling constants of the top, middle, and bottom panels in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' 3(c)-3(g) correspond to the black dots from close to the center toward the edge along the lines A-E [right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' 3(a)], respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' We find that Ij,A(B)(E) forms plateaus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' We mea- sure them in units of the lowest-energy plateau in the A sublattice, finding that the pronounced plateaus are described as 2n + 1 (n = 0, 1, 2, · · · ) on the A sublattice and 2n (n = 0, 1, 2, · · · ) on the B sublattice from the low energy in turn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' At or in the vicinity of the bound- ary between pronounced neighboring plateaus, ρj,A/B(E) reaches a peak, as indicated by the vertical dashed lines in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' 3(b)-3(g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' The peak at E = 0 generally ap- pears only on the A sublattice, and it is called the sub- lattice polarization40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' We then plot the peak energies, En (n = 0, 1, 2, · · · ), on the A sublattice (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' 4), whose coupling constants correspond to the black dots along the lines A-E [right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' 3(a)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' 4, En satisfies the relation En ∝ √n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' These results are char- acteristic of the Landau levels of massless Dirac fermions with time-reversal symmetry29,36,37,41;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' thus, the itinerant Majorana fermions under triaxial strain are quantized to the Landau levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Figure 3(c)-3(g) indicate that as the system leaves the isotropically interacting point, the Landau levels of Ma- jorana fermions are smeared at the higher energies and their number is reduces at the lower energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Within the shaded areas on the lines A-E in the right panel in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' 3(a), at least three Landau levels of Majorana fermions from n = 0 appear on the A sublattice, confirming the relation En ∝ √n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' From the permutation of Jx, Jy, and Jz, there are six equivalent regions in the phase diagram shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' 3(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' We apply our results to the five ad- ditional regions and summarize the results in the right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' 3(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' In the shaded area, the Landau lev- els of Majorana fermions emerge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' In the outer unshaded area, instead, one or two peaks appear at and next to E = 0 in ρj,A(E), and the sublattice polarization is sat- isfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' We perform the same calculations for R = 40, 45, and 50 while keeping CR = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' As R increases, the region where the Landau levels of Majorana fermions ap- pear expands toward the boundary between gapless and gapped phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' We, therefore, expect the Landau levels of Majorana fermions to emerge in the whole unstrained gapless phase, when the system becomes large enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' EFFECTIVE LOW-ENERGY THEORY A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Formulation, eigenenergy, and sublattice polarization We now discuss the low-energy states of the itiner- ant Majorana fermions on the triaxially strained honey- comb lattice through an analytical calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Following Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content='40,42, we adopt the effects of weak triaxial strain as Jα(r) = Jα � 1 + τ � r · rα/3a02�� , where rz = a0(0, 1), rx = a0(− √ 3/2, −1/2), and ry = a0( √ 3/2, −1/2) are vectors that connect the unstrained nearest-neighbor sites, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' 1(b), and τ controls the strain strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Also in the effective low-energy theory, the cou- pling constants in the unstrained lattice satisfy Jx+Jy + Jz = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' The strain effect is considered around the Dirac points, K and K′, of the isotropically interacting system in the unstrained honeycomb lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Therefore, this for- mulation is effective for a system with weakly anisotropic 4 A B C D E (a) Phase diagram (c) A (d) B (e) C (f) D (g) E (b) Isotropic point FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' 3: (Color online) (a) Phase diagram of the unstrained Kitaev model on the plane Jx +Jy +Jz = 1, where Jα (α = x, y, z) are the coupling constants of the unstrained system with Jα ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' In the left panel, the inner triangle represents the gapless phase and the three outer triangles are gapped phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' A gapless phase is enlarged in the right panel, where over three Landau levels of Majorana fermions appear in the inner shaded region and the sublattice polarization is satisfied for R = 60 and C = 1/300.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' In the outer unshaded area, one or two peaks appear at and next to E = 0 in ρj,A(E), and the sublattice polarization is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' (b)-(g) Local density of states, ρj,A/B(E), at the central hexagon of the system and the integral values, Ij,A/B(E) = � E 0 ρj,A/B(E′)dE′, for R = 60 and C = 1/300;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' (b) ρj,A/B(E) and Ij,A/B(E) for the isotropic interactions plotted using the small open circle in the right panel of (a);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' (c)-(g) the coupling constants of the top, middle, and bottom panels correspond to the black dots from close to the center toward the edge along the lines A-E [right panel of (a)], respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Kitaev interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' The pseudovector potential, Aξ = (ξAξ x, ξAξ y), induced by triaxial strain is given as Aξ x = vξ x −1 �� Jz − 1 2Jx − 1 2Jy � + (Jx − Jy) xτ 4 √ 3a0 + � Jz + 1 4Jx + 1 4Jy � yτ 3a0 � , (3) 5 (a) A (b) B (c) C (d) D (e) E FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' 4: (Color online) Peak energies, En (n = 0, 1, 2, · · · ), of ρj,A(E) as functions of √n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' The coupling constants in (a)-(e) correspond to those in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' 3(c)-3(g), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Aξ y = vξ y −1 √ 3 2 � (Jx − Jy) − (Jx + Jy) xτ 2 √ 3a0 − (Jx − Jy) yτ 6a0 � , (4) where ξ = ±1 for K/K′ and vξ x = √ 3a0 4ℏ �√ 3 (Jx + Jy) + iξ (Jx − Jy) � ≡ |vx|eiξφx, vξ y = √ 3a0 4ℏ � 1 √ 3 (3Jz + 1) − iξ (Jx − Jy) � ≡ |vy|eiξφy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' The pseudomagnetic field, Bξ = (0, 0, ξBξ z), is given as Bξ z = cos φx∂xAξ y−cos φy∂yAξ x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' The Hamiltonian around K and K′ reads Hξ = ξ � |vxvy| � 0 Πξ x ∗ − iΠξ y ∗ Πξ x + iΠξ y 0 � , (5) where Πξ x = � |vx/vy| � eiξφxpx + ξAξ x � and Πξ y = � |vy/vx| � eiξφypy + ξAξ y � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' By defining the annihilation operators as aK = lB √ 2ℏ � ΠK x ∗ − iΠK y ∗� , aK′ = lB √ 2ℏ � ΠK′ x + iΠK′ y � (6) with lB 2 = ℏ/|Bξ z|, the eigenenergy is obtained as En = � 2 √ 2ℏ/lB � � |vxvy|√n, (n = 0, 1, 2, · · · ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' The n = 0 eigenenstates are |Ψ+ 0 ⟩ = (0, |ψ0⟩)T for K and |Ψ− 0 ⟩ = (|ψ0⟩, 0)T for K′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Since the components of the two-dimensional spinor in the sublattice basis are ex- changed in K and K′, the eigenstates at n = 0 are nonzero only on the A sublattice (sublattice polariza- tion), while they are nonzero on both the A and B sub- lattices at n ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' These eigenenergy and eigenstate fea- tures agree with the numerical results, providing the evi- dence of the emergence of the Landau levels of Majorana fermions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Numerical results in terms of the effective low-energy theory FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' 5: (Color online) Peak energy, E1, of ρj,A(E) as a func- tion of √ C at the isotropically interacting system obtained in the numerical calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' R =40, 50, and 60 are set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Corre- sponding C is obtained for the fixed CR =1/25, 2/25, 3/25, 4/25, 5/25, 6/25, and 7/25, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Let us discuss the numerical results in terms of the effective low-energy theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' As the system leaves the isotropically interacting point in the numerical calcula- tion, the Landau levels of Majorana fermions are smeared at the higher energies and their number is reduced at the lower energies, as shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' 3(c)-3(g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' When the anisotropy of the interactions becomes strong, the Dirac points deviate considerably from those of the isotropi- cally interacting unstrained system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' This situation is op- 6 TABLE I: Parameters a and b to fit the coefficient (E1) of En ∝ √n using a linear regression, E1 = a √ 8C +b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' The coefficients (E1) are obtained by the numerical calculation for R =40, 50, and 60 at the fixed CR =1/25, 2/25, 3/25, 4/25, 5/25, 6/25, and 7/25, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' The coupling constants are for the isotropically interacting system and for the system closest to the isotropically interacting point on each line A-E in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' 3(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' The third line (below the results for a) denotes the ratios of a to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content='99951 at the isotropic point are denoted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' (Jx, Jy, Jz) ( 1 3, 1 3, 1 3) A ( 31 96, 31 96, 17 48) B ( 121 384, 127 384, 17 48) C ( 59 192, 65 192, 17 48) D ( 115 384, 133 384, 17 48) E ( 7 24, 17 48, 17 48) a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content='99951 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content='99966 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content='00109 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content='00420 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content='00720 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content='00989 Ration of a to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content='99951 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content='00015 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content='00016 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content='00047 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content='00769 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content='01039 b 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content='03667 × 10−5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content='48732 × 10−4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content='51854 × 10−5 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content='13822 × 10−4 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content='83096 × 10−4 −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content='69701 × 10−4 posed to the condition for the effective low-energy the- ory to hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Thus, as the system leaves the isotropically interacting point, the higher-order terms of Aξ become relevant, reducing the number of the Landau levels of Majorana fermions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' We next investigate the relation between the control parameters of the strain strength, C and τ, in the numer- ical and analytical calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' To this end, we evaluate the coefficient of En ∝ √n obtained in the numerical cal- culation for R =40, 50, and 60 at the seven values of the fixed CR within CR < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Figure 5 illustrates the coef- ficient (E1) as a function of √ C in the isotropically inter- acting system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' The data follows the line E1 = a √ 8C + b with a ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content='99951 and b ≈ 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content='03667×10−5, indicating that the relation, En = √ 8Cn, is deduced within our numer- ical accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' The coefficient E1 in the system closest to the isotropically interacting point on each line A-E is evaluated in the same way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' The evaluated a and b are summarized in Table I, indicating that a ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content='0 and b ≈ 0, and thus En ≈ √ 8Cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' The coefficient of En ∝ √n in the effective low-energy theory, � 2 √ 2ℏ/lB � � |vxvy|, is given by using τ and one of the three coupling constants, leading to the expression for En as En = � (Jz − 1)2 + � Jz + 1 3 �2� 1 2 � 3τn 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' (7) The eigenenergy in the isotropically interacting system is obtained as En = 2 � τn/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Comparing En in the numerical and analytical calculations for the isotropi- cally interacting system, the relation, τ = 6C, is de- rived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' This relation is consistent with that derived by comparing the pseudomagnetic field in the numerical43 and analytical42 calculations for the isotropically inter- acting system, where |Bξ z| = 4Cℏ/a02 and 2ℏτ/(3a02), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Since Jz’s in the anisotropically interacting system in Table I are the same, their coefficients of √τ take the same value, � 1025/768, according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' The ratio of this coefficient to that in the isotropically interacting system is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content='00049.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' This ratio is close to that denoted in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Therefore, the relation τ = 6C is approximately satisfied in the anisotropically interacting systems denoted in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' SUMMARY We have investigated the low-energy states of an anisotropically interacting Kitaev model under triaxial strain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Based on the numerical results, we argue that the Landau levels of Majorana fermions emerge in the wide gapless region around the isotropically interacting point in the phase diagram of the unstraind system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' The emer- gence of the strain-induced Landau levels of Majorana fermions in the anisotropically interacting Kitaev system is confirmed also by an analytical calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' When uni- axial strain is applied, the same features (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=', En ∝ √n and sublattice polarization) are expected to appear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' To observe the features of the strain-induced Landau lev- els of Majorana fermions, scanning tunneling microscopy (STM) is promising.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' According to the STM theory for the Kitaev model, the differential conductance through a single site provides direct information on the LDOS of Majorana fermions at low temperatures44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' In fact, the two features of the strain-induced Landau levels (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=', En ∝ √n and sublattice polarization) have been observed in artificial graphene under simulated triaxial strain eval- uated pseudomagnetic fields up to 60 T29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' The fabrica- tion of the α-RuCl3 thin films has been studied actively in recent years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' This situation favors experiments for in- vestigating whether the strain-induced Landau levels of Majorana fermions emerge in Kitaev candidate materials such as α-RuCl3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' We hope that our results contribute to such studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Acknowledgments We would like to thank T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Suzuki and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Taniguchi for valuable discussions.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Stoy- anova, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Kandpal, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Choi, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Coldea, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Rousochatzakis, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Hozoi, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' van den Brink, New J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' 16, 013056 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' 9 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Chun, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content='-W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Kim, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Kim, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Zheng, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Stoumpos, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Malliakas, J.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Jeschke, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Valent´ı, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' B 93, 214431 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' 11 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Kim, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Plumb, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Kim, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Kee, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Burch, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' B 93, 075144 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' 15 H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Kim, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Kee, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' B 93, 155143 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' 16 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Yadav, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Bogdanov, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Katukuri, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Nishimoto, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' van den Brink, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Hozoi, Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Chang, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Hur, Hi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Sato, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content='-G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Park, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Kim, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content='-D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Kim, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content='-W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Noh, Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Rep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' 6, 39544 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' 18 W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Wang, Z.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' B 96, 115103 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' 19 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Kitagawa, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Takayama, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Matsumoto, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Kato, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' 9, 17 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' 21 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Knolle and and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Moessner, Annu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Condens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Motome and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Nasu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Jpn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' 89, 1 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' 24 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Trebst and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Hickey, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Rep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' 950, 012002 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' 25 Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Kasahara, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Link, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Tom, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Zonno, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Michiardi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Schneider, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Zhdanovich, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Damascelli, Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' 5, eaaw5593 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' 33 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Rachel, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Fritz, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Vojta, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' 116, 167201 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' 34 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Perreault, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Rachel, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} 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(2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' 37 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Settnes, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Power, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content='-P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Jauho, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' B 93, 035456 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' 38 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Fremling and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Fritz, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' B 105, 085147 (2022).' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Venderbos and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Fu, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' B 93, 195126 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' 42 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Salerno, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' 44 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Udagawa, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Takayoshi, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Oka, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' 126, 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b/EtE3T4oBgHgl3EQfVQrm/content/tmp_files/2301.04459v1.pdf.txt @@ -0,0 +1,1679 @@ +ALGEBRAIC ACTIONS II. GROUPOID RIGIDITY +CHRIS BRUCE AND XIN LI +Abstract. We establish rigidity for partial transformation groupoids associated with algebraic actions +of semigroups: If two such groupoids (satisfying appropriate conditions) are isomorphic, then the +globalizations of the initial algebraic actions rationally embed in each other. +For specific example +classes arising for instance from toral endomorphisms, actions from rings, or actions from commutative +algebra, this mutual embedability can be improved in various ways to obtain surprisingly strong rigidity +phenomena. +This is witnessed in a particularly striking fashion for actions arising from algebraic +number theory: We prove that the groupoids associated with the action of the multiplicative monoid +of non-zero elements in a ring of algebraic integers on the additive group of the ring remembers the +initial algebraic action up to isomorphism, which in turn remembers the isomorphism class of the ring. +This resolves an open problem about isomorphisms of Cartan pairs and leads to a dynamical analogue +of the Neukirch–Uchida theorem using topological full groups. +1. Introduction +Algebraic actions of groups form an interesting and important class of dynamical systems which +provides a rich supply of actions of general groups (see, for instance, [49]). On the one hand, this +example class is interesting and exhibits new phenomena, and on the other hand, due to the particular +structure of algebraic actions, a variety of tools is available, allowing for a systematic and detailed +study. In [13], we initiated the study of one-sided or irreversible analogues, i.e., algebraic actions of +semigroups, which have not been studied in detail before in general (but see [30, 22] and the references +in [13] for special cases). An interesting new phenomenon that arises in this new setting is that actions +by non-invertible endomorphisms of a given group automatically produce a particular completion of +the group, and the original action induces a system of partial homeomorphisms on this completion. +The idea of [13] was to study the corresponding groupoids, which are interesting in their own right +but also give access to analyzing properties of C*-algebras generated by natural representations of the +initial algebraic action. Our goal now is to study the natural question of how much information the +groupoids constructed in [13] remember about the original algebraic actions. Surprisingly, we discover +the phenomenon of groupoid rigidity for a variety of example classes, i.e., our groupoids remember +more information than expected – in special cases, they even remember everything. +Let us now formulate our rigidity results. An algebraic action σ: S ↷ A consists of a monoid S, an +Abelian group A, and a semigroup homomorphism from S to injective endomorphisms of A, denoted +by S → End (A), s �→ σs. We will always assume our algebraic actions to be non-automorphic (i.e., +not all σs are automorphisms) and faithful (i.e., the map s �→ σs is injective). Let C be the collection +of subgroups of A which are of the form σ−1 +t1 σs1 · · · σ−1 +tm σsmA, where σ−1 +t X := {a ∈ A: σt(a) ∈ X} for +a subset X ⊆ A. In this paper, we will always assume that σ has the finite index property (FI), i.e., +#(A/C) < ∞ for all C ∈ C, or equivalently, #(A/σsA) < ∞ for all s ∈ S. In this case, the completion +of A mentioned above is given by A := lim +←−C∈C A/C, where C is partially ordered by inclusion. The +standing assumptions in this paper will furthermore include that σ: S ↷ A admits a globalization +(which is always assumed to be minimal in the sense of Remark 2.4), i.e., an embedding of S into a +group S and a group A containing A together with an algebraic action ˜σ: S ↷ A (necessarily by +automorphisms) such that ˜σs|A = σs for all s ∈ S. This allows us to form a partial action of S on +A by letting s ∈ S act via the restriction A ∩ ˜σ−1 +s A → ˜σsA ∩ A of ˜σs to A ∩ ˜σ−1 +s A. Similarly, we +Date: January 12, 2023. +2020 Mathematics Subject Classification. Primary 37A20, 37B99, 22A22; Secondary 20M18, 37A55, 46L05. +C. Bruce was supported by a Banting Fellowship administered by the Natural Sciences and Engineering Research +Council of Canada (NSERC) and has received funding from the European Union’s Horizon 2020 research and innovation +programme under the Marie Sklodowska-Curie grant agreement No 101022531. X. Li has received funding from the +European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant +agreement No. 817597). +1 +arXiv:2301.04459v1 [math.DS] 11 Jan 2023 + +also have the partial action of A on A by translation. In [13], we identified a condition, called (JF), +which ensures that these partial actions on A extend to a partial action of A ⋊ S on A by partial +homeomorphisms (see § 2.1 and [13, § 3.3] for details). In this paper, we will always assume that (JF) +holds. The associated partial transformation groupoid Gσ := (A ⋊S )⋉A is the groupoid constructed +in [13, § 3] arising from our algebraic action σ. We can now state our main rigidity result. +Theorem A (see Theorem 3.30). Let σ: S ↷ A and τ : T ↷ B be two algebraic actions of monoids +as above, with globalizations ˜σ: S ↷ A , ˜τ : T ↷ B and groupoids Gσ, Gτ. Assume that S and T +are Abelian, that A and B are torsion-free and finite rank, and that there exist s ∈ S and t ∈ T such +that idA − σs : A → A and idB − τt : B → B are injective. +If Gσ and Gτ are isomorphic as topological groupoids, then ˜σ: S ↷ A and ˜τ : T ↷ B embed rationally +into each other, i.e., there exist injective homomorphisms t: S �→ T , b: Q⊗A �→ Q⊗B, s: T �→ S , +and a: Q ⊗ B �→ Q ⊗ A such that b((idQ ⊗ ˜σs)(x)) = (idQ ⊗ ˜τt(s))(b(x)) and a((idQ ⊗ ˜τt)(y)) = +(idQ ⊗ ˜σs(t))(a(y)) for all s ∈ S , x ∈ Q ⊗ A , t ∈ T , and y ∈ Q ⊗ B. +We refer the reader to § 3, in particular § 3.5, for more general results and further explanations. +Let us now present a first class of algebraic actions where our general rigidity result applies. +Corollary B (see Corollary 4.2). Assume that S and T are Abelian, torsion-free monoids, that A +and B are torsion-free Abelian groups of finite rank, and that σ: S ↷ A and τ : T ↷ B are non- +automorphic faithful algebraic actions. Further suppose that the dual actions ˆσ and ˆτ are mixing. Let +˜σ: S−1S ↷ S−1A and ˜τ : T −1T ↷ T −1B be the canonical globalizations as in [13, Example 2.4], and +denote the groupoids attached to σ and τ by Gσ and Gτ, respectively. +If Gσ and Gτ are isomorphic as topological groupoids, then there exist injective homomorphisms +t: S−1S �→ T −1T and b: S−1A �→ T −1B such that b(˜σs(x)) = ˜τt(s)(b(x)) for all s ∈ S−1S and +x ∈ S−1A, injective homomorphisms s: T −1T �→ S−1S and a: T −1B �→ S−1A such that a(˜τt(y)) = +˜σs(t)(a(y)) for all t ∈ T −1T and y ∈ T −1B, and the images of b and a are finite index subgroups of +T −1B and S−1A, respectively. +The reader may consult § 4.1 for more explanations and details. The concrete case of toral endomor- +phisms is treated in Example 4.3. +Another motivating example class is given by the action of the monoid of non-zerodivisors of a ring on +the additive group of the ring by multiplication. For instance, torsion-free commutative rings which +are finitely generated as additive groups have received a great deal of attention because of Bhargava’s +work [5, 6, 7, 8, 9, 10]. In this setting, our rigidity result implies the following. +Theorem C (see Theorem 5.5). Let Ri, i = 1, 2, be finitely generated torsion-free commutative rings. +For i = 1, 2, let R× +i +be the monoid of non-zerodivisors in Ri and σi : R× +i ↷ Ri the algebraic action +given by multiplication. If the corresponding groupoids Gσ1 and Gσ2 are isomorphic, then Q ⊗ R1 and +Q ⊗ R2 are isomorphic as Q-algebras. +Somewhat surprisingly, using very different methods, we obtain a similar rigidity result for special +classes of non-commutative rings. +Theorem D (see Corollary 5.20). For i = 1, 2, let Ri be a ring whose additive group is finitely +generated and torsion-free. Let R× +i be the monoid of left regular elements (i.e., non-left-zerodivisors) +in Ri and σi : R× +i ↷ Ri the algebraic action given by left multiplication. Suppose that Q ⊗ R1 and +Q ⊗ R2 are semisimple Q-algebras. If the corresponding groupoids Gσ1 and Gσ2 are isomorphic, then +Q ⊗ R1 and Q ⊗ R2 are isomorphic as Q-algebras. +Examples of rings that are covered by Theorem D include integral group rings of finite groups and +rings of matrices over orders in algebraic number fields. +Let us now apply Theorem C to rings of algebraic integers in number fields. Let R be such a ring, +with quotient field K. Consider the algebraic action σ: R× ↷ R by multiplication. The completion +R is given in this case by the (additive group of the) integral adele ring, and the partial action from +above is given by the canonical partial action K ⋊ K× ↷ R. Moreover, the groupoid Gσ in this +2 + +case is the partial transformation groupoid (K ⋊ K×) ⋉ R, and its C*-algebra coincides with the +ring C*-algebra A[R], which has been introduced and studied in [19, 21, 34]. Since (K ⋊ K×) ⋉ R is +effective, A[R] contains a canonical Cartan subalgebra D[R]. Moreover, consider the topological full +group F ((K ⋊ K×) ⋉ R) of (K ⋊ K×) ⋉ R given by the group of global bisections (see for instance +[40, 42]) and its commutator subgroup D((K ⋊ K×) ⋉ R). +Corollary E (see Corollary 5.9). Let Ri, i = 1, 2, be rings of algebraic integers in number fields Ki. +With the notation introduced above, the following are equivalent: +(i) K1 and K2 are isomorphic; +(ii) (K1 ⋊ K× +1 ) ⋉ R1 and (K2 ⋊ K× +2 ) ⋉ R2 are isomorphic as topological groupoids; +(iii) K1 ⋊ K× +1 ↷ R1 and K2 ⋊ K× +2 ↷ R2 are continuously orbit equivalent in the sense of [37, 38]; +(iv) (A[R1], D[R1]) and (A[R2], D[R2]) are isomorphic as Cartan pairs; +(v) F ((K1 ⋊ K× +1 ) ⋉ R1) and F ((K2 ⋊ K× +2 ) ⋉ R2) are isomorphic as abstract groups; +(vi) D((K1 ⋊ K× +1 ) ⋉ R1) and D((K2 ⋊ K× +2 ) ⋉ R2) are isomorphic as abstract groups. +Remark 1.1. The equivalences of (i), (v), and (vi) in Corollary E gives dynamical analogues of +the Neukirch–Uchida theorem from anabelian geometry which says that the absolute Galois group of +a number field remembers the field up to isomorphism [43, 52]. +A different dynamical version of +the Neukirch–Uchida theorem is given in [14] using completely different techniques. Moreover, the +structure of our groups is much different from the absolute Galois groups or the topological full groups +from [14] since, e.g., D((K ⋊ K×) ⋉ R) is simple. +Remark 1.2. The equivalence of (i) and (iv) in Corollary E is in stark contrast with [39, Corol- +lary 1.3], which says that the ring C*-algebras A[R1] and A[R2] are always isomorphic. One conse- +quence of this is that A[Z] contains a family, parameterized by all number fields, of isomorphic but +non-conjugate Cartan subalgebras. +Semigroup C*-algebras C∗(R ⋊ R×) of ax + b-semigroups R ⋊ R× were studied in [20, 35, 36] for rings +of algebraic integers R in number fields K. C∗(R ⋊R×) has a canonical groupoid model (see [21, 35]), +and hence contains a canonical Cartan subalgebra D(R⋊R×). It was shown in [36] how to recover the +Dedekind zeta function and the ideal class group of K from the Cartan pair (C∗(R⋊R×), D(R⋊R×)). +However, the natural question of whether (C∗(R⋊R×), D(R⋊R×)) completely determines the number +field K has been left open in [36] (see the question at the end of [36, § 1]). Since (A[R], D[R]) can be +recovered from (C∗(R ⋊ R×), D(R ⋊ R×)), we are now able to answer this question. +Corollary F. Let Ri, i = 1, 2, be rings of algebraic integers in number fields Ki. Then K1 ∼= K2 if +and only if (C∗(R1 ⋊ R× +1 ), D(R1 ⋊ R× +1 )) and (C∗(R2 ⋊ R× +2 ), D(R2 ⋊ R× +2 )) are isomorphic as Cartan +pairs. +In fact, we completely resolve the more general problem left open in [12, § 5.2], see Remark 5.10. +Theorem D, applied to matrix algebras over rings of algebraic integers, yields the following analogous +result. +Corollary G (see Example 5.22). For i = 1, 2, let Ri be the ring of integers in a number field Ki, and +let ni be a positive integer. Consider the algebraic action σi : Mni(Ri)× ↷ Mni(Ri) by multiplication. +The groupoids Gσ1 and Gσ2 are isomorphic if and only if n1 = n2 and K1 ∼= K2. +We also have further equivalent statements analogous to (iii) – (vi) in Corollary E, and the analogue +of Corollary F holds as well. +We would like to point out that we obtain more general results than the ones presented in this +introduction (see § 3 for details). Thus, we can additionally treat the following example classes: +(a) Semigroups of canonical endomorphisms of finite rank torsion-free Abelian groups (§ 4.2); +(b) Actions form adding scalars to algebraic actions of subgroups of special linear groups (§ 4.3); +(c) Arithmetical S-integer dynamical systems (§ 4.4); +(d) Actions of congruence monoids on rings of algebraic integers (§ 5.2.2); +(e) Nd-actions from zero-dimensional ideals in commutative algebra (§ 5.2.3); +(f) Actions from integral group rings of finite groups (Example 5.23); +(g) Actions from orders in central simple algebras over number fields (Example 5.24). +3 + +The proofs of our rigidity results are inspired by continuous orbit equivalence rigidity for odometers +(see [16]). Given an algebraic action σ: S ↷ A satisfying our standing assumptions, the restriction of +the partial action of A ⋊ S ↷ A to A yields an odometer action A ↷ A. Our key insight is that a +careful analysis allows us to identify situations where rigidity can be upgraded from these odometer +actions to groupoid rigidity in our sense. For Abelian acting monoids, we take advantage of algebraic +identities in semidirect product groups arising from our algebraic actions. For non-Abelian acting +monoids, our rigidity results rely on the structure of nilpotent elements in certain matrix algebras. +2. Preliminaries +2.1. Standing assumptions. We first explain the standing assumptions on algebraic actions that +we will assume in this paper. +Let S be a non-trivial left cancellative monoid and A an Abelian +group, written additively, with identity element 0. Assume σ: S ↷ A is an algebraic S-action, i.e., +σ: S → End Z(A), s �→ σs is a monoid homomorphism such that σs is an injective endomorphism +A → A for all s ∈ S. Actions of this form are called algebraic actions (cf. [13]). Unless S is a group, +we shall assume that σ: S ↷ A is non-automorphic, i.e., there exists s ∈ S such that σsA ⊊ A. This +in particular implies that A is non-trivial. We will also always assume that the action σ: S ↷ A is +faithful, i.e., s �→ σs is injective. +Let C = CS↷A be the family of S-constructible subgroups of A, i.e., C is the smallest collection of +subgroups of A such that +• A ∈ C; +• s ∈ S and C ∈ C implies σsC, σ−1 +s C ∈ C. +It follows that C is closed under taking finite intersections (see [13, Proposition 3.9]. We say that +σ: S ↷ A satisfies the finite index property (see [13, Definition 7.1]) if +(FI) +#(A/σsA) < ∞ +for all s ∈ S, +If σ: S ↷ A satisfies (FI), then [13, Proposition 7.2] implies that every member of C is a finite +index subgroup of A. In this case, we get a compact group A := lim +←−C∈C A/C, and the canonical +homomorphism A → A has kernel � +C∈C C. Given C ∈ C, we denote by C the kernel of the canonical +projection A ↠ A/C, i.e., the preimage of C ∈ A/C in A. We have a canonical homeomorphism +C ∼= lim +←−D∈C, D⊆C C/D. +Remark 2.1. It suffices to check (FI) for generators of S: Say S is generated by S. Then we can +proceed inductively on the word length with respect to S of an element in S. Suppose #(A/σsA) < ∞ +for some s ∈ S, i.e., A = R + σsA for some finite set R. Moreover, for t ∈ S, write A = F + σtA for +some finite set F. Then A = R + σsA = R + σs(F + σσA) = R + σsF + σstA. Hence it follows that +#(A/σstA) < ∞. +In this paper, we will only consider algebraic actions σ: S ↷ A which satisfy (FI). Let ∂ �E be the +compact space of characters on the semilattice E := {b + C : C ∈ C, b ∈ A} ∪ {∅} (see [13, § 3.4]). +Each x = (xC + C)C ∈ A determines an element χx of ∂ �E by +χx(b + C) := +� +1 +if xC + C = b + C, +0 +if xC + C ̸= b + C. +Since (FI) is satisfied, it is not hard to see that the map A → ∂ �E given by x �→ χx is a homeomorphism. +In addition, we will always assume that σ: S ↷ A has a globalization ˜σ: S ↷ A , i.e., S embeds into +the group S , A is a subgroup of the group A , and ˜σ: S → Aut(A ) is an algebraic action such that +˜σs|A = σs for all s ∈ S. Then A ⋊ S acts on A by affine maps: (z, γ).x = z + ˜σγ(x). Reducing to +A ⊆ A , we get a partial action (in the sense of [25]) on the group A. Explicitly, g ∈ A ⋊ S acts by +the partial bijection +A ∩ g−1.A → (g.A) ∩ A, +x �→ g.x. +4 + +Note that s ∈ S acts via σs (where we view both maps as partial bijections on A). Consider the +condition +(JF) +C ⊆ ker (id − ˜σg) =⇒ g = 1 for all C ∈ C, g ∈ ⟨S⟩, +where ⟨S⟩ is the subgroup of S generated by S (cf. [13, § 3.3]). Moreover, our standing assumptions +in this paper include that (JF) is satisfied. In this case, the partial action A ⋊ S ↷ A extends +uniquely to a partial action A ⋊ S ↷ A. Given g ∈ A ⋊ S , let Ug−1 ⊆ A be the domain of g, and +for x ∈ Ug−1, we let g.x denote the image of x under g with respect to the action A ⋊ S ↷ A. The +associated transformation groupoid +(A ⋊ S ) ⋉ A := {(g, x) ∈ (A ⋊ S ) × A : x ∈ Ug−1, g.x ∈ A} +is canonically isomorphic to the groupoid Gσ = Iσ ⋉ ∂ �E from [13, § 3] (see [13, § 3.5]). Since (FI) is +satisfied, (A ⋊ S ) ⋉ A is minimal by [13, Corollary 7.4]. By [13, Theorem 4.14], (A ⋊ S ) ⋉ A is +effective if and only if σ: S ↷ A is exact in the sense of [13, Definition 4.11], i.e., � +C∈C C = {0}. +Remark 2.2. If S is left Ore, then we can replace (JF) by the condition that C ⊆ ker (σs − σs′) for +some C ∈ C implies that s = s′ (see [13, Example 3.17 (iii)]). +Remark 2.3. If S is left Ore, then there is a canonical partial action of G on A in general, without +the assumption that (JF) holds: We first construct the enveloping action as in [21]. σ extends to an +action of S of A, also denoted by σ. Then set S−1A := lim +−→S +� +A, σ +� +; this is a locally compact (non- +compact) topological group. Extend σ to σ : ⟨S⟩ ↷ S−1A. This way, we obtain a global dynamical +system S−1A ⋊ ⟨S⟩ ↷ S−1A. Then A is a clopen subset of S−1A, so that we obtain the desired partial +dynamical system by restricting S−1A ⋊ ⟨S⟩ ↷ S−1A to A. +However, note that (JF) holds automatically in this setting if A is torsion-free by [13, Proposition 7.5]. +Remark 2.4. We can and will always assume that S is generated by S, i.e., S = ⟨S⟩. Moreover, +by [13, Proposition 2.7], if S embeds into the group S , then we can always take A = ZS ⊗ZS A, and +the map A → ZS ⊗ZS A, a �→ 1 ⊗ a will always be injective if σ admits a globalization. In this case, +we have A = ⟨� +s∈S s.