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+arXiv:2301.00543v1  [math.GR]  2 Jan 2023
+ON PSEUDO-REAL FINITE SUBGROUPS OF PGL3(C)
+E. BADR AND A. ELGUINDY
+Abstract. Let G be a finite subgroup of PGL3(C), and let σ be the generator
+of Gal(C/R).
+We say that G has a real field of moduli if σG and G are
+PGL3(C)-conjugates, that is, if ∃ φ ∈ PGL3(C) such that φ−1 G φ =
+σG.
+Furthermore, we say that R is a field of definition for G or that G is definable
+over R if G is PGL3(C)-conjugate to some G′ ⊂ PGL3(R). In this situation,
+we call G′ a model for G over R. If G has R as a field of definition but is not
+definable over R, then we call G pseudo-real.
+In this paper, we first show that any finite cyclic subgroup G = Z/nZ in
+PGL3(C) has a real field of moduli and we provide a necessary and sufficient
+condition for G = Z/nZ to be definable over R; see Theorems 2.1, 2.2, and
+2.3. We also prove that any dihedral group D2n with n ≥ 3 in PGL3(C) is
+definable over R; see Theorem 2.4. Furthermore, we study all six classes of
+finite primitive subgroups of PGL3(C), and show that all of them except the
+icosahedral group A5 are pseudo-real; see Theorem 2.5, whereas A5 is definable
+over R. Finally, we explore the connection of these notions in group theory
+with their analogues in arithmetic geometry; see Theorem 2.6 and Example
+2.7.
+1. Introduction
+The projective general linear group over the complex numbers PGL3(C) is widely
+studied in several branches of mathematics for many reasons. Some of these mo-
+tivations come from algebraic geometry, arithmetic geometry, and also from group
+theory. We give some examples of such motivations.
+(1) In complex algebraic geometry, PGL3(C) can be viewed as the automorphism
+group Aut(P2(C)) of the complex projective plane P2(C), see [11, Example 7.1.1] for
+example. Moreover, any isomorphism between two smooth complex plane curves
+C and C′ of a fixed degree d ≥ 4 is induced by an element of PGL3(C), see [8,
+Theorem 1]. For such a curve we have the finiteness result | Aut(C)| < +∞ due to
+Hurwitz [19], hence we can view Aut(C) as a finite subgroup of PGL3(C) acting on
+a non-singular plane model F(X, Y, Z) = 0 for C inside P2(C). It is thus natural to
+classify finite subgroups G in PGL3(C). Based on geometrical methods, Mitchell
+[23] achieved such classification. Recently, Harui [12] made Mitchell’s classification
+more precise under the assumption that G = Aut(C) for some smooth plane curves
+C.
+However, some of these groups live in a short exact sequence, hence group
+extension problems arise, which can sometimes be hard to solve.
+Another parallel line of research is to obtain the stratification of C-isomorphism
+classes of smooth plane curves of a fixed degree d by their automorphism groups.
+Henn in his PhD dissertation [13] and Komiya-Kuribayashi [22] accomplished this
+task for smooth quartic curves (d = 4), Badr-Bars [3, 4, 5] for smooth quinitcs
+(d = 5) and for smooth sextics (d = 6).
+2020 Mathematics Subject Classification. 20G20, 14L35, 14H37, 22F50.
+Key words and phrases. Projective linear groups; Field of moduli; Fields of definitions; Pseudo-
+real; Smooth plane curves; Automorphism groups.
+1
+
+2
+E. BADR AND A. ELGUINDY
+(2) In complex arithmetic geometry, the problem of studying fields of definition
+versus fields of moduli for a Riemann surface S has attracted a lot of recent research.
+For example, we refer to [1, 2, 7, 9, 14, 15, 16, 18, 21].
+More precisely, a subfield K of C is called a field of definition for S if there exists
+a model of S defined by polynomials with coefficients in K. The field of moduli
+for S is the intersection of all fields of definition for S. The work of Koizumi [20]
+guarantee the existence of a model for S over a finite extension of its field of moduli.
+In this direction, the surface S is said to be pseudo-real if its field of moduli is a
+subfield of R, but S does not have R as a field of definition.
+The above aspects from algebraic geometry and arithmetic geometry are the
+main motivation for us to extend the notions of fields of definition, fields of moduli,
+pseudo-real, to the study of arithmetic groups. Indeed, there has been other in-
+stances in which it has been fruitful to translate concepts from arithmetic geometry
+to group theory, as we illustrate next.
+(3) In group theory, we can measure to which extent an infinite group Γ is
+similar to an abelian group by computing its Jordan constant, denoted by J(Γ).
+It is defined to be the smallest positive integer such that any finite subgroup of Γ
+has an abelian normal subgroup with index not exceeding J(Γ). This definition
+originated from the theory of abelian varieties, more specifically, [24, Definition
+2.1].
+Concerning the Jordan constant J(PGL3(K)), where K is a field of characteristic
+0, Hu [17] showed that it assumes only one of the values: 360, 168, 60, 24, 12, 6,
+depending on whether
+√
+5 or ζ3 belongs to K or not. Here ζ3 denotes a primitive 3rd
+root of unity in K, a fixed algebraic closure of K. In particular, J(PGL3(C)) = 360,
+see [17, Theorem 1.2] for full details.
+Notations. Throughout the paper, we use the following notations.
+• Norm(G, PGL3(C)) is the normalizer of G inside PGL3(C),
+• ζn = e
+2πi
+n , a fixed primitive nth root of unity in C.
+• We shall view C× as a subgroup of GL3(C) by identifying 0 ̸= c ∈ C with
+diag(c, c, c). If A is in GL3(C), we let π(A) denote its image under the
+canonical projection onto PGL3(C), namely π(A) is the coset (or equiva-
+lence class) C×A. To ease notation, we occasionally continue to use A in
+place of π(A) when the context is clear.
+• If A = (ai,j) ∈ GL3(C), then the projective linear transformation π(A) ∈
+PGL3(K) is sometimes written as
+[a1,1X + a1,2Y + a1,3Z : a2,1X + a2,2Y + a2,3Z : a3,1X + a3,2Y + a3,3Z].
+• The Galois group Gal(C/R)- action on PGL3(C) is a left action, denoted
+by σφ for any φ ∈ PGL3(C).
+• For c ∈ C, ℜ(c) and ℑ(c) denote the real and the imaginary parts of c
+respectively, and |c| denotes the absolute value of c.
+2. Main results
+Let G ⊂ PGL3(C) be cyclic of order 1 < n < +∞. Up to PGL3(C)-conjugation,
+such G is generated by a diagonal element A := diag(1, ζa
+n, ζb
+n), for some 0 ≤ a <
+b ≤ n − 1 such that gcd(a, b) = 1.
+Theorem 2.1. Let G = ⟨A⟩ ⊂ PGL3(C) be a cyclic group of order n as above.
+Then, we have that
+(1) G always has a real field of moduli.