(1 ⊗ A)⟩. Hence, for any globalization ˜σ : S ↷ A , we may and will always +assume that +(1) +A = ⟨� +s∈S ˜σs(A)⟩. +If S is left Ore, then we can always take S = S−1S, and in this case, we can and will always arrange +that A = � +r∈S ˜σ−1 +r (A). +2.2. Further properties. Let us now discuss a few properties which are not part of our standing +assumptions, but which we will assume for some of our results. +The principal S-constructible subgroups are cofinal in C if +(PC) +for every C ∈ C, there exists s ∈ S such that σsA ⊆ C. +Note that (PC) is satisfied if S is left reversible (see [13, Proposition 7.12]). +Consider the following condition on the algebraic action S ↷ A : +(F) +For all 1 ̸= s ∈ S , 1 − ˜σs := id − ˜σs : A → A is injective. +Condition (F) is a freeness condition, modulo the fact that in the linear setting 0 will always be a +fixed point. +3. From cocycles to embeddings +Let σ: S ↷ A and τ : T ↷ B be algebraic actions with globalizations ˜σ: S ↷ A and ˜τ : T ↷ B, +respectively, which satisfy all our standing assumptions from § 2.1 (and we use the same notation as +in § 2). We will often view S as a submonoid of A ⋊ S via S → A ⋊ S , s �→ (0, s) and A as a +subgroup of A ⋊ S via A → A ⋊ S , a �→ (a, 1). Moreover, we will use multiplicative notation for +A ⋊ S . +5 + +Suppose c: (A ⋊ S ) ⋉ A → B ⋊ T is a continuous cocycle (i.e., a groupoid homomorphism) such +that c−1(0, 1) = A. Note that c satisfies the cocycle identity c(gh, x) = c(g, h.x)c(h, x) whenever these +expressions make sense. We denote the map A ⋊ S → B ⋊ T , p �→ c(p, 0) again by c. +Lemma 3.1. For each p ∈ A ⋊ S, there exists C(p) ∈ C such that c(p, −): A → B ⋊ T is constant +on x + C(p) for every x ∈ A. +Proof. Since A is compact and c is continuous, the image of c(p, −) must be a finite set, which we +shall denote by F. For each f ∈ F, let Uf := c(g, −)−1({f}). Then each Uf is compact open, and we +have a partition A = � +f∈F Uf. Since the collection {x + C : C ∈ C, x ∈ A} forms a base consisting of +compact open sets for the topology of A, we can write Uf as a finite disjoint union Uf = � +i xi + Ci. +If we now set Cf := � +i Ci, then since Cf is a finite index subgroup of each Ci, we can even write Uf +as a disjoint union of the form � +i yi + Cf. Now set C(p) := � +f∈F Cf, so that A is a finite disjoint +union A = � +k xk + C(p). For all x ∈ A, there exists k such that x + C(p) = xk + C(p), and c(g, −) is +constant on xk + C(p). +□ +Lemma 3.2. For every finitely generated subgroup A of A, there exists CA ∈ C such that c(x, −) is +constant on a + CA for all a ∈ A and x ∈ A. +Proof. Let a ∈ A, and let {xi}i be a finite collection of elements in A that generate A as an additive +monoid. Set CA := � +i C(xi, 1) where the subgroups C(xi, 1) are provided by Lemma 3.1. We now +show that c((x, 1), −) is constant on a + CA by induction on the word length ℓ(x) of x with respect to +{xi}i. The induction base case ℓ(x) = 1 follows from Lemma 3.1. Now suppose the claim is true for +all x ∈ A with ℓ(x) = l. Given x ∈ A with ℓ(x) = l + 1, we can write x = xix′ for some index i and +x′ ∈ A with ℓ(x′) = l. Now we have for all a + x, a + y ∈ a + CA, +c((x, 1), a + x) += +c((xi, 1)(x′, 1), a + x) = c((xi, 1), (x′, 1).(a + x)) c((x′, 1), a + x) += +c((xi, 1), (x′, 1).(a + x)) c((x′, 1), a + y) = c((xi, 1), x′.(a + x)) c((x′, 1), a + y) += +c((xi, 1), x′.(a + y)) c((x′, 1), a + y) = c((x, 1), a + y). +Here, we used the induction hypothesis for the third equality and Lemma 3.1 for the fifth equality +(x′.(a + x) and x′.(a + y) both lie in x′ + a + CA). +□ +Remark 3.3. Lemma 3.2 can also be derived from the general results in [16], but we chose to give a +direct proof in our special setting. +Lemma 3.4. For every finitely generated subgroup A of A, the map A ∩ CA → B ⋊ T given by +x �→ c(x, 0) is an injective group homomorphism. +Proof. Additivity is easy to see. Now suppose we have x, y ∈ A ∩ CA with c(x, 0) = c(y, 0). Consider +the element (x, 0)(y, 0)−1 of the groupoid (A ⋊ S ) ⋉ A. +Since c is a groupoid homomorphism, +c((x, 0)(y, 0)−1) = (0, 1). Hence, using the assumption c−1(0, 1) = A, we conclude that (xy−1, 0) = +(x, 0)(y, 0)−1 ∈ A, which implies x = y. +□ +Note that A ∩ CA is a finite index subgroup of A. In particular, if A is finitely generated, then we get +an injective group homomorphism from a finite index subgroup of A into B ⋊ T . +3.1. The additive homomorphism. Given a finitely generated subgroup A of A, we now want to +find l ∈ Z>0 and C ∈ C together with a homomorphism b: C := l(A∩C) → B such that c(x) = (b(x), 1) +for all x ∈ C. +Proposition 3.5. Suppose that A ⊆ A is a finite rank subgroup and that s ∈ S satisfies σs(A) ⊆ A. +Let d be the degree of the polynomial det(z − ˙σs), where ˙σs := idQ ⊗ (σs|A): Q ⊗ A → Q ⊗ A, and let +κd ∈ Z>0 be the smallest positive integer such that p(z) := κd det(z − ˙σs) has integer coefficients, and +write p(z) = κdzd − κd−1zd−1 − . . . − κ1z − κ0 (for some κ• ∈ Z). +Then for every finitely generated subgroup A of A, there exists ˇC ∈ C depending on s such that +(i) The restriction of c to ˇC := A ∩ ˇC is an injective group homomorphism ˇC → B ⋊ T . +(ii) c(s)dc(x)κd = c(x)κ0c(s) · · · c(x)κd−1c(s) for all x ∈ ˇC. +(iii) The following holds for all x ∈ ˇC: If c(x) = (β, α) and c(s) = (δ, γ), then +(2) +γdακd = ακ0γακ1γ · · · ακd−1γ +in T . +6 + +Proof. Let us denote the restriction of σs to A again by σs. For all x ∈ A, the following holds in A⋊S +as κdσd +s(x) = κd−1σd−1 +s +(x) + . . . + κ0x in A: +sd(κdx) = κdσd +s(x)sd = (κ0x)s(κ1x)s · · · (κd−1x)s. +Given a finitely generated subgroup A of A, choose CA as in Lemma 3.4. Set +ˇC := C(s) ∩ σ−1 +s C(s) ∩ . . . ∩ σ−(d−1) +s +C(s) ∩ CA ∩ σ−1 +s CA ∩ . . . ∩ σ−(d−1) +s +CA. +Then (i) is satisfied because of Lemma 3.4. +We have for all x ∈ ˇC: +c(sd(κdx)) = c(sd(κdx), 0) = c(sd, κdx)c(κdx, 0) = c(sd−1, σs(κdx))c(s, κdx)c(κdx, 0) += . . . = c(s, σd−1 +s +(κdx))c(s, σd−2 +s +(κdx)) · · · c(s, κdx)c(κdx, 0) += c(s, 0)c(s, 0) · · · c(s, 0)c(κdx, 0) = c(s)dc(κdx). +Here we are allowed to replace σ? +s(x) by 0 because x lies in C(s) ∩ σ−1 +s C(s) ∩ . . . ∩ σ−(d−1) +s +C(s). +We also have for all x ∈ A ∩ ˇC: +c((κ0x)s(κ1x)s · · · (κd−1x)s) = c((κ0x)s(κ1x)s · · · (κd−1x)s, 0) += c((κ0x)s(κ1x)s · · · (κd−1x), 0)c(s, 0) = c((κ0x)s(κ1x)s · · · , κd−1x)c(κd−1x, 0)c(s, 0) += c(κ0x, σs(κ1x) + . . . + σd−1 +s +(κd−1x))c(s, (κ1x) + . . . + σd−2 +s +(κd−1x)) +· · · +c(κd−2x, σs(κd−1x))c(s, κd−1x) +c(κd−1x, 0)c(s, 0) += c(κ0x, 0)c(s, 0) +· · · +c(κd−2x, 0)c(s, 0) +c(κd−1x, 0)c(s, 0) += c(κ0x)c(s) · · · c(κd−1x)c(s). +Here we are allowed to replace the second arguments of c by 0 because x lies in C(s)∩. . .∩σ−(d−1) +s +C(s)∩ +A ∩ CA ∩ σ−1 +s CA ∩ . . . ∩ σ−(d−1) +s +CA. +Moreover, we have for all x ∈ ˇC and κ ∈ Z that c(κx) = c(x)κ because x lies in A ∩ CA. So all in all, +we have c(s)dc(x)κd = c(x)κ0c(s) · · · c(x)κd−1c(s) for all x ∈ ˇC = A ∩ ˇC. This shows (ii). +Now let us prove (iii). Suppose that c(x) = (β, α) and c(s) = (δ, γ). Comparing T -components, we +obtain γdακd = ακ0γακ1γ · · · ακd−1γ. +□ +Remark 3.6. Suppose we are in the setting of Proposition 3.5, and put ϵ := κd − κd−1 − . . . κ1 − κ0. +If T is Abelian, then (2) is equivalent to αϵ = 1. If p(z) = zd − κ0, then (2) is γdαγ−d = ακ0, i.e., +α and γd satisfy the defining relation for the Baumslag–Solitar group BS(1, κ0) ∼= Z[1/κ0] ⋊ Z. If γd +and α both have infinite order, then ⟨γd, α⟩ ∼= BS(1, κ0). +Lemma 3.7. With the same notation as in Proposition 3.5, set ϵ := κd − κd−1 − . . . κ1 − κ0. In +addition to the assumptions in Proposition 3.5, assume that 1 − σs is injective on A. Then we have +ϵ ̸= 0. If, in addition, every 2-generated subgroup of T is free or Abelian, then αϵ = 1. +Proof. The first claim follows from κd det(1 − ˙σs) = ϵ. For the second claim, our assumption implies +that the subgroup ⟨α, γ⟩ ⊆ ⟨T⟩ is free or Abelian. We claim that α and γ must commute. Indeed, it +suffices to treat the case that the subgroup is free. Because α and γ satisfy the non-trivial relation +(2), the group ⟨α, γ⟩ is either trivial or infinite cyclic; if it is trivial, we are done. Suppose ⟨α, γ⟩ is +cyclic. Then, in particular, αγ = γα, as desired. Now (2) becomes γdακd = γdακ0+κ1+···κd−1, which +implies that αϵ = 1. +□ +Example 3.8. Let us mention two classes of groups whose 2-generated subgroups are either free or +Abelian. A group is called semifree if it has a presentation where the only relators are of the form +7 + +[s, t], where s and t are generators. If s, t are elements in a semifree group and [s, t] ̸= 1, then {s, t} +is a basis for a free group by [1, Theorem 1.2]. +A group is 2-free if every subgroup generated by 2 elements is free. Given a non-empty set of prime +numbers ω, a Dω-free group is a group whose elements each have exactly one p-th root for all primes +p ∈ ω (see the introduction in [2]). By [3, § 8], every Dω-free group from [2] is 2-free. +Let us introduce the following conditions. +Definition 3.9. Let A ⊆ A be a subgroup of finite rank. Consider the following conditions: +(i1) There exists s ∈ S such that σs(A) ⊆ A, 1 − σs is injective on A, and every 2-generated +subgroup of T is free or Abelian. +(i2) There exists s ∈ S with σs = κ idA for some κ ∈ Z\{0, 1}, and for all α ∈ T , if α is conjugate +to ακ in T , then α must be torsion. +Corollary 3.10. If A ⊆ A is a subgroup of finite rank and (i1) or (i2) holds, then for all finite +generated subgroups A ⊆ A, there exist l ∈ Z>0 and C ∈ C such that, with C := l(A ∩ C), there exists +an injective homomorphism b : C → B such that c(x) = (b(x), 1) for all x ∈ C. +Proof. Let s ∈ S be as in (i1) or (i2), and ˇC, ˇC as in Proposition 3.5. Let cT be the composition +A → B ⋊ T ↠ T , where the first map is c and the second map is the canonical projection. Now +cT (ˇC) is finitely generated. Moreover, if (i1) holds, then Lemma 3.7 implies that cT (ˇC) is torsion, +hence finite. If (i2) holds, then Proposition 3.5 (iii) implies that cT (ˇC) is torsion, hence finite. Set +l := #cT (ˇC). Then for all x ∈ lˇC, cT (x) = 1, so that our claim follows (Lemma 3.4 gives injectivity of +b). +□ +Definition 3.11. Let A ⊆ A be a subgroup of finite rank. Consider the following conditions: +(ii1) For all torsion orders l > 1 of elements of T , there exists 1 ̸= sl ∈ S and coprime integers +µ, ν ∈ Z>0 such that σsl(A) ⊆ A and µ det(z− ˙σsl) = µzδ−ν with gcd(l, µ) > 1 or gcd(l, ν) > 1. +(ii2) Condition (F) holds for ˜τ. +Note that, in particular, (ii1) holds if T is torsion-free or if for all κ ∈ Z>0 there exists sκ ∈ S with +σsκ = κ idA. +Definition 3.12. We say that condition (III) holds if there exists s ∈ S such that σs = κ idA for some +κ ∈ Z \ {0, 1}, B is of finite rank, and condition (F) holds for ˜τ. +Corollary 3.13. Let A ⊆ A be a subgroup of finite rank. Assume that one of the following is true: +• (i1) or (i2) holds, and (ii1) is satisfied or (ii2) holds and A is torsion-free, +• (III) holds and A is torsion-free. +Then for all finitely generated subgroup A ⊆ A, there exists C ∈ C such that, with C := A ∩ C, there +exists an injective homomorphism b : C → B such that c(x) = (b(x), 1) for all x ∈ C. +Proof. To prove the first item, let s ∈ S be as in (i1) or (i2), and ˇC, ˇC as in Proposition 3.5. First +assume that (ii1) holds. Let cT be the composition A → B ⋊ T ↠ T , where the first map is c +and second map is the canonical projection. As cT (ˇC) is finitely generated, the set L of possible +non-trivial torsion orders of elements in cT (ˇC) is finite. For each l ∈ L, choose sl as in (ii1) and +set C := ˇC ∩ � +l∈L ˇC(sl). Applying Proposition 3.5 to sl, we obtain that for all x ∈ A ∩ C with +c(x) = (β, α), we have αµ and αν are conjugate in T . If α ̸= 1, then the torsion order of α is a +number l ∈ L, but (ii1) implies that αµ and αν have different torsion orders, which is absurd. Now +suppose that (ii2) holds. Take x ∈ ˇC and write c(x) = (β, α). Lemma 3.7 implies that αϵ = 1. Then +(β, α)ϵ = (β + ˜τα(β) + . . . + ˜τ ϵ−1 +α +(β), αϵ) = (β + ˜τα(β) + . . . + ˜τ ϵ−1 +α +(β), 1), +and +(1 − ˜τα)(β + ˜τα(β) + . . . + ˜τ ϵ−1 +α +(β)) = 0, +which implies β + ˜τα(β) + . . . + ˜τ ϵ−1 +α +(β) = 0 if α ̸= 1 by (F). So c(x)ϵ = 1 and hence c(ϵx) = 1 and +thus ϵx = 0. As A is torsion-free, this implies x = 0. So if x ̸= 0, we must have α = 1. +Now we prove the second item. As above, let s ∈ S be as in (III), and ˇC, ˇC as in Proposition 3.5. +Given x ∈ ˇC, write c(x) = (β, α) and c(s) = (δ, γ). Equation (2) implies that γα = ακγ, and therefore +α = γ−1ακγ. It follows that every eigenvalue of ˙τα is a root of unity. Indeed, take λ1 ∈ Sp ( ˙τα). Then +8 + +α = γ−1ακγ implies that there exists λ2 ∈ Sp ( ˙τα) with λ1 = λκ +2. Similarly, there exist λ3, λ4, . . . ∈ +Sp ( ˙τα) such that λi = λκ +i+1. As Sp ( ˙τα) is finite, we must have λi = λi+p for some i and p. It follows +that λi = λκp +i +and hence that λi is a root of unity. But then λ1 = λκi +i +implies that λ1 must be a root +of unity as well. Hence there exists m ∈ Z>0 such that 1 is an eigenvalue of ˙τ m +α . This implies that +1 − ˙τ m +α is not injective, so that ˙τ m +α = 1 by (F). Now argue as for the first item that this – together +with (F) – implies α = 1. +□ +Remark 3.14. The difference between Corollaries 3.10 and 3.13 is that in the latter, we may choose +l = 1. +Definition 3.15. Assume that A = � +n An for an increasing family of finite rank subgroups An. +Consider the following conditions: +(I) Condition (i1) holds for An for all n, or condition (i2) holds. +(II) Condition (ii1) holds for An for all n, or condition (ii2) holds and A is torsion-free. +Corollary 3.16. Suppose we can write A = � +n An for an increasing family An ⊆ A of finite rank +subgroups. +If (I) holds, then +(a) there is an increasing family of finitely generated subgroups Ak ⊆ A with A = � +k Ak, and for +any such Ak, there are lk ∈ Z>0 and Ck ∈ C such that, with Ck := lk(Ak ∩ Ck), there exists an +injective homomorphism bk : Ck → B such that c(x) = (b(x), 1) for all x ∈ Ck. +If (I) and (II) hold, or if (III) is satisfied and A is torsion-free, then +(a*) there is an increasing family of finitely generated subgroups Ak ⊆ A with A = � +k Ak, and +for any such Ak, there is Ck ∈ C such that, with Ck := Ak ∩ Ck, there exists an injective +homomorphism bk : Ck → B such that c(x) = (bk(x), 1) for all x ∈ Ck. +Proof. Since A = � +n An, we can find Ak with the desired properties. Moreover, for each k, since Ak +is finitely generated, and A = � +n An, we can find n such that Ak ⊆ An. Now apply Corollaries 3.10 +and 3.13. +□ +Remark 3.17. With the notation from Corollary 3.16, we have Ak/Ck �→ A/Ck, so that #(Ak/Ck) +is finite and divides #(A/Ck). +3.2. The multiplicative homomorphism. Define t: S → T to be the composition S → B⋊T ↠ +T , where the first arrow is given by c(−, 0), and the second arrow is the canonical projection. Clearly, +t is a homomorphism. +Lemma 3.18. +(i) If A is torsion-free and (a) holds, then t is injective. +(ii) Suppose that condition (F) is satisfied for ˜σ. +If there exist l ∈ Z>0, a non-zero, finitely +generated subgroup A ⊆ A, C ∈ C and an injective homomorphism b : C := l(A ∩ C) → B +such that c(x) = (b(x), 1) for all x ∈ C, then t is injective. +Proof. Suppose t(s) = t(s′). Then c(s−1s′, 0) = (χ, t(s−1s′)) = (χ, 1) for some χ ∈ B. Set ε := s−1s′. +Since 0 lies in the domain of ˜σε, and since c(ε, −) is locally constant, there must exist C(ε) ∈ C +such that C(ε) is contained in the domain of ˜σε and c(ε, −) is constant on C(ε). +Now suppose +that C ⊆ A is a non-zero subgroup such that there exists an injective homomorphism b : C → B +with c(x) = (b(x), 1) for all x ∈ C. +Then c(ε, 0) commutes with c(x, 0) for all x ∈ C. +We have +(0, ε)(x, 1) = (˜σϵ(x), ϵ) = (˜σε(x), 1)(0, ε). So if 0 ̸= x ∈ C ∩ C(ε), then +c((˜σε(x), ε), 0) = c(˜σε(x)ε, 0) = c(˜σε(x), 0)c(ε, 0). +At the same time, +c((˜σε(x), ϵ), 0) = c(εx, 0) = c(ε, x)c(x, 0) = c(ε, 0)c(x, 0) = c(x, 0)c(ε, 0). +Here we are allowed to replace x by 0 because x lies in C(ε). Moreover, we used that c(ε, 0) commutes +with c(x, 0). Therefore, a comparison yields c(˜σε(x), 0) = c(x, 0). It follows that ˜σε(x) = x. +For (i), write A = � +k Ak as in (a). Applying the above to C = Ck from (a), we obtain ˜σε(x) = x for +all x ∈ Ck ∩ C(ε) for all k. Now let y ∈ A; then there exists k with y ∈ Ak. Since Ck ∩ C(ε) is finite +9 + +index in Ak, there exists N ∈ Z>0 such that Ny ∈ Ck ∩ C(ε). We have N ˜σε(y) = ˜σε(Ny) = Ny, so +that, because A is torsion-free, ˜σε(y) = y. Now (JF) implies ε = 1. +For (ii), if ˜σε(x) = x for any x ̸= 0, then condition (F) implies that ε = 1. +□ +3.3. Equivariance. +Definition 3.19. We set I := {#(A/C): C ∈ C}. +Lemma 3.20. Assume that B is torsion-free. If (a) holds, then for all s ∈ S, x ∈ Ck, σs(x) ∈ Al +(k ≤ l) there exists n ∈ Z>0 such that nσs(x) = σs(nx) ∈ Cl, and we have b(σs(nx)) = ˜τt(s)(b(nx)). +If (a*) holds, then we may take n ∈ I in the statement above. +Proof. Since Cl is of finite index in Al (see Remark 3.17), there exists n ∈ Z>0 such that nσs(x) = +σs(nx) ∈ Cl (since l depends on s, n also depends on s). Since Ck ∩ C(s) is of finite index in Ck, we +can find N such that Nnx ∈ Ck ∩ C(s). Set y := Nnx. Let c(s) = (δ, t(s)). We have +(b(σs(y)) + δ, t(s)) = (b(σs(y)), 1)(δ, t(s)) = c(σs(y), 0)c(s, 0) = c((σs(y), s), 0) += c((0, s)(y, 0), 0) = c((0, s), y)c((y, 1), 0) = c((0, s), 0)c((y, 1), 0) += (δ, t(s))(b(y), 1) = (δ + ˜τt(s)(b(y)), t(s)), +where the sixth equality uses that y ∈ C(s). +Hence b(σs(y)) = ˜τt(s)(b(y)), i.e., Nb(σs(nx)) = +N ˜τt(s)(b(nx)). Since B is torsion-free, we obtain b(σs(nx)) = ˜τt(s)(b(nx)), as desired. +□ +3.4. Conclusion: The embedding theorem. Assume that σ: S ↷ A and τ : T ↷ B are algebraic +actions satisfying our standing assumptions from § 2. Let Z[I−1] be the subring of Q generated by Z +together with +� 1 +n: n ∈ I +� +. We start with the following observation. +Lemma 3.21. We have Z[I−1] ⊗ A = Z[I−1] ⊗ A , i.e., the canonical map Z[I−1] ⊗ A → Z[I−1] ⊗ +A , 1 ⊗ x �→ 1 ⊗ x is an isomorphism. +Recall that we always assume (1), i.e., A = ⟨� +s∈S ˜σs(A)⟩. Otherwise, Lemma 3.21 would not be true +in general. +Proof. Because of (1), it suffices to prove that for all x ∈ ˜σ−1 +t1 ˜σs1 . . . ˜σ−1 +tl ˜σslA (where t1, s1, . . . , tl, sl ∈ S) +there exists N in the multiplicative submonoid ⟨I⟩+ of Z× generated by I with Nx ∈ A. We proceed +inductively on l. For the case l = 1, observe that [˜σ−1 +t A : A] < ∞ for all t ∈ S because ˜σt induces a +bijection ˜σ−1 +t A/A ∼= A/σtA. Now suppose that x = ˜σ−1 +t +˜σs(y) for some y ∈ A with Ny ∈ A for some +N ∈ ⟨I⟩+. Since [˜σ−1 +t A : A] < ∞, there exists M ∈ ⟨I⟩+ with M ˜σ−1 +t (σs(Ny)) ∈ A. Hence MNx ∈ A, +as desired. +□ +Definition 3.22. We say that condition (∗) is satisfied if A = � +n An for an increasing family of finite +rank subgroups An, conditions (I) and (II) hold, or condition (III) holds; in addition, there exists an +increasing family of finitely generated subgroups Ak ⊆ A with A = � +k Ak such that σs(Ak) ⊆ Ak for +all s ∈ S and all k, S and T are right reversible, A = S−1A, S = S−1S, B = T −1B, T = T −1T, +and σ: S ↷ A satisfies (PC). +Now suppose c: (A ⋊ S ) ⋉ A → B ⋊ T is a continuous cocycle such that c−1(0, 1) = A. +Theorem 3.23. Suppose A and B are torsion-free. +(i) If (I) holds, then there exist injective homomorphisms t : S �→ T and ˙b: Q ⊗ A �→ Q ⊗ B +such that ˙b( ˙σs(x)) = ˙τt(s)(˙b(x)) for all s ∈ S and x ∈ A . +(ii) If (∗) holds, then there exist injective homomorphisms t : S �→ T and b′ : S−1A → T −1B +such that b′(σs(x)) = ˜τt(s)(b′(x)) for all x ∈ S−1A and s ∈ S . +Proof. (i): We proceed in several steps. First, we claim that for each k, bk : Ck → B has a unique +extension to an injective homomorphism ˜bk : Ak → Q ⊗ B, and that moreover, ˜bk satisfies +(3) +˜bk(x) = m−1bk(mx), +for any m ∈ Z>0 with mAk ⊆ Ck. To see this, choose m ∈ Z>0 such that mAk ⊆ Ck, and define +˜bk(x) := m−1b(mx) ∈ Q ⊗ B. It is easy to check that ˜bk is a group homomorphism (i.e., additive) +10 + +and that ˜b is injective (here we need that Ak is torsion-free). This is independent of the choice of m, +because if m′ ∈ Z>0 with m′Ak ⊆ Ck, then for x ∈ Ak, we have +m−1b(mx) = m−1(m′)−1b(m′mx) = (m′)−1m−1mb(m′x) = (m′)−1b(m′x). +Next, we claim that the maps ˜bk are compatible in the sense that ˜bl|Ak∩Al = ˜bk|Ak∩Al for all k, l ∈ Z>0. +Choose m large enough so that mAk ⊆ Ck and mAl ⊆ Cl. Then, using (3), we have for all x ∈ Ak ∩Al, +that ˜bl(x) = m−1bl(mx) = m−1bk(mx) = ˜bk(x). +It follows that we get a well-defined injective +homomorphism ˜b: A = � +k Ak → Q ⊗ B such that ˜b|Ak = ˜bk for all k. +Let us show that ˜b is equivariant. Let x ∈ A and s ∈ S. Then x ∈ Ak for some k, so by Lemma 3.20, we +can find n ∈ Z>0 large enough so that nx ∈ Ck, nσs(x) = σs(nx) ∈ Cl, and b(σs(nx)) = ˜τt(s)(b(nx)). +Now we have +˜b(σs(x)) = n−1b(σs(nx)) = n−1˜τt(s)(b(nx)) = ˙τt(s)(n−1b(nx)) = ˙τt(s)(˜b(x)). +Lastly, we extend ˜b to Q⊗A as follows: Given p +q ⊗x ∈ Q⊗A for p ∈ Z and q ∈ Z×, there exists a unique +element y ∈ B such that qy = ˜b(px) in B, and we set ˙b( p +q ⊗x) := y. Now it is straightforward to check +that this is independent of p, q and x, and that ˙b is an injective homomorphism Q ⊗ A �→ Q ⊗ B. +Moreover, equivariance of ˜b with respect to ˜σ and ˙τ implies equivariance of ˙b with respect to ˙σ and ˙τ. +(ii): First of all, every element of S−1A is of the form s−1a = ˜σ−1 +s a for some a ∈ Ck (for some k). +Indeed, the statement is clear if we just ask for a ∈ Ak for some k because of Remark 2.4. By (PC), +there exists ˙s ∈ S such that σ ˙s(a) ∈ Ck, so that σ ˙s(a) ∈ Ck because σ ˙s(Ak) ⊆ Ak, and we have +˜σ−1 +s a = ˜σ−1 +s ˜σ−1 +˙s (˜σ ˙sa), as desired. +Now given an element of S−1A of the form ˜σ−1 +s a for some a ∈ Ck (for some k), we claim that +b′(˜σ−1 +s a) := ˜τ −1 +t(s)(b(a)) is well defined and has the desired properties. To prove that it is well-defined, +assume that ˜σ−1 +s a = ˜σ−1 +t +˙a. Since S is right reversible, there exist u, v ∈ S with us = vt. Hence +˜σ−1 +s ˜σ−1 +u ˜σua = ˜σ−1 +s a = ˜σ−1 +t +˙a = ˜σ−1 +t +˜σ−1 +v ˜σv ˙a. It follows that ˜σua = ˜σv ˙a. Now choose an integer m such +that mσu(a), mσv(a) ∈ Ck and that equivariance holds (here, we use Lemma 3.20). Then we have +m˜τ −1 +t(s)b(a) = ˜τ −1 +t(s)b(ma) = ˜τ −1 +t(s)˜τ −1 +t(u)b(σu(ma)) = ˜τ −1 +t(t)˜τ −1 +t(v)b(σv(m˙a)) = ˜τ −1 +t(t)b(m˙a) = m˜τ −1 +t(t)b(˙a). +As B is torsion-free, we conclude that ˜τ −1 +t(s)b(a) = ˜τ −1 +t(t)b(˙a), as desired. It is easy to see that b′ is +additive. +To show equivariance, take r, s ∈ S. Since S is right reversible, there exist u, v ∈ S with ur = vs and +thus rs−1 = u−1v. Given a ∈ Ck, choose an integer m such that ˜σv(ma) ∈ Ck and equivariance holds +(here, we use Lemma 3.20). Then we have +mb′(˜σr˜σ−1 +s (a)) = b′(˜σ−1 +u ˜σv(ma)) = ˜τ −1 +t(u)b(˜σv(ma)) = ˜τ −1 +t(u)˜τt(v)b(ma) = m˜τt(r)˜τ −1 +t(s)b(a). +As B is torsion-free, we deduce that b′(˜σr˜σ−1 +s (a)) = ˜τt(r)(˜τ −1 +t(s)b(a)) = ˜τt(r)(b′(˜σ−1 +s a)), as desired. +□ +Remark 3.24. Actually we obtain an equivariant embedding Z[I−1] ⊗ A �→ Q ⊗ B. And if (I) and +(II) are satisfied, or (III) holds, then we even obtain an embedding Z[I−1] ⊗ A �→ Z[I−1] ⊗ B, using +Lemma 3.20, because (a*) holds by Corollary 3.16. +3.5. Consequences. Let us formulate symmetrized versions of our rigidity results. +Schmidt defined finite (algebraic) equivalence for algebraic actions of a fixed group (see, e.g., [50, +Definition 8.1]). The dual version of Schmidt’s notion will appear naturally in our setting. In order +to explain this, let us introduce some terminology. +Definition 3.25. +(i) An (algebraic) embedding of ˜σ: S ↷ A into ˜τ : T ↷ B consists of a pair +(t, b), where t: S �→ T and b: A �→ B are injective homomorphisms such that b(˜σs(x)) = +˜τt(s)(b(x)) for all s ∈ S and x ∈ A . The embedding (t, b) is called finite index if the image of +b has finite index in B, full if b is surjective, and strict if t is an isomorphism. +11 + +(ii) We say that ˜σ: S ↷ A and ˜τ : T ↷ B are mutually embeddable (written ˜σ: S ↷ A ∼ME +˜τ : T ↷ B) if each can be embedded into the other. If, in addition, the embeddings can be +chosen to be of finite index, then we write ˜σ: S ↷ A ∼MEF I ˜τ : T ↷ B. Given an Abelian +group Q, we write ˜σ: S ↷ A ∼MEQ ˜τ : T ↷ B if ˙σ: S ↷ Q ⊗ A ∼ME ˙τ : T ↷ Q ⊗ B, +where ˙σs = idQ ⊗ ˜σs and ˙τ := idQ ⊗ ˜τs. We write ˜σ: S ↷ A ∼MEQ∼ += ˜τ : T ↷ B if there +exist full embeddings of ˙σ: S ↷ Q ⊗ A into ˙τ : T ↷ Q ⊗ B and of ˙τ : T ↷ Q ⊗ B into +˙σ: S ↷ Q ⊗ A . +We say that ˜σ: S ↷ A and ˜τ : T ↷ B are strictly mutually embeddable (and we write +˜σ: S ↷ A ∼sME ˜τ : T ↷ B) if each can be strictly embedded into the other. If, in addition, +the strict embeddings can be chosen to be of finite index, then we write ˜σ: S ↷ A ∼sMEF I +˜τ : T ↷ B. Given an Abelian group Q, we write ˜σ: S ↷ A ∼sMEQ ˜τ : T ↷ B if ˙σ: S ↷ +Q ⊗ A ∼sME ˙τ : T ↷ Q ⊗ B. +We write ˜σ: S ↷ A ∼=Q ˜τ : T ↷ B, and call ˜σ: S ↷ A and ˜τ : T ↷ B isomorphic over +Q, if there exists a full and strict embedding of ˙σ: S ↷ Q ⊗ A into ˙τ : T ↷ Q ⊗ B. +(iii) We say that the algebraic actions σ: S ↷ A and τ : T ↷ B are isomorphic if there is a pair +(t, b), where t: S → T is an isomorphism of semigroups and b: A → B is a group isomorphism +such that b(σs(x)) = τt(s)(b(x)) for all s ∈ S and x ∈ A. +Remark 3.26. The dual notion of a strict (algebraic) embedding in our sense is an algebraic factor +map in the sense of [50, Definition 8.1]. +If (t, b) is a finite index embedding of ˜σ: S ↷ A into ˜τ : T ↷ B and A and B are torsion-free, +then A and B are quasi-isomorphic in the sense of [51, Definition 3.3]. +If ˜σ: S ↷ A ∼sMEF I ˜τ : T +↷ B, then we also call ˜σ: S ↷ A and ˜τ : T +↷ B finitely +algebraically equivalent (compare [50, Definition 8.1]). +We will mostly be interested in our notions involving an Abelian group Q when Q = Q. +If A and B are torsion-free, then ∼MEF I implies ∼MEQ∼ += and ∼sMEF I implies ∼=Q. +If A and B are torsion-free and of finite rank, then ∼ME implies ∼MEF I and ∼sME implies ∼sMEF I +(see, e.g., [27, Exercise 5 in § 92]). +Definition 3.27. Given algebraic actions σ: S ↷ A and τ : T ↷ B, let (Is) be the symmetrized +version of condition (I) from Definition 3.15, i.e., condition (I) holds and the analogue of (I) with +reversed roles for σ and τ holds as well. Similarly, let (IIs), (IIIs) and (∗s) be the symmetrized versions +of (II), (III) and (∗) from Definition 3.15, Definition (3.12), and Definition (3.22). +Definition 3.28. Suppose A is torsion-free and of finite rank. We say that ˜σ: S ↷ A is strongly +faithful if +(SF) +˙σs = ρ ˙σtρ−1 implies s = t for all s, t ∈ S and ρ ∈ Aut(Q ⊗ A ), +where ˙σs := idQ ⊗ ˜σs. +Remark 3.29. If det ◦ ˙σ: S → Q× is injective, then ˜σ: S ↷ A satisfies (SF). +We are now ready for the main result of this section. +Theorem 3.30. Assume that σ: S ↷ A and τ : T ↷ B are algebraic actions satisfying our standing +assumptions from § 2.1, with globalizations ˜σ: S ↷ A and ˜τ : T ↷ B. Suppose that A and B are +torsion-free. +(i) If (Is) holds and there exists an isomorphism of topological groupoids +(A ⋊ S ) ⋉ A ∼= (B ⋊ T ) ⋉ B, +then ˜σ: S ↷ A ∼MEQ ˜τ : T ↷ B. +(ii) If (∗s) holds and there exists an isomorphism of topological groupoids +(S−1A ⋊ S−1S) ⋉ A ∼= (T −1B ⋊ T −1T) ⋉ B, +then ˜σ: S−1S ↷ S−1A ∼ME ˜τ : T −1T ↷ T −1B. +If, in addition, A and B have finite rank, then we obtain ˜σ: S ↷ A ∼MEQ∼ += ˜τ : T ↷ B in (i) and +˜σ: S−1S ↷ S−1A ∼MEF I ˜τ : T −1T ↷ T −1B in (ii). +If, in addition, A and B have finite rank and ˜σ: S ↷ A , ˜τ : T ↷ B both satisfy (SF), then we +obtain ˜σ: S ↷ A ∼=Q ˜τ : T ↷ B in (i) and ˜σ: S−1S ↷ S−1A ∼sMEF I ˜τ : T −1T ↷ T −1B (i.e., +˜σ: S−1S ↷ S−1A and ˜τ : T −1T ↷ T −1B are finitely algebraically equivalent) in (ii). +12 + +Proof. Everything except the last claim follows from Theorem 3.23. +For the last claim, suppose +that (t, b) is an embedding of ˙σ: S ↷ Q ⊗ A into ˙τ : T ↷ Q ⊗ B and (s, a) is an embedding of +˙τ : T ↷ Q ⊗ B into ˙σ: S ↷ Q ⊗ A . For s ∈ S , we have (a ◦ b) ˙σt(a ◦ b)−1 = a ˙τt(s)a−1 = ˙τs◦t(s), so +that (SF) implies (s ◦ t)(s) = s. Hence, s ◦ t = idS , so by symmetry, t ◦ s = idT . Our assumptions +imply that Q ⊗ A and Q ⊗ B have the same dimension as rational vector spaces, so that the injective +maps a and b are invertible. The second part of the last claim is similar using that any injective +endomorphism of a torsion-free finite rank Abelian group necessarily has finite index image (see, e.g., +[27, Exercise 92.5]). +□ +Remark 3.31. In some of our examples, A will be finitely generated, in which case we take Ak = A +for all k in condition (∗), so that the requirement σs(Ak) ⊆ Ak in (∗) is automatic. +Remark 3.32. Suppose that both σ: S ↷ A and τ : T ↷ B are exact. +Then the corresponding +groupoids are effective and minimal by [13, Theorem 4.14] (see also [33, Lemma 2.23]) and [13, Corol- +lary 7.4]. Hence the following are equivalent: +(i) (A ⋊ S ) ⋉ A and (B ⋊ T ) ⋉ B are isomorphic as topological groupoids; +(ii) (Aσ, Dσ) and (Aτ, Dτ) are isomorphic as Cartan pairs, where Aσ and Aτ are as in [13, Defi- +nition 3.1], and Dσ and Dτ are as in [13, Proposition 3.30]; +(iii) F ((A ⋊ S ) ⋉ A) and F ((B ⋊ T ) ⋉ B) are isomorphic as abstract groups; +(iv) D((A ⋊ S ) ⋉ A) and D((B ⋊ T ) ⋉ B) are isomorphic as abstract groups. +This follows from [47] (see also [45]) and [48, Theorems 0.2 and 3.3] (or [41, Theorem 3.10]). +4. Algebraic actions on finite rank torsion-free Abelian groups +In this section, we apply our rigidity results to example classes of algebraic actions on finite rank +torsion-free Abelian groups. +4.1. Algebraic actions of torsion-free Abelian monoids whose dual actions are mixing. +Let σ: S ↷ A be an algebraic action, with A Abelian. Let ˆσ: S ↷ �A be the dual action as in [13, +Remark 2.2] and denote by µ the normalized Haar measure on �A. Recall (see, for instance, [49, § 1] +or [53, Definition 1.5]) that ˆσ is (strongly) mixing (with respect to µ) if for all Borel subsets X, Y of +�A we have +lim +s→∞ µ(X ∩ ˆσs(Y )) = µ(X)µ(Y ). +If S has no non-trivial finite subsemigroups, we have the following relation between the mixing property +of ˆσ and condition (F) for σ: +Remark 4.1. Assume that S has no non-trivial finite subsemigroups. Then ˆσ is mixing if and only +if we have, for all 0 ̸= a ∈ A and 1 ̸= s ∈ S, that σs(a) ̸= a, i.e., the analogue of condition (F) holds +for σ: S ↷ A. +Indeed, in general (without our assumption on S), ˆσ is mixing if and only if for all infinite sub- +semigroups S′ ⊆ S and 0 ̸= a ∈ A, we have that # {σs′(a): s′ ∈ S′} = ∞ (see [49, Theorem 1.6] +and also [4], in particular [4, Theorem 2.1]). It is now straightforward to see that, if S has no non- +trivial finite subsemigroups, the latter statement is equivalent to the condition that for all 0 ̸= a ∈ A, +we have # {s ∈ S: σs(a) = a} < ∞. This condition, in turn, is equivalent to the statement that we +have σs(a) ̸= a for all 0 ̸= a ∈ A and 1 ̸= s ∈ S (again assuming that S has no non-trivial finite +subsemigroups), as desired, because {s ∈ S: σs(a) = a} is always a subsemigroup of S. +Note that the condition that S has no non-trivial finite subsemigroups is in particular satisfied if S is +torsion-free, in the sense that for all s1, s2 ∈ S and i ∈ Z>0, si +1 = si +2 implies that s1 = s2. +With the help of this observation, let us now present the first example class to which we can apply +our general rigidity results. +Corollary 4.2. Assume that S and T are non-trivial, Abelian, cancellative and torsion-free monoids, +that A and B are torsion-free Abelian groups of finite rank, and that σ: S ↷ A and τ : T ↷ B are +non-automorphic faithful algebraic actions. Further suppose that the dual actions ˆσ and ˆτ are mixing. +Let ˜σ: S−1S ↷ S−1A and ˜τ : T −1T ↷ T −1B be the canonical globalizations as in [13, Example 2.4]. +13 + +If there exists an isomorphism of topological groupoids +(S−1A ⋊ S−1S) ⋉ A ∼= (T −1B ⋊ T −1T) ⋉ B, +then ˜σ: S−1S ↷ S−1A ∼MEF I ˜τ : T −1T ↷ T −1B. +Proof. First of all, note that condition (JF) is satisfied because of [13, Proposition 7.5] and (FI) holds +by [13, Example 7.6]. Thus σ and τ satisfy our standing assumptions from § 2.1. Moreover, it is +straightforward to check that S−1S and T −1T are torsion-free and that rkZS−1A = rkZA, rkZT −1B = +rkZB. Now our statement follows from Theorem 3.30 (ii) for the finite rank case because of Remark 4.1. +□ +Let us briefly explain the conclusion of our results for the case of (duals of) toral endomorphisms. +Example 4.3. Let a ∈ Mn(Z) and b ∈ Mm(Z) with | det(a)|, | det(b)| > 1, where n, m ∈ Z>0. If a and +b both have no roots of unity as eigenvalues, then the duals of the N-actions σ: N ↷ Zn and τ : N ↷ Zm +given by σk(v) = akv and τk(w) = bkw for k ∈ N, v ∈ Zn, and w ∈ Zm are mixing. Suppose that the +corresponding groupoids are isomorphic. Then, since (SF) holds in this case, Theorem 3.30 implies +that n = m and that the matrices a and b must be conjugate over Q, i.e., there exists c ∈ GLn(Q) +such that a = cbc−1. +4.2. Canonical endomorphisms of torsion-free finite rank Abelian groups. Let A ⊆ Qn be +a torsion-free Abelian group of rank n ∈ Z>0. The multiplicative monoid Z× := Z \ {0} acts on A by +multiplication: Each s ∈ Z× gives rise to the endomorphism σs : A → A given by σs(x) = sx. For any +submonoid M ⊆ Z×, the associated algebraic action σM : M ↷ A, where σM := σ|M, is faithful and +satisfies (FI). It is easy to see that σM : M ↷ A is non-automorphic if and only if there exists m ∈ M +such that A is not m-divisible, i.e., mA ⊊ A. Since det( ˙σs) = sn, we see that det ◦ ˙σ: Z>0 → Q× +is injective, so that σM : M ↷ A satisfies (SF) (see Remark 3.29). Since M is Abelian, we obtain +a globalization ˜σM : ⟨M⟩ ↷ M−1A, where ⟨M⟩ := M−1M ⊆ Q× acts on M−1A := � +s∈M +1 +sA by +multiplication. +It is easy to see that ˜σM : ⟨M⟩ ↷ M−1A satisfies (F). +For s, s′ ∈ Z>0, we have +sA ∩ s′A = lcm(s, s′)A (see, e.g., [26, § 20]). Hence, the M-constructible subgroups for M ↷ A are +given by {sA : s ∈ M}. From this, we see that ˜σM : ⟨M⟩ ↷ M−1A satisfies (JF). Let x1, ..., xn ∈ A +be rationally independent, and put A := spanZ({x1, ..., xn}) ∼= Zn ⊆ A. For each k ∈ Z>0, let +Ak := {x ∈ A : (k!)x ∈ A} = A ∩ (k!)−1A. +Each Ak is an Z×-invariant finitely generated subgroup of A, and we have A = � +k Ak. +We can now apply Theorem 3.30 to obtain the following result: +Corollary 4.4. Let A and B be torsion-free finite rank Abelian groups. Let M, N ⊆ Z× be submonoids +such that there exist m ∈ M and n ∈ N with mA ⊊ A and nB ⊊ B. If there is an isomorphism of +topological groupoids +(M−1A ⋊ ⟨M⟩) ⋉ A ∼= (N−1B ⋊ ⟨N⟩) ⋉ B, +then the actions ⟨M⟩ ↷ M−1A and ⟨N⟩ ↷ N−1B are finitely algebraically equivalent. +Remark 4.5. If Z>0 ⊆ M, then M ↷ A is exact if and only if the Ulm subgroup of A vanishes (cf. +[26, § 1.6]). +4.3. Actions adding scalars to algebraic actions of groups. Let Γ ⊆ SLn(Z) be any subgroup, +and let M ⊆ Z>0 a submonoid. Then, MΓ := {aγ : a ∈ M, γ ∈ Γ} is a submonoid of Mn(Z)× := {x ∈ +Mn(Z) : det(x) ̸= 0}, where we view M as a submonoid of Mn(Z) via the diagonal embedding. Since +Mn(Z)× acts canonically on Zn, we obtain a faithful algebraic action MΓ ↷ Zn. It is easy to see that +MΓ ↷ Zn is exact if and only if M is non-trivial. Note that ⟨M⟩Γ ↷ (M−1Z)n is a globalization for +MΓ ↷ Zn that satisfies (JF). Since Γ acts by automorphisms on Zn that commute with the action +of M, we have CMΓ↷Zn = CM↷Zn. Let Z +n +M denote the completion of Zn with respect to the family +CM↷Zn. +Remark 4.6. The globalization ⟨M⟩Γ ↷ (M−1Z)n often will not satisfy (F). For instance, take +M = Z>0 and Γ = SL2(Z). Then id − γ is not injective on M−1Z2 = Q2, where γ = +� 1 1 +0 1 +� +∈ SL2(Z). +14 + +In order to apply our rigidity result in this setting, we need an observation on subgroups of SLn(Z), +which comes from [23, Example 26.8 & Lemma 26.16]. +Lemma 4.7. Let Γ ⊆ SLn(Z) be a subgroup. +If γαγ−1 = ακ for α, γ ∈ Γ and κ ∈ Z>1, then +α ∈ tor(Γ). +Proof. Suppose γαγ−1 = ακ for α, γ ∈ Γ and κ ∈ Z>1. Let p be a prime divisor of κ. For l ≥ 1, +consider the congruence subgroup Γ(pl) := {a ∈ SLn(Z) : a ≡ In mod pl}, and put Γpl := Γ ∩ Γ(pl). +Since Γp is a finite index subgroup of Γ, we can find m ∈ Z>0 such that αm ∈ Γp. If αm ̸= In, then +since � +l Γpl = {In}, we can find l ≥ 1 with αm /∈ Γpl. Now αmΓpl and ακmΓpl have the same order in +the p-group Γp/Γpl because γαγ−1 = ακ, which is a contradiction since p | κ. +□ +Corollary 4.8. Let Γ, Λ ⊆ SLn(Z) be subgroups and M, N ⊆ Z>0 nontrivial submonoids such that for +every γ ∈ tor(Γ), there exists s ∈ M with gcd(ord(γ), s) > 1, and for every λ ∈ tor(Λ), there exists +t ∈ N with gcd(ord(λ), t) > 1. If there is an isomorphism of topological groupoids +(M−1Zn ⋊ ⟨M⟩Γ) ⋉ Z +n +M ∼= (N−1Zm ⋊ ⟨N⟩Λ) ⋉ Z +m +N, +then M = N and there exist g, h ∈ Mn(N−1Z) ∩ GLn(Q) such that Γ ⊆ gΛg−1 and hΛh−1 ⊆ Γ. +Proof. By Lemma 4.7, condition (∗s) holds, so this follows from Theorem 3.30. +□ +Remark 4.9. The assumptions on Γ, Λ and M, N in the statement Corollary 4.8 are satisfied, for +instance, if M = N = Z>0 or if Γ and Λ are torsion-free (and M, N ̸= {1}). +Remark 4.10. If in the statement of Corollary 4.8, Γ and Λ are not conjugate via an element in +GLn(Q) to any of their proper subgroups, then the conclusion can be strengthened to the following: +M = N and there exists g ∈ Mn(N−1Z) ∩ GLn(Q) such that Γ = gΛg−1. This holds, for instance, if +Γ and Λ are co-Hopfian. +4.4. Arithmetic dynamical systems. Let us consider the algebraic N-actions studied by Chothi, +Everest, and Ward in [15]. Let K be a number field with ring of integers R, and let PK denote the +set of non-zero prime ideals of R. For p ∈ PK, let vp and | · |p denote the associated additive and +multiplicative p-adic valuations on K, respectively. Given a subset S ⊆ PK, the corresponding ring +of S-integers is +RS := {x ∈ K : |x|p ≤ 1 for every p ∈ PK \ S}. +That is, RS consists of the elements of K that are p-adic integers for every p ∈ PK \ S. For ξ ∈ R× +S = +RS \ {0}, the map mξ : RS → RS given by mξ(x) = ξx is an injective endomorphism of the additive +group of RS. The dual action �mξ : N ↷ � +RS is called an arithmetic S-integer dynamical system, see +[15, § 2]. The group of units (i.e., invertible elements) in RS is R∗ +S = {x ∈ K∗ : |x|p = 1 for every p ∈ +PK \ S}. Note that RS is a proper subring of K if and only if S ⊊ PK. Also note that R∗ +S ⊊ R× +S +whenever RS ⊊ K. Let us record some basic observations about the algebraic action mξ : N ↷ RS. +Let ˜mξ : RS[1/ξ] → RS[1/ξ] be given by ˜mξ(x) = ξx. Then ˜mξ : Z ↷ RS[1/ξ] is a globalization of +mξ : N ↷ RS. +Lemma 4.11. +(i) mξ : N ↷ RS is faithful if and only if ξ is not a root of unity. +(ii) mξ : N ↷ RS is exact if and only if it is non-automorphic if and only if ξ is a non-unit. +(iii) ˜mξ : Z ↷ RS[1/ξ] satisfies (JF). +Proof. (i) and (iii) are obvious. For (ii), let ξ ∈ R× +S \ R∗ +S. Then there exists p ∈ PK \ S such that +|ξ|p < 1. If x ∈ RS lies in � +n≥0 ξnRS, then for each n ≥ 0, we can write x = ξnyn for some yn ∈ RS. +Now we have |x|p = |ξ|n +p |yn|p ≤ |ξ|n +p for every n, so that x = 0. The other implications in (ii) are easy +to see. +□ +Remark 4.12. An element x ∈ K is integral over Z if and only if Z[x] is finitely generated as a +Z-module, so the additive group of RS is not finitely generated whenever S ̸= ∅. +For each k ∈ Z>0, let Ak := RS ∩ (k!)