+
+ON PSEUDO-REAL FINITE SUBGROUPS IN PGL3(C)
+3
+(2) R is a field of definition for G if and only if A and A−1 are conjugates via a
+transformation of the shape φ σφ−1 for some φ ∈ PGL3(C). In this situation,
+φ−1 G φ would give a model for G over R.
+An homology of period n is a projective linear transformation of the plane P2(C),
+which is PGL3(C)-conjugate to diag(1, 1, ζn). Such a transformation fixes point-
+wise a projective line L, its axis, and a point P ∈ P2(C) − L, its center. In its
+canonical form, the line is L : Z = 0 and the point is P = (0 : 0 : 1). Otherwise, it
+is a non-homology.
+In particular, we have:
+Theorem 2.2. Let G = ⟨A⟩ ⊂ PGL3(C) be a cyclic group of order n as above.
+Then, there exists a model for G over R if and only if n = 2 or n > 2 such that
+a + b, a − 2b or 2a − b equals 0 mod n. In particular, any cyclic group generated by
+a homology of period n ≥ 3 is pseudo-real.
+Furthermore, we can get a model for G over R generated by
+φ−1 A φ =
+
+
+2ℑ(α β)
+0
+0
+0
+2ℑ(α β ζa
+n)
+2|β|2 sin(2πa/n)
+0
+−2|α|2 sin(2πa/n)
+2ℑ(α β ζ−a
+n )
+
+
+for some α, β ∈ C∗
+The above results can be reformulated using characteristic polynomials of lifts
+to B ∈ GL3(C). If we denote the characteristic polynomial of such B by fB(t),
+then it is straightforward to see that for c ∈ C∗
+fcB(t) = c3fB(t/c).
+(2.1)
+So while we can not attach a single polynomial as a characteristic polynomial
+to an element A ∈ PGL3(C), we can attach to such an A an equivalence class
+of polynomials in C[t] coming from the action given by (2.1).
+Such classes are
+preserved under conjugation in PGL3(C), and we can prove the following result.
+Corollary 2.3. A finite cyclic group G of order n ≥ 3 is definable over R if there
+exists A ∈ GL3(C) such that π(A) (the image of A in PGL3(C) under the natural
+projection) generates G in PGL3(C) and the characteristic polynomial fA(t) ∈ R[t].
+The converse is not necessarily true.
+For G = D2n, a dihedral group in PGL3(C), we prove:
+Theorem 2.4. Any dihedral group D2n of order 2n with n ≥ 3 in PGL3(C) is
+conjugate to ⟨B, π(A)⟩, where B = [X : Z : Y ] and A = diag(1, ζa
+n, ζ−a
+n ) for some
+integer a such that gcd(n, a) = 1. Moreover, we always can descend it to R as
+⟨ φ−1 B φ, φ−1 A φ⟩, where φ−1 A φ is as given in Theorem 2.2 and
+φ−1 B φ =
+
+
+2ℑ(α β)
+0
+0
+0
+−2ℑ(α β)
+−2ℑ(β2)
+0
+2ℑ(α2)
+2ℑ(α β)
+
+
+for some α, β ∈ C∗.
+When G is one of the finite primitive subgroup of PGL3(C), we show the follow-
+ing.
+Theorem 2.5. Any of the finite primitive subgroups namely, the Hessian groups
+Hess∗, for ∗ = 216, 72 and 36, the Klein group PSL(2, 7) of order 168, the icosa-
+hedral group A5 of order 60 and the alternating group A6 of order 360, has a real
+field of moduli. Moreover, none of them descends to R except A5. More concretely,
+
+4
+E. BADR AND A. ELGUINDY
+we always can descend A5 to R as φ−1 ⟨ A, B, C⟩ φ, such that φ−1 A φ and φ−1 B φ
+are as given in Theorem 2.4 with n = 5 and a = 4, and φ−1 C φ equals
+
+
+4ℑ(α β)
+8ℑ(α β) ℜ(α)
+8ℑ(α β) ℜ(β)
+2ℑ(β)
+2
+�
+cos(4π/5)ℑ(αβ) − cos(2π/5)ℑ(αβ)
+�
+−2 cos(2π/5)ℑ(β2)
+2ℑ(α)
+2 cos(2π/5)ℑ(α2)
+2
+�
+cos(4π/5)ℑ(αβ) + cos(2π/5)ℑ(αβ)
+�
+
+ ,
+for some α, β ∈ C∗.
+A connection with these notions in arithmetic geometry is described by the next
+result.
+Theorem 2.6. Let C : F(X, Y, Z) = 0 be a smooth plane curve over C. If C has a
+real field of moduli in the Arithmetic Geometry sense, then its automorphism group
+Aut(C) has a real field of moduli in the Group Theory sense.
+The converse of Theorem 2.6 is not necessarily true. Below is a counter example.
+Example 2.7. There are infinitely many smooth plane quintic curves defined over
+C by an equation of the form
+Cα,β : X5 + Y 5 + Z5 + αX(Y Z)2 + βX3(Y Z) = 0,
+such that the automorphism group Aut(Cα,β) = D10 has a real field of moduli, but
+Cα,β does not have a real field of moduli as its field of moduli.
+3. The case when G is cyclic
+Suppose that G = ⟨diag(1, ζa
+n, ζb
+n)⟩ in PGL3(C) such that 0 ≤ a < b ≤ n − 1 and
+gcd(a, b) = 1.
+Since the complex conjugation automorphism σ : C → C sends ζn �→ ζ−1
+n , then
+σG = ⟨diag(1, ζ−a
+n , ζ−b
+n )⟩ = G. In particular, G has a real field of moduli. This
+proves Theorem 2.1-(1).
+To prove Theorem 2.1-(2), we assume that G descends to R. That is, there exists
+φ ∈ PGL3(C) satisfying φ−1 A φ ∈ PGL3(R), where A = diag(1, ζa
+n, ζb
+n). This holds
+if and only if
+φ−1 A φ = σ �
+φ−1 A φ
+�
+= σφ−1 A−1 σφ,
+which we can read in two different ways. First as
+�
+φ σφ−1�−1 A
+�
+φ σφ−1�
+= A−1,
+which shows that A and A−1 are conjugates via φ σφ−1. Second as
+φ−1 A φ = σ �
+φ−1 A φ
+�
+,
+which shows that φ−1 A φ ∈ PGL3(R) as claimed.
+We need the following lemma to discuss Theorem 2.2.
+Lemma 3.1. Assume A and B are matrices in GL3(C) such that π(A) and π(B)
+are PGL3(C)-conjugates (where π denotes the natural projection from GL3(C) to
+PGL3(C)), then there is a constant c ∈ C∗ such that the eigenvalues of B are
+precisely cν1, cν2, cν3, where ν1, ν2, ν3 are the eigenvalues of A.
+Proof. Suppose that there is an ψ ∈ PGL3(C) such that ψ−1 π(A) ψ = π(B) in
+PGL3(C). Then, this equation corresponds to ψ−1 A ψ = (1/c)B in GL3(C) for
+some c ∈ C∗. Hence, A and (1/c)B are similar matrices in GL3(C), so by elementary
+linear algebra, we guarantee that their characteristic polynomials have the same
+roots, say ν1, ν2, ν3 . Therefore, the eigenvalues of B are cν1, cν2, cν3.