−1R = {x ∈ RS : (k!)x ∈ R}. Then RS = � +k Ak, and every Ak +is finitely generated and invariant under R×. +Corollary 4.13. Let K1 and K2 be number fields with rings of algebraic integers R1 and R2, respec- +tively, let S ⊊ PK and T ⊊ PL by proper subsets of primes, and let ξ ∈ R× +1 \ R∗ +1,S and η ∈ R× +2 \ R∗ +2,T . +If there is an isomorphism of topological groupoids +(R1,S[1/ξ] ⋊ ⟨ξ⟩) ⋉ R1,S ∼= (R2,T [1/η] ⋊ ⟨η⟩) ⋉ R2,T , +15 + +then ˜mξ ↷ R1,S[1/ξ] and ˜mη ↷ R2,T [1/η] are finitely algebraically equivalent. +Proof. Condition (∗s) is satisfied, so Theorem 3.30 yields the result. +□ +Remark 4.14. Given any ξ ∈ R× +S \R∗ +S, there exists l ∈ Z>0 such that lξ ∈ R, and then the pair (id, ml) +is a strict, finite index embedding of mξ : N ↷ RS into mlξ : N ↷ RS; in particular, mξ : N ↷ RS and +mlξ : N ↷ RS are isomorphic over Z[l−1]. Thus, up to inverting an integer, Corollary 4.13 applies to +all (faithful, exact) actions of the form mξ : N ↷ RS. +5. Algebraic actions from rings +The rank of a ring R is defined to be the rank of the additive group of R, that is, the dimension of +Q ⊗Z R as a vector space over Q. We shall say that R is torsion-free if the additive group of R is a +torsion-free (Abelian) group. Examples of torsion-free rings of finite rank include integral group rings +of finite groups and Rn or Mn(R), where R an order in a central simple algebra over an algebraic +number field. +5.1. General preparations. Let R be a unital torsion-free ring of finite rank n ∈ Z>0. Then Q⊗Z R +is an n-dimensional Q-algebra containing R as a full subring. The sum of two elementary tensors in +Q ⊗Z R is again an elementary tensor, so that Q ⊗Z R = QR := {q ⊗ x : q ∈ Q, x ∈ R}. Moreover, +if L ⊆ R is a full rank subgroup, then Q ⊗Z L = QL, and QL = QR. Each a ∈ QR gives rise to a +Q-linear map ˙σa : QR → QR given by x �→ ax. We let χa(t) denote the characteristic polynomial of +this map, and put N(a) := | det( ˙σa)|. For a ∈ R×, put σa := ˙σa|R. +Following the notation from [34], we let R× denote the multiplicative monoid of left regular elements +in R, i.e., R× consists of those a ∈ R such that σa is injective. Since R is a torsion-free ring of finite +rank, the element a ∈ R is left regular if and only if N(a) ̸= 0. The action of any submonoid M ⊆ R× +on (the additive group of) R by left multiplication is faithful, by injective endomorphisms, and satisfies +(FI) (see, e.g., [27, Exercise 92.5]). If L is a full rank additive subgroup of R that is invariant under +the action of M, then M also acts faithfully on L by injective endomorphisms and the action M ↷ L +satisfies (FI). Under the canonical inclusion R ⊆ QR, R× is carried into (QR)∗. If M ⊆ R× is a +submonoid, then we let ⟨M⟩ denote the subgroup of (QR)∗ generated by M. Since L is of full rank, +we have L ∩ R× ̸= ∅ and thus M ↷ L is faithful. +Proposition 5.1. For i = 1, 2, let Ri be a torsion-free ring of rank n and Mi ⊆ R× +i a submonoid. Let +L be a rank n subgroup of R1, and assume that spanZ(M1) has finite index in R1. If there is an injective +additive group homomorphism b: QL → QR2 and a group homomorphism t: ⟨M1⟩ → ⟨M2⟩ such that +b(ax) = t(a)b(x) for all a ∈ M1 and x ∈ QL, then there exists a unital Q-algebra isomorphism +ϕ: QR1 → QR2 such that ϕ|⟨M1⟩ = t. +Proof. First, we show that b(1) is invertible in QR2. For every a ∈ M1, we have b(a) = b(a1) = +t(a)b(1), so that b(x) lies in the Q-vector space (QR2)b(1) for all x ∈ spanZ(M1). Since b is injective +and rkZ(spanZ(M1)) = n, we have n = rkZ(im(b)) ≤ dimQ((QR2)b(1)) ≤ dimQQR2 = n. Hence, +dimQ((QR2)˜β(1)) = dimQ(QR2), which implies that b(1) is invertible. +We now define ϕ: QR1 → QR2 by ϕ(x) := b(x)b(1)−1. Clearly, ϕ is additive and ϕ(1) = 1. Since +b is injective and b−1 is invertible, we see that ϕ is also injective. Let a ∈ M1. We obtain t(a) = +b(a)b(1)−1 = ϕ(a). Thus, for a ∈ M1 and x ∈ R1, we have +ϕ(ax) = b(ax)b(1)−1 = t(a)b(x)b(1)−1 = t(a)ϕ(x) = ϕ(a)ϕ(x). +Set Mult(ϕ) := {a ∈ R1 : ϕ(ax) = ϕ(a)ϕ(x) for all x ∈ R1}. It is straightforward to see that Mult(ϕ) +is a subring of R1 containing Z and M1. Since spanZ(M1) is of finite index in R1, it follows that +Mult(ϕ) = R1. Hence ϕ is a ring homomorphism. Thus ϕ is a Q-algebra isomorphism QR1 → QR2 +satisfying ϕ|⟨M1⟩ = α. +□ +Given a torsion-free ring of finite rank R, we let O denote the integral closure of Z in QR. If R is +finitely generated, then R ⊆ O by [46, Theorem 1.10]. However, O may not be a subring if R is +non-commutative. Given a submonoid M ⊆ R×, let � +M := ⟨M⟩ ∩ O. Note that � +M need not be closed +under multiplication. +16 + +The following Corollary demonstrates criteria under which we can deduce rigidity results. +Corollary 5.2. For i = 1, 2, suppose Ri is a finitely generated torsion-free ring of finite rank and +that Mi ⊆ R× +i +is submonoid. +Assume there exist Q-algebra isomorphisms ϕ1 : QR1 → QR2 and +ϕ2 : QR2 → QR1 such that ϕ(⟨M1⟩) ⊆ ⟨M2⟩ and ϕ2(⟨M2⟩) ⊆ ⟨M1⟩. If +(S’) ψ(⟨M1⟩) ⊆ ⟨M1⟩ =⇒ ψ(⟨M1⟩) = ⟨M1⟩ for every ψ ∈ AutQ-alg(QR1), +then ϕ1(⟨M1⟩) = ⟨M2⟩ and ϕ2(⟨M2⟩) = ⟨M1⟩, so that ⟨M1⟩ ↷ QR1 and ⟨M2⟩ ↷ QR2 are isomor- +phic. +If +(N) Mi = � +Mi (for i = 1, 2), and +(S) ψ(M1) ⊆ M1 =⇒ ψ(M1) = M1 for every ψ ∈ AutQ-alg(QR1), +then ϕ1(M1) = M2 and ϕ2(M2) = M1; therefore, M1 ↷ spanZ(M1) and M2 ↷ spanZ(M2) are +isomorphic, and, if each Oi is closed under addition and Mi-invariant, then M1 ↷ O1 and M2 ↷ O2 +are isomorphic. +Proof. Let ψ := ϕ2 ◦ ϕ1 ∈ AutQ-alg(QR1). Then ψ(⟨M1⟩) = ϕ2(ϕ1(⟨M1⟩)) ⊆ ϕ2(⟨M2⟩) ⊆ ⟨M1⟩, so +that ψ(⟨M1⟩) = ⟨M1⟩ by (S’). Now we have ⟨M1⟩ = ψ(⟨M1⟩) = ϕ2(ϕ1(⟨M1⟩)) ⊆ ϕ2(⟨M2⟩) ⊆ ⟨M1⟩, so +that ϕ2(⟨M2⟩) = ⟨M1⟩. +Now assume that (N) and (S) hold. Since Q-algebra homomorphisms preserve integrality, we have +ϕ1(O1) = O2 and ϕ2(O2) = O1; moreover, we have Mi ⊆ Oi for i = 1, 2, so our assumption that +ϕ1(M1) ⊆ ⟨M2⟩ and ϕ2(M2) ⊆ ⟨M1⟩ forces ϕ1(M1) ⊆ O2 ∩ ⟨M2⟩ and ϕ2(M2) ⊆ O1 ∩ ⟨M1⟩. We have +ψ(M1) = ϕ2(ϕ1(M1)) ⊆ ϕ2(O2 ∩ ⟨M2⟩) +(N) += ϕ2(M2) ⊆ O2 ∩ ⟨M1⟩ +(N) += M1, +so that condition (S) forces ψ(M1) = M1, so that ϕ1(M1) = M2 and ϕ2(M2) = M1. +□ +5.2. Groupoid rigidity when the acting monoid is Abelian. In this section, we specialise to +the case where the acting monoids are Abelian. The following is an immediate consequence of Theo- +rem 3.23 and Corollary 5.2. +Theorem 5.3. For i = 1, 2, suppose Ri is a torsion-free finitely generated ring, Mi ⊆ R× +i an Abelian +submonoid such that spanZ(Mi) has finite index in Ri, and Li ⊆ Ri an Mi-invariant full rank subgroup. +Assume that there exists a ∈ M1 such that L1 → L1, x �→ (1 − a)x is injective, and similarly for M2. +If there is an isomorphism of topological groupoids (M−1 +1 L1⋊⟨M1⟩)⋉L1 ∼= (M−1 +2 L2⋊⟨M2⟩)⋉L2, then +there exist Q-algebra isomorphisms ϕ1 : QR1 +∼ += +→ QR2 and ϕ2 : QR2 +∼ += +→ QR1 such that ϕ1(M1) ⊆ � +M2 +and ϕ2(M2) ⊆ � +M1. +We obtain the following rigidity results. +Corollary 5.4. Suppose that, in addition to the assumptions in Theorem 5.3, conditions (N) and (S) +from Corollary 5.2 hold and that Li = spanZ(Mi) = Ri or Li = Oi, then the following statements are +equivalent: +(i) the algebraic actions M1 ↷ R1 and M2 ↷ R2 are isomorphic; +(ii) (M−1 +1 R1 ⋊ ⟨M1⟩) ⋉ L1 and (M−1 +2 R2 ⋊ ⟨M2⟩) ⋉ L2 are isomorphic as topological groupoids. +Note that Li = Oi requires that Oi is closed under addition and Mi-invariant, neither of which is +automatic. +5.2.1. Connection to Bhargava’s work. Torsion-free commutative rings whose additive groups are +finitely generated have received a great deal of attention recently [5, 6, 7, 8, 9, 10]. +Theorem 5.5. Let Ri, i = 1, 2, be finitely generated torsion-free rings. If the groupoids (QR1 ⋊ +(QR1)∗) ⋉ R1 and (QR2 ⋊ (QR2)∗) ⋉ R2 are isomorphic, then QR1 ∼= QR2 as Q-algebras. +Proof. This follows from Theorem 5.3, applied to Mi = R× +i and Li = Ri. Since Z ⊆ Ri, we just need +to show that spanZ(Mi) = Ri. Indeed, for every a ∈ Ri, there exists κ ∈ Z× such that a + κ ∈ R× +i : +As sp( ˙σa) is finite, we have 0 /∈ sp( ˙σa+κ) = sp( ˙σa + κ id) = sp( ˙σa) + κ for sufficiently big κ. +□ +17 + +5.2.2. Actions of congruence monoids on rings of algebraic integers. Let K be a number field with +ring of integers R, and let Rm,Γ ⊆ R× = R \ {0} be a congruence monoid as in [11, § 3], where +m = m∞m0 is a modulus for K and Γ is a group of residues modulo m. Let C∗ +λ(R ⋊ Rm,Γ) denote the +left regular C*-algebra of the monoid R ⋊ Rm,Γ and Dλ(R ⋊ Rm,Γ) the canonical Cartan subalgebra +of C∗ +λ(R ⋊ Rm,Γ) (see [12, § 2.2]). Using the results from [11, § 2], it is not difficult to show that +the family of constructible subgroups for the multiplication action Rm,Γ ↷ R is given by CRm,Γ↷R = +{(0) ̸= I � R : I is coprime with m}. In particular, the completion R of R with respect to CRm,Γ↷R +depends only on the prime divisors of m. +Lemma 5.6. The ring spanZ(Rm,Γ) is an order in R, i.e., it is of finite index in R. +Proof. Since Rm,Γ contains (1 + m0)+, it follows that the subring of R generated by Rm,Γ contains +(m0)+, the set of totally positive elements in the ideal m0. If x ∈ m0, choose k ∈ N× ∩ m0 such that +x + k is totally positive. Since x = (x + k) − k, we see that every element of m0 is a difference of +totally positive elements. Hence, the ring generated by Rm,Γ contains m0, which implies that it is of +finite index in R. +□ +Lemma 5.7. +(i) The monoid Rm,Γ satisfies conditions (N) and (S) from Corollary 5.2; +(ii) the action Rm,Γ ↷ R is exact. +Proof. We shall use the notation from [11]. (i): First, let us show condition (N). By Proposition [11, +Proposition 3.2], we have +⟨Rm,Γ⟩ = {x ∈ K× : vp(x) = 0 for all p | m0, [x]m ∈ Γ}, +from which we see that ⟨Rm,Γ⟩ ∩ R = Rm,Γ. +Now we verify that condition (S) holds. Let ψ ∈ Gal(K/Q). We have ψ(Rm,Γ) = Rψ(m),ψ(Γ) where +ψ(m) is the modulus defined by w | ψ(m)∞ if and only if w ◦ ψ | m∞ and ψ(m)0 := ψ(m0), and ψ(Γ) +is the image of Γ under the isomorphism (R/m)∗ ∼= (R/ψ(m))∗ given by [a]m �→ [ψ(a)]ψ(m). Since +Rψ(m),ψ(Γ) = ψ(Rm,Γ) ⊆ Rm,Γ, [11, Proposition 9.2(1)] implies that ψ(m) | m, i.e., ψ(m)0 | m0 and +ψ(m)∞(w) = 1 =⇒ m∞(w) = 1, and πm,ψ(m)(ψ(Γ)) ⊆ Γ, where πm,ψ(m) : (R/m)∗ → (R/ψ(m))∗ is the +canonical quotient map arising from the divisibility condition ψ(m) | m. Since ψ(m0) and m0 have the +same norm, ψ(m0) | m0 forces ψ(m0) = m0; since the finite sets supp(ψ(m)∞) and supp(m∞) have the +same cardinality, supp(ψ(m)∞) ⊆ supp(m∞) forces supp(ψ(m)∞) = supp(m∞), i.e., ψ(m)∞ = m∞. +Therefore, ψ(m) = m which implies that πm,ψ(m) = id, so that πm,ψ(m)(ψ(Γ)) ⊆ Γ becomes ψ(Γ) ⊆ Γ. +Since #ψ(Γ) = #Γ, we must have ψ(Γ) = Γ. Thus, we have Rψ(m),ψ(Γ) = Rm,Γ. +(ii): It is enough to show that Rm,Γ contains a non-unit a since then � +n≥0 anR = {0}. Observe that +Rm,Γ contains the set (1 + m0)+ of totally positive elements in 1 + m0. Since (1 + m0)+ contains +infinitely many positive integers, we see that Rm,Γ contains non-units. +□ +In this setting, we have the following complete rigidity theorem: +Theorem 5.8. For i = 1, 2, let Ki be a number field with ring of integers Ri, and suppose (Ri)mi,Γi ⊆ +R× +i is a congruence monoid as in [11, § 3]. The following statements are equivalent: +(i) the algebraic actions (R1)m1,Γ1 ↷ R1 and (R2)m2,Γ2 ↷ R2 are isomorphic; +(ii) ((R1)−1 +m1,Γ1R1 ⋊ ⟨(R1)m1,Γ1⟩) ⋉ R1 and ((R2)−1 +m2,Γ2R2 ⋊ ⟨(R2)m2,Γ2⟩) ⋉ R2 are isomorphic as +topological groupoids; +(iii) (R2)−1 +m2,Γ2R2 ⋊ ⟨(R2)m2,Γ2⟩ ↷ R1 and (R2)−1 +m2,Γ2R2 ⋊ ⟨(R2)m2,Γ2⟩× ↷ R2 are continuously orbit +equivalent in the sense of [37, 38]; +(iv) (A(R1)m1,Γ1↷R1, D(R1)m1,Γ1↷R1) and (A(R2)m2,Γ2↷R2, D(R2)m2,Γ2↷R2) are isomorphic as Cartan +pairs; +(v) (C∗ +λ(R1 ⋊ (R1)m1,Γ1), Dλ(R1 ⋊ (R1)m1,Γ1)) and (C∗ +λ(R2 ⋊ (R2)m2,Γ2), Dλ(R2 ⋊ (R2)m2,Γ2)) are +isomorphic as Cartan pairs; +(vi) F (((R1)−1 +m1,Γ1R1 ⋊ ⟨(R1)m1,Γ1⟩) ⋉ R1) and F (((R2)−1 +m2,Γ2R2 ⋊ ⟨(R2)m2,Γ2⟩) ⋉ R2) are isomorphic +as abstract groups; +(vii) D(((R1)−1 +m1,Γ1R1 ⋊ ⟨(R1)m1,Γ1⟩) ⋉ R1) and D(((R2)−1 +m2,Γ2R2 ⋊ ⟨(R2)m2,Γ2⟩) ⋉ R2) are isomorphic +as abstract groups. +Proof. (i)⇔(ii): Let 1 ̸= a ∈ Rmi,Γi. Then multiplication by 1 − a is injective on Ki = Q ⊗Z Ri, +and thus also injective on R−1 +mi,ΓiRi ⊆ Ki. Thus, by Lemma 5.7(i) and Lemma 5.6, we can apply +Corollary 5.4 to obtain the desired equivalence. +18 + +Equivalence of (ii), (iii), and (iv) follows from [38, Theorem 2.7]. +(v)⇒(iv) follows from the description of the primitive ideals of C∗ +λ(Ri ⋊(Ri)mi,Γi) in [11, Theorem 7.1] +combined with the observation that A(Ri)mi,Γi↷Ri is the unique simple quotient of C∗ +λ(Ri ⋊ (Ri)mi,Γi) +and the quotient map C∗ +λ(Ri ⋊(Ri)mi,Γi) → A(Ri)mi,Γi↷Ri carries Dλ(Ri ⋊(Ri)mi,Γi) onto D(Ri)mi,Γi↷Ri +(compare [11, § 8]). Clearly, (i)⇒(iv). +Equivalence of (vi), (vii), and (ii) follows from Lemma 5.7(ii) and Remark 3.32. +□ +Specalizing to the case where the moduli are trivial, and observing that the algebraic actions R× +1 ↷ R1 +and R× +2 ↷ R2 are isomorphic if and only if K1 ∼= K2, we obtain: +Corollary 5.9. For i = 1, 2, let Ki be a number field with rings of integers Ri, denote the corre- +sponding ring C*-algebras by A[Ri], and their canonical Cartan subalgebras by D[Ri]. The following +are equivalent: +(i) K1 and K2 are isomorphic; +(ii) (K1 ⋊ K× +1 ) ⋉ R1 and (K2 ⋊ K× +2 ) ⋉ R2 are isomorphic as topological groupoids; +(iii) K1 ⋊ K× +1 ↷ R1 and K2 ⋊ K× +2 ↷ R2 are continuously orbit equivalent in the sense of [37, 38]; +(iv) (A[R1], D[R1]) and (A[R2], D[R2]) are isomorphic as Cartan pairs; +(v) (C∗ +λ(R1 ⋊ R× +1 ), Dλ(R1 ⋊ R× +1 )) and (C∗ +λ(R2 ⋊ R× +2 ), Dλ(R2 ⋊ R× +2 )) are isomorphic as Cartan +pairs; +(vi) F ((K1 ⋊ K× +1 ) ⋉ R1) and F ((K2 ⋊ K× +2 ) ⋉ R2) are isomorphic as abstract groups; +(vii) D((K1 ⋊ K× +1 ) ⋉ R1) and D((K2 ⋊ K× +2 ) ⋉ R2) are isomorphic as abstract groups. +Remark 5.10. The equivalence of (i) and (v) in Theorem 5.8 completely answers the natural problem +left open in [12, § 5.2]: The Cartan pair (C∗ +λ(R ⋊ Rm,Γ), Dλ(R ⋊ Rm,Γ)) remembers the isomorphism +class of the semigroup R ⋊ Rm,Γ. The equivalences of (i), (iii), and (v) in Corollary 5.9, completely +answers the natural question left open in [36, § 1]. +5.2.3. Algebraic actions from commutative algebra. In this subsection, we analyze a class of algebraic +Nd-actions that are irreversible analogues of the algebraic Zd-actions studied in [49, Chapter II]. +We need the following observation. If R is a commutative finitely generated torsion-free ring of rank +n, then integral closure O of Z in QR is then a ring by [46, Corollary 1.11]. Since R ⊆ O, for each +element a ∈ R, the map ˙σa : QR → QR, ˙σa(x) = ax, leaves O invariant. Thus, N(a) = | det( ˙σa)| lies +in Z>0 for every a ∈ R×. +Lemma 5.11. Let R be a commutative finitely generated torsion-free ring of rank n and a1, ..., ak ∈ +R× \ R∗. In addition, assume that for every 1 ≤ i ≤ k, there exists a prime p such that p | N(ai) and +p ∤ N(aj) for j ̸= i. +(i) If ψ ∈ AutQ-alg(QR) is such that ψ(⟨a1, ..., ak⟩+) ⊆ ⟨a1, ..., ak⟩+, then ψ = id. +(ii) ⟨a1, ..., ak⟩ ∩ O = ⟨a1, ..., ak⟩+. +(iii) a1, ..., ak are multiplicatively independent, so that the canonical map Nk ↠ ⟨a1, ..., ak⟩+ is an +isomorphism. +Proof. (i): Suppose ψ ∈ AutQ-alg(QR) is such that ψ(⟨a1, ..., ak⟩+) ⊆ ⟨a1, ..., ak⟩+. Then for each +1 ≤ i ≤ k, there exist n1, ..., nk ∈ N such that ψ(ai) = an1 +1 · · · ank +k . Now, for a ∈ R×, we have ψ ◦ σa = +σψ(a)◦ψ. In particular, det(σψ(a)) = det(σa). Thus, we have N(ai) = N(ψ(ai)) = N(a1)n1 · · · N(ak)nk, +which, together with our assumption on N(ai), shows that ψ(ai) = ai. +(ii): Let x ∈ ⟨a1, ..., ak⟩ ∩ O. Then there exists n1, ..., nk ∈ Z with x = an1 +1 · · · ank +k . We need to show +that each ni is non-negative. By assumption, for each 1 ≤ i ≤ k, there exists a rational prime p +dividing N(ai) such that p ∤ N(aj) for j ̸= i. Thus, 0 ≤ vp(N(x)) = �k +j=1 njvp(N(aj)) = nivp(N(ai)), +which shows ni ≥ 0 (here, the first inequality uses that x lies in O). The containment “⊇” is obvious. +(iii): Suppose an1 +1 · · · ank +k += 1 for some n1, ..., nk ∈ Z. Fix 1 ≤ i ≤ k, and choose a rational prime p such +that vp(N(ai)) > 0 and vp(N(aj)) = 0 for j ̸= i. Now 0 = vp(N(a1)n1 · · · N(ak)nk) = nivp(N(ai)), so +that ni = 0. +□ +19 + +Let d ∈ Z>0 and denote by R+ +d := Z[u1, ..., ud] the ring of polynomials with integer coefficients in the +d variables u1, ..., ud. Let I �R+ +d be a non-zero ideal. By the Hilbert Basis Theorem (see, for instance, +[24, Theorem 1.2]), R+ +d is Noetherian, so that there exists f1, ..., fm ∈ R+ +d such that I is generated by +{f1, ..., fm}. Since we are only interested in the quotient ring R+ +d /I, let us assume that ui /∈ I for all +1 ≤ i ≤ d. Let +V (I) := {z ∈ Cd : f(z) = 0 for every f ∈ I} ⊆ Cd +be the complex variety defined by I. It follows from [18, Chapter 5, Theorem 6] that dimQQ ⊗Z +R+ +d /I < ∞ if and only if V (I) is a finite set. +If #V (I) < ∞, then C ⊗Z I is said to be zero- +dimensional, in which case there exists a basis for C ⊗Z R+ +d /I consisting of (cosets of) monomials (see +[18, Chapter 5, Proposition 4]), so that R+ +d /I is finitely generated. For the remainder of this section, +we shall assume #V (I) < ∞. +For f ∈ R+ +d , let σf denote the endomorphism of R given by left multiplication with the coset f + I. +Let χf(t) denote the characteristic polynomial of σf viewed as an endomorphism of C ⊗Z R+ +d . Let us +record the following properties of these endomorphisms: +Lemma 5.12. For f ∈ R+ +d , we have +(i) χf(t) = � +z∈V (I)(t−f(z))µ(z), where µ(z) := dimCOz/(C⊗ZI)Oz, where Oz is the localisation +of C ⊗Z R at the maximal ideal mz := {g ∈ C ⊗Z R : g(z) = 0}; +(ii) σf is injective if and only if f(z) ̸= 0 for every z ∈ V (I); +(iii) id − σf = σ1−f is injective if and only if f(z) ̸= 1 for every z ∈ V (I); +(iv) if for all F ⊆ V (I), � +z∈F f(z) ̸= ±1, then χf is not divisible by any unimodular polynomial. +Proof. (i) follows from [17, Chapter 4, Proposition 2.7], and the other parts are consequence of this. +□ +If N ∈ Z>0, then it follows from part (ii) of Lemma 5.12 that the endomorphism σN is injective, so +we see that R+ +d /I is torsion-free. +We let ˙ui denote the image of ui modulo I. Note that ˙ui ̸= 0 by our assumption that ui /∈ I. If zi ̸= 0 +for all z ∈ V (I), then σui is an injective endomorphism of R+ +d /I by Lemma 5.12, and we obtain an +algebraic action ⟨ ˙u1, ..., ˙ud⟩+ ↷ R+ +d /I, which satisfies (FI). Let Rd := Z[u±1 +1 , ..., u±1 +d ] be the ring of +Laurent polynomials in the variables u1, ..., ud; then R+ +d /I embeds in Rd/IRd, and the multiplicative +group ⟨ ˙u1, ..., ˙ud⟩ acts on Rd/IRd by multiplication. It is easy to see that ⟨ ˙u1, ..., ˙ud⟩ ↷ Rd/IRd is +a globalization of ⟨ ˙u1, ..., ˙ud⟩+ ↷ R+ +d /I. Let ΩI denote the completion of R+ +d /I with respect to the +family of ⟨ ˙u1, ..., ˙ud⟩+-constructible subgroups. +Theorem 5.13. For i = 1, 2, let di ∈ Z>0 and let Ii be a non-zero ideal of Z[u1, ..., udi]. Assume that +for i = 1, 2, +(a) #V (Ii) < ∞ and uk /∈ Ii for all k = 1, ..., di; +(b) zk ̸= 0 for every z ∈ V (Ii) and k = 1, ..., di; +(c) there exists a monomial f in u1, . . . , ud such that f(z) ̸= 1 for all z ∈ V (Ii); +(d) for each 1 ≤ j ≤ di, there exists a rational prime p with p | N( ˙uj) and p ∤ N( ˙uk) for k ̸= j. +Then the following statements are equivalent: +(i) the algebraic actions Nd1 ↷ R+ +d1/I1 and Nd2 ↷ R+ +d2/I2 are isomorphic; +(ii) (Rd1/I1Rd1 ⋊ Zd1) ⋉ ΩI1 and (Rd2/I2Rd2 ⋊ Zd2) ⋉ ΩI2 are isomorphic as topological groupoids. +Proof. First, note that (d) implies |N( ˙uj)| > 1, so that ˙uj is a non-unit for all 1 ≤ j ≤ di. Conditions +(a) and (b) ensure that R+ +di/Ii is a finitely generated torsion-free ring and that the action Ndi ↷ R+ +di/Ii +is by injective group endomorphisms whose images all have finite index. We are in the situation of +Corollary 5.4, so we only need to show that (N) and (S) are satisfied. That (N) and (S) are satisfied +follows from parts (ii) and (i) of Lemma 5.11, respectively. +□ +Remark 5.14. In the situation of Theorem 5.13, Lemma 5.11(iii) implies that Ndi ∼= ⟨ ˙u1, ..., ˙udi⟩+. +Remark 5.15. Every finitely generated commutative ring of finite rank is isomorphic to a ring of +the form Z[u1, ..., ud]/I, where I is zero-dimensional. However, the isomorphism will typically not be +canonical, e.g., if R is the ring of algebraic integers in a number field K, then any choice of Z-basis +{x1, ..., xd} for R gives rise to a surjective homomorphism Z[u1, ..., ud] → R, whose kernel must be a +zero-dimensional ideal. +20 + +Let us explain two concrete example classes that are covered by Theorem 5.13. +Example 5.16 (Principal algebraic N-actions). A proper ideal I � Z[u] satisfies #V (I) < ∞ if and +only if I = Z[u]f for a non-constant monic polynomial f ∈ Z[u] (see, e.g., [24, Proposition 4.1.a]). +The action σu : N ↷ Z[u]/Z[u]f is called a principal algebraic N-action. When f is non-constant +and monic, the cosets of 1, u, ..., un−1 form a Z-basis for Z[u]/Z[u]f, where n = deg(f). The matrix +for σu with respect to this basis is equal to the companion matrix Cf of f, so σu is injective if and +only if f(0) = ± det(Cf) is non-zero. All in all, since V (Z[u]f) is the set of zeros of f, we see that +σu : N ↷ Z[u]/Z[u]f satisfies conditions (a)-(d) in Theorem 5.13 if and if f is non-contant, monic, +and |f(0)| > 1, and f(1) ̸= 0. +Remark 5.17. It follows from [32, Theorem] that σu : N ↷ Z[u]/Z[u]f is exact if and only if no +unimodular polynomial divides f (in Q[u]). +Example 5.18 (Algebraic Nd-actions defined by a point). A special class of Nd-actions arises from +d-tuples of algebraic integers. These are the irreversible analogues of the algebraic Zd-actions from +[49, § 7]. Suppose that c = (c1, ..., cd) ∈ (Z +×)d, where Z denotes the ring of all algebraic integers, +and let pc denote the kernel of the evaluation at c map R+ +d → Z[c1, ..., cd] ⊆ Q(c1, ..., cd). Then pc +is a prime ideal of R+ +d , and we can characterize when the action Nd ↷ R+ +d /pc satisfies conditions +(a)-(d) in Theorem 5.13 in terms of c. First, identify V (pc) with the set Hom (Q(c1, ..., cd), Q) of field +embeddings of Q(c1, ..., cd) into Q, the algebraic closure of Q in C. Explicitly, this identification is +given by sending z = (z1, ..., zd) ∈ V (pc) to the embedding Q(c1, ..., cd) �→ Q determined by ci �→ zi. +From this, it is easy to see that conditions (a) and (b) from Theorem 5.13 are satisfied if and only if +ci ̸= 0 for all i, and condition (c) from Theorem 5.13 is satisfied if and only if there exists a finite non- +empty set F ⊆ {1, ..., d} such that � +i∈F ci ̸= 1; to see this, note that for any z = (z1, ..., zd) ∈ V (pc), +we have � +i∈F zi ̸= 1 if and only if � +i∈F ci ̸= 1. Condition (d) from Theorem 5.13 is satisfied if and +only if for each 1 ≤ j ≤ di, there exists a rational prime p with p | N(cj) and p ∤ N(ck) for k ̸= j. +5.3. Algebraic actions from rings: The non-commutative case. For i = 1, 2, let Ri be a ring +whose additive group is finitely generated and torsion-free, let Li ⊆ Ri be full rank subgroup, and let +Mi ⊆ R× +i a submonoid such that Li is Mi-invariant. Then, Mi ↷ Li is faithful since Li has finite +index in Ri and Ri is torsion-free. Let Li denote the completion of Li with respect to the family Ci of +Mi-constructible subgroups of Li. Our goal now is to establish the following rigidity result: +Theorem 5.19. Continue with the notation and assumptions above, with the additional assumptions +that, for i = 1, 2, spanZ(Mi) has finite index in Ri, that there exists κi ∈ Mi for some κi ∈ Z \ {0, 1}, +and that QRi is a semisimple Q-algebra, i.e., the (Jacobson) radical of QRi is trivial (see, e.g., [31, +Part II, § 1]). If there is an isomorphism of topological groupoids +(QR1 ⋊ ⟨M1⟩) ⋉ L1 ∼= (QR2 ⋊ ⟨M2⟩) ⋉ L2, +then there are Q-algebra isomorphisms ϕ1 : QR1 +∼ += +→ QR2 and ϕ2 : QR2 +∼ += +→ QR1 such that ϕ1(⟨M1⟩) ⊆ +⟨M2⟩ and ϕ2(⟨M2⟩) ⊆ ⟨M1⟩. +Taking Mi = R× +i and Li = Ri in Theorem 5.19 yields the following: +Corollary 5.20. If QR1 and QR2 are semisimple Q-algebras and there is an isomorphism of topo- +logical groupoids +(QR1 ⋊ (QR1)∗) ⋉ R1 ∼= (QR2 ⋊ (QR2)∗) ⋉ R2, +then there is a Q-algebra isomorphism QR1 ∼= QR2. +Proof. Observe that spanZ(R× +i ) = Ri as shown in Theorem 5.5. The claim now follows from Theo- +rem 5.19. +□ +Remark 5.21. The actions R× +i ↷ Ri in Corollary 5.20 are exact, so Remark 3.32 applies here. +Before proceeding to the proof, let us explain several example classes to which our results apply. +Example 5.22 (Matrices over orders in number fields). Let n ∈ Z>0. Let R be an order in a number +field K, and let I�R be a nonzero ideal. Then, Mn(I) ⊆ Mn(R) is invariant under the canonical action +Mn(R)× ↷ Mn(R), so we get an algebraic action Mn(R)× ↷ Mn(I). We have QMn(I) = Mn(K), +so Mn(I) has full rank in Mn(R). We have Z1n ⊆ Mn(R)×, and spanZ(Mn(R)×) has finite index +21 + +in Mn(R) by the proof of Corollary 5.20. Thus, the hypotheses of Theorem 5.19 are satisfied (for +L = Mn(I) and M = Mn(R)×). Moreover, ⟨Mn(R)×⟩ = GLn(K). Therefore, if K1 and K2 are +number fields with rings of algebraic integers R1 and R2, respectively, I1 � R1, I2 � R2 are non-zero +ideals, n1, n2 ∈ Z>0, and if (Mn1(K1) ⋊ GLn1(K1)) ⋉ Mn(I1) ∼= (Mn2(K2) ⋊ GLn2(K2)) ⋉ Mn(I2) as +topological groupoids, then Mn1(K1) ∼= Mn2(K2). +In particular, Theorem 5.19 implies that the groupoids (Mn1(K) ⋊ GLn1(K)) ⋉ Mn1(R1) and +(Mn2(K2) ⋊ GLn2(K2)) ⋉ Mn2(R2) are isomorphic if and only if n1 = n2 and K1 ∼= K2. +Example 5.23 (Group rings of finite groups). Let F1 and F2 be finite groups. By Maschke’s Theorem +(see, e.g., [31, Theorem 25]), QFi is a semisimple Q-algebra (i = 1, 2), so Corollary 5.20 implies +the following: If there is an isomorphism (QF1 ⋊ (QF1)∗) ⋉ ZF1 ∼= (QF2 ⋊ (QF2)∗) ⋉ ZF2, then +QF1 ∼= QF2. +Note that if F1, F2 are Abelian, then QF1 ∼= QF2 if and only if F1 ∼= F2 by [44, +Corollary 1 & Theorem 3]. It is a non-trivial result that there exist finite non-Abelian groups F1, F2 +with ZF1 ∼= ZF2 and F1 ̸∼= F2 (see [29]). +Example 5.24 (Central simple algebras over number fields). Let A be a central simple algebra over +the number fields K, i.e., A is a finite-dimensional simple K-algebra whose centre is precisely K, +and let O be an order in A. By the Wedderburn Structure Theorem, there exists a (central) division +algebra D over K and µ ∈ Z>0 such that A ∼= Mµ(D). Thus, the algebraic action O× ↷ O fits into +the setting of Corollary 5.20. Thus if (A1 ⋊ A∗ +1) ⋉ O1 ∼= (A2 ⋊ A∗ +2) ⋉ O2 as topological groupoids, then +A1 ∼= A2. +Now our goal is to prove Theorem 5.19. For the remainder of this section, we shall work with the +assumptions and notation from Theorem 5.19. We need some preparations. +Lemma 5.25. Let R be ring whose additive group is torsion-free and of rank n, and let M ⊆ R× be +a submonoid. Suppose that α ∈ ⟨M⟩ satisfies α = γακγ−1 for some γ ∈ (QR)∗ and κ ∈ Z>1. Set +m := κ(dimQQR)! − 1. Then there exists a nilpotent element ηα ∈ QR such that αm = 1 + ηα (i.e., +αm is a unipotent element of the Q-algebra QR). +Proof. The map π: QR → End C(CR) ∼= Mn(C) given by π(q ⊗ a)(z ⊗ b) = qz ⊗ ab is an injective +Q-algebra homomorphism. +The equation α = γακγ−1 implies that π(α) = π(γ)π(α)κπ(γ)−1 in +Mn(C). It follows that sp(π(α)) = sp(π(α)κ) = sp(π(α))κ := {λκ : λ ∈ sp(π(α))}, where sp(π(α)) +is the spectrum of π(α). Thus, the map sp(π(α)) → sp(π(α)) given by λ �→ λk is bijective; write +sp(π(α)) = {λ1, ..., λj}, and let ρ be the permutation of sp(π(α)) determined by λi = λκ +ρ(i) for all +1 ≤ i ≤ j. Since j ≤ dimQQR, we have ρ(dimQQR)! = id. Thus, λi = λκ(dimQQR!) +ρ(dimQQR)!(i) = λκ(dimQQR)! +i +, +so that λκ(dimQQR)!−1 +i += 1. We now see that 1 is the only eigenvalue of π(α)m. By considering the +Jordan Normal Form of π(α)m, it follows that there exists a nilpotent matrix Nα ∈ Mn(C) such that +π(α)m = 1 + Nα. Since Nα = π(α)m − 1 = π(αm − 1), we see, by injectivity of π, that ηα := αm − 1 +is nilpotent. +□ +Given a division algebra D, we shall regard Dn as an Mn(D)-D-bimodule in the usual way. Thus, a +basis for Dn will always mean a right D-basis. We shall use a subscript D on the right to indicate +that we are viewing something as a right D-vector space. +Lemma 5.26. Let D be a finite dimensional division algebra over Q and n ∈ Z>0. Suppose that Σ is +a non-trivial finitely generated Abelian subgroup of GLn(D) consisting of unipotent matrices, so that +every α ∈ Σ is of the form α = 1 + ηα, where ηα ∈ Mn(D) is a nilpotent matrix. Let k := � +α∈Σ ker ηα +and k := dimkD. We have rkZΣ < n · (n − k)[D : Q]. +Proof. Let N := spanQ{ηα : α ∈ Σ} ⊆ Mn(D). For α, α′ ∈ Σ, we have +1 + ηαα′ = αα′ = (1 + ηα)(1 + ηα′) = 1 + ηα + ηα′ + ηαηα′, +so that ηαηα′ = ηαα′ − ηα − ηα′ lies in N. From this, we see that N is a non-unital commutative +sub-Q-algebra of Mn(D). +For α ∈ Σ, let log α := �∞ +i=1(−1)i−1 (1−α)i +i +; this is a finite sum because 1 − α is nilpotent. Note that +log α lies in N. Since Σ is Abelian, log defines an injective group homomorphism (with inverse given +22 + +by exp, see for instance [28, § 2.10, Exercise 8]) from Σ into N, so that rkZΣ ≤ dimQN. Thus, we will +be done once we show that dimQN < n · (n − k)[D : Q]. +Since N is closed under multiplication and consists of nilpotent elements, [31, Part II, § 5, Theorem 35] +asserts that there exists a right D-basis w1, ..., wn for Dn such that N is strictly upper triangular with +respect to this basis; this means that if we define Wi := span{w1, ..., wi}D for i = 1, .., n, then ηw1 = 0 +and for each i ≥ 2, ηWi ⊆ Wi−1 for all η ∈ N. Let W ⊆ {w1, ..., wn} be a subset such that W together +with a basis for k is a basis for Dn, and put W = span(W)D. Since η|k ≡ 0 for all η ∈ N, the map +N → Hom (W, Dn)D, η �→ η|W is injective; moreover, since wn ̸∈ NDn, we see that it is not surjective. +Hence, dimQN < dimQHom (W, Dn)D = dimQMn×(n−k)(D) = n · (n − k)[D : Q]. +□ +Proof of Theorem 5.19. Let c be the cocycle defined by the composition +(QR1 ⋊ ⟨M1⟩) ⋉ L1 ∼= (QR2 ⋊ ⟨M2⟩) ⋉ L2 +(h,y)�→h +−→ +QR2 ⋊ ⟨M2⟩, +where the second map is the canonical cocycle obtained by projecting onto the group component. +Lemma 3.4 produces a (finite index) subgroup C ∈ C1 such that g(x) := c(x, 0) defines an injective +group homomorphism from C into QR2⋊⟨M2⟩. Let T ⊆ ⟨M2⟩ be the image of C under the composition +(4) +C +g→ QR2 ⋊ ⟨M2⟩ +(b,t)�→t +→ +⟨M2⟩. +By assumption, there exists κ ∈ M1 with κ ∈ Z \ {0, 1}. Let m := κ(dimQQR2)! − 1. Then we claim +that for every α ∈ Tm, there exists a nilpotent element ηα ∈ QR2 such that α = 1 + ηα. Indeed, +as κ ∈ M2, Proposition 3.5 implies that there exists γ ∈ ⟨M2⟩ such that α = γακγ−1 for all α ∈ T. +The result now follows from Lemma 5.25. In particular, Tm is torsion-free. The subgroup mC ⊆ C +is mapped onto Tm under the projection in (4), so Tm is moreover finitely generated. We will show +that Tm is trivial, which will imply that there exists an injective group homomorphism b: mC → QR2 +such that g(x) = (b(x), 1) for all x ∈ mC. Since Tm is free abelian, there exists a subgroup CT ⊆ mC +such that mC = C′ ⊕ CT, where C′ = {x ∈ mC : g(x) = (y, 1) for some y ∈ QR2}, and CT is mapped +isomorphically onto Tm under the map in (4). +Let B be the be the image of C′ under the composition mC +g→ QR2 ⋊ ⟨M2⟩ +(b,t)�→b +→ +QR2. Note that B +is a subgroup of QR2 because the second map is a homomorphism on QR2 ⋊ {1}, which contains the +image of C′. +Let us now show that the group Tm is trivial. +Since QR2 is a semisimple Q-algebra, the Artin– +Wedderburn theorem implies that there exists a decomposition of Q-algebras QR2 = �r +i=1 Mni(Di), +where r, n1, ..., nr ∈ Z>0 and each Di is a finite-dimensional division algebra over Q. Thus, ⟨M2⟩ ⊆ +�r +i=1 GLni(Di). +For each i = 1, ..., r, let Bi be the image of B under the canonical projection +QR2 → Mni(Di). For each i = 1, ..., r, let Ti be the image of Tm under the canonical projection +⟨M2⟩ → GLni(Di). +Then Ti = {1 + ηi +α : α ∈ Tm}, where ηi +α denotes the i-th coordinate of ηα +(which come from Lemma 5.25); in particular, the group Ti consists of unipotent elements. +Let +ki := � +α∈Tm ker (ηi +α) and ki := dim(ki)D. +Take x′ ∈ C′ and write g(x′) = (β′, 1). If x ∈ CT, we can write g(x) = (β, α). Then g(x′)g(x) = +(β′ + β, α), whereas g(x)g(x′) = (β + αβ′, α). Thus, αβ′ = β′, i.e., ηαβ′ = 0. It follows that ηi +αβ = 0 +for all α ∈ Tm and β ∈ Bi, so that im(β) ⊆ ki for all β ∈ Bi. From this, we see that im(β) ⊆ ki for all +β ∈ span(Bi)Di := {� +j βjdj : βj ∈ Bi, dj ∈ Di}. Hence, dim(span(Bi)Di)Di ≤ ni · ki. Now we have +rkZBi = dimQ(Q ⊗ Bi) ≤ dimQ(span(Bi)Di) ≤ ni · ki · [Di : Q]. +Assume for the sake of contradiction that Tm is non-trivial, so that Ti is non-trivial for some i (i.e., +ki < ni). By Lemma 5.26, we have rkZTi < ni · (ni − ki)[Di : Q], where ki := dim(ki)Di. Since +rkZTm ≤ �r +i=1 rkZTi and rkZB ≤ �r +i=1 rkZBi, we obtain +dimQQR1 = rkZC = rkZC′ + rkZCT = rkZB + rkZTm ≤ +r +� +i=1 +rkZTi + +r +� +i=1 +rkZBi +< +r +� +i=1 +n2 +i · [Di : Q] = dimQR2, +23 + +where the strict inequality uses our assumption that ki < ni for some i. By symmetry, we also get +dimQQR2 < dimQQR1, which is a contradiction. Thus, ki = ni for all i. 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Walters, An introduction to ergodic theory, Graduate Texts in Mathematics, 79, Springer-Verlag, New York- +Berlin, 1982. +School of Mathematics and Statistics, University of Glasgow, University Place, Glasgow G12 8QQ, +United Kingdom +Email address: Chris.Bruce@glasgow.ac.uk +School of Mathematics and Statistics, University of Glasgow, University Place, Glasgow G12 8QQ, +United Kingdom +Email address: Xin.Li@glasgow.ac.uk +25 + diff --git a/EtE3T4oBgHgl3EQfVQrm/content/tmp_files/load_file.txt b/EtE3T4oBgHgl3EQfVQrm/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..1065aed384571e10cec0e80f2eea0d95770db5ca --- /dev/null +++ b/EtE3T4oBgHgl3EQfVQrm/content/tmp_files/load_file.txt @@ -0,0 +1,1865 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf,len=1864 +page_content='ALGEBRAIC ACTIONS II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' GROUPOID RIGIDITY CHRIS BRUCE AND XIN LI Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' We establish rigidity for partial transformation groupoids associated with algebraic actions of semigroups: If two such groupoids (satisfying appropriate conditions) are isomorphic, then the globalizations of the initial algebraic actions rationally embed in each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' For specific example classes arising for instance from toral endomorphisms, actions from rings, or actions from commutative algebra, this mutual embedability can be improved in various ways to obtain surprisingly strong rigidity phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' This is witnessed in a particularly striking fashion for actions arising from algebraic number theory: We prove that the groupoids associated with the action of the multiplicative monoid of non-zero elements in a ring of algebraic integers on the additive group of the ring remembers the initial algebraic action up to isomorphism, which in turn remembers the isomorphism class of the ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' This resolves an open problem about isomorphisms of Cartan pairs and leads to a dynamical analogue of the Neukirch–Uchida theorem using topological full groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Introduction Algebraic actions of groups form an interesting and important class of dynamical systems which provides a rich supply of actions of general groups (see, for instance, [49]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' On the one hand, this example class is interesting and exhibits new phenomena, and on the other hand, due to the particular structure of algebraic actions, a variety of tools is available, allowing for a systematic and detailed study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' In [13], we initiated the study of one-sided or irreversible analogues, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', algebraic actions of semigroups, which have not been studied in detail before in general (but see [30, 22] and the references in [13] for special cases).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' An interesting new phenomenon that arises in this new setting is that actions by non-invertible endomorphisms of a given group automatically produce a particular completion of the group, and the original action induces a system of partial homeomorphisms on this completion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' The idea of [13] was to study the corresponding groupoids, which are interesting in their own right but also give access to analyzing properties of C*-algebras generated by natural representations of the initial algebraic action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Our goal now is to study the natural question of how much information the groupoids constructed in [13] remember about the original algebraic actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Surprisingly, we discover the phenomenon of groupoid rigidity for a variety of example classes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', our groupoids remember more information than expected – in special cases, they even remember everything.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Let us now formulate our rigidity results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' An algebraic action σ: S ↷ A consists of a monoid S, an Abelian group A, and a semigroup homomorphism from S to injective endomorphisms of A, denoted by S → End (A), s �→ σs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' We will always assume our algebraic actions to be non-automorphic (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', not all σs are automorphisms) and faithful (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', the map s �→ σs is injective).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Let C be the collection of subgroups of A which are of the form σ−1 t1 σs1 · · · σ−1 tm σsmA, where σ−1 t X := {a ∈ A: σt(a) ∈ X} for a subset X ⊆ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' In this paper, we will always assume that σ has the finite index property (FI), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', #(A/C) < ∞ for all C ∈ C, or equivalently, #(A/σsA) < ∞ for all s ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' In this case, the completion of A mentioned above is given by A := lim ←−C∈C A/C, where C is partially ordered by inclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' The standing assumptions in this paper will furthermore include that σ: S ↷ A admits a globalization (which is always assumed to be minimal in the sense of Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='4), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', an embedding of S into a group S and a group A containing A together with an algebraic action ˜σ: S ↷ A (necessarily by automorphisms) such that ˜σs|A = σs for all s ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' This allows us to form a partial action of S on A by letting s ∈ S act via the restriction A ∩ ˜σ−1 s A → ˜σsA ∩ A of ˜σs to A ∩ ˜σ−1 s A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Similarly, we Date: January 12, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' 2020 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Primary 37A20, 37B99, 22A22;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Secondary 20M18, 37A55, 46L05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Bruce was supported by a Banting Fellowship administered by the Natural Sciences and Engineering Research Council of Canada (NSERC) and has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No 101022531.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Li has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' 817597).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='04459v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='DS] 11 Jan 2023 also have the partial action of A on A by translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' In [13], we identified a condition, called (JF), which ensures that these partial actions on A extend to a partial action of A ⋊ S on A by partial homeomorphisms (see § 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='1 and [13, § 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='3] for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' In this paper, we will always assume that (JF) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' The associated partial transformation groupoid Gσ := (A ⋊S )⋉A is the groupoid constructed in [13, § 3] arising from our algebraic action σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' We can now state our main rigidity result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Theorem A (see Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Let σ: S ↷ A and τ : T ↷ B be two algebraic actions of monoids as above, with globalizations ˜σ: S ↷ A , ˜τ : T ↷ B and groupoids Gσ, Gτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Assume that S and T are Abelian, that A and B are torsion-free and finite rank, and that there exist s ∈ S and t ∈ T such that idA − σs : A → A and idB − τt : B → B are injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' If Gσ and Gτ are isomorphic as topological groupoids, then ˜σ: S ↷ A and ˜τ : T ↷ B embed rationally into each other, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', there exist injective homomorphisms t: S �→ T , b: Q⊗A �→ Q⊗B, s: T �→ S , and a: Q ⊗ B �→ Q ⊗ A such that b((idQ ⊗ ˜σs)(x)) = (idQ ⊗ ˜τt(s))(b(x)) and a((idQ ⊗ ˜τt)(y)) = (idQ ⊗ ˜σs(t))(a(y)) for all s ∈ S , x ∈ Q ⊗ A , t ∈ T , and y ∈ Q ⊗ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' We refer the reader to § 3, in particular § 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='5, for more general results and further explanations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Let us now present a first class of algebraic actions where our general rigidity result applies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Corollary B (see Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Assume that S and T are Abelian, torsion-free monoids, that A and B are torsion-free Abelian groups of finite rank, and that σ: S ↷ A and τ : T ↷ B are non- automorphic faithful algebraic actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Further suppose that the dual actions ˆσ and ˆτ are mixing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Let ˜σ: S−1S ↷ S−1A and ˜τ : T −1T ↷ T −1B be the canonical globalizations as in [13, Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='4], and denote the groupoids attached to σ and τ by Gσ and Gτ, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' If Gσ and Gτ are isomorphic as topological groupoids, then there exist injective homomorphisms t: S−1S �→ T −1T and b: S−1A �→ T −1B such that b(˜σs(x)) = ˜τt(s)(b(x)) for all s ∈ S−1S and x ∈ S−1A, injective homomorphisms s: T −1T �→ S−1S and a: T −1B �→ S−1A such that a(˜τt(y)) = ˜σs(t)(a(y)) for all t ∈ T −1T and y ∈ T −1B, and the images of b and a are finite index subgroups of T −1B and S−1A, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' The reader may consult § 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='1 for more explanations and details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' The concrete case of toral endomor- phisms is treated in Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Another motivating example class is given by the action of the monoid of non-zerodivisors of a ring on the additive group of the ring by multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' For instance, torsion-free commutative rings which are finitely generated as additive groups have received a great deal of attention because of Bhargava’s work [5, 6, 7, 8, 9, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' In this setting, our rigidity result implies the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Theorem C (see Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Let Ri, i = 1, 2, be finitely generated torsion-free commutative rings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' For i = 1, 2, let R× i be the monoid of non-zerodivisors in Ri and σi : R× i ↷ Ri the algebraic action given by multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' If the corresponding groupoids Gσ1 and Gσ2 are isomorphic, then Q ⊗ R1 and Q ⊗ R2 are isomorphic as Q-algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Somewhat surprisingly, using very different methods, we obtain a similar rigidity result for special classes of non-commutative rings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Theorem D (see Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' For i = 1, 2, let Ri be a ring whose additive group is finitely generated and torsion-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Let R× i be the monoid of left regular elements (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', non-left-zerodivisors) in Ri and σi : R× i ↷ Ri the algebraic action given by left multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Suppose that Q ⊗ R1 and Q ⊗ R2 are semisimple Q-algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' If the corresponding groupoids Gσ1 and Gσ2 are isomorphic, then Q ⊗ R1 and Q ⊗ R2 are isomorphic as Q-algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Examples of rings that are covered by Theorem D include integral group rings of finite groups and rings of matrices over orders in algebraic number fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Let us now apply Theorem C to rings of algebraic integers in number fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Let R be such a ring, with quotient field K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Consider the algebraic action σ: R× ↷ R by multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' The completion R is given in this case by the (additive group of the) integral adele ring, and the partial action from above is given by the canonical partial action K ⋊ K× ↷ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Moreover, the groupoid Gσ in this 2 case is the partial transformation groupoid (K ⋊ K×) ⋉ R, and its C*-algebra coincides with the ring C*-algebra A[R], which has been introduced and studied in [19, 21, 34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Since (K ⋊ K×) ⋉ R is effective, A[R] contains a canonical Cartan subalgebra D[R].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Moreover, consider the topological full group F ((K ⋊ K×) ⋉ R) of (K ⋊ K×) ⋉ R given by the group of global bisections (see for instance [40, 42]) and its commutator subgroup D((K ⋊ K×) ⋉ R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Corollary E (see Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Let Ri, i = 1, 2, be rings of algebraic integers in number fields Ki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' With the notation introduced above, the following are equivalent: (i) K1 and K2 are isomorphic;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' (ii) (K1 ⋊ K× 1 ) ⋉ R1 and (K2 ⋊ K× 2 ) ⋉ R2 are isomorphic as topological groupoids;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' (iii) K1 ⋊ K× 1 ↷ R1 and K2 ⋊ K× 2 ↷ R2 are continuously orbit equivalent in the sense of [37, 38];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' (iv) (A[R1], D[R1]) and (A[R2], D[R2]) are isomorphic as Cartan pairs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' (v) F ((K1 ⋊ K× 1 ) ⋉ R1) and F ((K2 ⋊ K× 2 ) ⋉ R2) are isomorphic as abstract groups;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' (vi) D((K1 ⋊ K× 1 ) ⋉ R1) and D((K2 ⋊ K× 2 ) ⋉ R2) are isomorphic as abstract groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' The equivalences of (i), (v), and (vi) in Corollary E gives dynamical analogues of the Neukirch–Uchida theorem from anabelian geometry which says that the absolute Galois group of a number field remembers the field up to isomorphism [43, 52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' A different dynamical version of the Neukirch–Uchida theorem is given in [14] using completely different techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Moreover, the structure of our groups is much different from the absolute Galois groups or the topological full groups from [14] since, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', D((K ⋊ K×) ⋉ R) is simple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' The equivalence of (i) and (iv) in Corollary E is in stark contrast with [39, Corol- lary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='3], which says that the ring C*-algebras A[R1] and A[R2] are always isomorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' One conse- quence of this is that A[Z] contains a family, parameterized by all number fields, of isomorphic but non-conjugate Cartan subalgebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Semigroup C*-algebras C∗(R ⋊ R×) of ax + b-semigroups R ⋊ R× were studied in [20, 35, 36] for rings of algebraic integers R in number fields K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' C∗(R ⋊R×) has a canonical groupoid model (see [21, 35]), and hence contains a canonical Cartan subalgebra D(R⋊R×).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' It was shown in [36] how to recover the Dedekind zeta function and the ideal class group of K from the Cartan pair (C∗(R⋊R×), D(R⋊R×)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' However, the natural question of whether (C∗(R⋊R×), D(R⋊R×)) completely determines the number field K has been left open in [36] (see the question at the end of [36, § 1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Since (A[R], D[R]) can be recovered from (C∗(R ⋊ R×), D(R ⋊ R×)), we are now able to answer this question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Corollary F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Let Ri, i = 1, 2, be rings of algebraic integers in number fields Ki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Then K1 ∼= K2 if and only if (C∗(R1 ⋊ R× 1 ), D(R1 ⋊ R× 1 )) and (C∗(R2 ⋊ R× 2 ), D(R2 ⋊ R× 2 )) are isomorphic as Cartan pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' In fact, we completely resolve the more general problem left open in [12, § 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='2], see Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Theorem D, applied to matrix algebras over rings of algebraic integers, yields the following analogous result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Corollary G (see Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' For i = 1, 2, let Ri be the ring of integers in a number field Ki, and let ni be a positive integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Consider the algebraic action σi : Mni(Ri)× ↷ Mni(Ri) by multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' The groupoids Gσ1 and Gσ2 are isomorphic if and only if n1 = n2 and K1 ∼= K2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' We also have further equivalent statements analogous to (iii) – (vi) in Corollary E, and the analogue of Corollary F holds as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' We would like to point out that we obtain more general results than the ones presented in this introduction (see § 3 for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Thus, we can additionally treat the following example classes: (a) Semigroups of canonical endomorphisms of finite rank torsion-free Abelian groups (§ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' (b) Actions form adding scalars to algebraic actions of subgroups of special linear groups (§ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='3);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' (c) Arithmetical S-integer dynamical systems (§ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='4);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' (d) Actions of congruence monoids on rings of algebraic integers (§ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' (e) Nd-actions from zero-dimensional ideals in commutative algebra (§ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='3);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' (f) Actions from integral group rings of finite groups (Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='23);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' (g) Actions from orders in central simple algebras over number fields (Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' 3 The proofs of our rigidity results are inspired by continuous orbit equivalence rigidity for odometers (see [16]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Given an algebraic action σ: S ↷ A satisfying our standing assumptions, the restriction of the partial action of A ⋊ S ↷ A to A yields an odometer action A ↷ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Our key insight is that a careful analysis allows us to identify situations where rigidity can be upgraded from these odometer actions to groupoid rigidity in our sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' For Abelian acting monoids, we take advantage of algebraic identities in semidirect product groups arising from our algebraic actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' For non-Abelian acting monoids, our rigidity results rely on the structure of nilpotent elements in certain matrix algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Preliminaries 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Standing assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' We first explain the standing assumptions on algebraic actions that we will assume in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Let S be a non-trivial left cancellative monoid and A an Abelian group, written additively, with identity element 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Assume σ: S ↷ A is an algebraic S-action, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', σ: S → End Z(A), s �→ σs is a monoid homomorphism such that σs is an injective endomorphism A → A for all s ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Actions of this form are called algebraic actions (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' [13]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Unless S is a group, we shall assume that σ: S ↷ A is non-automorphic, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', there exists s ∈ S such that σsA ⊊ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' This in particular implies that A is non-trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' We will also always assume that the action σ: S ↷ A is faithful, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', s �→ σs is injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Let C = CS↷A be the family of S-constructible subgroups of A, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', C is the smallest collection of subgroups of A such that A ∈ C;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' s ∈ S and C ∈ C implies σsC, σ−1 s C ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' It follows that C is closed under taking finite intersections (see [13, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' We say that σ: S ↷ A satisfies the finite index property (see [13, Definition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='1]) if (FI) #(A/σsA) < ∞ for all s ∈ S, If σ: S ↷ A satisfies (FI), then [13, Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='2] implies that every member of C is a finite index subgroup of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' In this case, we get a compact group A := lim ←−C∈C A/C, and the canonical homomorphism A → A has kernel � C∈C C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Given C ∈ C, we denote by C the kernel of the canonical projection A ↠ A/C, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', the preimage of C ∈ A/C in A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' We have a canonical homeomorphism C ∼= lim ←−D∈C, D⊆C C/D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' It suffices to check (FI) for generators of S: Say S is generated by S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Then we can proceed inductively on the word length with respect to S of an element in S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Suppose #(A/σsA) < ∞ for some s ∈ S, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', A = R + σsA for some finite set R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Moreover, for t ∈ S, write A = F + σtA for some finite set F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Then A = R + σsA = R + σs(F + σσA) = R + σsF + σstA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Hence it follows that #(A/σstA) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' In this paper, we will only consider algebraic actions σ: S ↷ A which satisfy (FI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Let ∂ �E be the compact space of characters on the semilattice E := {b + C : C ∈ C, b ∈ A} ∪ {∅} (see [13, § 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Each x = (xC + C)C ∈ A determines an element χx of ∂ �E by χx(b + C) := � 1 if xC + C = b + C, 0 if xC + C ̸= b + C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Since (FI) is satisfied, it is not hard to see that the map A → ∂ �E given by x �→ χx is a homeomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' In addition, we will always assume that σ: S ↷ A has a globalization ˜σ: S ↷ A , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', S embeds into the group S , A is a subgroup of the group A , and ˜σ: S → Aut(A ) is an algebraic action such that ˜σs|A = σs for all s ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Then A ⋊ S acts on A by affine maps: (z, γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='x = z + ˜σγ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Reducing to A ⊆ A , we get a partial action (in the sense of [25]) on the group A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Explicitly, g ∈ A ⋊ S acts by the partial bijection A ∩ g−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='A → (g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='A) ∩ A, x �→ g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' 4 Note that s ∈ S acts via σs (where we view both maps as partial bijections on A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Consider the condition (JF) C ⊆ ker (id − ˜σg) =⇒ g = 1 for all C ∈ C, g ∈ ⟨S⟩, where ⟨S⟩ is the subgroup of S generated by S (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' [13, § 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Moreover, our standing assumptions in this paper include that (JF) is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' In this case, the partial action A ⋊ S ↷ A extends uniquely to a partial action A ⋊ S ↷ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Given g ∈ A ⋊ S , let Ug−1 ⊆ A be the domain of g, and for x ∈ Ug−1, we let g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='x denote the image of x under g with respect to the action A ⋊ S ↷ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' The associated transformation groupoid (A ⋊ S ) ⋉ A := {(g, x) ∈ (A ⋊ S ) × A : x ∈ Ug−1, g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='x ∈ A} is canonically isomorphic to the groupoid Gσ = Iσ ⋉ ∂ �E from [13, § 3] (see [13, § 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Since (FI) is satisfied, (A ⋊ S ) ⋉ A is minimal by [13, Corollary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' By [13, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='14], (A ⋊ S ) ⋉ A is effective if and only if σ: S ↷ A is exact in the sense of [13, Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='11], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', � C∈C C = {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' If S is left Ore, then we can replace (JF) by the condition that C ⊆ ker (σs − σs′) for some C ∈ C implies that s = s′ (see [13, Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='17 (iii)]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' If S is left Ore, then there is a canonical partial action of G on A in general, without the assumption that (JF) holds: We first construct the enveloping action as in [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' σ extends to an action of S of A, also denoted by σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Then set S−1A := lim −→S � A, σ � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' this is a locally compact (non- compact) topological group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Extend σ to σ : ⟨S⟩ ↷ S−1A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' This way, we obtain a global dynamical system S−1A ⋊ ⟨S⟩ ↷ S−1A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Then A is a clopen subset of S−1A, so that we obtain the desired partial dynamical system by restricting S−1A ⋊ ⟨S⟩ ↷ S−1A to A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' However, note that (JF) holds automatically in this setting if A is torsion-free by [13, Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' We can and will always assume that S is generated by S, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', S = ⟨S⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Moreover, by [13, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='7], if S embeds into the group S , then we can always take A = ZS ⊗ZS A, and the map A → ZS ⊗ZS A, a �→ 1 ⊗ a will always be injective if σ admits a globalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' In this case, we have A = ⟨� s∈S s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='(1 ⊗ A)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Hence, for any globalization ˜σ : S ↷ A , we may and will always assume that (1) A = ⟨� s∈S ˜σs(A)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' If S is left Ore, then we can always take S = S−1S, and in this case, we can and will always arrange that A = � r∈S ˜σ−1 r (A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Further properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Let us now discuss a few properties which are not part of our standing assumptions, but which we will assume for some of our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' The principal S-constructible subgroups are cofinal in C if (PC) for every C ∈ C, there exists s ∈ S such that σsA ⊆ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Note that (PC) is satisfied if S is left reversible (see [13, Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='12]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Consider the following condition on the algebraic action S ↷ A : (F) For all 1 ̸= s ∈ S , 1 − ˜σs := id − ˜σs : A → A is injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Condition (F) is a freeness condition, modulo the fact that in the linear setting 0 will always be a fixed point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' From cocycles to embeddings Let σ: S ↷ A and τ : T ↷ B be algebraic actions with globalizations ˜σ: S ↷ A and ˜τ : T ↷ B, respectively, which satisfy all our standing assumptions from § 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='1 (and we use the same notation as in § 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' We will often view S as a submonoid of A ⋊ S via S → A ⋊ S , s �→ (0, s) and A as a subgroup of A ⋊ S via A → A ⋊ S , a �→ (a, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Moreover, we will use multiplicative notation for A ⋊ S .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' 5 Suppose c: (A ⋊ S ) ⋉ A → B ⋊ T is a continuous cocycle (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', a groupoid homomorphism) such that c−1(0, 1) = A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Note that c satisfies the cocycle identity c(gh, x) = c(g, h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='x)c(h, x) whenever these expressions make sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' We denote the map A ⋊ S → B ⋊ T , p �→ c(p, 0) again by c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' For each p ∈ A ⋊ S, there exists C(p) ∈ C such that c(p, −): A → B ⋊ T is constant on x + C(p) for every x ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Since A is compact and c is continuous, the image of c(p, −) must be a finite set, which we shall denote by F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' For each f ∈ F, let Uf := c(g, −)−1({f}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Then each Uf is compact open, and we have a partition A = � f∈F Uf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Since the collection {x + C : C ∈ C, x ∈ A} forms a base consisting of compact open sets for the topology of A, we can write Uf as a finite disjoint union Uf = � i xi + Ci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' If we now set Cf := � i Ci, then since Cf is a finite index subgroup of each Ci, we can even write Uf as a disjoint union of the form � i yi + Cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Now set C(p) := � f∈F Cf, so that A is a finite disjoint union A = � k xk + C(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' For all x ∈ A, there exists k such that x + C(p) = xk + C(p), and c(g, −) is constant on xk + C(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' □ Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' For every finitely generated subgroup A of A, there exists CA ∈ C such that c(x, −) is constant on a + CA for all a ∈ A and x ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Let a ∈ A, and let {xi}i be a finite collection of elements in A that generate A as an additive monoid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Set CA := � i C(xi, 1) where the subgroups C(xi, 1) are provided by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' We now show that c((x, 1), −) is constant on a + CA by induction on the word length ℓ(x) of x with respect to {xi}i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' The induction base case ℓ(x) = 1 follows from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Now suppose the claim is true for all x ∈ A with ℓ(x) = l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Given x ∈ A with ℓ(x) = l + 1, we can write x = xix′ for some index i and x′ ∈ A with ℓ(x′) = l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Now we have for all a + x, a + y ∈ a + CA, c((x, 1), a + x) = c((xi, 1)(x′, 1), a + x) = c((xi, 1), (x′, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' (a + x)) c((x′, 1), a + x) = c((xi, 1), (x′, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' (a + x)) c((x′, 1), a + y) = c((xi, 1), x′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' (a + x)) c((x′, 1), a + y) = c((xi, 1), x′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' (a + y)) c((x′, 1), a + y) = c((x, 1), a + y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Here, we used the induction hypothesis for the third equality and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='1 for the fifth equality (x′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' (a + x) and x′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' (a + y) both lie in x′ + a + CA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' □ Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='2 can also be derived from the general results in [16], but we chose to give a direct proof in our special setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' For every finitely generated subgroup A of A, the map A ∩ CA → B ⋊ T given by x �→ c(x, 0) is an injective group homomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Additivity is easy to see.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Now suppose we have x, y ∈ A ∩ CA with c(x, 0) = c(y, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Consider the element (x, 0)(y, 0)−1 of the groupoid (A ⋊ S ) ⋉ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Since c is a groupoid homomorphism, c((x, 0)(y, 0)−1) = (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Hence, using the assumption c−1(0, 1) = A, we conclude that (xy−1, 0) = (x, 0)(y, 0)−1 ∈ A, which implies x = y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' □ Note that A ∩ CA is a finite index subgroup of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' In particular, if A is finitely generated, then we get an injective group homomorphism from a finite index subgroup of A into B ⋊ T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' The additive homomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Given a finitely generated subgroup A of A, we now want to find l ∈ Z>0 and C ∈ C together with a homomorphism b: C := l(A∩C) → B such that c(x) = (b(x), 1) for all x ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Suppose that A ⊆ A is a finite rank subgroup and that s ∈ S satisfies σs(A) ⊆ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Let d be the degree of the polynomial det(z − ˙σs), where ˙σs := idQ ⊗ (σs|A): Q ⊗ A → Q ⊗ A, and let κd ∈ Z>0 be the smallest positive integer such that p(z) := κd det(z − ˙σs) has integer coefficients, and write p(z) = κdzd − κd−1zd−1 − .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' − κ1z − κ0 (for some κ• ∈ Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Then for every finitely generated subgroup A of A, there exists ˇC ∈ C depending on s such that (i) The restriction of c to ˇC := A ∩ ˇC is an injective group homomorphism ˇC → B ⋊ T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' (ii) c(s)dc(x)κd = c(x)κ0c(s) · · · c(x)κd−1c(s) for all x ∈ ˇC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' (iii) The following holds for all x ∈ ˇC: If c(x) = (β, α) and c(s) = (δ, γ), then (2) γdακd = ακ0γακ1γ · · · ακd−1γ in T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' 6 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Let us denote the restriction of σs to A again by σs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' For all x ∈ A, the following holds in A⋊S as κdσd s(x) = κd−1σd−1 s (x) + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' + κ0x in A: sd(κdx) = κdσd s(x)sd = (κ0x)s(κ1x)s · · · (κd−1x)s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Given a finitely generated subgroup A of A, choose CA as in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Set ˇC := C(s) ∩ σ−1 s C(s) ∩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' ∩ σ−(d−1) s C(s) ∩ CA ∩ σ−1 s CA ∩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' ∩ σ−(d−1) s CA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Then (i) is satisfied because of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' We have for all x ∈ ˇC: c(sd(κdx)) = c(sd(κdx), 0) = c(sd, κdx)c(κdx, 0) = c(sd−1, σs(κdx))c(s, κdx)c(κdx, 0) = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' = c(s, σd−1 s (κdx))c(s, σd−2 s (κdx)) · · · c(s, κdx)c(κdx, 0) = c(s, 0)c(s, 0) · · · c(s, 0)c(κdx, 0) = c(s)dc(κdx).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Here we are allowed to replace σ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' s(x) by 0 because x lies in C(s) ∩ σ−1 s C(s) ∩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' ∩ σ−(d−1) s C(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' We also have for all x ∈ A ∩ ˇC: c((κ0x)s(κ1x)s · · · (κd−1x)s) = c((κ0x)s(κ1x)s · · · (κd−1x)s, 0) = c((κ0x)s(κ1x)s · · · (κd−1x), 0)c(s, 0) = c((κ0x)s(κ1x)s · · · , κd−1x)c(κd−1x, 0)c(s, 0) = c(κ0x, σs(κ1x) + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' + σd−1 s (κd−1x))c(s, (κ1x) + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' + σd−2 s (κd−1x)) · · c(κd−2x, σs(κd−1x))c(s, κd−1x) c(κd−1x, 0)c(s, 0) = c(κ0x, 0)c(s, 0) · · c(κd−2x, 0)c(s, 0) c(κd−1x, 0)c(s, 0) = c(κ0x)c(s) · · · c(κd−1x)c(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Here we are allowed to replace the second arguments of c by 0 because x lies in C(s)∩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='∩σ−(d−1) s C(s)∩ A ∩ CA ∩ σ−1 s CA ∩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' ∩ σ−(d−1) s CA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Moreover, we have for all x ∈ ˇC and κ ∈ Z that c(κx) = c(x)κ because x lies in A ∩ CA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' So all in all, we have c(s)dc(x)κd = c(x)κ0c(s) · · · c(x)κd−1c(s) for all x ∈ ˇC = A ∩ ˇC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' This shows (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Now let us prove (iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Suppose that c(x) = (β, α) and c(s) = (δ, γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Comparing T -components, we obtain γdακd = ακ0γακ1γ · · · ακd−1γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' □ Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Suppose we are in the setting of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='5, and put ϵ := κd − κd−1 − .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' κ1 − κ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' If T is Abelian, then (2) is equivalent to αϵ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' If p(z) = zd − κ0, then (2) is γdαγ−d = ακ0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', α and γd satisfy the defining relation for the Baumslag–Solitar group BS(1, κ0) ∼= Z[1/κ0] ⋊ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' If γd and α both have infinite order, then ⟨γd, α⟩ ∼= BS(1, κ0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' With the same notation as in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='5, set ϵ := κd − κd−1 − .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' κ1 − κ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' In addition to the assumptions in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='5, assume that 1 − σs is injective on A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Then we have ϵ ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' If, in addition, every 2-generated subgroup of T is free or Abelian, then αϵ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' The first claim follows from κd det(1 − ˙σs) = ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' For the second claim, our assumption implies that the subgroup ⟨α, γ⟩ ⊆ ⟨T⟩ is free or Abelian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' We claim that α and γ must commute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Indeed, it suffices to treat the case that the subgroup is free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Because α and γ satisfy the non-trivial relation (2), the group ⟨α, γ⟩ is either trivial or infinite cyclic;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' if it is trivial, we are done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Suppose ⟨α, γ⟩ is cyclic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Then, in particular, αγ = γα, as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Now (2) becomes γdακd = γdακ0+κ1+···κd−1, which implies that αϵ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' □ Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Let us mention two classes of groups whose 2-generated subgroups are either free or Abelian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' A group is called semifree if it has a presentation where the only relators are of the form 7 [s, t], where s and t are generators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' If s, t are elements in a semifree group and [s, t] ̸= 1, then {s, t} is a basis for a free group by [1, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' A group is 2-free if every subgroup generated by 2 elements is free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Given a non-empty set of prime numbers ω, a Dω-free group is a group whose elements each have exactly one p-th root for all primes p ∈ ω (see the introduction in [2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' By [3, § 8], every Dω-free group from [2] is 2-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Let us introduce the following conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Let A ⊆ A be a subgroup of finite rank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Consider the following conditions: (i1) There exists s ∈ S such that σs(A) ⊆ A, 1 − σs is injective on A, and every 2-generated subgroup of T is free or Abelian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' (i2) There exists s ∈ S with σs = κ idA for some κ ∈ Z\\{0, 1}, and for all α ∈ T , if α is conjugate to ακ in T , then α must be torsion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' If A ⊆ A is a subgroup of finite rank and (i1) or (i2) holds, then for all finite generated subgroups A ⊆ A, there exist l ∈ Z>0 and C ∈ C such that, with C := l(A ∩ C), there exists an injective homomorphism b : C → B such that c(x) = (b(x), 1) for all x ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Let s ∈ S be as in (i1) or (i2), and ˇC, ˇC as in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Let cT be the composition A → B ⋊ T ↠ T , where the first map is c and the second map is the canonical projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Now cT (ˇC) is finitely generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Moreover, if (i1) holds, then Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='7 implies that cT (ˇC) is torsion, hence finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' If (i2) holds, then Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='5 (iii) implies that cT (ˇC) is torsion, hence finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Set l := #cT (ˇC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Then for all x ∈ lˇC, cT (x) = 1, so that our claim follows (Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='4 gives injectivity of b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' □ Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Let A ⊆ A be a subgroup of finite rank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Consider the following conditions: (ii1) For all torsion orders l > 1 of elements of T , there exists 1 ̸= sl ∈ S and coprime integers µ, ν ∈ Z>0 such that σsl(A) ⊆ A and µ det(z− ˙σsl) = µzδ−ν with gcd(l, µ) > 1 or gcd(l, ν) > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' (ii2) Condition (F) holds for ˜τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Note that, in particular, (ii1) holds if T is torsion-free or if for all κ ∈ Z>0 there exists sκ ∈ S with σsκ = κ idA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' We say that condition (III) holds if there exists s ∈ S such that σs = κ idA for some κ ∈ Z \\ {0, 1}, B is of finite rank, and condition (F) holds for ˜τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Let A ⊆ A be a subgroup of finite rank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Assume that one of the following is true: (i1) or (i2) holds, and (ii1) is satisfied or (ii2) holds and A is torsion-free, (III) holds and A is torsion-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Then for all finitely generated subgroup A ⊆ A, there exists C ∈ C such that, with C := A ∩ C, there exists an injective homomorphism b : C → B such that c(x) = (b(x), 1) for all x ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' To prove the first item, let s ∈ S be as in (i1) or (i2), and ˇC, ˇC as in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' First assume that (ii1) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Let cT be the composition A → B ⋊ T ↠ T , where the first map is c and second map is the canonical projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' As cT (ˇC) is finitely generated, the set L of possible non-trivial torsion orders of elements in cT (ˇC) is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' For each l ∈ L, choose sl as in (ii1) and set C := ˇC ∩ � l∈L ˇC(sl).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Applying Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='5 to sl, we obtain that for all x ∈ A ∩ C with c(x) = (β, α), we have αµ and αν are conjugate in T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' If α ̸= 1, then the torsion order of α is a number l ∈ L, but (ii1) implies that αµ and αν have different torsion orders, which is absurd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Now suppose that (ii2) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Take x ∈ ˇC and write c(x) = (β, α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='7 implies that αϵ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Then (β, α)ϵ = (β + ˜τα(β) + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' + ˜τ ϵ−1 α (β), αϵ) = (β + ˜τα(β) + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' + ˜τ ϵ−1 α (β), 1), and (1 − ˜τα)(β + ˜τα(β) + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' + ˜τ ϵ−1 α (β)) = 0, which implies β + ˜τα(β) + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' + ˜τ ϵ−1 α (β) = 0 if α ̸= 1 by (F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' So c(x)ϵ = 1 and hence c(ϵx) = 1 and thus ϵx = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' As A is torsion-free, this implies x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' So if x ̸= 0, we must have α = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Now we prove the second item.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' As above, let s ∈ S be as in (III), and ˇC, ˇC as in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Given x ∈ ˇC, write c(x) = (β, α) and c(s) = (δ, γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Equation (2) implies that γα = ακγ, and therefore α = γ−1ακγ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' It follows that every eigenvalue of ˙τα is a root of unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Indeed, take λ1 ∈ Sp ( ˙τα).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Then 8 α = γ−1ακγ implies that there exists λ2 ∈ Sp ( ˙τα) with λ1 = λκ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Similarly, there exist λ3, λ4, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' ∈ Sp ( ˙τα) such that λi = λκ i+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' As Sp ( ˙τα) is finite, we must have λi = λi+p for some i and p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' It follows that λi = λκp i and hence that λi is a root of unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' But then λ1 = λκi i implies that λ1 must be a root of unity as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Hence there exists m ∈ Z>0 such that 1 is an eigenvalue of ˙τ m α .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' This implies that 1 − ˙τ m α is not injective, so that ˙τ m α = 1 by (F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Now argue as for the first item that this – together with (F) – implies α = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' □ Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' The difference between Corollaries 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='10 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='13 is that in the latter, we may choose l = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Assume that A = � n An for an increasing family of finite rank subgroups An.