+□
+We now present the proof of Theorem 2.2.
+
+ON PSEUDO-REAL FINITE SUBGROUPS IN PGL3(C)
+5
+Proof. (of the necessity direction) First, assume that G is generated by a homology
+A = diag(1, 1, ζn). Since {c, c, c ζn} ̸= {1, 1, ζ−1
+n } for any c ∈ C∗ unless n = 2, then
+A and A−1 are never PGL3(C)-conjugates for n ≥ 3 by Lemma 3.1. In particular,
+G does not have a model over R by Theorem 2.1.
+Secondly, assume that G is generated by a non-homology A = diag(1, ζa
+n, ζb
+n)
+such that {c, c ζa
+n, c ζb
+n} = {1, ζ−a
+n , ζ−b
+n } for some c ∈ C∗. Then, c is either 1, ζ−a
+n
+or
+ζ−b
+n . Moreover,
+- if c = 1, then ζa
+n = ζ−a
+n , ζb
+n = ζ−b
+n
+or ζa
+n = ζ−b
+n . That is, 2a = 2b = 0 mod n or
+a + b = 0 mod n. We discard the case 2a = 2b = 0 mod n as it implies that n or
+n/2 would divide gcd(a, b) = 1, a contradiction because n ≥ 3. This leaves us with
+a + b = 0 mod n.
+- if c = ζ−a
+n , then ζb−a
+n
+= ζ−b
+n , and n | a − 2b = 0 mod n.
+- if c = ζ−b
+n , then ζa−b
+n
+= ζ−a
+n , and 2a − b = 0 mod n.
+This completes the necessity part.
+□
+Proof. (of the sufficiency direction) If G is cyclic generated by a homology of period
+2, then G is PGL3(C)-conjugate to ⟨diag(1, 1, −1)⟩ in PGL3(R), and we are done.
+Otherwise, G is generated by a non-homology A = diag(1, ζa
+n, ζb
+n) of order n ≥ 3
+such that a + b, a − 2b or 2a − b equals 0 mod n. First, we show that any of the
+last two situation can be reduced to the first one. Indeed, if A = diag(1, ζ2b
+n , ζb
+n),
+then one can take ψ = [Y : Z : X] so that
+ψ−1 A ψ = diag(ζb
+n, 1, ζ2b
+n ) = diag(1, ζ−b
+n , ζb
+n) = diag(1, ζa′
+n , ζ−a′
+n
+) in PGL3(C),
+where a′ := −b. Similarly, if A = diag(1, ζa
+n, ζ2a
+n ), then take ψ = [Z : X : Y ] to get
+ψ−1 A ψ = diag(ζa
+n, ζ2a
+n , 1) = diag(1, ζa
+n, ζ−a
+n ) in PGL3(C).
+Now we are going to handle the situation when n divides a + b. Take
+φ =
+
+
+1
+0
+0
+0
+α
+β
+0
+α
+β
+
+ ∈ PGL3(C).
+One easily verifies that φ σφ−1 = [X : Z : Y ] ∈ Norm(G, PGL3(C)) such that
+[X : Z : Y ] A [X : Z : Y ] = A−1. In particular, we deduce by Theorem 2.1 that
+φ−1 G φ ≤ PGL3(R) is a model of G over R. More specifically,
+φ−1 A φ
+=
+
+
+2ℑ(α β) i
+0
+0
+0
+β
+−β
+0
+−α
+α
+
+ diag(1, ζa
+n, ζ−a
+n )
+
+
+1
+0
+0
+0
+α
+β
+0
+α
+β
+
+
+=
+
+
+2ℑ(α β) i
+0
+0
+0
+ζa
+n β
+−ζ−a
+n
+β
+0
+−ζa
+n α
+ζ−a
+n
+α
+
+
+
+
+1
+0
+0
+0
+α
+β
+0
+α
+β
+
+
+=
+
+
+2ℑ(α β) i
+0
+0
+0
+2ℑ(α β ζa
+n) i
+2|β|2 sin(2πa/n) i
+0
+−2|α|2 sin(2πa/n) i
+2ℑ(α β ζ−a
+n ) i
+
+
+=
+
+
+2ℑ(α β)
+0
+0
+0
+2ℑ(α β ζa
+n)
+2|β|2 sin(2πa/n)
+0
+−2|α|2 sin(2πa/n)
+2ℑ(α β ζ−a
+n )
+
+ ∈ PGL3(R).
+This completes the proof of Theorem 2.2.
+□
+Next, assume that G is generated by a non-homology π(A) ∈ PGL3(C) of order
+n ≥ 3. As a consequence Theorem 2.2, we can say that fA(t) ∈ R[t] is a sufficient
+(rather than necessary) condition for G to descend to R.
+
+6
+E. BADR AND A. ELGUINDY
+Proof. (of Corollary 2.3) By Lemma 3.1, there exists c ∈ C∗ such that
+fA(t) = (t − c)(t − cζa
+n)(t − cζb
+n) ∈ R[t].
+Moreover, the roots c, c ζa
+n, c ζb
+n of fA(t) are pairwise distinct, since π(A) is a non-
+homology in PGL3(C) by assumption.
+Now, the coefficients c3ζa+b
+n
+, c(1+ζa
+n +ζb
+n), c2(ζa+b
+n
++ζa
+n +ζb
+n) belong to R. Thus
+there are r, r′ ∈ R such that ζa+b
+n
+= r/c3 and ζa
+n + ζb
+n = r′/c− 1. Consequently, the
+last condition becomes c2(r/c3+r′/c−1) ∈ R, in other words, c3−r′c2+r′′c−r = 0
+for some r, r′, r′′ ∈ R. This means that c ∈ C is algebraic over R of degree dividing
+3. Since C/R is a field extension of degree 2, then c must be algebraic over R of
+degree 1. Therefore, c ∈ R, which in turns implies that ζa+b
+n
+, ζa
+n + ζb
+n ∈ R.
+Clearly, ζa+b
+n
+∈ R only if a+b = k( n
+2 ) with k = 1, 2 or 3, since 3 ≤ a+b ≤ 2n−3.
+If k = 1 or 3, then ζa+b
+n
+= −1 and ζa
+n + ζb
+n = ζa
+n − ζ−a
+n
+= 2 sin(2π a/n) i /∈ R, a
+contradiction. Hence k = 1 and a + b = 0 mod n. By Theorem 2.2 we deduce that
+G descends to R, which was to be shown.
+To see that the converse does not hold in general, take A = diag(ζ3
+5, ζ4
+5, ζ2
+5)
+in GL3(C). Clearly, fA(t) /∈ R[t]. However, G = ⟨π(A)⟩ is definable over R by
+Theorem 2.2, since π(A) = diag(1, ζ5, ζ−1
+5 ) = diag(1, ζa
+n, ζb
+n) with n | a + b.
+□
+4. The case when G is a Dihedral group
+Suppose that G = ⟨A, B : An = B2 = 1, BAB = A−1⟩ is a dihedral group D2n
+in PGL3(C) with n ≥ 3. There is no loss of generality to take A = diag(1, ζa
+n, ζb
+n)
+up to conjugation and projective equivalence.
+Since A and A−1 are PGL3(C)-
+conjugates via B, then, by Theorem 2.2, A must be a non-homology. Moreover,
+we can always reduce to the case b = −a modulo n. Furthermore, we can assume
+by [18, Lemma 2.3.7] that B belongs to PBD(2, 1). Since BAB = A−1, we obtain
+B = [X : νZ : ν−1Y ] for some ν ∈ C∗.
+Through a projective transformation
+ψ = diag(1, λν, λ), which is in Norm (⟨A⟩, PGL3(C)), we can further reduce to
+ν = 1. Eventually, we conclude:
+Lemma 4.1. For each fixed integer n ≥ 3, there is, up to PGL3(C)-conjugation, a
+unique dihedral group D2n of order 2n. More precisely, any such group is conjugate
+to the group generated by B = [X : Z : Y ] and A = diag(1, ζn, ζ−1
+n ).
+Now, we will prove that a dihedral group G = ⟨ A, B⟩ as above has a real field
+of moduli, moreover, it descends to R.
+Proof. Since σ A = A−1 and σ B = B−1, then σG = G and G has a real field of
+moduli.
+On the other hand, we have seen in Theorem 2.2 that φ−1 A φ ∈ PGL3(R)
+through a projective transformation φ of the shape:
+φ =
+
+
+1
+0
+0
+0
+α
+β
+0
+α
+β
+
+ .
+It remains to see that φ−1 B φ ∈ PGL3(R) so that φ−1 G φ is a model of G over R.
+Indeed, we have
+
+ON PSEUDO-REAL FINITE SUBGROUPS IN PGL3(C)
+7
+φ−1 B φ
+=
+
+
+2ℑ(α β) i
+0
+0
+0
+β
+−β
+0
+−α
+α
+
+ [X : Z : Y ]
+
+
+1
+0
+0
+0
+α
+β
+0
+α
+β
+
+
+=
+
+
+2ℑ(α β)
+0
+0
+0
+−β
+β
+0
+α
+−α
+
+
+
+
+1
+0
+0
+0
+α
+β
+0
+α
+β
+
+
+=
+
+
+2ℑ(α β) i
+0
+0
+0
+−2ℑ(α β) i
+−2ℑ(β2) i
+0
+2ℑ(α2) i
+2ℑ(α β) i
+
+
+=
+
+
+2ℑ(α β)
+0
+0
+0
+−2ℑ(α β)
+−2ℑ(β2)
+0
+2ℑ(α2)
+2ℑ(α β)
+
+ ∈ PGL3(R).
+□
+This completes the proof of Theorem 2.4.
+5. The cases when G is a finite primitive subgroup of PGL3(C)
+Recall that the finite primitive subgroups PGL3(C) are the Hessian groups Hess∗,
+for ∗ = 216, 72, 36, the alternating groups A∗, for ∗ = 5, 6, and the Klein group
+PSL(2, 7) of order 168. We study their definability over R in this section.
+5.1. The Hessian groups Hess∗. The Hessian group of order 216, denoted by
+Hess216, is unique up to conjugation in PGL3(C). See [23, p. 217] or [18, Lemma
+2.3.7] for more details. For instance, we fix Hess216 = ⟨S, T, U, V ⟩ where
+S = diag(1, ζ3, ζ−1
+3 ), U = diag(1, 1, ζ3), V =
+
+
+1
+1
+1
+1
+ζ3
+ζ−1
+3
+1
+ζ−1
+3
+ζ3
+
+ , T = [Y : Z : X].
+Also, we consider the Hessian subgroup of order 72, Hess72 = ⟨S, T, V, UV U −1⟩,
+and the Hessian subgroup of order 36, Hess36 = ⟨S, T, V ⟩.
+Concerning the Hessian groups Hess∗, for ∗ ∈ {36, 72, 216}. We first show
+Proposition 5.1. Any of the Hessian groups Hess∗ has a real field of moduli.
+Proof. It is easy to see that σS = S−1, σU = U −1, and σT = T . Furthermore
+σV = 3V −1 in GL3(C), hence we also have σV = V −1 in PGL3(C). It follows that
+σ Hess∗ = Hess∗ if ∗ = 216 or 36. So Hess216 and Hess36 indeed have a real field of
+moduli. For Hess72, we get σ Hess72 = ⟨S, T, V, U −1V −1U⟩ ⊂ Hess216; another copy
+of Hess72 inside Hess216. The Group structure of Hess216 [10] assures that all copies
+of Hess72 are Hess216-conjugates, that is to say, there is a projective transformation
+ψ ∈ Hess216 such that ψ−1 Hess72 ψ = σ Hess72. From this we obtain that Hess72
+has a real field of moduli as well.
+□
+As a consequence,
+Corollary 5.2. The Hessian groups Hess∗ for ∗ = 216, 72 and 36 are all pseudo-
+real.
+Proof. It is easy to see that ST = T S, so ⟨S, T ⟩ is isomorphic to C3 × C3. By [17,
+Lemma 5.2] (see also [25, Section 4]), C3 ×C3 is a subgroup of PGL3(K) if and only
+if the field K contains a nontrivial cube root of unity. Since ζ3 /∈ R, we can’t reduce
+⟨S, T ⟩ to a subgroup of PGL3(R) as ζ3 /∈ R. In particular, φ−1 Hess∗ φ ⊈ PGL3(R)
+for any φ ∈ PGL3(C). Combining with Proposition 5.1, we conclude that Hess∗ is
+pseudo-real for ∗ = 216, 72 and 36 as claimed.
+□
+
+8
+E. BADR AND A. ELGUINDY
+5.2. The alternating groups A5 and A6. We first note that PGL3(C) possesses
+a single conjugacy class isomorphic to each of A5 and A6, see [23, p. 224, 225] or [18,
+Lemma 2.3.7]. Therefore, for i ∈ {5, 6} Ai and σ Ai must be PGL3(C)-conjugates.
+In other words, Ai has a real field of moduli.
+Since A6 contains C3 × C3 as a subgroup, then we can use the same argument
+as in Corollary 5.2 to deduce the following.
+Corollary 5.3. The alternating group A6 is pseudo-real.
+For the icosahedral group A5, the situation is different. To study it we fix the
+copy G := ⟨A, B, C⟩ in PGL3(C), where
+A = diag(1, ζ−1
+5 , ζ5), B = [X : Z : Y ], C =
+
+
+2
+2
+2
+1
+cos(4π/5)
+cos(2π/5)
+1
+cos(2π/5)
+cos(4π/5)
+
+ .