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Consider the following conditions: (I) Condition (i1) holds for An for all n, or condition (i2) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' (II) Condition (ii1) holds for An for all n, or condition (ii2) holds and A is torsion-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Suppose we can write A = � n An for an increasing family An ⊆ A of finite rank subgroups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' If (I) holds, then (a) there is an increasing family of finitely generated subgroups Ak ⊆ A with A = � k Ak, and for any such Ak, there are lk ∈ Z>0 and Ck ∈ C such that, with Ck := lk(Ak ∩ Ck), there exists an injective homomorphism bk : Ck → B such that c(x) = (b(x), 1) for all x ∈ Ck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' If (I) and (II) hold, or if (III) is satisfied and A is torsion-free, then (a*) there is an increasing family of finitely generated subgroups Ak ⊆ A with A = � k Ak, and for any such Ak, there is Ck ∈ C such that, with Ck := Ak ∩ Ck, there exists an injective homomorphism bk : Ck → B such that c(x) = (bk(x), 1) for all x ∈ Ck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Since A = � n An, we can find Ak with the desired properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Moreover, for each k, since Ak is finitely generated, and A = � n An, we can find n such that Ak ⊆ An.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Now apply Corollaries 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='10 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' □ Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' With the notation from Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='16, we have Ak/Ck �→ A/Ck, so that #(Ak/Ck) is finite and divides #(A/Ck).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' The multiplicative homomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Define t: S → T to be the composition S → B⋊T ↠ T , where the first arrow is given by c(−, 0), and the second arrow is the canonical projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Clearly, t is a homomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' (i) If A is torsion-free and (a) holds, then t is injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' (ii) Suppose that condition (F) is satisfied for ˜σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' If there exist l ∈ Z>0, a non-zero, finitely generated subgroup A ⊆ A, C ∈ C and an injective homomorphism b : C := l(A ∩ C) → B such that c(x) = (b(x), 1) for all x ∈ C, then t is injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Suppose t(s) = t(s′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Then c(s−1s′, 0) = (χ, t(s−1s′)) = (χ, 1) for some χ ∈ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Set ε := s−1s′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Since 0 lies in the domain of ˜σε, and since c(ε, −) is locally constant, there must exist C(ε) ∈ C such that C(ε) is contained in the domain of ˜σε and c(ε, −) is constant on C(ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Now suppose that C ⊆ A is a non-zero subgroup such that there exists an injective homomorphism b : C → B with c(x) = (b(x), 1) for all x ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Then c(ε, 0) commutes with c(x, 0) for all x ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' We have (0, ε)(x, 1) = (˜σϵ(x), ϵ) = (˜σε(x), 1)(0, ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' So if 0 ̸= x ∈ C ∩ C(ε), then c((˜σε(x), ε), 0) = c(˜σε(x)ε, 0) = c(˜σε(x), 0)c(ε, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' At the same time, c((˜σε(x), ϵ), 0) = c(εx, 0) = c(ε, x)c(x, 0) = c(ε, 0)c(x, 0) = c(x, 0)c(ε, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Here we are allowed to replace x by 0 because x lies in C(ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Moreover, we used that c(ε, 0) commutes with c(x, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Therefore, a comparison yields c(˜σε(x), 0) = c(x, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' It follows that ˜σε(x) = x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' For (i), write A = � k Ak as in (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Applying the above to C = Ck from (a), we obtain ˜σε(x) = x for all x ∈ Ck ∩ C(ε) for all k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Now let y ∈ A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' then there exists k with y ∈ Ak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Since Ck ∩ C(ε) is finite 9 index in Ak, there exists N ∈ Z>0 such that Ny ∈ Ck ∩ C(ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' We have N ˜σε(y) = ˜σε(Ny) = Ny, so that, because A is torsion-free, ˜σε(y) = y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Now (JF) implies ε = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' For (ii), if ˜σε(x) = x for any x ̸= 0, then condition (F) implies that ε = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Equivariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' We set I := {#(A/C): C ∈ C}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Assume that B is torsion-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' If (a) holds, then for all s ∈ S, x ∈ Ck, σs(x) ∈ Al (k ≤ l) there exists n ∈ Z>0 such that nσs(x) = σs(nx) ∈ Cl, and we have b(σs(nx)) = ˜τt(s)(b(nx)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' If (a*) holds, then we may take n ∈ I in the statement above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Since Cl is of finite index in Al (see Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='17), there exists n ∈ Z>0 such that nσs(x) = σs(nx) ∈ Cl (since l depends on s, n also depends on s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Since Ck ∩ C(s) is of finite index in Ck, we can find N such that Nnx ∈ Ck ∩ C(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Set y := Nnx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Let c(s) = (δ, t(s)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' We have (b(σs(y)) + δ, t(s)) = (b(σs(y)), 1)(δ, t(s)) = c(σs(y), 0)c(s, 0) = c((σs(y), s), 0) = c((0, s)(y, 0), 0) = c((0, s), y)c((y, 1), 0) = c((0, s), 0)c((y, 1), 0) = (δ, t(s))(b(y), 1) = (δ + ˜τt(s)(b(y)), t(s)), where the sixth equality uses that y ∈ C(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Hence b(σs(y)) = ˜τt(s)(b(y)), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', Nb(σs(nx)) = N ˜τt(s)(b(nx)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Since B is torsion-free, we obtain b(σs(nx)) = ˜τt(s)(b(nx)), as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Conclusion: The embedding theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Assume that σ: S ↷ A and τ : T ↷ B are algebraic actions satisfying our standing assumptions from § 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Let Z[I−1] be the subring of Q generated by Z together with � 1 n: n ∈ I � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' We start with the following observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' We have Z[I−1] ⊗ A = Z[I−1] ⊗ A , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', the canonical map Z[I−1] ⊗ A → Z[I−1] ⊗ A , 1 ⊗ x �→ 1 ⊗ x is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Recall that we always assume (1), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', A = ⟨� s∈S ˜σs(A)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Otherwise, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='21 would not be true in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Because of (1), it suffices to prove that for all x ∈ ˜σ−1 t1 ˜σs1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' ˜σ−1 tl ˜σslA (where t1, s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' , tl, sl ∈ S) there exists N in the multiplicative submonoid ⟨I⟩+ of Z× generated by I with Nx ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' We proceed inductively on l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' For the case l = 1, observe that [˜σ−1 t A : A] < ∞ for all t ∈ S because ˜σt induces a bijection ˜σ−1 t A/A ∼= A/σtA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Now suppose that x = ˜σ−1 t ˜σs(y) for some y ∈ A with Ny ∈ A for some N ∈ ⟨I⟩+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Since [˜σ−1 t A : A] < ∞, there exists M ∈ ⟨I⟩+ with M ˜σ−1 t (σs(Ny)) ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Hence MNx ∈ A, as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' □ Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' We say that condition (∗) is satisfied if A = � n An for an increasing family of finite rank subgroups An, conditions (I) and (II) hold, or condition (III) holds;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' in addition, there exists an increasing family of finitely generated subgroups Ak ⊆ A with A = � k Ak such that σs(Ak) ⊆ Ak for all s ∈ S and all k, S and T are right reversible, A = S−1A, S = S−1S, B = T −1B, T = T −1T, and σ: S ↷ A satisfies (PC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Now suppose c: (A ⋊ S ) ⋉ A → B ⋊ T is a continuous cocycle such that c−1(0, 1) = A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Suppose A and B are torsion-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' (i) If (I) holds, then there exist injective homomorphisms t : S �→ T and ˙b: Q ⊗ A �→ Q ⊗ B such that ˙b( ˙σs(x)) = ˙τt(s)(˙b(x)) for all s ∈ S and x ∈ A .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' (ii) If (∗) holds, then there exist injective homomorphisms t : S �→ T and b′ : S−1A → T −1B such that b′(σs(x)) = ˜τt(s)(b′(x)) for all x ∈ S−1A and s ∈ S .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' (i): We proceed in several steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' First, we claim that for each k, bk : Ck → B has a unique extension to an injective homomorphism ˜bk : Ak → Q ⊗ B, and that moreover, ˜bk satisfies (3) ˜bk(x) = m−1bk(mx), for any m ∈ Z>0 with mAk ⊆ Ck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' To see this, choose m ∈ Z>0 such that mAk ⊆ Ck, and define ˜bk(x) := m−1b(mx) ∈ Q ⊗ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' It is easy to check that ˜bk is a group homomorphism (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', additive) 10 and that ˜b is injective (here we need that Ak is torsion-free).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' This is independent of the choice of m, because if m′ ∈ Z>0 with m′Ak ⊆ Ck, then for x ∈ Ak, we have m−1b(mx) = m−1(m′)−1b(m′mx) = (m′)−1m−1mb(m′x) = (m′)−1b(m′x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Next, we claim that the maps ˜bk are compatible in the sense that ˜bl|Ak∩Al = ˜bk|Ak∩Al for all k, l ∈ Z>0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Choose m large enough so that mAk ⊆ Ck and mAl ⊆ Cl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Then, using (3), we have for all x ∈ Ak ∩Al, that ˜bl(x) = m−1bl(mx) = m−1bk(mx) = ˜bk(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' It follows that we get a well-defined injective homomorphism ˜b: A = � k Ak → Q ⊗ B such that ˜b|Ak = ˜bk for all k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Let us show that ˜b is equivariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Let x ∈ A and s ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Then x ∈ Ak for some k, so by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='20, we can find n ∈ Z>0 large enough so that nx ∈ Ck, nσs(x) = σs(nx) ∈ Cl, and b(σs(nx)) = ˜τt(s)(b(nx)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Now we have ˜b(σs(x)) = n−1b(σs(nx)) = n−1˜τt(s)(b(nx)) = ˙τt(s)(n−1b(nx)) = ˙τt(s)(˜b(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Lastly, we extend ˜b to Q⊗A as follows: Given p q ⊗x ∈ Q⊗A for p ∈ Z and q ∈ Z×, there exists a unique element y ∈ B such that qy = ˜b(px) in B, and we set ˙b( p q ⊗x) := y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Now it is straightforward to check that this is independent of p, q and x, and that ˙b is an injective homomorphism Q ⊗ A �→ Q ⊗ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Moreover, equivariance of ˜b with respect to ˜σ and ˙τ implies equivariance of ˙b with respect to ˙σ and ˙τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' (ii): First of all, every element of S−1A is of the form s−1a = ˜σ−1 s a for some a ∈ Ck (for some k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Indeed, the statement is clear if we just ask for a ∈ Ak for some k because of Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' By (PC), there exists ˙s ∈ S such that σ ˙s(a) ∈ Ck, so that σ ˙s(a) ∈ Ck because σ ˙s(Ak) ⊆ Ak, and we have ˜σ−1 s a = ˜σ−1 s ˜σ−1 ˙s (˜σ ˙sa), as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Now given an element of S−1A of the form ˜σ−1 s a for some a ∈ Ck (for some k), we claim that b′(˜σ−1 s a) := ˜τ −1 t(s)(b(a)) is well defined and has the desired properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' To prove that it is well-defined, assume that ˜σ−1 s a = ˜σ−1 t ˙a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Since S is right reversible, there exist u, v ∈ S with us = vt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Hence ˜σ−1 s ˜σ−1 u ˜σua = ˜σ−1 s a = ˜σ−1 t ˙a = ˜σ−1 t ˜σ−1 v ˜σv ˙a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' It follows that ˜σua = ˜σv ˙a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Now choose an integer m such that mσu(a), mσv(a) ∈ Ck and that equivariance holds (here, we use Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Then we have m˜τ −1 t(s)b(a) = ˜τ −1 t(s)b(ma) = ˜τ −1 t(s)˜τ −1 t(u)b(σu(ma)) = ˜τ −1 t(t)˜τ −1 t(v)b(σv(m˙a)) = ˜τ −1 t(t)b(m˙a) = m˜τ −1 t(t)b(˙a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' As B is torsion-free, we conclude that ˜τ −1 t(s)b(a) = ˜τ −1 t(t)b(˙a), as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' It is easy to see that b′ is additive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' To show equivariance, take r, s ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Since S is right reversible, there exist u, v ∈ S with ur = vs and thus rs−1 = u−1v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Given a ∈ Ck, choose an integer m such that ˜σv(ma) ∈ Ck and equivariance holds (here, we use Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Then we have mb′(˜σr˜σ−1 s (a)) = b′(˜σ−1 u ˜σv(ma)) = ˜τ −1 t(u)b(˜σv(ma)) = ˜τ −1 t(u)˜τt(v)b(ma) = m˜τt(r)˜τ −1 t(s)b(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' As B is torsion-free, we deduce that b′(˜σr˜σ−1 s (a)) = ˜τt(r)(˜τ −1 t(s)b(a)) = ˜τt(r)(b′(˜σ−1 s a)), as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' □ Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Actually we obtain an equivariant embedding Z[I−1] ⊗ A �→ Q ⊗ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' And if (I) and (II) are satisfied, or (III) holds, then we even obtain an embedding Z[I−1] ⊗ A �→ Z[I−1] ⊗ B, using Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='20, because (a*) holds by Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Consequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Let us formulate symmetrized versions of our rigidity results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Schmidt defined finite (algebraic) equivalence for algebraic actions of a fixed group (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', [50, Definition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' The dual version of Schmidt’s notion will appear naturally in our setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' In order to explain this, let us introduce some terminology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' (i) An (algebraic) embedding of ˜σ: S ↷ A into ˜τ : T ↷ B consists of a pair (t, b), where t: S �→ T and b: A �→ B are injective homomorphisms such that b(˜σs(x)) = ˜τt(s)(b(x)) for all s ∈ S and x ∈ A .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' The embedding (t, b) is called finite index if the image of b has finite index in B, full if b is surjective, and strict if t is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' 11 (ii) We say that ˜σ: S ↷ A and ˜τ : T ↷ B are mutually embeddable (written ˜σ: S ↷ A ∼ME ˜τ : T ↷ B) if each can be embedded into the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' If, in addition, the embeddings can be chosen to be of finite index, then we write ˜σ: S ↷ A ∼MEF I ˜τ : T ↷ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Given an Abelian group Q, we write ˜σ: S ↷ A ∼MEQ ˜τ : T ↷ B if ˙σ: S ↷ Q ⊗ A ∼ME ˙τ : T ↷ Q ⊗ B, where ˙σs = idQ ⊗ ˜σs and ˙τ := idQ ⊗ ˜τs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' We write ˜σ: S ↷ A ∼MEQ∼ = ˜τ : T ↷ B if there exist full embeddings of ˙σ: S ↷ Q ⊗ A into ˙τ : T ↷ Q ⊗ B and of ˙τ : T ↷ Q ⊗ B into ˙σ: S ↷ Q ⊗ A .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' We say that ˜σ: S ↷ A and ˜τ : T ↷ B are strictly mutually embeddable (and we write ˜σ: S ↷ A ∼sME ˜τ : T ↷ B) if each can be strictly embedded into the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' If, in addition, the strict embeddings can be chosen to be of finite index, then we write ˜σ: S ↷ A ∼sMEF I ˜τ : T ↷ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Given an Abelian group Q, we write ˜σ: S ↷ A ∼sMEQ ˜τ : T ↷ B if ˙σ: S ↷ Q ⊗ A ∼sME ˙τ : T ↷ Q ⊗ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' We write ˜σ: S ↷ A ∼=Q ˜τ : T ↷ B, and call ˜σ: S ↷ A and ˜τ : T ↷ B isomorphic over Q, if there exists a full and strict embedding of ˙σ: S ↷ Q ⊗ A into ˙τ : T ↷ Q ⊗ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' (iii) We say that the algebraic actions σ: S ↷ A and τ : T ↷ B are isomorphic if there is a pair (t, b), where t: S → T is an isomorphism of semigroups and b: A → B is a group isomorphism such that b(σs(x)) = τt(s)(b(x)) for all s ∈ S and x ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' The dual notion of a strict (algebraic) embedding in our sense is an algebraic factor map in the sense of [50, Definition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' If (t, b) is a finite index embedding of ˜σ: S ↷ A into ˜τ : T ↷ B and A and B are torsion-free, then A and B are quasi-isomorphic in the sense of [51, Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' If ˜σ: S ↷ A ∼sMEF I ˜τ : T ↷ B, then we also call ˜σ: S ↷ A and ˜τ : T ↷ B finitely algebraically equivalent (compare [50, Definition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' We will mostly be interested in our notions involving an Abelian group Q when Q = Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' If A and B are torsion-free, then ∼MEF I implies ∼MEQ∼ = and ∼sMEF I implies ∼=Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' If A and B are torsion-free and of finite rank, then ∼ME implies ∼MEF I and ∼sME implies ∼sMEF I (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', [27, Exercise 5 in § 92]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Given algebraic actions σ: S ↷ A and τ : T ↷ B, let (Is) be the symmetrized version of condition (I) from Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='15, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', condition (I) holds and the analogue of (I) with reversed roles for σ and τ holds as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Similarly, let (IIs), (IIIs) and (∗s) be the symmetrized versions of (II), (III) and (∗) from Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='15, Definition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='12), and Definition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Suppose A is torsion-free and of finite rank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' We say that ˜σ: S ↷ A is strongly faithful if (SF) ˙σs = ρ ˙σtρ−1 implies s = t for all s, t ∈ S and ρ ∈ Aut(Q ⊗ A ), where ˙σs := idQ ⊗ ˜σs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' If det ◦ ˙σ: S → Q× is injective, then ˜σ: S ↷ A satisfies (SF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' We are now ready for the main result of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Assume that σ: S ↷ A and τ : T ↷ B are algebraic actions satisfying our standing assumptions from § 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='1, with globalizations ˜σ: S ↷ A and ˜τ : T ↷ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Suppose that A and B are torsion-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' (i) If (Is) holds and there exists an isomorphism of topological groupoids (A ⋊ S ) ⋉ A ∼= (B ⋊ T ) ⋉ B, then ˜σ: S ↷ A ∼MEQ ˜τ : T ↷ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' (ii) If (∗s) holds and there exists an isomorphism of topological groupoids (S−1A ⋊ S−1S) ⋉ A ∼= (T −1B ⋊ T −1T) ⋉ B, then ˜σ: S−1S ↷ S−1A ∼ME ˜τ : T −1T ↷ T −1B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' If, in addition, A and B have finite rank, then we obtain ˜σ: S ↷ A ∼MEQ∼ = ˜τ : T ↷ B in (i) and ˜σ: S−1S ↷ S−1A ∼MEF I ˜τ : T −1T ↷ T −1B in (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' If, in addition, A and B have finite rank and ˜σ: S ↷ A , ˜τ : T ↷ B both satisfy (SF), then we obtain ˜σ: S ↷ A ∼=Q ˜τ : T ↷ B in (i) and ˜σ: S−1S ↷ S−1A ∼sMEF I ˜τ : T −1T ↷ T −1B (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', ˜σ: S−1S ↷ S−1A and ˜τ : T −1T ↷ T −1B are finitely algebraically equivalent) in (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' 12 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Everything except the last claim follows from Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' For the last claim, suppose that (t, b) is an embedding of ˙σ: S ↷ Q ⊗ A into ˙τ : T ↷ Q ⊗ B and (s, a) is an embedding of ˙τ : T ↷ Q ⊗ B into ˙σ: S ↷ Q ⊗ A .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' For s ∈ S , we have (a ◦ b) ˙σt(a ◦ b)−1 = a ˙τt(s)a−1 = ˙τs◦t(s), so that (SF) implies (s ◦ t)(s) = s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Hence, s ◦ t = idS , so by symmetry, t ◦ s = idT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Our assumptions imply that Q ⊗ A and Q ⊗ B have the same dimension as rational vector spaces, so that the injective maps a and b are invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' The second part of the last claim is similar using that any injective endomorphism of a torsion-free finite rank Abelian group necessarily has finite index image (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', [27, Exercise 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' □ Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' In some of our examples, A will be finitely generated, in which case we take Ak = A for all k in condition (∗), so that the requirement σs(Ak) ⊆ Ak in (∗) is automatic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Suppose that both σ: S ↷ A and τ : T ↷ B are exact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Then the corresponding groupoids are effective and minimal by [13, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='14] (see also [33, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='23]) and [13, Corol- lary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Hence the following are equivalent: (i) (A ⋊ S ) ⋉ A and (B ⋊ T ) ⋉ B are isomorphic as topological groupoids;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' (ii) (Aσ, Dσ) and (Aτ, Dτ) are isomorphic as Cartan pairs, where Aσ and Aτ are as in [13, Defi- nition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='1], and Dσ and Dτ are as in [13, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='30];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' (iii) F ((A ⋊ S ) ⋉ A) and F ((B ⋊ T ) ⋉ B) are isomorphic as abstract groups;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' (iv) D((A ⋊ S ) ⋉ A) and D((B ⋊ T ) ⋉ B) are isomorphic as abstract groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' This follows from [47] (see also [45]) and [48, Theorems 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='3] (or [41, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='10]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Algebraic actions on finite rank torsion-free Abelian groups In this section, we apply our rigidity results to example classes of algebraic actions on finite rank torsion-free Abelian groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Algebraic actions of torsion-free Abelian monoids whose dual actions are mixing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Let σ: S ↷ A be an algebraic action, with A Abelian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Let ˆσ: S ↷ �A be the dual action as in [13, Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='2] and denote by µ the normalized Haar measure on �A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Recall (see, for instance, [49, § 1] or [53, Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='5]) that ˆσ is (strongly) mixing (with respect to µ) if for all Borel subsets X, Y of �A we have lim s→∞ µ(X ∩ ˆσs(Y )) = µ(X)µ(Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' If S has no non-trivial finite subsemigroups, we have the following relation between the mixing property of ˆσ and condition (F) for σ: Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Assume that S has no non-trivial finite subsemigroups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Then ˆσ is mixing if and only if we have, for all 0 ̸= a ∈ A and 1 ̸= s ∈ S, that σs(a) ̸= a, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', the analogue of condition (F) holds for σ: S ↷ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Indeed, in general (without our assumption on S), ˆσ is mixing if and only if for all infinite sub- semigroups S′ ⊆ S and 0 ̸= a ∈ A, we have that # {σs′(a): s′ ∈ S′} = ∞ (see [49, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='6] and also [4], in particular [4, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' It is now straightforward to see that, if S has no non- trivial finite subsemigroups, the latter statement is equivalent to the condition that for all 0 ̸= a ∈ A, we have # {s ∈ S: σs(a) = a} < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' This condition, in turn, is equivalent to the statement that we have σs(a) ̸= a for all 0 ̸= a ∈ A and 1 ̸= s ∈ S (again assuming that S has no non-trivial finite subsemigroups), as desired, because {s ∈ S: σs(a) = a} is always a subsemigroup of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Note that the condition that S has no non-trivial finite subsemigroups is in particular satisfied if S is torsion-free, in the sense that for all s1, s2 ∈ S and i ∈ Z>0, si 1 = si 2 implies that s1 = s2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' With the help of this observation, let us now present the first example class to which we can apply our general rigidity results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Assume that S and T are non-trivial, Abelian, cancellative and torsion-free monoids, that A and B are torsion-free Abelian groups of finite rank, and that σ: S ↷ A and τ : T ↷ B are non-automorphic faithful algebraic actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Further suppose that the dual actions ˆσ and ˆτ are mixing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Let ˜σ: S−1S ↷ S−1A and ˜τ : T −1T ↷ T −1B be the canonical globalizations as in [13, Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' 13 If there exists an isomorphism of topological groupoids (S−1A ⋊ S−1S) ⋉ A ∼= (T −1B ⋊ T −1T) ⋉ B, then ˜σ: S−1S ↷ S−1A ∼MEF I ˜τ : T −1T ↷ T −1B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' First of all, note that condition (JF) is satisfied because of [13, Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='5] and (FI) holds by [13, Example 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Thus σ and τ satisfy our standing assumptions from § 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Moreover, it is straightforward to check that S−1S and T −1T are torsion-free and that rkZS−1A = rkZA, rkZT −1B = rkZB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Now our statement follows from Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='30 (ii) for the finite rank case because of Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' □ Let us briefly explain the conclusion of our results for the case of (duals of) toral endomorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Let a ∈ Mn(Z) and b ∈ Mm(Z) with | det(a)|, | det(b)| > 1, where n, m ∈ Z>0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' If a and b both have no roots of unity as eigenvalues, then the duals of the N-actions σ: N ↷ Zn and τ : N ↷ Zm given by σk(v) = akv and τk(w) = bkw for k ∈ N, v ∈ Zn, and w ∈ Zm are mixing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Suppose that the corresponding groupoids are isomorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Then, since (SF) holds in this case, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='30 implies that n = m and that the matrices a and b must be conjugate over Q, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', there exists c ∈ GLn(Q) such that a = cbc−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Canonical endomorphisms of torsion-free finite rank Abelian groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Let A ⊆ Qn be a torsion-free Abelian group of rank n ∈ Z>0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' The multiplicative monoid Z× := Z \\ {0} acts on A by multiplication: Each s ∈ Z× gives rise to the endomorphism σs : A → A given by σs(x) = sx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' For any submonoid M ⊆ Z×, the associated algebraic action σM : M ↷ A, where σM := σ|M, is faithful and satisfies (FI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' It is easy to see that σM : M ↷ A is non-automorphic if and only if there exists m ∈ M such that A is not m-divisible, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', mA ⊊ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Since det( ˙σs) = sn, we see that det ◦ ˙σ: Z>0 → Q× is injective, so that σM : M ↷ A satisfies (SF) (see Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='29).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Since M is Abelian, we obtain a globalization ˜σM : ⟨M⟩ ↷ M−1A, where ⟨M⟩ := M−1M ⊆ Q× acts on M−1A := � s∈M 1 sA by multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' It is easy to see that ˜σM : ⟨M⟩ ↷ M−1A satisfies (F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' For s, s′ ∈ Z>0, we have sA ∩ s′A = lcm(s, s′)A (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', [26, § 20]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Hence, the M-constructible subgroups for M ↷ A are given by {sA : s ∈ M}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' From this, we see that ˜σM : ⟨M⟩ ↷ M−1A satisfies (JF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Let x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', xn ∈ A be rationally independent, and put A := spanZ({x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', xn}) ∼= Zn ⊆ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' For each k ∈ Z>0, let Ak := {x ∈ A : (k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' )x ∈ A} = A ∩ (k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' )−1A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Each Ak is an Z×-invariant finitely generated subgroup of A, and we have A = � k Ak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' We can now apply Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='30 to obtain the following result: Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Let A and B be torsion-free finite rank Abelian groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Let M, N ⊆ Z× be submonoids such that there exist m ∈ M and n ∈ N with mA ⊊ A and nB ⊊ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' If there is an isomorphism of topological groupoids (M−1A ⋊ ⟨M⟩) ⋉ A ∼= (N−1B ⋊ ⟨N⟩) ⋉ B, then the actions ⟨M⟩ ↷ M−1A and ⟨N⟩ ↷ N−1B are finitely algebraically equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' If Z>0 ⊆ M, then M ↷ A is exact if and only if the Ulm subgroup of A vanishes (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' [26, § 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='6]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Actions adding scalars to algebraic actions of groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Let Γ ⊆ SLn(Z) be any subgroup, and let M ⊆ Z>0 a submonoid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Then, MΓ := {aγ : a ∈ M, γ ∈ Γ} is a submonoid of Mn(Z)× := {x ∈ Mn(Z) : det(x) ̸= 0}, where we view M as a submonoid of Mn(Z) via the diagonal embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Since Mn(Z)× acts canonically on Zn, we obtain a faithful algebraic action MΓ ↷ Zn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' It is easy to see that MΓ ↷ Zn is exact if and only if M is non-trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Note that ⟨M⟩Γ ↷ (M−1Z)n is a globalization for MΓ ↷ Zn that satisfies (JF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Since Γ acts by automorphisms on Zn that commute with the action of M, we have CMΓ↷Zn = CM↷Zn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Let Z n M denote the completion of Zn with respect to the family CM↷Zn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' The globalization ⟨M⟩Γ ↷ (M−1Z)n often will not satisfy (F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' For instance, take M = Z>0 and Γ = SL2(Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Then id − γ is not injective on M−1Z2 = Q2, where γ = � 1 1 0 1 � ∈ SL2(Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' 14 In order to apply our rigidity result in this setting, we need an observation on subgroups of SLn(Z), which comes from [23, Example 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='8 & Lemma 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Let Γ ⊆ SLn(Z) be a subgroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' If γαγ−1 = ακ for α, γ ∈ Γ and κ ∈ Z>1, then α ∈ tor(Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Suppose γαγ−1 = ακ for α, γ ∈ Γ and κ ∈ Z>1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Let p be a prime divisor of κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' For l ≥ 1, consider the congruence subgroup Γ(pl) := {a ∈ SLn(Z) : a ≡ In mod pl}, and put Γpl := Γ ∩ Γ(pl).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Since Γp is a finite index subgroup of Γ, we can find m ∈ Z>0 such that αm ∈ Γp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' If αm ̸= In, then since � l Γpl = {In}, we can find l ≥ 1 with αm /∈ Γpl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Now αmΓpl and ακmΓpl have the same order in the p-group Γp/Γpl because γαγ−1 = ακ, which is a contradiction since p | κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' □ Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Let Γ, Λ ⊆ SLn(Z) be subgroups and M, N ⊆ Z>0 nontrivial submonoids such that for every γ ∈ tor(Γ), there exists s ∈ M with gcd(ord(γ), s) > 1, and for every λ ∈ tor(Λ), there exists t ∈ N with gcd(ord(λ), t) > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' If there is an isomorphism of topological groupoids (M−1Zn ⋊ ⟨M⟩Γ) ⋉ Z n M ∼= (N−1Zm ⋊ ⟨N⟩Λ) ⋉ Z m N, then M = N and there exist g, h ∈ Mn(N−1Z) ∩ GLn(Q) such that Γ ⊆ gΛg−1 and hΛh−1 ⊆ Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='7, condition (∗s) holds, so this follows from Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' □ Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' The assumptions on Γ, Λ and M, N in the statement Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='8 are satisfied, for instance, if M = N = Z>0 or if Γ and Λ are torsion-free (and M, N ̸= {1}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' If in the statement of Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='8, Γ and Λ are not conjugate via an element in GLn(Q) to any of their proper subgroups, then the conclusion can be strengthened to the following: M = N and there exists g ∈ Mn(N−1Z) ∩ GLn(Q) such that Γ = gΛg−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' This holds, for instance, if Γ and Λ are co-Hopfian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Arithmetic dynamical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Let us consider the algebraic N-actions studied by Chothi, Everest, and Ward in [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Let K be a number field with ring of integers R, and let PK denote the set of non-zero prime ideals of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' For p ∈ PK, let vp and | · |p denote the associated additive and multiplicative p-adic valuations on K, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Given a subset S ⊆ PK, the corresponding ring of S-integers is RS := {x ∈ K : |x|p ≤ 1 for every p ∈ PK \\ S}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' That is, RS consists of the elements of K that are p-adic integers for every p ∈ PK \\ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' For ξ ∈ R× S = RS \\ {0}, the map mξ : RS → RS given by mξ(x) = ξx is an injective endomorphism of the additive group of RS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' The dual action �mξ : N ↷ � RS is called an arithmetic S-integer dynamical system, see [15, § 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' The group of units (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', invertible elements) in RS is R∗ S = {x ∈ K∗ : |x|p = 1 for every p ∈ PK \\ S}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Note that RS is a proper subring of K if and only if S ⊊ PK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Also note that R∗ S ⊊ R× S whenever RS ⊊ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Let us record some basic observations about the algebraic action mξ : N ↷ RS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Let ˜mξ : RS[1/ξ] → RS[1/ξ] be given by ˜mξ(x) = ξx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Then ˜mξ : Z ↷ RS[1/ξ] is a globalization of mξ : N ↷ RS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' (i) mξ : N ↷ RS is faithful if and only if ξ is not a root of unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' (ii) mξ : N ↷ RS is exact if and only if it is non-automorphic if and only if ξ is a non-unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' (iii) ˜mξ : Z ↷ RS[1/ξ] satisfies (JF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' (i) and (iii) are obvious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' For (ii), let ξ ∈ R× S \\ R∗ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Then there exists p ∈ PK \\ S such that |ξ|p < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' If x ∈ RS lies in � n≥0 ξnRS, then for each n ≥ 0, we can write x = ξnyn for some yn ∈ RS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Now we have |x|p = |ξ|n p |yn|p ≤ |ξ|n p for every n, so that x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' The other implications in (ii) are easy to see.