+According to [18, Lemma 2.3.7 ], G is PGL3(C)-conjugate to A5. Any subgroup of
+PGL3(C) isomorphic to A5 is PGL3(C) conjugate to G.
+Now, we are going to construct an explicit model for G over R.
+Recall, from our study above of the Dihedral group in §4, that ⟨ A, B⟩ descends to
+R via a transformation of the shape
+φ =
+
+
+1
+0
+0
+0
+α
+β
+0
+α
+β
+
+ ∈ PGL3(C).
+Moreover, one can check that φ−1 C φ equals
+
+
+4ℑ(α β)
+8ℑ(α β) ℜ(α)
+8ℑ(α β) ℜ(β)
+2ℑ(β)
+2
+�
+cos(4π/5)ℑ(αβ) − cos(2π/5)ℑ(αβ)
+�
+−2 cos(2π/5)ℑ(β2)
+2ℑ(α)
+2 cos(2π/5)ℑ(α2)
+2
+�
+cos(4π/5)ℑ(αβ) + cos(2π/5)ℑ(αβ)
+�
+
+ ,
+in PGL3(R). Thus all generators of G when conjugated by the same φ become in
+PGL3(R), hence the same is true for the whole group and the result follows.
+5.3. The Klein group PSL(2, 7). Again, there is a single conjugacy class of
+PSL(2, 7) in PGL3(C). Thus it has a real field of moduli. Also, we know by [18,
+Lemma 2.3.7] that a representative of such a class contains the element diag(1, ζ7, ζ3
+7).
+Theorem 2.2 applies to n = 7, a = 1, b = 3 to conclude that PSL(2, 7) is not defin-
+able over R.
+6. Connection to Arithmetic Geometry
+Let C : F(X, Y, Z) = 0 be a non-singular plane curve defined over C with non-
+trivial automorphism group Aut(C) in PGL3(C),
+Lemma 6.1. We have Aut(σC) = σ Aut(C)
+Proof. For any φ ∈ Aut(C), φF(X, Y, Z) = cF(X, Y, Z) for some c ∈ C∗. Applying
+σ to both sides yields
+σ(c) σF(X, Y, Z) = σ �φF(X, Y, Z)
+�
+=
+σφ (σF(X, Y, Z)) .
+That is, σφ leaves invariant σC : σF(X, Y, Z) = 0. Equivalently, σφ ∈ Aut(σC),
+hence σ Aut(C) ⊆ Aut(σC).
+By a similar argument we can show the other inclusion.
+□
+Theorem 6.2. Let C : F(X, Y, Z) = 0 be a smooth plane curve over C. If C has
+a real field of moduli in the Arithmetic Geometry sense, then Aut(C) has
+a real
+field of moduli in the Group Theory sense.
+The converse need not be true.
+
+ON PSEUDO-REAL FINITE SUBGROUPS IN PGL3(C)
+9
+Proof. Since C : F(X, Y, Z) = 0 has a real field of moduli, then it must be the case
+that σC : σF(X, Y, Z) = 0 and C : F(X, Y, Z) = 0 are C-projectively equivalent
+(isomorphic over C). Moreover, any isomorphism between complex non-singular
+plane curves C and C′ is always given by a projective transformation φ ∈ PGL3(C)
+such that their automorphism groups are conjugates via this φ. As a consequence,
+we obtain that φ−1 Aut(C) φ = Aut(σC), which equals σ Aut(C) by Lemma 6.1.
+Thus Aut(C) has a real field of moduli as claimed.
+To complete the argument, Example 6.3 below provides infinitely many counter
+examples that Aut(C) can descend R, but C : F(X, Y, Z) = 0 does not even have
+a real field of moduli.
+□
+Example 6.3. Consider the two-dimensional family Ca,b of smooth plane quintic
+curves given by
+Ca,b : X5 + Y 5 + Z5 + iaX(Y Z)2 + ibX3(Y Z),
+where a, b ∈ R∗ such that a/b ̸= (c5 − 3)c2
+2c5 − 1 ζm
+10 for any c ∈ C∗ and m ∈ {±1, ±3, 5}.
+• Non-singularity. We first note that no singular points lie over Y = 0.
+Indeed, if C has singularity at (α : 0 : β), then α and β must be 0 in
+order to satisfy FX = FZ = 0, a contradiction. Second, the resultant of
+f1(X, Z) := FY (X, 1, Z) and f2(X, Z) := FZ(X, 1, Z) with respect to X is
+given by
+ResX(f1, f2) = −125 i b3 (Z5 − 1)3.
+Using MATHEMATICA, one can verify that we have singular points over
+Z5 = 1 only if a/b = (c5 − 3)c2
+2c5 − 1 ζm
+10 for some c ∈ C∗ and m ∈ {±1, ±3, 5},
+which is absurd by assumption.
+• Automorphism group. The stratification of smooth plane quintics by
+their automorphism groups in [3, 6] assures that the group D10 gener-
+ated by ρ1 = diag(1, ζ5, ζ−1
+5 ) and ρ2 = [X : Z : Y ] is a always a sub-
+group of automorphisms for Ca,b. Moreover, if Ca,b admits a larger auto-
+morphism group, then it should be GAP(150, 5) = (Z/5Z)2 ⋊ S3, where
+in this situation Ca,b is K-isomorphic to the Fermat quintic curve F5;
+the most symmetric smooth quintic curve.
+In particular, there must
+be an extra automorphism ρ3 /∈ ⟨ρ1⟩ of order 5 that commutes with
+ρ1 as any Z/5Z inside (Z/5Z)2 ⋊ S3 is contained in a (Z/5Z)2.
+See
+Group Structure of (Z/5Z)2 ⋊ S3 [10]. Straightforward calculations in PGL3(C)
+lead to ρ3 = diag(1, α, β) with α5 = β5 = 1. Checking the action of such
+an automorphism on the defining equation of Ca,b tells us that a = b = 0
+or ρ3 ∈ ⟨ρ1⟩. Therefore, Aut(Ca,b) = D10 = ⟨ρ1, ρ2⟩.
+Now, we conclude by Theorem 2.4 that Aut(Ca,b) descends to R.
+• Ca,b does not have a real field of moduli. Suppose that C is a member
+of the family Ca,b such that C has a real field of moduli. Hence C and σC
+are C-projectively equivalent via some φ ∈ PGL3(C). Since C and σC
+belong to the same family Ca,b, we have σ Aut(C) = Aut(C) = ⟨ρ1, ρ2⟩.
+In particular, φ should be in the normalizer of ⟨ρ1, ρ2⟩ in PGL3(C). We
+reduce to the case φ−1ρ1φ = ρ1 or ρ−1 as {c, cζ5, cζ−1
+5 } ̸= {1, ζ2
+5, ζ−2
+5 } or
+{1, ζ3
+5, ζ−3
+5 } for any c ∈ C∗ by Lemma 3.1. Consequently, φ = diag(1, α, β)
+or [X : αZ : βY ] for some α, β ∈ C∗. Because φC = σC, we must have
+α5 = β5 = 1 and αβ = (αβ)2 = −1. The last condition is inconsistent,
+which means that C and σC are never C-isomorphic.
+
+10
+E. BADR AND A. ELGUINDY
+References
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+[22] A. Kuribayashi and K. Komiya, On Weierstrass points and automorphisms of curves of genus
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+[25] E. Yasinsky, The Jordan constant for Cremona group of rank 2, Korean Math. Soc. 54, no.
+5 (2017), 1859-1871.
+• Eslam Badr
+Mathematics Department, Faculty of Science, Cairo University, Giza-Egypt
+Email address: eslam@sci.cu.edu.eg
+Mathematics and Actuarial Science Department (MACT), American University in
+Cairo (AUC), New Cairo-Egypt
+Email address: eslammath@aucegypt.edu
+• Ahmad El-Guindy
+
+ON PSEUDO-REAL FINITE SUBGROUPS IN PGL3(C)
+11
+Mathematics Department, Faculty of Science, Cairo University, Giza, Egypt
+Email address: aelguindy@sci.cu.edu.eg
+

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@@ -0,0 +1,1737 @@
+Convergence of Adaptive Mixed Interior Penalty Dis-
+continuous Galerkin Methods for H(cur l)-Elliptic
+Problems
+K. Liu1, M. Tang2,, X. Q. Xing2 and L. Q. Zhong2
+1 School of Sciece, East China University of Technology, Nanchang, 330013, China
+2 School of Mathematical Sciences, South China Normal University, Guangzhou,
+510631, China
+Abstract. In this paper, we study the convergence of adaptive mixed interior penalty
+discontinuous Galerkin method for H(cur l)-elliptic problems. We first get the mixed
+model of H(cur l)-elliptic problem by introducing a new intermediate variable. Then
+we discuss the continuous variational problem and discrete variational problem, which
+based on interior penalty discontinuous Galerkin approximation. Next, we construct the
+corresponding posteriori error indicator, and prove the contraction of the summation of
+the energy error and the scaled error indicator. At last, we confirm and illustrate the
+theoretical result through some numerical experiments.
+AMS subject classifications: 65M15,65N12,65N30
+Key words: Adaptive mixed interior penalty discontinuous Galerkin methods, Convergence, H(cur l)-
+elliptic problems.
+1. Introduction
+Let Ω ⊂ �3 be Lipschitz bounded polygonal domain with a single connected boundary
+∂ Ω. We consider the following H(cur l)-elliptic problem
+∇ × µ∇ × u + κu = f
+in
+Ω,
+(1.1)
+u × n = 0
+on
+∂ Ω,
+(1.2)
+where n is the unit normal vector of the boundary ∂ Ω, f ∈ L2(Ω), µ and κ are piecewise
+constants is consistent with the initial partition �0 for Ω and satisfy µ1 < µ < µ2 and
+κ1 < κ < κ2, here, µi and κi(i = 1,2) are positive constants. By introducing an auxiliary
+∗Corresponding author. Email addresses: liukai@ecut.edu.cn (K. Liu), mingtang@m.scnu.edu.cn (M.
+Tang),xingxq@scnu.edu.cn(X. Q. Xing), zhong@m.scnu.edu.cn (L. Q. Zhong)
+1
+arXiv:2301.01439v1  [math.NA]  4 Jan 2023
+
+2
+K Liu et al.
+variable p = µ∇ × u, then we get the mixed scheme with the boundary value problem
+(1.1)-(1.2)
+p = µ∇ × u
+in
+Ω,
+(1.3)
+∇ × p + κu = f
+in
+Ω,
+(1.4)
+u × n = 0
+on
+∂ Ω.
+(1.5)
+The mixed finite element method is very convenient for processing high-order equations
+and equations containing two or more unknown functions, which has attracted widespread
+attention. For mixed finite element method, there are only few research results for Maxwell
+problem [13] and Maxwell’s eigenvalue problem [12,14,15].
+Adaptive finite element method automatically refines and optimizes meshes accord-
+ing to the singularity of solutions. It is a highly reliable and efficient numerical calculation
+method. At present, the convergence analysis research of the adaptive mixed finite element
+method for the elliptic equation is relatively complete. Chen, Holst and Xu [7] proved the
+convergence analysis of the adaptive mixed finite element algorithm for elliptic equations.
+Du and Xie [10] proved the convergence analysis of the adaptive mixed finite element
+algorithm for the convection diffusion equation. However, there are only few research
+results on the posterior error estimator of Maxwell’s equations for the adaptive mixed fi-
+nite element method. For example, Carstensen and Ma [5] establishes the convergence of
+adaptive mixed finite element methods for second-order linear non-self-adjoint indefinite
+elliptic problems. Carstensen, Hoppe, Sharma and Warburton [4] designs and analyzes
+the posterior error estimation of the adaptive hybrid conforming finite element method of
+H(cur l)-elliptic problem. Recently, Chung, Yuen and Zhong [8] present a-posteriori error
+analysis for the staggered discontinuous Galerkin method. As far as we know, there are not
+any published literatures for the convergence analysis of the adaptive mixed finite element
+method for the boundary value problem(1.3)-(1.5). Our contributions in this paper are to
+• construct a new error estimator, which does not include the negative power of the
+local mesh size in the jump term for the traditional DG method;
+• get the convergence of the Adaptive Mixed Interior Penalty Discontinuous Galerkin
+(AMIPDG) method by using the similar technique used in [2]. However, this tech-
+nique in [2] can not be used directly for mixed forms.
+We present our main result in the following theorem.
+Theorem 1.1. Let {�k,Uk,Qk, uk, pk,η(uk, pk;�k)}k≥0 be the sequence of meshes, finite
+element space, mixed discrete solution and posterior error estimate indicator produced by the
+AMIPDG algorithm. Then there exist constants ρ > 0 and δ ∈ (0,1), which depend on
+marking parameter and the shape regularity of the initial mesh �0, such that
+∥|u − uk+1|∥2
+k+1 + ρη2(uk+1, pk+1;�k+1) ≤ δ
+�
+∥|u − uk|∥2
+k + ρη2(uk, pk;�k)
+�
+.
+Therefore, for a given precision, the AMIPDG method will terminate after a finite number of
+operations.
+
+Convergence of AMIPDG methods for H(cur l)-elliptic problems
+3
+For convenience, we let C denote a generic positive constant which may be different
+at different occurrences and adopt the following notation. The subscripted constant Ci
+represents a particularly important constant. a ≲ b means a ≤ C b for some constants C
+which are independent of mesh sizes.
+The rest of this paper is organized as follows. In Section 2, we first present the contin-
+uous variational problem, the discrete variational problem, and the procedure of AMIPDG.
+In Section 3, we first show the upper bound estimate of the error, which is key to the con-
+vergence analysis, then we prove the indicator reduction and the convergence of AMIPDG
+algorithm. In Section 4, we provide some numerical experiments to illustrate the effective-
+ness of the AMIPDG.