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' □ Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' An element x ∈ K is integral over Z if and only if Z[x] is finitely generated as a Z-module, so the additive group of RS is not finitely generated whenever S ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' For each k ∈ Z>0, let Ak := RS ∩ (k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' )−1R = {x ∈ RS : (k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' )x ∈ R}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Then RS = � k Ak, and every Ak is finitely generated and invariant under R×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Let K1 and K2 be number fields with rings of algebraic integers R1 and R2, respec- tively, let S ⊊ PK and T ⊊ PL by proper subsets of primes, and let ξ ∈ R× 1 \\ R∗ 1,S and η ∈ R× 2 \\ R∗ 2,T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' If there is an isomorphism of topological groupoids (R1,S[1/ξ] ⋊ ⟨ξ⟩) ⋉ R1,S ∼= (R2,T [1/η] ⋊ ⟨η⟩) ⋉ R2,T , 15 then ˜mξ ↷ R1,S[1/ξ] and ˜mη ↷ R2,T [1/η] are finitely algebraically equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Condition (∗s) is satisfied, so Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='30 yields the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' □ Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Given any ξ ∈ R× S \\R∗ S, there exists l ∈ Z>0 such that lξ ∈ R, and then the pair (id, ml) is a strict, finite index embedding of mξ : N ↷ RS into mlξ : N ↷ RS;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' in particular, mξ : N ↷ RS and mlξ : N ↷ RS are isomorphic over Z[l−1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Thus, up to inverting an integer, Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='13 applies to all (faithful, exact) actions of the form mξ : N ↷ RS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Algebraic actions from rings The rank of a ring R is defined to be the rank of the additive group of R, that is, the dimension of Q ⊗Z R as a vector space over Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' We shall say that R is torsion-free if the additive group of R is a torsion-free (Abelian) group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Examples of torsion-free rings of finite rank include integral group rings of finite groups and Rn or Mn(R), where R an order in a central simple algebra over an algebraic number field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' General preparations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Let R be a unital torsion-free ring of finite rank n ∈ Z>0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Then Q⊗Z R is an n-dimensional Q-algebra containing R as a full subring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' The sum of two elementary tensors in Q ⊗Z R is again an elementary tensor, so that Q ⊗Z R = QR := {q ⊗ x : q ∈ Q, x ∈ R}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Moreover, if L ⊆ R is a full rank subgroup, then Q ⊗Z L = QL, and QL = QR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Each a ∈ QR gives rise to a Q-linear map ˙σa : QR → QR given by x �→ ax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' We let χa(t) denote the characteristic polynomial of this map, and put N(a) := | det( ˙σa)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' For a ∈ R×, put σa := ˙σa|R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Following the notation from [34], we let R× denote the multiplicative monoid of left regular elements in R, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', R× consists of those a ∈ R such that σa is injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Since R is a torsion-free ring of finite rank, the element a ∈ R is left regular if and only if N(a) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' The action of any submonoid M ⊆ R× on (the additive group of) R by left multiplication is faithful, by injective endomorphisms, and satisfies (FI) (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', [27, Exercise 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' If L is a full rank additive subgroup of R that is invariant under the action of M, then M also acts faithfully on L by injective endomorphisms and the action M ↷ L satisfies (FI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Under the canonical inclusion R ⊆ QR, R× is carried into (QR)∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' If M ⊆ R× is a submonoid, then we let ⟨M⟩ denote the subgroup of (QR)∗ generated by M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Since L is of full rank, we have L ∩ R× ̸= ∅ and thus M ↷ L is faithful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' For i = 1, 2, let Ri be a torsion-free ring of rank n and Mi ⊆ R× i a submonoid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Let L be a rank n subgroup of R1, and assume that spanZ(M1) has finite index in R1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' If there is an injective additive group homomorphism b: QL → QR2 and a group homomorphism t: ⟨M1⟩ → ⟨M2⟩ such that b(ax) = t(a)b(x) for all a ∈ M1 and x ∈ QL, then there exists a unital Q-algebra isomorphism ϕ: QR1 → QR2 such that ϕ|⟨M1⟩ = t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' First, we show that b(1) is invertible in QR2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' For every a ∈ M1, we have b(a) = b(a1) = t(a)b(1), so that b(x) lies in the Q-vector space (QR2)b(1) for all x ∈ spanZ(M1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Since b is injective and rkZ(spanZ(M1)) = n, we have n = rkZ(im(b)) ≤ dimQ((QR2)b(1)) ≤ dimQQR2 = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Hence, dimQ((QR2)˜β(1)) = dimQ(QR2), which implies that b(1) is invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' We now define ϕ: QR1 → QR2 by ϕ(x) := b(x)b(1)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Clearly, ϕ is additive and ϕ(1) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Since b is injective and b−1 is invertible, we see that ϕ is also injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Let a ∈ M1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' We obtain t(a) = b(a)b(1)−1 = ϕ(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Thus, for a ∈ M1 and x ∈ R1, we have ϕ(ax) = b(ax)b(1)−1 = t(a)b(x)b(1)−1 = t(a)ϕ(x) = ϕ(a)ϕ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Set Mult(ϕ) := {a ∈ R1 : ϕ(ax) = ϕ(a)ϕ(x) for all x ∈ R1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' It is straightforward to see that Mult(ϕ) is a subring of R1 containing Z and M1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Since spanZ(M1) is of finite index in R1, it follows that Mult(ϕ) = R1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Hence ϕ is a ring homomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Thus ϕ is a Q-algebra isomorphism QR1 → QR2 satisfying ϕ|⟨M1⟩ = α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' □ Given a torsion-free ring of finite rank R, we let O denote the integral closure of Z in QR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' If R is finitely generated, then R ⊆ O by [46, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' However, O may not be a subring if R is non-commutative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Given a submonoid M ⊆ R×, let � M := ⟨M⟩ ∩ O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Note that � M need not be closed under multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' 16 The following Corollary demonstrates criteria under which we can deduce rigidity results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' For i = 1, 2, suppose Ri is a finitely generated torsion-free ring of finite rank and that Mi ⊆ R× i is submonoid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Assume there exist Q-algebra isomorphisms ϕ1 : QR1 → QR2 and ϕ2 : QR2 → QR1 such that ϕ(⟨M1⟩) ⊆ ⟨M2⟩ and ϕ2(⟨M2⟩) ⊆ ⟨M1⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' If (S’) ψ(⟨M1⟩) ⊆ ⟨M1⟩ =⇒ ψ(⟨M1⟩) = ⟨M1⟩ for every ψ ∈ AutQ-alg(QR1), then ϕ1(⟨M1⟩) = ⟨M2⟩ and ϕ2(⟨M2⟩) = ⟨M1⟩, so that ⟨M1⟩ ↷ QR1 and ⟨M2⟩ ↷ QR2 are isomor- phic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' If (N) Mi = � Mi (for i = 1, 2), and (S) ψ(M1) ⊆ M1 =⇒ ψ(M1) = M1 for every ψ ∈ AutQ-alg(QR1), then ϕ1(M1) = M2 and ϕ2(M2) = M1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' therefore, M1 ↷ spanZ(M1) and M2 ↷ spanZ(M2) are isomorphic, and, if each Oi is closed under addition and Mi-invariant, then M1 ↷ O1 and M2 ↷ O2 are isomorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Let ψ := ϕ2 ◦ ϕ1 ∈ AutQ-alg(QR1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Then ψ(⟨M1⟩) = ϕ2(ϕ1(⟨M1⟩)) ⊆ ϕ2(⟨M2⟩) ⊆ ⟨M1⟩, so that ψ(⟨M1⟩) = ⟨M1⟩ by (S’).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Now we have ⟨M1⟩ = ψ(⟨M1⟩) = ϕ2(ϕ1(⟨M1⟩)) ⊆ ϕ2(⟨M2⟩) ⊆ ⟨M1⟩, so that ϕ2(⟨M2⟩) = ⟨M1⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Now assume that (N) and (S) hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Since Q-algebra homomorphisms preserve integrality, we have ϕ1(O1) = O2 and ϕ2(O2) = O1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' moreover, we have Mi ⊆ Oi for i = 1, 2, so our assumption that ϕ1(M1) ⊆ ⟨M2⟩ and ϕ2(M2) ⊆ ⟨M1⟩ forces ϕ1(M1) ⊆ O2 ∩ ⟨M2⟩ and ϕ2(M2) ⊆ O1 ∩ ⟨M1⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' We have ψ(M1) = ϕ2(ϕ1(M1)) ⊆ ϕ2(O2 ∩ ⟨M2⟩) (N) = ϕ2(M2) ⊆ O2 ∩ ⟨M1⟩ (N) = M1, so that condition (S) forces ψ(M1) = M1, so that ϕ1(M1) = M2 and ϕ2(M2) = M1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' □ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Groupoid rigidity when the acting monoid is Abelian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' In this section, we specialise to the case where the acting monoids are Abelian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' The following is an immediate consequence of Theo- rem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='23 and Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' For i = 1, 2, suppose Ri is a torsion-free finitely generated ring, Mi ⊆ R× i an Abelian submonoid such that spanZ(Mi) has finite index in Ri, and Li ⊆ Ri an Mi-invariant full rank subgroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Assume that there exists a ∈ M1 such that L1 → L1, x �→ (1 − a)x is injective, and similarly for M2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' If there is an isomorphism of topological groupoids (M−1 1 L1⋊⟨M1⟩)⋉L1 ∼= (M−1 2 L2⋊⟨M2⟩)⋉L2, then there exist Q-algebra isomorphisms ϕ1 : QR1 ∼ = → QR2 and ϕ2 : QR2 ∼ = → QR1 such that ϕ1(M1) ⊆ � M2 and ϕ2(M2) ⊆ � M1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' We obtain the following rigidity results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Suppose that, in addition to the assumptions in Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='3, conditions (N) and (S) from Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='2 hold and that Li = spanZ(Mi) = Ri or Li = Oi, then the following statements are equivalent: (i) the algebraic actions M1 ↷ R1 and M2 ↷ R2 are isomorphic;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' (ii) (M−1 1 R1 ⋊ ⟨M1⟩) ⋉ L1 and (M−1 2 R2 ⋊ ⟨M2⟩) ⋉ L2 are isomorphic as topological groupoids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Note that Li = Oi requires that Oi is closed under addition and Mi-invariant, neither of which is automatic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Connection to Bhargava’s work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Torsion-free commutative rings whose additive groups are finitely generated have received a great deal of attention recently [5, 6, 7, 8, 9, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Let Ri, i = 1, 2, be finitely generated torsion-free rings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' If the groupoids (QR1 ⋊ (QR1)∗) ⋉ R1 and (QR2 ⋊ (QR2)∗) ⋉ R2 are isomorphic, then QR1 ∼= QR2 as Q-algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' This follows from Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='3, applied to Mi = R× i and Li = Ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Since Z ⊆ Ri, we just need to show that spanZ(Mi) = Ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Indeed, for every a ∈ Ri, there exists κ ∈ Z× such that a + κ ∈ R× i : As sp( ˙σa) is finite, we have 0 /∈ sp( ˙σa+κ) = sp( ˙σa + κ id) = sp( ˙σa) + κ for sufficiently big κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' □ 17 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Actions of congruence monoids on rings of algebraic integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Let K be a number field with ring of integers R, and let Rm,Γ ⊆ R× = R \\ {0} be a congruence monoid as in [11, § 3], where m = m∞m0 is a modulus for K and Γ is a group of residues modulo m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Let C∗ λ(R ⋊ Rm,Γ) denote the left regular C*-algebra of the monoid R ⋊ Rm,Γ and Dλ(R ⋊ Rm,Γ) the canonical Cartan subalgebra of C∗ λ(R ⋊ Rm,Γ) (see [12, § 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Using the results from [11, § 2], it is not difficult to show that the family of constructible subgroups for the multiplication action Rm,Γ ↷ R is given by CRm,Γ↷R = {(0) ̸= I � R : I is coprime with m}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' In particular, the completion R of R with respect to CRm,Γ↷R depends only on the prime divisors of m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' The ring spanZ(Rm,Γ) is an order in R, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', it is of finite index in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Since Rm,Γ contains (1 + m0)+, it follows that the subring of R generated by Rm,Γ contains (m0)+, the set of totally positive elements in the ideal m0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' If x ∈ m0, choose k ∈ N× ∩ m0 such that x + k is totally positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Since x = (x + k) − k, we see that every element of m0 is a difference of totally positive elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Hence, the ring generated by Rm,Γ contains m0, which implies that it is of finite index in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' □ Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' (i) The monoid Rm,Γ satisfies conditions (N) and (S) from Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' (ii) the action Rm,Γ ↷ R is exact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' We shall use the notation from [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' (i): First, let us show condition (N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' By Proposition [11, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='2], we have ⟨Rm,Γ⟩ = {x ∈ K× : vp(x) = 0 for all p | m0, [x]m ∈ Γ}, from which we see that ⟨Rm,Γ⟩ ∩ R = Rm,Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Now we verify that condition (S) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Let ψ ∈ Gal(K/Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' We have ψ(Rm,Γ) = Rψ(m),ψ(Γ) where ψ(m) is the modulus defined by w | ψ(m)∞ if and only if w ◦ ψ | m∞ and ψ(m)0 := ψ(m0), and ψ(Γ) is the image of Γ under the isomorphism (R/m)∗ ∼= (R/ψ(m))∗ given by [a]m �→ [ψ(a)]ψ(m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Since Rψ(m),ψ(Γ) = ψ(Rm,Γ) ⊆ Rm,Γ, [11, Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='2(1)] implies that ψ(m) | m, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', ψ(m)0 | m0 and ψ(m)∞(w) = 1 =⇒ m∞(w) = 1, and πm,ψ(m)(ψ(Γ)) ⊆ Γ, where πm,ψ(m) : (R/m)∗ → (R/ψ(m))∗ is the canonical quotient map arising from the divisibility condition ψ(m) | m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Since ψ(m0) and m0 have the same norm, ψ(m0) | m0 forces ψ(m0) = m0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' since the finite sets supp(ψ(m)∞) and supp(m∞) have the same cardinality, supp(ψ(m)∞) ⊆ supp(m∞) forces supp(ψ(m)∞) = supp(m∞), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', ψ(m)∞ = m∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Therefore, ψ(m) = m which implies that πm,ψ(m) = id, so that πm,ψ(m)(ψ(Γ)) ⊆ Γ becomes ψ(Γ) ⊆ Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Since #ψ(Γ) = #Γ, we must have ψ(Γ) = Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Thus, we have Rψ(m),ψ(Γ) = Rm,Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' (ii): It is enough to show that Rm,Γ contains a non-unit a since then � n≥0 anR = {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Observe that Rm,Γ contains the set (1 + m0)+ of totally positive elements in 1 + m0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Since (1 + m0)+ contains infinitely many positive integers, we see that Rm,Γ contains non-units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' □ In this setting, we have the following complete rigidity theorem: Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' For i = 1, 2, let Ki be a number field with ring of integers Ri, and suppose (Ri)mi,Γi ⊆ R× i is a congruence monoid as in [11, § 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' The following statements are equivalent: (i) the algebraic actions (R1)m1,Γ1 ↷ R1 and (R2)m2,Γ2 ↷ R2 are isomorphic;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' (ii) ((R1)−1 m1,Γ1R1 ⋊ ⟨(R1)m1,Γ1⟩) ⋉ R1 and ((R2)−1 m2,Γ2R2 ⋊ ⟨(R2)m2,Γ2⟩) ⋉ R2 are isomorphic as topological groupoids;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' (iii) (R2)−1 m2,Γ2R2 ⋊ ⟨(R2)m2,Γ2⟩ ↷ R1 and (R2)−1 m2,Γ2R2 ⋊ ⟨(R2)m2,Γ2⟩× ↷ R2 are continuously orbit equivalent in the sense of [37, 38];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' (iv) (A(R1)m1,Γ1↷R1, D(R1)m1,Γ1↷R1) and (A(R2)m2,Γ2↷R2, D(R2)m2,Γ2↷R2) are isomorphic as Cartan pairs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' (v) (C∗ λ(R1 ⋊ (R1)m1,Γ1), Dλ(R1 ⋊ (R1)m1,Γ1)) and (C∗ λ(R2 ⋊ (R2)m2,Γ2), Dλ(R2 ⋊ (R2)m2,Γ2)) are isomorphic as Cartan pairs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' (vi) F (((R1)−1 m1,Γ1R1 ⋊ ⟨(R1)m1,Γ1⟩) ⋉ R1) and F (((R2)−1 m2,Γ2R2 ⋊ ⟨(R2)m2,Γ2⟩) ⋉ R2) are isomorphic as abstract groups;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' (vii) D(((R1)−1 m1,Γ1R1 ⋊ ⟨(R1)m1,Γ1⟩) ⋉ R1) and D(((R2)−1 m2,Γ2R2 ⋊ ⟨(R2)m2,Γ2⟩) ⋉ R2) are isomorphic as abstract groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' (i)⇔(ii): Let 1 ̸= a ∈ Rmi,Γi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Then multiplication by 1 − a is injective on Ki = Q ⊗Z Ri, and thus also injective on R−1 mi,ΓiRi ⊆ Ki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Thus, by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='7(i) and Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='6, we can apply Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='4 to obtain the desired equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' 18 Equivalence of (ii), (iii), and (iv) follows from [38, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' (v)⇒(iv) follows from the description of the primitive ideals of C∗ λ(Ri ⋊(Ri)mi,Γi) in [11, Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='1] combined with the observation that A(Ri)mi,Γi↷Ri is the unique simple quotient of C∗ λ(Ri ⋊ (Ri)mi,Γi) and the quotient map C∗ λ(Ri ⋊(Ri)mi,Γi) → A(Ri)mi,Γi↷Ri carries Dλ(Ri ⋊(Ri)mi,Γi) onto D(Ri)mi,Γi↷Ri (compare [11, § 8]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Clearly, (i)⇒(iv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Equivalence of (vi), (vii), and (ii) follows from Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='7(ii) and Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' □ Specalizing to the case where the moduli are trivial, and observing that the algebraic actions R× 1 ↷ R1 and R× 2 ↷ R2 are isomorphic if and only if K1 ∼= K2, we obtain: Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' For i = 1, 2, let Ki be a number field with rings of integers Ri, denote the corre- sponding ring C*-algebras by A[Ri], and their canonical Cartan subalgebras by D[Ri].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' The following are equivalent: (i) K1 and K2 are isomorphic;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' (ii) (K1 ⋊ K× 1 ) ⋉ R1 and (K2 ⋊ K× 2 ) ⋉ R2 are isomorphic as topological groupoids;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' (iii) K1 ⋊ K× 1 ↷ R1 and K2 ⋊ K× 2 ↷ R2 are continuously orbit equivalent in the sense of [37, 38];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' (iv) (A[R1], D[R1]) and (A[R2], D[R2]) are isomorphic as Cartan pairs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' (v) (C∗ λ(R1 ⋊ R× 1 ), Dλ(R1 ⋊ R× 1 )) and (C∗ λ(R2 ⋊ R× 2 ), Dλ(R2 ⋊ R× 2 )) are isomorphic as Cartan pairs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' (vi) F ((K1 ⋊ K× 1 ) ⋉ R1) and F ((K2 ⋊ K× 2 ) ⋉ R2) are isomorphic as abstract groups;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' (vii) D((K1 ⋊ K× 1 ) ⋉ R1) and D((K2 ⋊ K× 2 ) ⋉ R2) are isomorphic as abstract groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' The equivalence of (i) and (v) in Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='8 completely answers the natural problem left open in [12, § 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='2]: The Cartan pair (C∗ λ(R ⋊ Rm,Γ), Dλ(R ⋊ Rm,Γ)) remembers the isomorphism class of the semigroup R ⋊ Rm,Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' The equivalences of (i), (iii), and (v) in Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='9, completely answers the natural question left open in [36, § 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Algebraic actions from commutative algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' In this subsection, we analyze a class of algebraic Nd-actions that are irreversible analogues of the algebraic Zd-actions studied in [49, Chapter II].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' We need the following observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' If R is a commutative finitely generated torsion-free ring of rank n, then integral closure O of Z in QR is then a ring by [46, Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Since R ⊆ O, for each element a ∈ R, the map ˙σa : QR → QR, ˙σa(x) = ax, leaves O invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Thus, N(a) = | det( ˙σa)| lies in Z>0 for every a ∈ R×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Let R be a commutative finitely generated torsion-free ring of rank n and a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', ak ∈ R× \\ R∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' In addition, assume that for every 1 ≤ i ≤ k, there exists a prime p such that p | N(ai) and p ∤ N(aj) for j ̸= i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' (i) If ψ ∈ AutQ-alg(QR) is such that ψ(⟨a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', ak⟩+) ⊆ ⟨a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', ak⟩+, then ψ = id.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' (ii) ⟨a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', ak⟩ ∩ O = ⟨a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', ak⟩+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' (iii) a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', ak are multiplicatively independent, so that the canonical map Nk ↠ ⟨a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', ak⟩+ is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' (i): Suppose ψ ∈ AutQ-alg(QR) is such that ψ(⟨a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', ak⟩+) ⊆ ⟨a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', ak⟩+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Then for each 1 ≤ i ≤ k, there exist n1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', nk ∈ N such that ψ(ai) = an1 1 · · · ank k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Now, for a ∈ R×, we have ψ ◦ σa = σψ(a)◦ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' In particular, det(σψ(a)) = det(σa).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Thus, we have N(ai) = N(ψ(ai)) = N(a1)n1 · · · N(ak)nk, which, together with our assumption on N(ai), shows that ψ(ai) = ai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' (ii): Let x ∈ ⟨a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', ak⟩ ∩ O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Then there exists n1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', nk ∈ Z with x = an1 1 · · · ank k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' We need to show that each ni is non-negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' By assumption, for each 1 ≤ i ≤ k, there exists a rational prime p dividing N(ai) such that p ∤ N(aj) for j ̸= i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Thus, 0 ≤ vp(N(x)) = �k j=1 njvp(N(aj)) = nivp(N(ai)), which shows ni ≥ 0 (here, the first inequality uses that x lies in O).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' The containment “⊇” is obvious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' (iii): Suppose an1 1 · · · ank k = 1 for some n1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', nk ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Fix 1 ≤ i ≤ k, and choose a rational prime p such that vp(N(ai)) > 0 and vp(N(aj)) = 0 for j ̸= i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Now 0 = vp(N(a1)n1 · · · N(ak)nk) = nivp(N(ai)), so that ni = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' □ 19 Let d ∈ Z>0 and denote by R+ d := Z[u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', ud] the ring of polynomials with integer coefficients in the d variables u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', ud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Let I �R+ d be a non-zero ideal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' By the Hilbert Basis Theorem (see, for instance, [24, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='2]), R+ d is Noetherian, so that there exists f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', fm ∈ R+ d such that I is generated by {f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', fm}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Since we are only interested in the quotient ring R+ d /I, let us assume that ui /∈ I for all 1 ≤ i ≤ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Let V (I) := {z ∈ Cd : f(z) = 0 for every f ∈ I} ⊆ Cd be the complex variety defined by I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' It follows from [18, Chapter 5, Theorem 6] that dimQQ ⊗Z R+ d /I < ∞ if and only if V (I) is a finite set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' If #V (I) < ∞, then C ⊗Z I is said to be zero- dimensional, in which case there exists a basis for C ⊗Z R+ d /I consisting of (cosets of) monomials (see [18, Chapter 5, Proposition 4]), so that R+ d /I is finitely generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' For the remainder of this section, we shall assume #V (I) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' For f ∈ R+ d , let σf denote the endomorphism of R given by left multiplication with the coset f + I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Let χf(t) denote the characteristic polynomial of σf viewed as an endomorphism of C ⊗Z R+ d .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Let us record the following properties of these endomorphisms: Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' For f ∈ R+ d , we have (i) χf(t) = � z∈V (I)(t−f(z))µ(z), where µ(z) := dimCOz/(C⊗ZI)Oz, where Oz is the localisation of C ⊗Z R at the maximal ideal mz := {g ∈ C ⊗Z R : g(z) = 0};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' (ii) σf is injective if and only if f(z) ̸= 0 for every z ∈ V (I);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' (iii) id − σf = σ1−f is injective if and only if f(z) ̸= 1 for every z ∈ V (I);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' (iv) if for all F ⊆ V (I), � z∈F f(z) ̸= ±1, then χf is not divisible by any unimodular polynomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' (i) follows from [17, Chapter 4, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='7], and the other parts are consequence of this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' □ If N ∈ Z>0, then it follows from part (ii) of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='12 that the endomorphism σN is injective, so we see that R+ d /I is torsion-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' We let ˙ui denote the image of ui modulo I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Note that ˙ui ̸= 0 by our assumption that ui /∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' If zi ̸= 0 for all z ∈ V (I), then σui is an injective endomorphism of R+ d /I by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='12, and we obtain an algebraic action ⟨ ˙u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', ˙ud⟩+ ↷ R+ d /I, which satisfies (FI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Let Rd := Z[u±1 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', u±1 d ] be the ring of Laurent polynomials in the variables u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', ud;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' then R+ d /I embeds in Rd/IRd, and the multiplicative group ⟨ ˙u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', ˙ud⟩ acts on Rd/IRd by multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' It is easy to see that ⟨ ˙u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', ˙ud⟩ ↷ Rd/IRd is a globalization of ⟨ ˙u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', ˙ud⟩+ ↷ R+ d /I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Let ΩI denote the completion of R+ d /I with respect to the family of ⟨ ˙u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', ˙ud⟩+-constructible subgroups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' For i = 1, 2, let di ∈ Z>0 and let Ii be a non-zero ideal of Z[u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', udi].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Assume that for i = 1, 2, (a) #V (Ii) < ∞ and uk /∈ Ii for all k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', di;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' (b) zk ̸= 0 for every z ∈ V (Ii) and k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', di;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' (c) there exists a monomial f in u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' , ud such that f(z) ̸= 1 for all z ∈ V (Ii);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' (d) for each 1 ≤ j ≤ di, there exists a rational prime p with p | N( ˙uj) and p ∤ N( ˙uk) for k ̸= j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Then the following statements are equivalent: (i) the algebraic actions Nd1 ↷ R+ d1/I1 and Nd2 ↷ R+ d2/I2 are isomorphic;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' (ii) (Rd1/I1Rd1 ⋊ Zd1) ⋉ ΩI1 and (Rd2/I2Rd2 ⋊ Zd2) ⋉ ΩI2 are isomorphic as topological groupoids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' First, note that (d) implies |N( ˙uj)| > 1, so that ˙uj is a non-unit for all 1 ≤ j ≤ di.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Conditions (a) and (b) ensure that R+ di/Ii is a finitely generated torsion-free ring and that the action Ndi ↷ R+ di/Ii is by injective group endomorphisms whose images all have finite index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' We are in the situation of Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='4, so we only need to show that (N) and (S) are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' That (N) and (S) are satisfied follows from parts (ii) and (i) of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='11, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' □ Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' In the situation of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='13, Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='11(iii) implies that Ndi ∼= ⟨ ˙u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', ˙udi⟩+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Every finitely generated commutative ring of finite rank is isomorphic to a ring of the form Z[u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', ud]/I, where I is zero-dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' However, the isomorphism will typically not be canonical, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', if R is the ring of algebraic integers in a number field K, then any choice of Z-basis {x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', xd} for R gives rise to a surjective homomorphism Z[u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', ud] → R, whose kernel must be a zero-dimensional ideal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' 20 Let us explain two concrete example classes that are covered by Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='16 (Principal algebraic N-actions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' A proper ideal I � Z[u] satisfies #V (I) < ∞ if and only if I = Z[u]f for a non-constant monic polynomial f ∈ Z[u] (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', [24, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='a]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' The action σu : N ↷ Z[u]/Z[u]f is called a principal algebraic N-action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' When f is non-constant and monic, the cosets of 1, u, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', un−1 form a Z-basis for Z[u]/Z[u]f, where n = deg(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' The matrix for σu with respect to this basis is equal to the companion matrix Cf of f, so σu is injective if and only if f(0) = ± det(Cf) is non-zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' All in all, since V (Z[u]f) is the set of zeros of f, we see that σu : N ↷ Z[u]/Z[u]f satisfies conditions (a)-(d) in Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='13 if and if f is non-contant, monic, and |f(0)| > 1, and f(1) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' It follows from [32, Theorem] that σu : N ↷ Z[u]/Z[u]f is exact if and only if no unimodular polynomial divides f (in Q[u]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='18 (Algebraic Nd-actions defined by a point).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' A special class of Nd-actions arises from d-tuples of algebraic integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' These are the irreversible analogues of the algebraic Zd-actions from [49, § 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Suppose that c = (c1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', cd) ∈ (Z ×)d, where Z denotes the ring of all algebraic integers, and let pc denote the kernel of the evaluation at c map R+ d → Z[c1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', cd] ⊆ Q(c1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', cd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Then pc is a prime ideal of R+ d , and we can characterize when the action Nd ↷ R+ d /pc satisfies conditions (a)-(d) in Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='13 in terms of c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' First, identify V (pc) with the set Hom (Q(c1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', cd), Q) of field embeddings of Q(c1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', cd) into Q, the algebraic closure of Q in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Explicitly, this identification is given by sending z = (z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', zd) ∈ V (pc) to the embedding Q(c1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', cd) �→ Q determined by ci �→ zi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' From this, it is easy to see that conditions (a) and (b) from Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='13 are satisfied if and only if ci ̸= 0 for all i, and condition (c) from Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='13 is satisfied if and only if there exists a finite non- empty set F ⊆ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', d} such that � i∈F ci ̸= 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' to see this, note that for any z = (z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', zd) ∈ V (pc), we have � i∈F zi ̸= 1 if and only if � i∈F ci ̸= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Condition (d) from Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='13 is satisfied if and only if for each 1 ≤ j ≤ di, there exists a rational prime p with p | N(cj) and p ∤ N(ck) for k ̸= j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Algebraic actions from rings: The non-commutative case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' For i = 1, 2, let Ri be a ring whose additive group is finitely generated and torsion-free, let Li ⊆ Ri be full rank subgroup, and let Mi ⊆ R× i a submonoid such that Li is Mi-invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Then, Mi ↷ Li is faithful since Li has finite index in Ri and Ri is torsion-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Let Li denote the completion of Li with respect to the family Ci of Mi-constructible subgroups of Li.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Our goal now is to establish the following rigidity result: Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Continue with the notation and assumptions above, with the additional assumptions that, for i = 1, 2, spanZ(Mi) has finite index in Ri, that there exists κi ∈ Mi for some κi ∈ Z \\ {0, 1}, and that QRi is a semisimple Q-algebra, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', the (Jacobson) radical of QRi is trivial (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', [31, Part II, § 1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' If there is an isomorphism of topological groupoids (QR1 ⋊ ⟨M1⟩) ⋉ L1 ∼= (QR2 ⋊ ⟨M2⟩) ⋉ L2, then there are Q-algebra isomorphisms ϕ1 : QR1 ∼ = → QR2 and ϕ2 : QR2 ∼ = → QR1 such that ϕ1(⟨M1⟩) ⊆ ⟨M2⟩ and ϕ2(⟨M2⟩) ⊆ ⟨M1⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Taking Mi = R× i and Li = Ri in Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='19 yields the following: Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' If QR1 and QR2 are semisimple Q-algebras and there is an isomorphism of topo- logical groupoids (QR1 ⋊ (QR1)∗) ⋉ R1 ∼= (QR2 ⋊ (QR2)∗) ⋉ R2, then there is a Q-algebra isomorphism QR1 ∼= QR2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Observe that spanZ(R× i ) = Ri as shown in Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' The claim now follows from Theo- rem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' □ Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' The actions R× i ↷ Ri in Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='20 are exact, so Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='32 applies here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Before proceeding to the proof, let us explain several example classes to which our results apply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='22 (Matrices over orders in number fields).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Let n ∈ Z>0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Let R be an order in a number field K, and let I�R be a nonzero ideal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Then, Mn(I) ⊆ Mn(R) is invariant under the canonical action Mn(R)× ↷ Mn(R), so we get an algebraic action Mn(R)× ↷ Mn(I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' We have QMn(I) = Mn(K), so Mn(I) has full rank in Mn(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' We have Z1n ⊆ Mn(R)×, and spanZ(Mn(R)×) has finite index 21 in Mn(R) by the proof of Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Thus, the hypotheses of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='19 are satisfied (for L = Mn(I) and M = Mn(R)×).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Moreover, ⟨Mn(R)×⟩ = GLn(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Therefore, if K1 and K2 are number fields with rings of algebraic integers R1 and R2, respectively, I1 � R1, I2 � R2 are non-zero ideals, n1, n2 ∈ Z>0, and if (Mn1(K1) ⋊ GLn1(K1)) ⋉ Mn(I1) ∼= (Mn2(K2) ⋊ GLn2(K2)) ⋉ Mn(I2) as topological groupoids, then Mn1(K1) ∼= Mn2(K2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' In particular, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='19 implies that the groupoids (Mn1(K) ⋊ GLn1(K)) ⋉ Mn1(R1) and (Mn2(K2) ⋊ GLn2(K2)) ⋉ Mn2(R2) are isomorphic if and only if n1 = n2 and K1 ∼= K2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='23 (Group rings of finite groups).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Let F1 and F2 be finite groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' By Maschke’s Theorem (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', [31, Theorem 25]), QFi is a semisimple Q-algebra (i = 1, 2), so Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='20 implies the following: If there is an isomorphism (QF1 ⋊ (QF1)∗) ⋉ ZF1 ∼= (QF2 ⋊ (QF2)∗) ⋉ ZF2, then QF1 ∼= QF2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Note that if F1, F2 are Abelian, then QF1 ∼= QF2 if and only if F1 ∼= F2 by [44, Corollary 1 & Theorem 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' It is a non-trivial result that there exist finite non-Abelian groups F1, F2 with ZF1 ∼= ZF2 and F1 ̸∼= F2 (see [29]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='24 (Central simple algebras over number fields).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Let A be a central simple algebra over the number fields K, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', A is a finite-dimensional simple K-algebra whose centre is precisely K, and let O be an order in A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' By the Wedderburn Structure Theorem, there exists a (central) division algebra D over K and µ ∈ Z>0 such that A ∼= Mµ(D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Thus, the algebraic action O× ↷ O fits into the setting of Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Thus if (A1 ⋊ A∗ 1) ⋉ O1 ∼= (A2 ⋊ A∗ 2) ⋉ O2 as topological groupoids, then A1 ∼= A2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Now our goal is to prove Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' For the remainder of this section, we shall work with the assumptions and notation from Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' We need some preparations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Let R be ring whose additive group is torsion-free and of rank n, and let M ⊆ R× be a submonoid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Suppose that α ∈ ⟨M⟩ satisfies α = γακγ−1 for some γ ∈ (QR)∗ and κ ∈ Z>1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Set m := κ(dimQQR)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Then there exists a nilpotent element ηα ∈ QR such that αm = 1 + ηα (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', αm is a unipotent element of the Q-algebra QR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' The map π: QR → End C(CR) ∼= Mn(C) given by π(q ⊗ a)(z ⊗ b) = qz ⊗ ab is an injective Q-algebra homomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' The equation α = γακγ−1 implies that π(α) = π(γ)π(α)κπ(γ)−1 in Mn(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' It follows that sp(π(α)) = sp(π(α)κ) = sp(π(α))κ := {λκ : λ ∈ sp(π(α))}, where sp(π(α)) is the spectrum of π(α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Thus, the map sp(π(α)) → sp(π(α)) given by λ �→ λk is bijective;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' write sp(π(α)) = {λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', λj}, and let ρ be the permutation of sp(π(α)) determined by λi = λκ ρ(i) for all 1 ≤ i ≤ j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Since j ≤ dimQQR, we have ρ(dimQQR)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' = id.