+2. Adaptive Mixed interior penalty discontinuous Galerkin method
+In this section, we introduce the continuous variational problem, the discrete variational
+problem of mixed internal penalty discontinuous finite element method, and the procedure
+of AMIPDG.
+2.1. Continuous variational problem
+For an open and connected bounded domain D ⊂ �3, we denote by L2(D) (resp.
+L2(D) := (L2(D))3) the spaces of square-integrable functions (resp. vector fields) on D
+with inner product (·,·)0,D. We define the spaces
+H(cur l; D) = {u ∈ L2(D) : ∇ × u ∈ L2(D)},
+H(div; D) = {u ∈ L2(D) : ∇ · u ∈ L2(D)},
+with
+(u, v)cur l,D := (u, v)0,D + (∇ × u,∇ × v)0,D,
+∀u, v ∈ H(cur l; D),
+(u, v)div,D := (u, v)0,D + (∇ · u,∇ · v)0,D,
+∀u, v ∈ H(div; D),
+and the induced norm as:
+∥u∥2
+cur l,D := ∥u∥2
+0,D + ∥∇ × u∥2
+0,D, ∀u ∈ H(cur l, D),
+∥u∥2
+div,D := ∥u∥2
+0,D + ∥∇ · u∥2
+0,D,
+∀u ∈ H(div, D),
+respectively, where ∥ · ∥L2(D) := (·,·)1/2
+D
+denotes the norm of the space L2(D) or L2(D). We
+also define H0(cur l; D) = {v ∈ H(cur l; D) : v × n = 0 on ∂ D} in the trace sense.
+Next, we first define two space U := H0(curl;Ω),Q := L2(Ω). Then, the mixed vari-
+ational problem of the mixed boundary value problem (1.3)-(1.5) reads as: find (u, p) ∈
+U × Q such that:
+a(p,q) − b(u,q) = ℓ1(q),
+∀q ∈ Q,
+(2.1)
+d(v, p) + c(u, v) = ℓ2(v),
+∀v ∈ U.
+(2.2)
+
+4
+K Liu et al.
+The bilinear forms a, b, c and the functionals ℓ1(·),ℓ2(·) are given by
+a(p,q) := (p,q),
+(2.3)
+b(u,q) := (µ∇ × u,q),
+(2.4)
+c(u, v) := (κu, v),
+(2.5)
+d(v, p) := (∇ × v, p)
+(2.6)
+ℓ1(q) := 0,
+(2.7)
+ℓ2(v) := ( f , v).
+(2.8)
+The operator-theoretic framework involves operator � : (U × Q) → (U × Q)∗ defined
+by
+(� (u, p))(v,q) := a(p,q) − b(u,q) + d(v, p) + c(u, v),∀u, v ∈ U, p,q ∈ Q,
+(2.9)
+where (Q × U)∗ is the dual spaces of (Q × U). Then we can rewrite (2.1)-(2.2) as
+(� (u, p))(v,q) = ℓ(v,q),
+(2.10)
+with ℓ(v,q) = ℓ1(q) + ℓ2(v), and ℓi are given by (2.7)-(2.8).
+Then, we state the well-posedness of the variational problem (2.1)-(2.2) in the follow-
+ing lemma, and it can be found in section 3 of [3].
+Lemma 2.1. Under the assumptions on the problem of (1.1)-(1.2), � is a continuous and
+bijective linear operator. Hence, for any ℓ = (ℓ1,ℓ2) ∈ (Q×U)∗, the mixed variational problem
+(2.1)-(2.2) has a unique solution (u, p) ∈ (U × Q), which satisfy the following continuously
+∥(u, p)∥U×Q := (∥u∥2
+curl,Ω + ∥p∥2
+0)1/2 ≲ ∥ℓ1∥Q∗ + ∥ℓ2∥U∗.
+(2.11)
+2.2. Discrete variational problem
+We suppose that �h is a family of shape regularity, quasi-uniform and conform tetrahe-
+dral generation on Ω. Let hτ = |τ|1/3 denote the mesh size with |τ| being the volume of
+τ ∈ �h.
+Define the discontinuous finite element function space �(�h) as:
+�(�h) = {v ∈ L2(Ω) : vτ = v|τ ∈ (Pl(τ))3,
+∀τ ∈ �h},
+where Pl(τ) is the set of polynomials defined in the volume τ whose degree does not exceed
+l, where l ≥ 1 is an integer.
+Let �h, � 0
+h and � ∂
+h denote the set of the all faces of its volumes, and the set of internal
+faces, and the set of boundary faces, respectively. Thus, �h = � 0
+h
+�
+� ∂
+h . Let H1(Ω;�h) be
+the space of piecewise Sobolev functions defined by
+H1(Ω;�h) =
+�
+v ∈ L2(Ω) : vτ = v|τ ∈ H1(τ),
+∀ τ ∈ �h
+�
+.
+
+Convergence of AMIPDG methods for H(cur l)-elliptic problems
+5
+and H1(Ω;�h) = (H1(Ω;�h))3. Let L2(�h) be the set of L2 functions defined on �h. More-
+over, we define the following inner products
+(v, w)� ′
+h
+=
+�
+τ∈� ′
+h
+�
+τ
+v · wdx,
+∀v, w ∈ L2(Ω), ∀�
+′
+h ⊂ �h,
+< v, w >� ′
+h
+=
+�
+f ∈� ′
+h
+�
+f
+v · wds,
+∀v, w ∈ L2(�h), ∀�
+′
+h ⊂ �h.
+For f ∈ � 0
+h , we have τi ∈ �h(i = 1,2), such that f = ∂ τ1 ∩ ∂ τ2. Then we denote the
+jump and average of v as:
+[[v]]
+=
+v1 × n1 + v2 × n2,
+∀v ∈ H1(Ω;�h),
+{{v}}
+=
+v1 + v2
+2
+,
+∀v ∈ H1(Ω;�h),
+where v i denote the values of v on v|τi(i = 1,2) and ni denote the out unit normal vectors
+on f exterior v|τi.
+For f ∈ � ∂
+h , we have τ ∈ �h, such that f = ∂ τ ∩ ∂ Ω. Then we denote the jump and
+average of v as:
+[[v]] = vτ × n∂ Ω, {{v}} = vτ.
+(2.12)
+Next, we give the corresponding discrete scheme of (2.1)-(2.2). Firstly, we define the
+corresponding discrete space as follow
+Uh := {vh ∈ �(�h)|
+[[vh]]|f = 0,∀f ∈ � ∂
+h },
+Qh := �(�h).