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Thus, λi = λκ(dimQQR!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=') ρ(dimQQR)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' (i) = λκ(dimQQR)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' i , so that λκ(dimQQR)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='−1 i = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' We now see that 1 is the only eigenvalue of π(α)m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' By considering the Jordan Normal Form of π(α)m, it follows that there exists a nilpotent matrix Nα ∈ Mn(C) such that π(α)m = 1 + Nα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Since Nα = π(α)m − 1 = π(αm − 1), we see, by injectivity of π, that ηα := αm − 1 is nilpotent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' □ Given a division algebra D, we shall regard Dn as an Mn(D)-D-bimodule in the usual way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Thus, a basis for Dn will always mean a right D-basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' We shall use a subscript D on the right to indicate that we are viewing something as a right D-vector space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Let D be a finite dimensional division algebra over Q and n ∈ Z>0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Suppose that Σ is a non-trivial finitely generated Abelian subgroup of GLn(D) consisting of unipotent matrices, so that every α ∈ Σ is of the form α = 1 + ηα, where ηα ∈ Mn(D) is a nilpotent matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Let k := � α∈Σ ker ηα and k := dimkD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' We have rkZΣ < n · (n − k)[D : Q].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Let N := spanQ{ηα : α ∈ Σ} ⊆ Mn(D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' For α, α′ ∈ Σ, we have 1 + ηαα′ = αα′ = (1 + ηα)(1 + ηα′) = 1 + ηα + ηα′ + ηαηα′, so that ηαηα′ = ηαα′ − ηα − ηα′ lies in N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' From this, we see that N is a non-unital commutative sub-Q-algebra of Mn(D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' For α ∈ Σ, let log α := �∞ i=1(−1)i−1 (1−α)i i ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' this is a finite sum because 1 − α is nilpotent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Note that log α lies in N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Since Σ is Abelian, log defines an injective group homomorphism (with inverse given 22 by exp, see for instance [28, § 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='10, Exercise 8]) from Σ into N, so that rkZΣ ≤ dimQN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Thus, we will be done once we show that dimQN < n · (n − k)[D : Q].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Since N is closed under multiplication and consists of nilpotent elements, [31, Part II, § 5, Theorem 35] asserts that there exists a right D-basis w1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', wn for Dn such that N is strictly upper triangular with respect to this basis;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' this means that if we define Wi := span{w1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', wi}D for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='., n, then ηw1 = 0 and for each i ≥ 2, ηWi ⊆ Wi−1 for all η ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Let W ⊆ {w1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', wn} be a subset such that W together with a basis for k is a basis for Dn, and put W = span(W)D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Since η|k ≡ 0 for all η ∈ N, the map N → Hom (W, Dn)D, η �→ η|W is injective;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' moreover, since wn ̸∈ NDn, we see that it is not surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Hence, dimQN < dimQHom (W, Dn)D = dimQMn×(n−k)(D) = n · (n − k)[D : Q].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' □ Proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Let c be the cocycle defined by the composition (QR1 ⋊ ⟨M1⟩) ⋉ L1 ∼= (QR2 ⋊ ⟨M2⟩) ⋉ L2 (h,y)�→h −→ QR2 ⋊ ⟨M2⟩, where the second map is the canonical cocycle obtained by projecting onto the group component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='4 produces a (finite index) subgroup C ∈ C1 such that g(x) := c(x, 0) defines an injective group homomorphism from C into QR2⋊⟨M2⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Let T ⊆ ⟨M2⟩ be the image of C under the composition (4) C g→ QR2 ⋊ ⟨M2⟩ (b,t)�→t → ⟨M2⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' By assumption, there exists κ ∈ M1 with κ ∈ Z \\ {0, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Let m := κ(dimQQR2)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Then we claim that for every α ∈ Tm, there exists a nilpotent element ηα ∈ QR2 such that α = 1 + ηα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Indeed, as κ ∈ M2, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='5 implies that there exists γ ∈ ⟨M2⟩ such that α = γακγ−1 for all α ∈ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' The result now follows from Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' In particular, Tm is torsion-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' The subgroup mC ⊆ C is mapped onto Tm under the projection in (4), so Tm is moreover finitely generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' We will show that Tm is trivial, which will imply that there exists an injective group homomorphism b: mC → QR2 such that g(x) = (b(x), 1) for all x ∈ mC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Since Tm is free abelian, there exists a subgroup CT ⊆ mC such that mC = C′ ⊕ CT, where C′ = {x ∈ mC : g(x) = (y, 1) for some y ∈ QR2}, and CT is mapped isomorphically onto Tm under the map in (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Let B be the be the image of C′ under the composition mC g→ QR2 ⋊ ⟨M2⟩ (b,t)�→b → QR2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Note that B is a subgroup of QR2 because the second map is a homomorphism on QR2 ⋊ {1}, which contains the image of C′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Let us now show that the group Tm is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Since QR2 is a semisimple Q-algebra, the Artin– Wedderburn theorem implies that there exists a decomposition of Q-algebras QR2 = �r i=1 Mni(Di), where r, n1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', nr ∈ Z>0 and each Di is a finite-dimensional division algebra over Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Thus, ⟨M2⟩ ⊆ �r i=1 GLni(Di).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' For each i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', r, let Bi be the image of B under the canonical projection QR2 → Mni(Di).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' For each i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', r, let Ti be the image of Tm under the canonical projection ⟨M2⟩ → GLni(Di).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Then Ti = {1 + ηi α : α ∈ Tm}, where ηi α denotes the i-th coordinate of ηα (which come from Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='25);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' in particular, the group Ti consists of unipotent elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Let ki := � α∈Tm ker (ηi α) and ki := dim(ki)D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Take x′ ∈ C′ and write g(x′) = (β′, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' If x ∈ CT, we can write g(x) = (β, α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Then g(x′)g(x) = (β′ + β, α), whereas g(x)g(x′) = (β + αβ′, α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Thus, αβ′ = β′, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', ηαβ′ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' It follows that ηi αβ = 0 for all α ∈ Tm and β ∈ Bi, so that im(β) ⊆ ki for all β ∈ Bi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' From this, we see that im(β) ⊆ ki for all β ∈ span(Bi)Di := {� j βjdj : βj ∈ Bi, dj ∈ Di}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Hence, dim(span(Bi)Di)Di ≤ ni · ki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Now we have rkZBi = dimQ(Q ⊗ Bi) ≤ dimQ(span(Bi)Di) ≤ ni · ki · [Di : Q].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Assume for the sake of contradiction that Tm is non-trivial, so that Ti is non-trivial for some i (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', ki < ni).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' By Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='26, we have rkZTi < ni · (ni − ki)[Di : Q], where ki := dim(ki)Di.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Since rkZTm ≤ �r i=1 rkZTi and rkZB ≤ �r i=1 rkZBi, we obtain dimQQR1 = rkZC = rkZC′ + rkZCT = rkZB + rkZTm ≤ r � i=1 rkZTi + r � i=1 rkZBi < r � i=1 n2 i · [Di : Q] = dimQR2, 23 where the strict inequality uses our assumption that ki < ni for some i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' By symmetry, we also get dimQQR2 < dimQQR1, which is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Thus, ki = ni for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Hence Tm is indeed trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' We conclude that there exists a group homomorphism b: mC → QR2 such that g(x) = (b(x), 1) for every x ∈ mC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Let t be the composition ⟨M1⟩ g→ QR2 ⋊ ⟨M2⟩ ↠ ⟨M2⟩, where the second arrow is the canonical projection map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' It is easy to check that t is a group homomorphism, and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='18 shows that t is injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Now Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='20 shows that for all s ∈ M1 and x ∈ mC, we have b((σ1)s(x)) = (˜σ2)t(s)(b(x)), where σ1 is the algebraic action M1 ↷ L1 and ˜σ2 is the algebraic action ⟨M2⟩ ↷ QR2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Hence the same proof as for Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='23 (i) produces an injective homomorphism ˙b: QR1 �→ QR2 which is equivariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Applying Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='1 yields the desired result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' □ References [1] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Baudisch, Subgroups of semifree groups, Acta Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Acad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Hungar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' 38 (1981), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' 1-4, 19–28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' [2] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Baumslag, Some aspects of groups with unique roots, Acta Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' 104 (1960), 217–303.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' [3] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Baumslag, Generalized free products whose two-generator subgroups are free, J.' metadata={'source': 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I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' A new view on Gauss composition, and quadratic generalizations, Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' of Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' (2) 159 (2004), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' 1, 217–250.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' [6] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Bhargava, Higher composition laws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' On cubic analogues of Gauss composition, Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' of Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' (2) 159 (2004), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' 2, 865–886.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' [7] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} 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Statistics, University of Glasgow, University Place, Glasgow G12 8QQ, United Kingdom Email address: Xin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='Li@glasgow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='uk 25' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} diff --git a/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf b/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..e4853e1f2e3b3c22e4c5e29df8d60b0cac0a609b --- /dev/null +++ b/I9AzT4oBgHgl3EQfx_6k/content/2301.01747v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid 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a/IdE4T4oBgHgl3EQfIQyU/content/tmp_files/2301.04911v1.pdf.txt b/IdE4T4oBgHgl3EQfIQyU/content/tmp_files/2301.04911v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..7a8912fd0eec9b09c8aab38bbd648edfabe6213f --- /dev/null +++ b/IdE4T4oBgHgl3EQfIQyU/content/tmp_files/2301.04911v1.pdf.txt @@ -0,0 +1,2480 @@ +arXiv:2301.04911v1 [math.AP] 12 Jan 2023 +Multi-bubble nodal solutions to slightly subcritical +elliptic problems with Hardy terms in symmetric +domains +Thomas Bartsch∗, Qianqiao Guo† +Abstract We consider the slightly subcritical elliptic problem with Hardy term + + + + + +−∆u − µ u +|x|2 = |u|2∗−2−εu +in Ω ⊂ RN, +u = 0 +on ∂Ω, +where 0 ∈ Ω and Ω is invariant under the subgroup SO(2) × {±EN−2} ⊂ O(N); here En denots the n × n +identity matrix. If µ = µ0εα with µ0 > 0 fixed and α > N−4 +N−2 the existence of nodal solutions that blow up, +as ε → 0+, positively at the origin and negatively at a different point in a general bounded domain has been +proved in [5]. Solutions with more than two blow-up points have not been found so far. In the present paper +we obtain the existence of nodal solutions with a positive blow-up point at the origin and k = 2 or k = 3 +negative blow-up points placed symmetrically in Ω ∩ (R2 × {0}) around the origin provided a certain function +fk : R+ ×R+ ×I → R has stable critical points; here I = {t > 0 : (t, 0, . . . , 0) ∈ Ω}. If Ω = B(0, 1) ⊂ RN is the +unit ball centered at the origin we obtain two solutions for k = 2 and N ≥ 7, or k = 3 and N large. The result +is optimal in the sense that for Ω = B(0, 1) there cannot exist solutions with a positive blow-up point at the +origin and four negative blow-up points placed on the vertices of a square centered at the origin. Surprisingly +there do exist solutions on Ω = B(0, 1) with a positive blow-up point at the origin and four blow-up points +on the vertices of a square with alternating positive and negative signs. The results of our paper show that +the structure of the set of blow-up solutions of the above problem offers fascinating features and is not well +understood. +2010 Mathematics Subject Classification: 35B44, 35B33, 35J60. +Key words: Hardy term; Critical exponent; Slightly subcritical problems; Nodal solutions; Multi-bubble +solutions. +∗Mathematisches Institut, Justus-Liebig-Universit¨at Giessen, Arndtstr. 2, 35392 Giessen, Germany +†School of Mathematics and Statistics, Northwestern Polytechnical University, 710129 Xi’an, China +1 + +2 +T. Bartsch, Q. Guo +1 +Introduction +The paper is concerned with the semilinear singular problem +(1.1) + + + + + +−∆u − µ u +|x|2 = |u|2∗−2−εu +in Ω, +u = 0 +on ∂Ω, +where Ω ⊂ RN, N ≥ 7, is a smooth bounded domain with 0 ∈ Ω; 2∗ := +2N +N−2 is the critical Sobolev exponent. +In [5], we obtained the existence of two-bubble nodal solutions to problem (1.1) that blow up positively at the +origin and negatively at a different point in a general bounded domain, as ε → 0+ and µ = µ0εα with µ0 > 0 +and α > N−4 +N−2. The location of the negative blow-up point is determined by the geometry of the domain. +The existence of nodal bubble tower solutions has been proved in [6]. These are superpositions of positive +and negative bubbles with different scalings, all blowing up at the origin. It seems to be a difficult and open +problem whether solutions with a blow-up point at the origin and more than one blow-up point outside the +origin exist in a general domain. In the present paper we investigate this problem in symmetric domains, in +particular for the model case of the ball Ω = B(0, 1). +In the case of a ball it is natural to place one blow-up point, say positive, at the origin and k blow-up +points, say negative, at the vertices of a regular k-gon with center at the origin. We shall prove that solutions +of this shape exist for 2 ≤ k ≤ 3 but, somewhat surprisingly, not for k = 4. On the other hand, we prove the +existence of solutions with four blow-up points, two positive and two negative ones, at the vertices of a square, +centered symmetrically around the positive blow-up point at the origin. Our results show that the existence +of solutions of (1.1) with three or more blow-up points is interesting and far from being understood. +When µ = 0 the blow-up phenomenon for positive and for nodal solutions to problem (1.1) has been +studied extensively, see for instance [2–4, 7, 9, 14, 19, 22, 24, 26, 28–31] and the references therein. +However +for µ ̸= 0, few results are known about the existence of positive or nodal solutions with multiple bubbles to +problem (1.1). Positive solutions have been obtained in [12]. Related results, though for different equations, +can be found in [16,17,27]. We also want to mention the papers [10,11,15,18,21,23,25,32,33,35] dealing with +the critical exponent, i.e. ε = 0. +An important role will be played by the limit problem +(1.2) + + + + + +−∆u − µ u +|x|2 = |u|2∗−2u +in RN, +u → 0 +as |x| → ∞ +which has been investigated in [13,35]. Positive solutions of (1.2) in the range 0 ≤ µ < µ := (N−2)2 +4 +are given +by +Vµ,σ(x) = Cµ +� +σ +σ2|x|β1 + |x|β2 +� N−2 +2 + +Nodal solutions to problems with Hardy terms +3 +with σ > 0, β1 := (√µ − √µ − µ)/√µ, β2 := (√µ + √µ − µ)/√µ, and Cµ := +� +4N(µ−µ) +N−2 +� N−2 +4 . These solutions +minimize +Sµ := +min +u∈D1,2(RN )\{0} +� +RN(|∇u|2 − µ u2 +|x|2 )dx +( +� +RN |u|2∗dx)2/2∗ +, +and there holds +� +RN +� +|∇Vµ,σ|2 − µ|Vµ,σ|2 +|x|2 +� +dx = +� +RN |Vµ,σ|2∗dx = S +N +2 +µ . +In the range 0 < µ < µ these are all positive solutions of (1.2). In the case µ = 0 these are all solutions with +maximum at x = 0. Of course, if µ = 0 also translates of Vµ,σ are solutions of +(1.3) + + + +−∆u = |u|2∗−2u +in RN, +u → 0 +as |x| → ∞. +We will write +Uδ,ξ(x) = C0 +� +δ +δ2 + |x − ξ|2 +� N−2 +2 +for the solutions of (1.3) where δ > 0, ξ ∈ RN and C0 := (N(N − 2)) +N−2 +4 . +These are the well known +Aubin-Talenti instantons (see [1,34]). +Now we state our main results. We consider domains satisfying the condition +(A1) Ω ⊂ RN is a bounded domain with 0 ∈ Ω, and it is invariant under the subgroup SO(2) × {±EN−2} ⊂ +O(N). +We use the notation x = (x′, x′′) ∈ Ω ⊂ R2 × RN−2 and write A(x′, x′′) = (Ax′, x′′) for A ∈ SO(2). For k ∈ N +let Rk = + +cos 2π +k +− sin 2π +k +sin 2π +k +cos 2π +k + + ∈ SO(2), and set I = {t > 0 : (t, 0, . . . , 0) ∈ Ω} ⊂ R. +Our first results are concerned with the existence of nodal solutions with k +1 bubbles, one being positive +and k being negative. Let G(x, y) = +1 +|x−y|N−2 − H(x, y) be the Green function (up to a coefficient involving +the volume of the unit ball) for the Dirichlet Laplace operator in Ω with regular part H. For k = 2, 3 we define +the function fk : R+ × R+ × I → R by +fk(λ0, λ1, t) := b1 +� +H(0, 0)λN−2 +0 ++ kH(ξ(t), ξ(t))λN−2 +1 ++ 2kG(ξ(t), 0)λ +N−2 +2 +0 +λ +N−2 +2 +1 +− 2 +�k +2 +� +G(ξ(t), Rkξ(t))λN−2 +1 +� +− b2 +N − 2 +2 +ln +� +λ0λk +1 +� +. +where ξ(t) = (t, 0, . . . , 0) and +b1 = 1 +2C0 +� +RN U 2∗−1 +1,0 +and +b2 = 1 +2∗ +� +RN U 2∗ +1,0. +Finally we call a critical point of fk stable if it is isolated and has nontrivial local degree. This is the case, for +instance, if it is non-degenerate or an isolated local maximum or minimum. + +4 +T. Bartsch, Q. Guo +Theorem 1.1. Suppose Ω satisfies (A1), and suppose (λ0, λ1, t) ∈ R+ × R+ × I is a stable critical point of +fk, k = 2 or k = 3. Let µ0 > 0 and α > N−4 +N−2 be fixed. Then there exists ε0 > 0 such that for every ε ∈ (0, ε0) +problem (1.1) with µ = µε = µ0εα has a pair of solutions ±uε satisfying +(1.4) +uε(x) = Vµε,σε(x) − +k +� +i=1 +Uδε,Ri +kξ(tε)(x) + o(1) += Cµε +� +σε +(σε)2|x|β1 + |x|β2 +� N−2 +2 +− C0 +k +� +i=1 +� +δε +(δε)2 + |x − Ri +kξ(tε)|2 +� N−2 +2 ++ o(1), +where +σε = +� +λ0 + o(1) +� +ε +1 +N−2 , δε = +� +λ1 + o(1) +� +ε +1 +N−2 , ξ(tε) = (tε, 0, . . . , 0) = (t + o(1), 0, . . . , 0) +as ε → 0. +These solutions satisfy the following symmetries: +(1.5) +uε(x′, x′′) = uε(x′, −x′′) = uε(Rkx′, x′′) +for (x′, x′′) ∈ Ω ⊂ R2 × RN−2. +As Proposition 1.4 below shows, for k = 4 a family uε as in Theorem 1.1 need not exist even in the case +of the ball. It is a challenging problem to find critical points of fk for general domains satisfying (A1). We +consider the special case where Ω = B(0, 1) ⊂ RN is the unit ball. +Theorem 1.2. If Ω = B(0, 1) ⊂ RN, k = 2 and N ≥ 7, or k = 3 and N is large enough, then fk has two stable +critical points, one is a local minimum, the other a mountain pass point with Morse index 1. As a consequence, +problem (1.1) has two families of solutions ±u1 +ε, ±u2 +ε as in Theorem 1.1. They have the additional symmetry +uε(x′, x′′) = uε(x′, Ax′′) +for all A ∈ SO(N − 2). +Remark 1.3. a) In the proof of the case k = 3 we provide an explicit inequality for N, so that the solutions +as in Theorem 1.2 exist if this inequality holds. Numerical computations show that this inequality is not +satisfied for N = 7. We do not know the optimal value for N such that Theorem 1.2 is true for k = 3; see also +Remark 4.2. +b) We conjecture that Theorem 1.2 holds for other domains satisying (A1), for instance for Ω = B2(0, 1)× +Ω′ ⊂ R2 × RN−2 with Ω′ = −Ω′ ⊂ RN−2 a bounded symmetric neighborhood of 0. Our proof of Theorem 1.2 +uses the explicit knowledge of the Green function for the ball, hence it does not extend immmediately to other +domains. +The next result shows that Theorem 1.2 is optimal. +Proposition 1.4. For Ω = B(0, 1) ⊂ RN, N ≥ 7 and k = 4 there does not exist a family of solutions ±uε as +in Theorem 1.1. +Remark 1.5. a) We conjecture that Proposition 1.4 can be generalized to k ≥ 4. + +Nodal solutions to problems with Hardy terms +5 +b) We cannot exclude the existence of solutions with a positive bubble at the origin and k = 4 negative +bubbles placed somewhere in the ball and with possibly different blow-up parameters δ. However, we can show +that there do not exist solutions with four negative bubbles at the vertices of a square centered at the origin +even if one allows different blow-up speeds, i.e. if one replaces the δε in (1.4) by δi,ε, i = 1, . . . , 4. In fact, +it is not difficult to prove that the blow-up parameters δi,ε have to be independent of i if the vertices are at +Ri +4ξ(tε), i = 1, . . . , 4. +Considering Proposition 1.4 the following existence results of nodal solutions with five bubbles, three +being positive and two being negative, is somewhat surprising. +Theorem 1.6. Let Ω = B(0, 1) ⊂ RN, N ≥ 7, µ0 > 0 and α > N−4 +N−2 be fixed. Then there exists ε0 > 0 such +that for any ε ∈ (0, ε0), there exist a pair of 5-bubble solutions ±uε to problem (1.1) with µ = µε = µ0εα of +the shape +uε(x) = Vµε,σε(x) + +4 +� +i=1 +(−1)iUδi,ε,Ri +4ξ(tε)(x) + o(1) += Cµε +� +σε +(σε)2|x|β1 + |x|β2 +� N−2 +2 ++ C0 +4 +� +i=1 +(−1)i +� +δi,ε +(δi,ε)2 + |x − Ri +4ξ(tε)|2 +� N−2 +2 ++ o(1), +where σε = +� +λ0 + o(1) +� +ε +1 +N−2 , δ1,ε = δ3,ε = +� +λ1 + o(1) +� +ε +1 +N−2 , δ2,ε = δ4,ε = +� +λ2 + o(1) +� +ε +1 +N−2 , ξ(tε) = +(tε, 0, . . . , 0) = (t + o(1), 0 . . . , 0) as ε → 0, for some λ0, λ1, λ2 > 0, t ∈ (0, 1). +These solutions satisfy +the symmetries: +(1.6) +uε(x′, x′′) = uε(x′, Ax′′) = −uε(R4x′, x′′) +for (x′, x′′) ∈ B(0, 1) ⊂ R2 × RN−2, A ∈ SO(N − 2). +Remark 1.7. a) The parameters (λ0, λ1, λ2, t) ∈ R+ × R+ × R+ × (0, 1) in Theorem 1.6 are obtained as a +critical point of a suitable limit function f5. We conjecture that there exists a second solution in Theorem 1.6 +but the computations for finding a second critical point of f5 are intimidating. +b) It seems that for k = 2 in Theorem 1.2, it is still possible to obtain the information on the nodal sets +of the solutions as in [4]. For k = 3 in Theorem 1.2 and for Theorem 1.6, it is an interesting problem to study +the profile of the nodal sets of the solutions. +c) As stated in [5], the assumption α > N−4 +N−2 is essential in our theorems. +The paper is organized as follows. In Section 2, we collect some notations and preliminary results. Section 3 +is devoted to the method of finite dimensional reduction. Section 4 contains the proof of Theorems 1.1 and +1.2. Proposition 1.4 is proved in Section 5, and finally Theorem 1.6 is proved in Section 6. + +6 +T. Bartsch, Q. Guo +2 +Notations and preliminary results +Throughout this paper, positive constants will be denoted by C, c. By Hardy’s inequality the norm +∥u∥µ := +�� +Ω +(|∇u|2 − µ u2 +|x|2 )dx +� 1 +2 +is equivalent to the norm ∥u∥0 = +�� +Ω |∇u|2dx +�1/2 on H1 +0(Ω) provided 0 ≤ µ < µ. This inequality is of course +satisfied for µ = µ0εα with α > 0 and ε > 0 small. We write Hµ(Ω) for the Hilbert space consisting of the +H1 +0(Ω) functions with the inner product +(u, v)µ := +� +Ω +� +∇u∇v − µ uv +|x|2 +� +dx. +As in [5, 16] let ι∗ +µ : L2N/(N+2)(Ω) → Hµ(Ω) be the adjoint operator of the inclusion ιµ : Hµ(Ω) → +L2N/(N−2)(Ω), that is, +ι∗ +µ(u) = v +⇐⇒ +(v, φ)µ = +� +Ω +u(x)φ(x)dx, +for all φ ∈ Hµ(Ω). +There exists c > 0 such that +∥ι∗ +µ(u)∥µ ≤ c∥u∥2N/(N+2). +Then problem (1.1) is equivalent to the fixed point problem +u = ι∗ +µ(fε(u)), +u ∈ Hµ(Ω), +where fε(s) = |s|2∗−2−εs. +The following proposition is from [5, Proposition 3.1]. +Proposition 2.1. Let 0 < µ < µ, and let Λi, i = 1, 2, . . . , be the eigenvalues of + + + + + +−∆u − µ u +|x|2 = Λ|Vσ|2∗−2u +in RN, +|u| → 0 +as |x| → +∞ +in increasing order. Then Λ1 = 1 with eigenfunction Vσ, Λ2 = 2∗ − 1 with eigenfunction ∂Vσ +∂σ . +Our main results will be proved using variational and singular limit methods applied to the energy func- +tional +Jε(u) := 1 +2 +� +Ω +� +|∇u|2 − µ u2 +|x|2 +� +dx − +1 +2∗ − ε +� +Ω +|u|2∗−εdx +defined on Hµ(Ω). +Let us also recall that the Green’s function of the Dirichlet Laplacian G(x, y) = +1 +|x−y|N−2 − H(x, y) and +its regular part H are symmetric: G(x, y) = G(y, x) and H(x, y) = H(y, x). If Ω is invariant under some +A ∈ O(N) then G(Ax, Ay) = G(x, y), and the same holds for H. + +Nodal solutions to problems with Hardy terms +7 +3 +The finite dimensional reduction +First we recall some notation from [5]. +Fix µ0 > 0, α > +N−4 +N−2 and an integer k ≥ 0. +For λ = +(λ0, λ1, . . . , λk) ∈ Rk+1 ++ +and ξ = (ξ1, ξ2, . . . , ξk) ∈ Ωk we define +Wε,λ,ξ := +k +� +i=1 +Ker +� +−∆ − (2∗ − 1)U 2∗−2 +δi,ξi +� ++ Ker +� +−∆ − µε +|x|2 − (2∗ − 1)V 2∗−2 +µε,σε +� +⊂ H1(RN) +where δi = λiε +1 +N−2 , µε = µ0εα, σε = λ0ε +1 +N−2 . By Proposition 2.1 and [8] we know that +Wε,λ,ξ = span +� +Ψj +i, Ψ0 +i , Ψ0, i = 1, 2, . . . , k, j = 1, 2, . . ., N +� +, +where for i = 1, 2, . . . , k and j = 1, 2, . . . , N: +Ψj +i := ∂Uδi,ξi +∂ξi,j +, +Ψ0 +i := ∂Uδi,ξi +∂δi +, +Ψ0 := ∂Vµε,σε +∂σ +with ξi,j the j-th component of ξi. For η ∈ (0, 1) we define +Oη := +� +(λ, ξ) ∈ Rk+1 ++ +× Ωk : λi ∈ (η, η−1) for i = 0, . . . , k, dist(ξi, ∂Ω) > η, +|ξi| > η, |ξi1 − ξi2| > η, for i, i1, i2 = 1, . . . , k, i1 ̸= i2 +� +. +The projection P : H1(RN) → H1 +0(Ω) is defined by ∆Pu = ∆u in Ω and Pu = 0 on ∂Ω. We also need +the spaces +Kε,λ,ξ := PWε,λ,ξ +and +K⊥ +ε,λ,ξ := {φ ∈ Hµ(Ω) : (φ, PΨ)µε = 0, for all Ψ ∈ Wε,λ,ξ}, +as well as the (·, ·)µε-orthogonal projections +Πε,λ,ξ : Hµε(Ω) → Kε,λ,ξ +and +Π⊥ +ε,λ,ξ := Id − Πε,λ,ξ : Hµε(Ω) → K⊥ +ε,λ,ξ. +Then solving problem (1.1) is equivalent to finding η > 0, ε > 0, (λ, ξ) ∈ Oη and φε,λ,ξ ∈ K⊥ +ε,λ,ξ such that: +(3.1) +Π⊥ +ε,λ,ξ +� +Vε,λ,ξ + φε,λ,ξ − ι∗ +µ(fε(Vε,λ,ξ + φε,λ,ξ)) +� += 0, +and +Πε,λ,ξ +� +Vε,λ,ξ + φε,λ,ξ − ι∗ +µ(fε(Vε,λ,ξ + φε,λ,ξ)) +� += 0, +where in the case of Theorem 1.2 +(3.2) +Vε,λ,ξ = − +k +� +i=1 +PUδi,ξi + PVµε,σε + +8 +T. Bartsch, Q. Guo +with k = 2, 3, and in the case of Theorem 1.6 +(3.3) +Vε,λ,ξ = +k +� +i=1 +(−1)iPUδi,ξi + PVµε,σε +with k = 4. +The following two propositions have been proved in [5]. +Proposition 3.1. For every η > 0 there exist ε0 > 0 and c0 > 0 with the following property. For every +(λ, ξ) ∈ Oη and for every ε ∈ (0, ε0) there exists a unique solution φε,λ,ξ ∈ K⊥ +ε,λ,ξ of equation (3.1) satisfying +∥φε,λ,ξ∥µε ≤ c0 +� +ε +N+2 +2(N−2) + ε +1+2α +4 +� +. +The map Φε : Oη → K⊥ +ε,λ,ξ defined by Φε(λ, ξ) := φε,λ,ξ is C1. +Now we can define the reduced functional +Iε : Oη → R, +Iε(λ, ξ) := Jε(Vε,λ,ξ + φε,λ,ξ). +Proposition 3.2. If (λ, ξ) ∈ Oη is a critical point of Iε then Vε,λ,ξ + φε,λ,ξ is a solution to problem (1.1) for +ε > 0 small. +So far everything works on a general bounded domain. +Now we will use the invariance of Iε under +certain symmetries for further reductions. For A ∈ O(N), ξ = (ξ1, . . . , ξk) ∈ (RN)k and u ∈ Lp(RN) we set +Aξ := (Aξ1, . . . , Aξk) and A ∗ u := u ◦ A−1. This induces isometric actions of O(N) on (RN)k as well as on +Lp(RN) and, if AΩ = Ω, on Lp(Ω) and on Hµ(Ω) such that ιµ and ι∗ +µ are equivariant. Moreover we have +Uδ,Aξ = A ∗ Uδ,ξ +and +Wε,λ,Aξ = {A ∗ u : u ∈ Wε,λ,ξ}, +and analogously for Kε,λ,ξ, Πε,λ,ξ, Vε,λ,ξ. +As a consequence, the uniqueness statement in Proposition 3.1 +implies +(3.4) +φε,λ,Aξ = A ∗ φε,λ,ξ, +hence Iε is invariant with respect to the action A ∗ (λ, ξ) = (λ, Aξ) of O(N) on Oη: +Iε(λ, Aξ) = Iε(λ, ξ). +Now we apply the principle of symmetric criticality using the matrix AN := + +E2 +0 +0 +−EN−2 + + ∈ O(N). By +assumption AN(Ω) = Ω, hence AN acts on Oη as above leaving Iε invariant. The principle of symmetric +criticality implies that critical points of Iε constrained to the fixed point set +OAN +η += {(λ, ξ) ∈ Oη : ANξ = ξ} = {(λ, ξ) ∈ Oη : ξi = (ξ′ +i, 0) ∈ R2 × RN−2, i = 1, . . . , k} + +Nodal solutions to problems with Hardy terms +9 +are critical points of Iε. We also need the invariance of Iε with respect to permutations of the blow-up points. +Here we need to distinguish between the cases where Vε,λ,ξ is of the form (3.2) or of the form (3.3). Let Sk +denote the group of permutations of {1, . . ., k}. For π ∈ Sk and (λ, ξ) ∈ Rk+1 × (RN)k we define +π ∗ (λ0, λ1, . . . , λk) := (λ0, λπ(1), . . . , λπ(k)) +and +π ∗ (ξ1, . . . , ξk) := (ξπ(1), . . . , ξπ(k)). +In the case when Vε,λ,ξ is of the form (3.2) it is obvious that +Iε(π ∗ λ, π ∗ ξ) = Iε(λ, ξ) +for all π ∈ Sk, (λ, ξ) ∈ Oη. +It follows that Iε is invariant under the map +τ : OAN +η +→ OAN +η +, +τ(λ0, λ1, . . . , λk, ξ1, . . . , ξk) := (λ0, λk, λ1, . . . , λk−1, Rkξk, Rkξ1, . . . , Rkξk−1), +which induces an action of Z/kZ on OAN +η +; here Rk(ξ′, ξ′′) := (Rkξ′, ξ′′) where Rk ∈ SO(2) is the rotation from +Theorem 1.2. Therefore critical points of Iε constrained to the fixed point set of the above map, i.e. to +OAN,τ +η += {(λ, ξ) ∈ OAN +η +: λi = · · · = λ1, ξi = Ri−1 +k +ξ1, i = 2, . . . , k}, +are critical points of Iε. +In conclusion, for the proofs of Theorems 1.1 and 1.2 it remains to find critical points of Iε constrained +to OAN ,τ for ε > 0 small. +The additional symmetry of the solutions stated in Theorem 1.2, and also in +Theorem 1.6, is obtained as follows. Since the ball is invariant under the action of A ∈ SO(N − 2) defined by +A(x′, x′′) := (x′, Ax′′) and since A ∗ (λ, ξ) = (λ, Aξ) = (λ, ξ) for (λ, ξ) ∈ OAN +η +it follows from (3.4) that +A ∗ φε,λ,ξ = φε,λ,Aξ = φε,λ,ξ +for all A ∈ SO(N − 2), +hence uε = Vε,λ,ξ + φε,λ,ξ satisfies A ∗ uε = uε, i.e. uε(x′, Ax′′) = uε(x′, x′′), for all A ∈ SO(N − 2). +In Theorem 1.6 Vε,λ,ξ is of the form (3.3) with k = 4. Here Iε is invariant under the map +�τ(λ1, λ2, λ3, λ4, λ0, ξ1, ξ2, ξ3, ξ4) = (λ3, λ4, λ1, λ2, R4ξ4, R4ξ1, R4ξ2, R4ξ3), +so, applying the principle of symmetric criticality once more, a critical point of Iε constrained to the set +OAN ,�τ +η += {(λ, ξ) ∈ OAN +η +: λ1 = λ3, λ2 = λ4, ξi = Ri−1 +k +ξ1, i = 2, 3, 4} +is an unconstrained critical point of Iε. This can of course be generalized to any even integer k ≥ 4. +4 +Proof of Theorems 1.1 and 1.2 +In this section we consider Vε,λ,ξ = − +k� +i=1 +PUδi,ξi + PVµε,σε for k = 2 and k = 3. The reduced energy is +expanded as follows; see [5, Proposition 5.1]. + +10 +T. Bartsch, Q. Guo +Lemma 4.1. For ε → 0+ there holds +Iε(λ, ξ) = a1 + a2ε − a3εα − a4ε ln ε + ψ(λ, ξ)ε + o(ε) +C1-uniformly with respect to (λ, ξ) in compact sets of Oη. The constants are given by +a1 = 1 +N (k + 1)S +N +2 +0 , a2 = (k + 1) +2∗ +� +RN U 2∗ +1,0 ln U1,0 − k + 1 +(2∗)2 S +N +2 +0 , a3 = 1 +2S +N−2 +2 +0 +Sµ0, a4 = k + 1 +2 · 2∗ +� +RN U 2∗ +1,0, +where S > 0 is defined by Sµ = S0 − Sµ + O(µ2); see [5, Lemma A.10]. The function ψ : Oη → R is given by +ψ(λ, ξ) = b1 +� +H(0, 0)λN−2 +0 ++ +k +� +i=1 +H(ξi, ξi)λN−2 +i ++ 2 +k +� +i=1 +G(ξi, 0)λ +N−2 +2 +i +λ +N−2 +2 +0 +− 2 +k +� +i,j=1,i 0 small, and that (λε, ξε) → (λ, ξ) as ε → 0. This applies in particular if (λ, ξ) is an isolated critical +point of ψ with nontrivial local degree. +Proof of Theorem 1.1. Since the symmetries of Iε carry over to ψ, for the existence of solutions uε as stated +in Theorem 1.1 it is sufficient to find stable critical points of ψ constrained to +OAN ,τ +η += {(λ, ξ) ∈ OAN +η +: λi = λ1, ξi = Ri−1 +k +ξ1, i = 2, . . . , k}, +where k = 2, 3. Observe that Iε and ψ are also invariant with respect to the action of A ∈ SO(2) given by +(x′, 0) �→ (Ax′, 0) acting on the ξi. Therefore in the case k = 2, setting ξ1 = (t, 0, . . . , 0) for 0 < t < 1 and +ξ2 = R2ξ1 = −ξ1, it is sufficient to find stable critical points of the function f2 : R+ × R+ × (0, 1) → R defined +by +f2(λ0, λ1, t) = ψ(λ0, λ1, λ1, ξ1, −ξ1) += b1 +� +H(0, 0)λN−2 +0 ++ 2H(ξ1, ξ1)λN−2 +1 ++ 4G(ξ1, 0)λ +N−2 +2 +1 +λ +N−2 +2 +0 +− 2G(ξ1, −ξ1)λN−2 +1 +� +− b2 +N − 2 +2 +ln +� +λ2 +1λ0 +� +. +This proves Theorem 1.1 in the case k = 2. For k = 3 we set ξ1 = (t, 0, . . . , 0) for 0 < t < 1, ξ2 = R3ξ1 = +� +− t +2, +√ +3t +2 , 0, . . . , 0 +� +, ξ3 = R3ξ2 = +� +− t +2, − +√ +3t +2 , 0, . . . , 0 +� +. As above it is sufficient to find stable critical points +of the function f3 : R+ × R+ × (0, 1) → R defined by +f3(λ0, λ1, t) = ψ(λ0, λ1, λ1, λ1, ξ1, ξ2, ξ3) += b1 +� +H(0, 0)λN−2 +0 ++ 3H(ξ1, ξ1)λN−2 +1 ++ 6G(ξ1, 0)λ +N−2 +2 +0 +λ +N−2 +2 +1 +− 6G(ξ1, ξ2)λN−2 +1 +� +− b2 ln +� +λ3 +1λ0 +� N−2 +2 +. + +Nodal solutions to problems with Hardy terms +11 +Here we used that G(ξ1, 0) = G(ξ2, 0) = G(ξ3, 0) and G(ξ1, ξ2) = G(ξ1, ξ3) = G(ξ2, ξ3), as well as H(ξ1, ξ1) = +H(ξ2, ξ2) = H(ξ3, ξ3). This proves Theorem 1.1 also in the case k = 3. +□ +Proof of Theorem 1.2. Here we need to find stable critical points of f2 and f3 if G is the Green function of +the Dirichlet Laplace operator in the unit ball in RN. Our proof uses the explicit knowledge of G. In the case +k = 2 the proof of [4, Lemma 3.1] applies almost verbatim and shows that f2 has two isolated critical points: +a local saddle point with Morse index 1, hence with local degree −1, and an isolated local minimum, hence +with local degree 1. These are stable critical points, giving rise to critical points of Iε for ε > 0 small. +In the case k = 3 we set +γ1(t) := H(ξ1, ξ1) − 2G(ξ1, ξ2) = +1 +(1 − t2)N−2 − +2 +( +√ +3t)N−2 + +2 +(t4 + t2 + 1) +N−2 +2 +and +(4.1) +τ1(t) := G(ξ1, 0) = +1 +tN−2 − 1 +so that +f3(λ0, λ1, t) = b1 +� +H(0, 0)λN−2 +0 ++ 3γ1(t)λN−2 +1 ++ 6τ1(t)λ +N−2 +2 +0 +λ +N−2 +2 +1 +� +− b2 +N − 2 +2 +ln +� +λ3 +1λ0 +� +. +One easily checks that γ′ +1(t) > 0, γ1(t) → −∞ as t → 0+, and γ1( 1 +2) > 0. Thus there exists t∗ ∈ (0, 1 +2) such +that +γ1(t∗) = 0 +and +γ1(t) > 0 for all t ∈ (t∗, 1). +A direct computation shows that for t ∈ (t∗, 1) there exist unique λ0(t), λ1(t) such that +∂f3(λ0(t), λ1(t), t) +∂λ0 += 0 +and +∂f3(λ0(t), λ1(t), t) +∂λ1 += 0. +In fact one obtains +(4.2) +λ0(t) +N−2 +2 += α(ξ1, ξ2)λ1(t) +N−2 +2 +and +λ1(t) +N−2 +2 += +� +1 +β(ξ1, ξ2) · b2 +2b1 +, +where +α(x, y) = −2G(x, 0) + +� +4G2(x, 0) + 4H(0, 0)(H(x, x) − 2G(x, y)) +2H(0, 0) +and +β(x, y) = H(x, x) − 2G(x, y) + G(x, 0)α(x, y). + +12 +T. Bartsch, Q. Guo +Moreover, continuing the computation one obtains +∂2f3(λ0(t), λ1(t), t) +∂λ2 +1 += +3(N − 2)b1 +� +(N − 3)γ1(t)λN−4 +1 ++ N − 4 +2 +τ1(t)λ +N−6 +2 +0 +λ +N−2 +2 +1 +� ++ 3(N − 2)b2 +2λ2 +1 += +3(N − 2)b1 +� +(N − 2)γ1(t)λN−4 +1 ++ N − 2 +2 +τ1(t)λ +N−6 +2 +0 +λ +N−2 +2 +1 +� +, +∂2f3(λ0(t), λ1(t), t) +∂λ2 +0 += +(N − 2)b1 +� +(N − 3)H(0, 0)λN−4 +0 ++ 3(N − 4) +2 +τ1(t)λ +N−2 +2 +0 +λ +N−6 +2 +1 +� ++(N − 2)b2 +2λ2 +0 += +(N − 2)b1 +� +(N − 2)H(0, 0)λN−4 +0 ++ 3(N − 2) +2 +τ1(t)λ +N−2 +2 +0 +λ +N−6 +2 +1 +� +, +∂2f3(λ0(t), λ1(t), t) +∂λ0∂λ1 += +3(N − 2)2 +2 +b1τ1(t)λ +N−4 +2 +0 +λ +N−4 +2 +1 +. +It follows that the Hessian matrix D2 +λ0,λ1f3(λ0(t), λ1(t), t) is positive definite, hence nondegenerate. Therefore +it is sufficient to find stable critical points of the function +ν1(t) := f3 (λ0(t), λ1(t), t) = 2b2 − b2 +N − 2 +2 +ln +� +λ3 +1(t)λ0(t) +� +. +As in [4, (3.4)] one sees that +(4.3) +lim +t→(t∗)+ ν1(t) = −∞ +and +lim +t→1− ν1(t) = +∞. +Now we prove ν′ +1( 1 +2) < 0 for N large. Set +α(t) := α(ξ1, ξ2) = −τ1(t) + +� +τ 2 +1 (t) + γ1(t), +where we used H(0, 0) = 1. We obtain +ν′ +1(t) = ∂f3(λ0(t), λ1(t), t) +∂t += 3b1 +� +γ′ +1(t) + 2α(t)τ ′ +1(t) +� +λN−2 +1 +. +Then setting ι1(t) := γ′ +1(t) + 2α(t)τ ′ +1(t), it is enough to show ι1 +� 1 +2 +� +< 0 for N large. In fact, since γ1( 1 +2 ) +τ 2 +1 ( 1 +2 ) < 1 +for N large we see as in [4, (3.9)] that +ι1( 1 +2) ≤ γ′ +1( 1 +2) + 4γ1( 1 +2) +5τ1( 1 +2) τ ′ +1( 1 +2). +A direct computation gives for N large the inequalites +γ′ +1( 1 +2) = (N − 2) +� +( 4 +3)N−1 + 4( 2 +√ +3)N−2 − +3 +2 +( 1 +16 + 1 +4 + 1) +N +2 +� +< 11(N − 2) +10 +( 4 +3)N−1 +and +τ ′ +1( 1 +2) = −(N − 2)2N−1 +and +γ1( 1 +2) +τ1( 1 +2) = +( 4 +3)N−2 − 2( 2 +√ +3)N−2 + +2 +( 1 +16 + 1 +4 +1) +N−2 +2 +2N−2 − 1 +> 11 +12 · ( 4 +3)N−2 +2N−2 + +Nodal solutions to problems with Hardy terms +13 +which yield ι1( 1 +2) < 0, hence ν′ +1( 1 +2) < 0 for N large enough. This together with (4.3) implies that ν1 has a +local maximum t1 ∈ (t∗, 1 +2) and a local minimum t2 ∈ ( 1 +2, ∞). These are nondegenerate because ν1 is analytic. +In conclusion, f3 has two critical points: (λ0(t1), λ1(t1), t1) with Morse index 1 and (λ0(t2), λ1(t2), t2) with +Morse index 0. This concludes the proof of Theorem 1.2. +□ +Remark 4.2. a) For k = 3, N = 7, numerical computations show that one cannot find t0 ∈ (t∗, 1) such that +ν′ +1(t0) = 0. Therefore it is necessary to assume N large here. +b) For k = 4, the idea above cannot give the existence of nodal solutions with five bubbles, one positive +at the origin and four negative as in Theorem 1.1. This is the content of Proposition 1.4. +5 +Proof of Proposition 1.4 +It follows from Lemma 4.1 that Iε does not have critical points for ε > 0 small if ψ does not have +critical points. This also holds if we constrain Iε and ψ to OAN,τ +η +. Setting ξ1 = (t, 0, . . . , 0) for 0 < t < 1, +ξ2 = R4ξ1 = (0, t, 0, . . . , 0), ξ3 = R4ξ2 = (−t, 0, . . . , 0), and ξ4 = R4ξ3 = (0, −t, . . . , 0) we need to consider the +function +f4(λ0, λ1, t) := ψ(λ0, λ1, λ1, λ1, λ1, ξ1, ξ2, ξ3, ξ4) += b1 +� +H(0, 0)λN−2 +0 ++ 4H(ξ1, ξ1)λN−2 +1 ++ 8G(ξ1, 0)λ +N−2 +2 +0 +λ +N−2 +2 +1 +− 8G(ξ1, ξ2)λN−2 +1 +− 4G(ξ1, ξ3)λN−2 +1 +� +− b2 +N − 2 +2 +ln(λ4 +1λ0), +where we use +H(ξ1, ξ1) = H(ξ2, ξ2) = H(ξ3, ξ3) = H(ξ4, ξ4) +and +G(ξ1, 0) = G(ξ2, 0) = G(ξ3, 0) = G(ξ4, 0) +as well as +G(ξ1, ξ2) = G(ξ2, ξ3) = G(ξ3, ξ4) = G(ξ4, ξ1), +G(ξ1, ξ3) = G(ξ2, ξ4). +Proposition 1.4 follows if we can prove that f4 does not have critical points. Let τ1(t) be as in (4.1) and define +γ2(t) +:= +H(ξ1, ξ1) − 2G(ξ1, ξ2) − G(ξ1, ξ3) += +1 +(1 − t2)N−2 − +2 +( +√ +2t)N−2 + +2 +(t4 + 1) +N−2 +2 +− +1 +(2t)N−2 + +1 +(t2 + 1)N−2 +so that +f4(λ0, λ1, t) = b1 +� +H(0, 0)λN−2 +0 ++ 4γ2(t)λN−2 +1 ++ 8τ1(t)λ +N−2 +2 +0 +λ +N−2 +2 +1 +� +− b2 +N − 2 +2 +ln +� +λ4 +1λ0 +� +. +A direct computation shows that +γ′ +2(t) = (N − 2) +� +2t +(1 − t2)N−1 + +2 +( +√ +2)N−2tN−1 − +4t3 +(t4 + 1) +N +2 + +1 +2N−2tN−1 − +2t +(t2 + 1)N−1 +� +> 0. + +14 +T. Bartsch, Q. Guo +Clearly γ2(t) → −∞ as t → 0+, and γ2 +� +1 +√ +2 +� +> 0. Then there exists t∗ ∈ (0, +1 +√ +2) such that +(5.1) +γ2(t∗) = 0 +and +γ2(t) > 0 for all t ∈ (t∗, 1). +Notice that +λ +N−2 +2 +0 += α1(ξ1, ξ2, ξ3)λ +N−2 +2 +1 +, +λ +N−2 +2 +1 += +� +1 +β1(ξ1, ξ2, ξ3) · b2 +2b1 +, +H(ξ1, ξ1) − 2G(ξ1, ξ2) − G(ξ1, ξ3) > 0, +where +α1(x, y, z) = −3G(x, 0) + +� +9G2(x, 0) + 4H(0, 0)(H(x, x) − 2G(x, y) − G(x, z)) +2H(0, 0) +, +and +β1(x, y, z) = H(x, x) − 2G(x, y) − G(x, z) + G(x, 0)α1(x, y, z). +Setting α1(t) := α1(ξ1, ξ2, ξ3) = +−3τ1(t)+√ +9τ 2 +1 (t)+4γ2(t) +2 +and ι2(t) := γ′ +2(t) + 2α1(t)τ ′ +1(t), a similar argument as +above shows that problem (1.1) admits a solution with 5 bubbles, one positive at the origin and 4 negative as +in Theorem 1.1 only if ι2(t) has a zero in (t∗, 1). The following claim implies that this is not the case. +Claim: If N ≥ 7 then ι2(t) > 0 for any t ∈ (t∗, 1). +We first show that t∗ > +√ +6− +√ +2 +2 +, where t∗ is from (5.1). In order to see this, it is enough to prove γ2 +� √ +6− +√ +2 +2 +� +< +0. Since 22/5 · 2( +√ +6− +√ +2 +2 +)2 < 1 < ( +√ +6− +√ +2 +2 +)4 + 1, we have +1 +( +√ +2 · +√ +6− +√ +2 +2 +)N−2 > +2 +(( +√ +6− +√ +2 +2 +)4 + 1) +N−2 +2 +, +for all N ≥ 7. +On the other hand, it is easy to see that +1 +(1 − ( +√ +6− +√ +2 +2 +)2)N−2 = +1 +( +√ +2( +√ +6− +√ +2 +2 +))N−2 +and +1 +(2( +√ +6− +√ +2 +2 +))N−2 > +1 +(( +√ +6− +√ +2 +2 +)2 + 1)N−2 . +It follows that γ2( +√ +6− +√ +2 +2 +) < 0. +Now we prove ι2(t) > 0 for t ∈ (t∗, 1) ⊂ ( +√ +6− +√ +2 +2 +, 1). It is easy to see that +γ′ +2(t) ≥ (N − 2) · +2t +(1 − t2)N−1 +and +γ2(t) ≤ +1 +(1 − t2)N−2 +for all t ∈ +�√ +6 − +√ +2 +2 +, 1 +� +. +Then we have for all t ∈ (t∗, 1) and N ≥ 7: +ι2(t) +N − 2 +≥ +2t +(1 − t2)N−1 − 3 +� +1 +tN−2 − 1 +� +· +1 +tN−1 + + + +� +� +� +�1 + +4 · +1 +(1−t2)N−2 +9( +1 +tN−2 − 1)2 − 1 + + + +≥ +2t +(1 − t2)N−1 − +3 +t2N−3 · +�� +1 + 4 +9( +t2 +1 − t2 )N−2 · +1 +(1 − tN−2)2 − 1 +� +. +Setting T := +t2 +1−t2 it is enough to prove that +2 +3 · T N−1 + 1 > +� +1 + 4 +9 · T N−2 · +1 +(1 − tN−2)2 + +Nodal solutions to problems with Hardy terms +15 +which is equivalent to +(5.2) +(T N + 3T ) · (1 − tN−2)2 > 1. +It is obvious that if t ∈ [ 1 +√ +2, 4 +5), then +(5.3) +3T · (1 − tN−2)2 ≥ 3(1 − t5)2 > 1, +and if t ∈ [ 4 +5, 1), then +(5.4) +T N · (1 − tN−2)2 ≥ T N · (1 − t)2 = +t4 +(1 + t)2 · T N−2 > ( 4 +5)4 +4 +( +( 4 +5)2 +1 − ( 4 +5)2 )5 > 1. +Now we are left to prove (5.2) for t ∈ +� +t∗, +1 +√ +2 +� +. First of all, if t ∈ +� √ +6− +√ +2 +2 +, +1 +√ +2 +� +, then T ∈ +� √ +3−1 +2 +, 1 +� +. Setting +f(T ) := 3T +� +1 − tN−2�2 = 3T +� +1 − +� +T +1 + T +� N−2 +2 �2 +, +a direct computation shows that +f ′(T ) += +� +1 − +� +T +1 + T +� N−2 +2 � � +3 − 3 +� +T +1 + T +� N−2 +2 +− 3(N − 2) +� +T +1 + T +� N−2 +2 +1 +1 + T +� +≥ +� +1 − +�1 +2 +� N−2 +2 � � +3 − 3 +�1 +2 +� N−2 +2 +− 3(N − 2) +�1 +2 +� N−2 +2 +1 +1 + +√ +3−1 +2 +� +≥ +� +1 − +�1 +2 +� N−2 +2 � � +3 − 3 +�1 +2 +� 5 +2 +− 15 +�1 +2 +� 5 +2 +1 +1 + +√ +3−1 +2 +� +> 0, +where in the second inequality we use the fact that 3 − 3( 1 +2) +N−2 +2 +− 3(N − 2)( 1 +2) +N−2 +2 +1 +1+ +√ +3−1 +2 +is increasing in N. +Now we conclude that +f(T ) > 3 · +√ +3 − 1 +2 + +1 − +� +√ +3−1 +2 +1 + +√ +3−1 +2 +� 5 +2  + +2 +> 1 +for all N ≥ 7. +(5.5) +The claim, hence Proposition 1.4, follows combining (5.2), (5.3), (5.4), and (5.5). +6 +Proof of Theorem 1.6 +In this section we consider solutions of the form Vε,λ,ξ = +k� +i=1 +(−1)iPUδi,ξi + PVσ. +Then the reduced +function in Lemma 4.1 becomes +�ψ(λ, ξ) = b1 +� +H(0, 0)λN−2 +0 ++ +k +� +i=1 +H(ξi, ξi)λN−2 +i ++ 2 +k +� +i=1 +(−1)i−1G(ξi, 0)λ +N−2 +2 +0 +λ +N−2 +2 +i ++2 +k +� +i,j=1,i 0, γ3(t) → −∞ as t → 0+, γ3(t) → +∞ as t → 1−, and +γ3( 1 +2) > 0. Thus there exists t∗ +1 ∈ (0, 1 +2) such that +γ3(t∗ +1) = 0 +and +γ3(t) < 0 for all t ∈ (0, t∗ +1). +On the other hand, +� +γ3(t) − 2τ 2 +1 (t) +�′ > 0, γ3(t) − 2τ 2 +1 (t) → −∞ as t → 0+, γ3(t) − 2τ 2 +1 (t) → +∞ as t → 1−, +and γ3( 1 +2) − 2τ 2 +1 ( 1 +2) < 0. Thus there exists t∗ +2 ∈ ( 1 +2, 1) such that +γ3(t∗ +2) − 2τ 2 +1 (t∗ +2) = 0 +and +γ3(t) − 2τ 2 +1 (t) > 0 for all t ∈ (t∗ +2, 1). +It follows that for every t ∈ (0, t∗ +1) ∪ (t∗ +2, 1) there exist unique λ0(t), λ1(t), λ2(t) such that +∇λ0,λ1,λ2f5(λ0(t), λ1(t), λ2(t), t) = 0, +where λ0(t), λ1(t), λ2(t) satisfy (6.5), (6.6), (6.7) and (6.8). This proves Claim 1. +Claim 2: The Hessian matrix D2 +λ0,λ1,λ2f5(λ0(t), λ1(t), λ2(t), t) is nondegenerate for any t ∈ (0, t∗ +1) ∪ (t∗ +2, 1). + +18 +T. Bartsch, Q. Guo +A direct computation using (6.2), (6.3), and (6.4) shows that, writing λi instead of λi(t), +∂2f5(λ0, λ1, λ2, t) +∂λ2 +0 += (N − 2)b1 +� +(N − 3)H(0, 0)λN−4 +0 ++ (N − 4)τ1(t)λ +N−6 +2 +0 +� +λ +N−2 +2 +1 +− λ +N−2 +2 +2 +� � ++ (N − 2)b2 +2λ2 +0 += (N − 2)2b1 +� +H(0, 0)λN−4 +0 ++ τ1(t)λ +N−6 +2 +0 +� +λ +N−2 +2 +1 +− λ +N−2 +2 +2 +� � +, +∂2f5(λ0, λ1, λ2, t) +∂λ2 +1 += (N − 2)b1 +� +2(N − 3)γ3(t)λN−4 +1 ++ (N − 4)τ1(t)λ +N−2 +2 +0 +λ +N−6 +2 +1 ++ 2(N − 4)γ4(t)λ +N−6 +2 +1 +λ +N−2 +2 +2 +� ++ (N − 2)b2 +λ2 +1 += (N − 2)2b1 +� +2γ3(t)λN−4 +1 ++ τ1(t)λ +N−2 +2 +0 +λ +N−6 +2 +1 ++ 2γ4(t)λ +N−6 +2 +1 +λ +N−2 +2 +2 +� +, +∂2f5(λ0, λ1, λ2, t) +∂λ2 +2 += (N − 2)b1 +� +2(N − 3)γ3(t)λN−4 +2 +− (N − 4)τ1(t)λ +N−2 +2 +0 +λ +N−6 +2 +2 ++ 2(N − 4)γ4(t)λ +N−6 +2 +1 +λ +N−2 +2 +2 +� ++ (N − 2)b2 +λ2 +2 += (N − 2)2b1 +� +2γ3(t)λN−4 +2 +− τ1(t)λ +N−2 +2 +0 +λ +N−6 +2 +2 ++ 2γ4(t)λ +N−2 +2 +1 +λ +N−6 +2 +2 +� +, +∂2f4(λ0, λ1, λ2, t) +∂λ0∂λ1 += (N − 2)2b1τ1(t)λ +N−4 +2 +0 +λ +N−4 +2 +1 +, +∂2f4(λ1, λ2, λ0, t) +∂λ0∂λ2 += −(N − 2)2b1τ1(t)λ +N−4 +2 +0 +λ +N−4 +2 +2 +, +∂2f4(λ1, λ2, λ0, t) +∂λ1∂λ2 += 2(N − 2)2b1γ4(t)λ +N−4 +2 +1 +λ +N−4 +2 +2 +. +For simplicity, we introduce the notation +X := λ +N−2 +2 +0 +, +Y := λ +N−2 +2 +1 +, +Z := λ +N−2 +2 +2 +. +In order to prove that D2 +λ0,λ1,λ2f5(λ0(t), λ1(t), λ2(t), t) is nondegenerate for any t ∈ (0, t∗ +1) ∪ (t∗ +2, 1), it suffices +to show that the matrix + + + + +X + τ1(t)(Y − Z) +τ1(t)X +2 +N−2 Y +N−4 +N−2 +−τ1(t)X +2 +N−2 Z +N−4 +N−2 +τ1(t)X +N−4 +N−2 Y +2 +N−2 +2γ3(t)Y + τ1(t)X + 2γ4(t)Z +2γ4(t)Y +2 +N−2 Z +N−4 +N−2 +−τ1(t)X +N−4 +N−2 Z +2 +N−2 +2γ4(t)Y +N−4 +N−2 Z +2 +N−2 +2γ3(t)Z − τ1(t)X + 2γ4(t)Y + + + + +is nondegenerate. Using (6.2), (6.3) and (6.4) this is equivalent to showing that the matrix + + + + +X +2 + b2 +4b1 · 1 +X +τ1(t)X +2 +N−2 Y +N−4 +N−2 +−τ1(t)X +2 +N−2 Z +N−4 +N−2 +τ1(t)X +N−4 +N−2 Y +2 +N−2 +γ3(t)Y + b2 +2b1 · 1 +Y +2γ4(t)Y +2 +N−2 Z +N−4 +N−2 +−τ1(t)X +N−4 +N−2 Z +2 +N−2 +2γ4(t)Y +N−4 +N−2 Z +2 +N−2 +γ3(t)Z + b2 +2b1 · 1 +Z + + + + +is nondegenerate. A direct computation, using (6.7), shows that the determinant of the above matrix has the +same sign as γ3(t), hence is nontrivial, proving Claim 2. +Theorem 1.6 now follows from + +Nodal solutions to problems with Hardy terms +19 +Claim 3: The function ν2(t) := f5 +� +λ0(t), λ1(t), λ2(t), t +� +has a critical point t1 ∈ (0, t∗ +1). +Observe that, writing again λi instead of λi(t), +ν′ +2(t) = ∂f4(λ0(t), λ1(t), λ2(t), t) +∂t += 2b1 +� +γ′ +3(t) +� +λN−2 +1 ++ λN−2 +2 +� ++ 2τ ′ +1(t)λ +N−2 +2 +0 +� +λ +N−2 +2 +1 +− λ +N−2 +2 +2 +� ++ 4γ′ +4(t)λ +N−2 +2 +1 +λ +N−2 +2 +2 +� +, +where λ0, λ1, λ2 satisfy (6.5), (6.6), (6.7) and (6.8). Therefore, ν′ +2(t) = 0 for t ∈ (0, t∗ +1) is equivalent to +ι3(t) := γ′ +3(t) +� +2γ3(t)(γ3(t) − 2τ 2 +1 (t)) + τ2 +1 (t)(γ3(t) + 2γ4(t)) +� +− 2τ ′ +1(t)τ1(t)γ3(t) +� +γ3(t) + 2γ4(t) +� ++ 4γ′ +4(t)γ3(t) +� +γ3(t) − 2τ 2 +1 (t) +� += 0. +It is easy to check that ι3(t) → −∞ as t → 0+ and ι3(t∗ +1) > 0 because γ′ +3(t∗ +1) > 0, γ4(t∗ +1) > 0 and γ3(t∗ +1) = 0. +Hence there exists t1 ∈ (0, t∗ +1) such that ι3(t1) = 0. Claim 3 follows, finishing the proof of Theorem 1.6. +□ +Remark 6.1. We conjecture that there should also exist t2 ∈ (t∗ +2, 1) such that ι3(t2) = 0. This is not considered +here because the computations get enormous. +Acknowledgements: The authors would like to thank Professor Daomin Cao for many helpful discussions +during the preparation of this paper. This work was carried out while Qianqiao Guo was visiting Justus- +Liebig-Universit¨at Gießen, to which he would like to express his gratitude for their warm hospitality. +Funding: Qianqiao Guo was supported by the National Natural Science Foundation of China (Grant No. +11971385) and the Natural Science Basic Research Plan in Shaanxi Province of China (Grant No. 2019JM275). +References +[1] Th. Aubin, Probl`emes isop´erim´etriques et espaces de Sobolev, J. Differential Geometry 11 (1976), 573-598. +[2] A. Bahri, Y. Li, O. Rey, On a variational problem with lack of compactness: the topological effect of the +critical points at infinity, Calc. Var. Part. Diff. Equ. 3 (1995), 67-93. +[3] T. Bartsch, T. D’Aprile, A. Pistoia, Multi-bubble nodal solutions for slightly subcritical elliptic problems +in domains with symmetries, Ann. Inst. H. Poincar´e Anal. Non Lin´eaire 30 (2013), 1027-1047. +[4] T. Bartsch, T. D’Aprile, A. 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Pistoia, Tower of bubbles for almost critical problems in general domains, J. Math. Pures +Appl. 93 (2010), 1-40. +[27] M. Musso, J. Wei, Nonradial solutions to critical elliptic equations of Caffarelli-Kohn-Nirenberg type, Int. +Math. Res. Not. 18 (2012), 4120-4162. +[28] A. Pistoia, T. Weth, Sign changing bubble tower solutions in a slightly subcritical semilinear Dirichlet +problem, Ann. Inst. H. Poincar´e Anal. Non Lin´eaire 24 (2007), 325-340. +[29] O. Rey, Proof of two conjectures of H. Br´ezis and L.A. Peletier, Manuscripta Math. 65 (1989), 19-37. +[30] O. Rey, The role of the Green’s function in a nonlinear elliptic equation involving the critical Sobolev +exponent, J. Functional Analysis 89 (1990), 1-52. +[31] O. Rey, Blow-up points of solutions to elliptic equations with limiting nonlinearity, Differential Integral +Equations 4 (1991), 1155-1167. +[32] D. Ruiz, M. Willem, Elliptic problems with critical exponents and Hardy potentials, J. Diff. Equ. 190 +(2003), 524-538. +[33] D. Smets, Nonlinear Schr¨odinger equations with Hardy potential and critical nonlinearities, Trans. Amer. +Math. Soc. 357 (2005), 2909-2938. +[34] G. Talenti, Best constant in Sobolev inequality, Ann. Mat. Pura Appl. 110 (1976), 353-372. +[35] S.Terracini, On positive entire solutions to a class of equations with a singular coefficient and critical +exponent, Adv. Diff. Equations 2 (1996), 241-264. +E-mail: + +22 +T. Bartsch, Q. Guo +thomas.bartsch@math.uni-giessen.de +gqianqiao@nwpu.edu.cn + diff --git a/IdE4T4oBgHgl3EQfIQyU/content/tmp_files/load_file.txt b/IdE4T4oBgHgl3EQfIQyU/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..fb4b999deb9efdd62288092a2661e63a10616e07 --- /dev/null +++ b/IdE4T4oBgHgl3EQfIQyU/content/tmp_files/load_file.txt @@ -0,0 +1,885 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf,len=884 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content='04911v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content='AP] 12 Jan 2023 Multi-bubble nodal solutions to slightly subcritical elliptic problems with Hardy terms in symmetric domains Thomas Bartsch∗, Qianqiao Guo† Abstract We consider the slightly subcritical elliptic problem with Hardy term \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 −∆u − µ u |x|2 = |u|2∗−2−εu in Ω ⊂ RN, u = 0 on ∂Ω, where 0 ∈ Ω and Ω is invariant under the subgroup SO(2) × {±EN−2} ⊂ O(N);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' here En denots the n × n identity matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' If µ = µ0εα with µ0 > 0 fixed and α > N−4 N−2 the existence of nodal solutions that blow up, as ε → 0+, positively at the origin and negatively at a different point in a general bounded domain has been proved in [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' Solutions with more than two blow-up points have not been found so far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' In the present paper we obtain the existence of nodal solutions with a positive blow-up point at the origin and k = 2 or k = 3 negative blow-up points placed symmetrically in Ω ∩ (R2 × {0}) around the origin provided a certain function fk : R+ ×R+ ×I → R has stable critical points;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' here I = {t > 0 : (t, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' , 0) ∈ Ω}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' If Ω = B(0, 1) ⊂ RN is the unit ball centered at the origin we obtain two solutions for k = 2 and N ≥ 7, or k = 3 and N large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' The result is optimal in the sense that for Ω = B(0, 1) there cannot exist solutions with a positive blow-up point at the origin and four negative blow-up points placed on the vertices of a square centered at the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' Surprisingly there do exist solutions on Ω = B(0, 1) with a positive blow-up point at the origin and four blow-up points on the vertices of a square with alternating positive and negative signs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' The results of our paper show that the structure of the set of blow-up solutions of the above problem offers fascinating features and is not well understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' 2010 Mathematics Subject Classification: 35B44, 35B33, 35J60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' Key words: Hardy term;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' Critical exponent;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' Slightly subcritical problems;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' Nodal solutions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' Multi-bubble solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' ∗Mathematisches Institut, Justus-Liebig-Universit¨at Giessen, Arndtstr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' 2, 35392 Giessen, Germany †School of Mathematics and Statistics, Northwestern Polytechnical University, 710129 Xi’an, China 1 2 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' Bartsch, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' Guo 1 Introduction The paper is concerned with the semilinear singular problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content='1) \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 −∆u − µ u |x|2 = |u|2∗−2−εu in Ω, u = 0 on ∂Ω, where Ω ⊂ RN, N ≥ 7, is a smooth bounded domain with 0 ∈ Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' 2∗ := 2N N−2 is the critical Sobolev exponent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' In [5], we obtained the existence of two-bubble nodal solutions to problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content='1) that blow up positively at the origin and negatively at a different point in a general bounded domain, as ε → 0+ and µ = µ0εα with µ0 > 0 and α > N−4 N−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' The location of the negative blow-up point is determined by the geometry of the domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' The existence of nodal bubble tower solutions has been proved in [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' These are superpositions of positive and negative bubbles with different scalings, all blowing up at the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' It seems to be a difficult and open problem whether solutions with a blow-up point at the origin and more than one blow-up point outside the origin exist in a general domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' In the present paper we investigate this problem in symmetric domains, in particular for the model case of the ball Ω = B(0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' In the case of a ball it is natural to place one blow-up point, say positive, at the origin and k blow-up points, say negative, at the vertices of a regular k-gon with center at the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' We shall prove that solutions of this shape exist for 2 ≤ k ≤ 3 but, somewhat surprisingly, not for k = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' On the other hand, we prove the existence of solutions with four blow-up points, two positive and two negative ones, at the vertices of a square, centered symmetrically around the positive blow-up point at the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' Our results show that the existence of solutions of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content='1) with three or more blow-up points is interesting and far from being understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' When µ = 0 the blow-up phenomenon for positive and for nodal solutions to problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content='1) has been studied extensively, see for instance [2–4, 7, 9, 14, 19, 22, 24, 26, 28–31] and the references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' However for µ ̸= 0, few results are known about the existence of positive or nodal solutions with multiple bubbles to problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' Positive solutions have been obtained in [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' Related results, though for different equations, can be found in [16,17,27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' We also want to mention the papers [10,11,15,18,21,23,25,32,33,35] dealing with the critical exponent, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' ε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' An important role will be played by the limit problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content='2) \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 −∆u − µ u |x|2 = |u|2∗−2u in RN, u → 0 as |x| → ∞ which has been investigated in [13,35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' Positive solutions of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content='2) in the range 0 ≤ µ < µ := (N−2)2 4 are given by Vµ,σ(x) = Cµ � σ σ2|x|β1 + |x|β2 � N−2 2 Nodal solutions to problems with Hardy terms 3 with σ > 0, β1 := (√µ − √µ − µ)/√µ, β2 := (√µ + √µ − µ)/√µ, and Cµ := � 4N(µ−µ) N−2 � N−2 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' These solutions minimize Sµ := min u∈D1,2(RN )\\{0} � RN(|∇u|2 − µ u2 |x|2 )dx ( � RN |u|2∗dx)2/2∗ , and there holds � RN � |∇Vµ,σ|2 − µ|Vµ,σ|2 |x|2 � dx = � RN |Vµ,σ|2∗dx = S N 2 µ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' In the range 0 < µ < µ these are all positive solutions of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' In the case µ = 0 these are all solutions with maximum at x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' Of course, if µ = 0 also translates of Vµ,σ are solutions of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content='3) \uf8f1 \uf8f2 \uf8f3 −∆u = |u|2∗−2u in RN, u → 0 as |x| → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' We will write Uδ,ξ(x) = C0 � δ δ2 + |x − ξ|2 � N−2 2 for the solutions of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content='3) where δ > 0, ξ ∈ RN and C0 := (N(N − 2)) N−2 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' These are the well known Aubin-Talenti instantons (see [1,34]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' Now we state our main results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' We consider domains satisfying the condition (A1) Ω ⊂ RN is a bounded domain with 0 ∈ Ω, and it is invariant under the subgroup SO(2) × {±EN−2} ⊂ O(N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' We use the notation x = (x′, x′′) ∈ Ω ⊂ R2 × RN−2 and write A(x′, x′′) = (Ax′, x′′) for A ∈ SO(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' For k ∈ N let Rk = \uf8eb \uf8edcos 2π k − sin 2π k sin 2π k cos 2π k \uf8f6 \uf8f8 ∈ SO(2), and set I = {t > 0 : (t, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' , 0) ∈ Ω} ⊂ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' Our first results are concerned with the existence of nodal solutions with k +1 bubbles, one being positive and k being negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' Let G(x, y) = 1 |x−y|N−2 − H(x, y) be the Green function (up to a coefficient involving the volume of the unit ball) for the Dirichlet Laplace operator in Ω with regular part H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' For k = 2, 3 we define the function fk : R+ × R+ × I → R by fk(λ0, λ1, t) := b1 � H(0, 0)λN−2 0 + kH(ξ(t), ξ(t))λN−2 1 + 2kG(ξ(t), 0)λ N−2 2 0 λ N−2 2 1 − 2 �k 2 � G(ξ(t), Rkξ(t))λN−2 1 � − b2 N − 2 2 ln � λ0λk 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' where ξ(t) = (t, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' , 0) and b1 = 1 2C0 � RN U 2∗−1 1,0 and b2 = 1 2∗ � RN U 2∗ 1,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' Finally we call a critical point of fk stable if it is isolated and has nontrivial local degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' This is the case, for instance, if it is non-degenerate or an isolated local maximum or minimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' 4 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' Bartsch, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' Guo Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' Suppose Ω satisfies (A1), and suppose (λ0, λ1, t) ∈ R+ × R+ × I is a stable critical point of fk, k = 2 or k = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' Let µ0 > 0 and α > N−4 N−2 be fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' Then there exists ε0 > 0 such that for every ε ∈ (0, ε0) problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content='1) with µ = µε = µ0εα has a pair of solutions ±uε satisfying (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content='4) uε(x) = Vµε,σε(x) − k � i=1 Uδε,Ri kξ(tε)(x) + o(1) = Cµε � σε (σε)2|x|β1 + |x|β2 � N−2 2 − C0 k � i=1 � δε (δε)2 + |x − Ri kξ(tε)|2 � N−2 2 + o(1), where σε = � λ0 + o(1) � ε 1 N−2 , δε = � λ1 + o(1) � ε 1 N−2 , ξ(tε) = (tε, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' , 0) = (t + o(1), 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' , 0) as ε → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' These solutions satisfy the following symmetries: (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content='5) uε(x′, x′′) = uε(x′, −x′′) = uε(Rkx′, x′′) for (x′, x′′) ∈ Ω ⊂ R2 × RN−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' As Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content='4 below shows, for k = 4 a family uε as in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content='1 need not exist even in the case of the ball.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' It is a challenging problem to find critical points of fk for general domains satisfying (A1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' We consider the special case where Ω = B(0, 1) ⊂ RN is the unit ball.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' If Ω = B(0, 1) ⊂ RN, k = 2 and N ≥ 7, or k = 3 and N is large enough, then fk has two stable critical points, one is a local minimum, the other a mountain pass point with Morse index 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' As a consequence, problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content='1) has two families of solutions ±u1 ε, ±u2 ε as in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' They have the additional symmetry uε(x′, x′′) = uε(x′, Ax′′) for all A ∈ SO(N − 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' a) In the proof of the case k = 3 we provide an explicit inequality for N, so that the solutions as in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content='2 exist if this inequality holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' Numerical computations show that this inequality is not satisfied for N = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' We do not know the optimal value for N such that Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content='2 is true for k = 3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' see also Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' b) We conjecture that Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content='2 holds for other domains satisying (A1), for instance for Ω = B2(0, 1)× Ω′ ⊂ R2 × RN−2 with Ω′ = −Ω′ ⊂ RN−2 a bounded symmetric neighborhood of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' Our proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content='2 uses the explicit knowledge of the Green function for the ball, hence it does not extend immmediately to other domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' The next result shows that Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content='2 is optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' For Ω = B(0, 1) ⊂ RN, N ≥ 7 and k = 4 there does not exist a family of solutions ±uε as in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' a) We conjecture that Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content='4 can be generalized to k ≥ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' Nodal solutions to problems with Hardy terms 5 b) We cannot exclude the existence of solutions with a positive bubble at the origin and k = 4 negative bubbles placed somewhere in the ball and with possibly different blow-up parameters δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' However, we can show that there do not exist solutions with four negative bubbles at the vertices of a square centered at the origin even if one allows different blow-up speeds, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' if one replaces the δε in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content='4) by δi,ε, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' , 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' In fact, it is not difficult to prove that the blow-up parameters δi,ε have to be independent of i if the vertices are at Ri 4ξ(tε), i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' , 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' Considering Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content='4 the following existence results of nodal solutions with five bubbles, three being positive and two being negative, is somewhat surprising.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' Let Ω = B(0, 1) ⊂ RN, N ≥ 7, µ0 > 0 and α > N−4 N−2 be fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' Then there exists ε0 > 0 such that for any ε ∈ (0, ε0), there exist a pair of 5-bubble solutions ±uε to problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content='1) with µ = µε = µ0εα of the shape uε(x) = Vµε,σε(x) + 4 � i=1 (−1)iUδi,ε,Ri 4ξ(tε)(x) + o(1) = Cµε � σε (σε)2|x|β1 + |x|β2 � N−2 2 + C0 4 � i=1 (−1)i � δi,ε (δi,ε)2 + |x − Ri 4ξ(tε)|2 � N−2 2 + o(1), where σε = � λ0 + o(1) � ε 1 N−2 , δ1,ε = δ3,ε = � λ1 + o(1) � ε 1 N−2 , δ2,ε = δ4,ε = � λ2 + o(1) � ε 1 N−2 , ξ(tε) = (tε, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' , 0) = (t + o(1), 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' , 0) as ε → 0, for some λ0, λ1, λ2 > 0, t ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' These solutions satisfy the symmetries: (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content='6) uε(x′, x′′) = uε(x′, Ax′′) = −uε(R4x′, x′′) for (x′, x′′) ∈ B(0, 1) ⊂ R2 × RN−2, A ∈ SO(N − 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' a) The parameters (λ0, λ1, λ2, t) ∈ R+ × R+ × R+ × (0, 1) in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content='6 are obtained as a critical point of a suitable limit function f5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' We conjecture that there exists a second solution in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content='6 but the computations for finding a second critical point of f5 are intimidating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' b) It seems that for k = 2 in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content='2, it is still possible to obtain the information on the nodal sets of the solutions as in [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' For k = 3 in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content='2 and for Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content='6, it is an interesting problem to study the profile of the nodal sets of the solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' c) As stated in [5], the assumption α > N−4 N−2 is essential in our theorems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' In Section 2, we collect some notations and preliminary results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' Section 3 is devoted to the method of finite dimensional reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' Section 4 contains the proof of Theorems 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content='1 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content='4 is proved in Section 5, and finally Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content='6 is proved in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' 6 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' Bartsch, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' Guo 2 Notations and preliminary results Throughout this paper, positive constants will be denoted by C, c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' By Hardy’s inequality the norm ∥u∥µ := �� Ω (|∇u|2 − µ u2 |x|2 )dx � 1 2 is equivalent to the norm ∥u∥0 = �� Ω |∇u|2dx �1/2 on H1 0(Ω) provided 0 ≤ µ < µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' This inequality is of course satisfied for µ = µ0εα with α > 0 and ε > 0 small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' We write Hµ(Ω) for the Hilbert space consisting of the H1 0(Ω) functions with the inner product (u, v)µ := � Ω � ∇u∇v − µ uv |x|2 � dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' As in [5, 16] let ι∗ µ : L2N/(N+2)(Ω) → Hµ(Ω) be the adjoint operator of the inclusion ιµ : Hµ(Ω) → L2N/(N−2)(Ω), that is, ι∗ µ(u) = v ⇐⇒ (v, φ)µ = � Ω u(x)φ(x)dx, for all φ ∈ Hµ(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' There exists c > 0 such that ∥ι∗ µ(u)∥µ ≤ c∥u∥2N/(N+2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' Then problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content='1) is equivalent to the fixed point problem u = ι∗ µ(fε(u)), u ∈ Hµ(Ω), where fε(s) = |s|2∗−2−εs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' The following proposition is from [5, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' Let 0 < µ < µ, and let Λi, i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' , be the eigenvalues of \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 −∆u − µ u |x|2 = Λ|Vσ|2∗−2u in RN, |u| → 0 as |x| → +∞ in increasing order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' Then Λ1 = 1 with eigenfunction Vσ, Λ2 = 2∗ − 1 with eigenfunction ∂Vσ ∂σ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' Our main results will be proved using variational and singular limit methods applied to the energy func- tional Jε(u) := 1 2 � Ω � |∇u|2 − µ u2 |x|2 � dx − 1 2∗ − ε � Ω |u|2∗−εdx defined on Hµ(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' Let us also recall that the Green’s function of the Dirichlet Laplacian G(x, y) = 1 |x−y|N−2 − H(x, y) and its regular part H are symmetric: G(x, y) = G(y, x) and H(x, y) = H(y, x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' If Ω is invariant under some A ∈ O(N) then G(Ax, Ay) = G(x, y), and the same holds for H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' Nodal solutions to problems with Hardy terms 7 3 The finite dimensional reduction First we recall some notation from [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' Fix µ0 > 0, α > N−4 N−2 and an integer k ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' For λ = (λ0, λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' , λk) ∈ Rk+1 + and ξ = (ξ1, ξ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' , ξk) ∈ Ωk we define Wε,λ,ξ := k � i=1 Ker � −∆ − (2∗ − 1)U 2∗−2 δi,ξi � + Ker � −∆ − µε |x|2 − (2∗ − 1)V 2∗−2 µε,σε � ⊂ H1(RN) where δi = λiε 1 N−2 , µε = µ0εα, σε = λ0ε 1 N−2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' By Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content='1 and [8] we know that Wε,λ,ξ = span � Ψj i, Ψ0 i , Ψ0, i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' , k, j = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=', N � , where for i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' , k and j = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' , N: Ψj i := ∂Uδi,ξi ∂ξi,j , Ψ0 i := ∂Uδi,ξi ∂δi , Ψ0 := ∂Vµε,σε ∂σ with ξi,j the j-th component of ξi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' For η ∈ (0, 1) we define Oη := � (λ, ξ) ∈ Rk+1 + × Ωk : λi ∈ (η, η−1) for i = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' , k, dist(ξi, ∂Ω) > η, |ξi| > η, |ξi1 − ξi2| > η, for i, i1, i2 = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' , k, i1 ̸= i2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' The projection P : H1(RN) → H1 0(Ω) is defined by ∆Pu = ∆u in Ω and Pu = 0 on ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' We also need the spaces Kε,λ,ξ := PWε,λ,ξ and K⊥ ε,λ,ξ := {φ ∈ Hµ(Ω) : (φ, PΨ)µε = 0, for all Ψ ∈ Wε,λ,ξ}, as well as the (·, ·)µε-orthogonal projections Πε,λ,ξ : Hµε(Ω) → Kε,λ,ξ and Π⊥ ε,λ,ξ := Id − Πε,λ,ξ : Hµε(Ω) → K⊥ ε,λ,ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' Then solving problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content='1) is equivalent to finding η > 0, ε > 0, (λ, ξ) ∈ Oη and φε,λ,ξ ∈ K⊥ ε,λ,ξ such that: (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content='1) Π⊥ ε,λ,ξ � Vε,λ,ξ + φε,λ,ξ − ι∗ µ(fε(Vε,λ,ξ + φε,λ,ξ)) � = 0, and Πε,λ,ξ � Vε,λ,ξ + φε,λ,ξ − ι∗ µ(fε(Vε,λ,ξ + φε,λ,ξ)) � = 0, where in the case of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content='2 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content='2) Vε,λ,ξ = − k � i=1 PUδi,ξi + PVµε,σε 8 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' Bartsch, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' Guo with k = 2, 3, and in the case of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content='6 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content='3) Vε,λ,ξ = k � i=1 (−1)iPUδi,ξi + PVµε,σε with k = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' The following two propositions have been proved in [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' For every η > 0 there exist ε0 > 0 and c0 > 0 with the following property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' For every (λ, ξ) ∈ Oη and for every ε ∈ (0, ε0) there exists a unique solution φε,λ,ξ ∈ K⊥ ε,λ,ξ of equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content='1) satisfying ∥φε,λ,ξ∥µε ≤ c0 � ε N+2 2(N−2) + ε 1+2α 4 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' The map Φε : Oη → K⊥ ε,λ,ξ defined by Φε(λ, ξ) := φε,λ,ξ is C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' Now we can define the reduced functional Iε : Oη → R, Iε(λ, ξ) := Jε(Vε,λ,ξ + φε,λ,ξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' If (λ, ξ) ∈ Oη is a critical point of Iε then Vε,λ,ξ + φε,λ,ξ is a solution to problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content='1) for ε > 0 small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' So far everything works on a general bounded domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' Now we will use the invariance of Iε under certain symmetries for further reductions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' For A ∈ O(N), ξ = (ξ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' , ξk) ∈ (RN)k and u ∈ Lp(RN) we set Aξ := (Aξ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' , Aξk) and A ∗ u := u ◦ A−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' This induces isometric actions of O(N) on (RN)k as well as on Lp(RN) and, if AΩ = Ω, on Lp(Ω) and on Hµ(Ω) such that ιµ and ι∗ µ are equivariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' Moreover we have Uδ,Aξ = A ∗ Uδ,ξ and Wε,λ,Aξ = {A ∗ u : u ∈ Wε,λ,ξ}, and analogously for Kε,λ,ξ, Πε,λ,ξ, Vε,λ,ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' As a consequence, the uniqueness statement in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content='1 implies (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content='4) φε,λ,Aξ = A ∗ φε,λ,ξ, hence Iε is invariant with respect to the action A ∗ (λ, ξ) = (λ, Aξ) of O(N) on Oη: Iε(λ, Aξ) = Iε(λ, ξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' Now we apply the principle of symmetric criticality using the matrix AN := \uf8eb \uf8edE2 0 0 −EN−2 \uf8f6 \uf8f8 ∈ O(N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' By assumption AN(Ω) = Ω, hence AN acts on Oη as above leaving Iε invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' The principle of symmetric criticality implies that critical points of Iε constrained to the fixed point set OAN η = {(λ, ξ) ∈ Oη : ANξ = ξ} = {(λ, ξ) ∈ Oη : ξi = (ξ′ i, 0) ∈ R2 × RN−2, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' , k} Nodal solutions to problems with Hardy terms 9 are critical points of Iε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' We also need the invariance of Iε with respect to permutations of the blow-up points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' Here we need to distinguish between the cases where Vε,λ,ξ is of the form (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content='2) or of the form (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' Let Sk denote the group of permutations of {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=', k}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' For π ∈ Sk and (λ, ξ) ∈ Rk+1 × (RN)k we define π ∗ (λ0, λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' , λk) := (λ0, λπ(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' , λπ(k)) and π ∗ (ξ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' , ξk) := (ξπ(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' , ξπ(k)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' In the case when Vε,λ,ξ is of the form (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content='2) it is obvious that Iε(π ∗ λ, π ∗ ξ) = Iε(λ, ξ) for all π ∈ Sk, (λ, ξ) ∈ Oη.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' It follows that Iε is invariant under the map τ : OAN η → OAN η , τ(λ0, λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' , λk, ξ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' , ξk) := (λ0, λk, λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' , λk−1, Rkξk, Rkξ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' , Rkξk−1), which induces an action of Z/kZ on OAN η ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' here Rk(ξ′, ξ′′) := (Rkξ′, ξ′′) where Rk ∈ SO(2) is the rotation from Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' Therefore critical points of Iε constrained to the fixed point set of the above map, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' to OAN,τ η = {(λ, ξ) ∈ OAN η : λi = · · · = λ1, ξi = Ri−1 k ξ1, i = 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' , k}, are critical points of Iε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' In conclusion, for the proofs of Theorems 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content='1 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content='2 it remains to find critical points of Iε constrained to OAN ,τ for ε > 0 small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' The additional symmetry of the solutions stated in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content='2, and also in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content='6, is obtained as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' Since the ball is invariant under the action of A ∈ SO(N − 2) defined by A(x′, x′′) := (x′, Ax′′) and since A ∗ (λ, ξ) = (λ, Aξ) = (λ, ξ) for (λ, ξ) ∈ OAN η it follows from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content='4) that A ∗ φε,λ,ξ = φε,λ,Aξ = φε,λ,ξ for all A ∈ SO(N − 2), hence uε = Vε,λ,ξ + φε,λ,ξ satisfies A ∗ uε = uε, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' uε(x′, Ax′′) = uε(x′, x′′), for all A ∈ SO(N − 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' In Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content='6 Vε,λ,ξ is of the form (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content='3) with k = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' Here Iε is invariant under the map �τ(λ1, λ2, λ3, λ4, λ0, ξ1, ξ2, ξ3, ξ4) = (λ3, λ4, λ1, λ2, R4ξ4, R4ξ1, R4ξ2, R4ξ3), so, applying the principle of symmetric criticality once more, a critical point of Iε constrained to the set OAN ,�τ η = {(λ, ξ) ∈ OAN η : λ1 = λ3, λ2 = λ4, ξi = Ri−1 k ξ1, i = 2, 3, 4} is an unconstrained critical point of Iε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' This can of course be generalized to any even integer k ≥ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' 4 Proof of Theorems 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content='1 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content='2 In this section we consider Vε,λ,ξ = − k� i=1 PUδi,ξi + PVµε,σε for k = 2 and k = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' The reduced energy is expanded as follows;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' see [5, Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' 10 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' Bartsch, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' Guo Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' For ε → 0+ there holds Iε(λ, ξ) = a1 + a2ε − a3εα − a4ε ln ε + ψ(λ, ξ)ε + o(ε) C1-uniformly with respect to (λ, ξ) in compact sets of Oη.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' The constants are given by a1 = 1 N (k + 1)S N 2 0 , a2 = (k + 1) 2∗ � RN U 2∗ 1,0 ln U1,0 − k + 1 (2∗)2 S N 2 0 , a3 = 1 2S N−2 2 0 Sµ0, a4 = k + 1 2 · 2∗ � RN U 2∗ 1,0, where S > 0 is defined by Sµ = S0 − Sµ + O(µ2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' see [5, Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content='10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfIQyU/content/2301.04911v1.pdf'} +page_content=' The function ψ : Oη → R is given by ψ(λ, ξ) = b1 � H(0, 0)λN−2 0 + k � i=1 H(ξi, ξi)λN−2 i + 2 k � i=1 G(ξi, 0)λ N−2 2 i λ N−2 2 0 − 2 k � i,j=1,i