+Then, the formulation of the discrete Mixed Interior Penalty Discontinuous Galerkin (MIPDG)
+method reads: find (uh, ph) ∈ (Uh,Qh) such that
+ah(ph,qh) − bh(uh,qh) = ℓ1,h(qh) + d1,h(uh,qh),
+∀qh ∈ Qh,
+(2.13)
+dh(vh, ph) + ch(uh, vh) = ℓ2,h(vh) + d2,h(uh, vh),
+∀vh ∈ Uh,
+(2.14)
+where
+ah(ph,qh) := (ph,qh)�h,
+bh(uh,qh) := (µ∇ × uh,qh)�h,
+ch(uh, vh) := (κuh, vh)�h,
+dh(vh, ph) := (∇ × vh, ph)�h,
+ℓ1,h(qh) := 0,
+ℓ2,h(vh) := ( f , vh)�h,
+d1,h(uh,qh) := − < {{µqh}},[[uh]] >�h,
+d2,h(uh, vh) :=< ({{µ∇ × uh}} − αh−1
+f [[uh]]),[[vh]] >�h,
+
+6
+K Liu et al.
+here the constant α > 0 denote the penalty parameter, hf denote the diameter of the
+circumcircle of f . Thus hτ ≈ hf .
+Remark 2.1. The calculation of ∇ × uh in the bilinear terms are piecewise derivations.
+The standard symmetric Interior Penalty Discontinuous Galerkin (IPDG) method of the
+boundary value problem (1.1)-(1.2) is to find uh ∈ Uh, such that
+aIP(uh, vh)
+:= (κuh, vh)�h + (µ∇ × uh,∇ × vh)�h− < {{µ∇ × vh}},[[uh]] >�h
+− < {{µ∇ × uh}},[[vh]] >�h +αh−1
+f
+< [[uh]],[[vh]] >�h
+(2.15)
+= ( f , vh)�h.
+The following lemma shows that the discrete variational problems (2.13)-(2.14) and (2.15)
+are equivalent.
+Lemma 2.2. [ [3], Theorem 4.1] The formulations (2.13)-(2.14) and (2.15) are formally
+equivalent in the following sense. If (uh, ph) ∈ (Uh,Qh) are the solution of discrete variational
+problem (2.13)-(2.14), then uh ∈ Uh solves (2.15). Conversely, if uh ∈ Uh solves (2.15), then
+there exists some ph ∈ Qh such that (uh, ph) ∈ (Uh,Qh) are the solution of (2.13)-(2.14).
+Ayuso de Dios, Hiptmair and Pagliantini proved the well-posedness of (2.15) in section
+2 of [1]. Therefore, by combining Lemma 2.2, we obtain the well-posedness of discrete
+variational problems (2.13)-(2.14).
+2.3. Adaptive Mixed Interior Penalty Discontinuous Galerkin method(AMIPDG)
+Our adaptive cycle can be implemented by the following algorithm:
+Next, we will discuss each step in AEFEM in detail.
+2.3.1. Procedure SOLVE
+For f ∈ L2(Ω), and a shape regular mesh �k, Let (uk, pk) be the exact MIPDG solution of
+(2.13)-(2.14). Here, we assume that the solutions (uk, pk) can be solved accurately.
+2.3.2. Procedure ESTIMATE
+A posteriori error indicator is an essential ingredient of adaptivity. They are computable
+quantities depending on the computed solution(s) and data that provide information about
+the quality of approximation and may consequently be used to make judicious mesh modi-
+fications. Here, we design a new posteriori error estimation indicator for equations (2.13)-
+(2.14), which is similar to that in [20]. For τ ∈ �h, f ∈ �h and (vh,qh) ∈ Uh × Qh, the
+residual a posteriori error estimator for the symmetric AMIPDG method is given by
+η2(vh,qh;τ) :
+=
+∥R1(vh,qh)∥2
+L2(τ) + h2
+τ
+�
+∥R2(vh,qh)∥2
+L2(τ) + ∥R3(vh)∥2
+L2(τ)
+�
++
+�
+f ∈∂ τ
+hf
+�
+∥J1(qh)∥2
+L2(f ) + ∥J2(vh)∥2
+L2(f )
+�
+.
+(2.16)
+
+Convergence of AMIPDG methods for H(cur l)-elliptic problems
+7
+Algorithm 2.1 Adaptive Mixed Interior Penalty Discontinuous Galerkin Method (AMIPDG)
+cycle
+Input initial triangulation �0; data f ; tolerance tol; marking parameter θ ∈ (0,1).
+Output a triangulation �J; MIPDG solution (uJ, pJ).
+η = 1; k = 0;
+while η ≥ tol
+SOLVE solve discrete varational problem (2.13)-(2.14) on �k to get the solution (uk, pk);
+ESTIMATE compute the posterior error estimator η = η(uk, pk,�k) by using (2.17);
+MARK seek a minimum cardinality �k ⊂ �k such that
+η2 �
+uk, pk,�k
+�
+≥ θη2 �
+uk, pk,�k
+�
+;
+REFINE bisect elements in �k and the neighboring elements to form a conforming �k+1;
+k = k + 1;
+end
+uJ = uk; pJ = pk; �J = �k;
+They consist of the element residuals and face jump residuals as
+R1(vh,qh)|τ := qh|τ − µ∇ × vh|τ,
+R2(vh,qh)|τ := f |τ − (∇ × qh + κvh)|τ,
+R3(vh)|τ := ∇ · ( f |τ − κvh|τ),
+J1(qh)|f := [[qh]],
+J2(vh)|f := [[(f − κvh)]].
+where hf denote the diameter of the circumcircle of f , and hτ ≈ hf .
+For any set � ′
+h ⊆ �h, the error indicator is defined as
+η2(vh,qh;� ′
+h ) =
+�
+τ∈� ′
+h
+η2(vh,qh;τ).
+(2.17)
+2.3.3. Procedure MARK
+We use the Dörfler mark which was proposed by Dörfler [9]. Set marking parameter θ ∈
+(0,1), the module MARK outputs a subset of marked elements �k ⊂ �k with minimal
+cardinality, such that
+η2(v k,q k;�k) ≥ θη2(v k,q k;�k).
+(2.18)
+2.3.4. Procedure REFINE
+Our implementation of REFINE uses the longest edge bisection strategy. A detailed intro-
+duction about the longest edge bisection strategy was provided in [6]. To avoid confusion,
+the relationship between the two tetrahedral meshes �h and �H that are nested into each
+
+8
+K Liu et al.
+other is defined as: �h is the new mesh division of �H after one cycle of the above cycle
+process, abbreviated as �H ≤ �h.
+3. Convergence of AMIPDG algorithm
+In this section, we establish the upper bound estimate of the error. Subsequently, we
+demonstrate that the sum of the energy error and the error estimator between two consec-
+utive adaptive loops is a contraction. Finally, we proof that the AMIPDG is convergence.
+3.1. The upper bound estimate of the error
+In this subsection, before establishing the reliability of a posteriori error estimator, we
+need to define the corresponding DG norm, for any (v,q) ∈ U × Q and (vh,qh) ∈ Uh × Qh,
+∥(v,q)
+−
+(vh,qh)∥2
+DG := ∥q − qh∥2
+L2(Ω) + ∥κ(v − vh)∥2
+L2(Ω)
++
+�
+τ∈�h
+∥µ∇ × (v − vh)∥2
+L2(τ) +
+�
+f ∈�h
+αh−1
+f
+< [[vh]],[[vh]] >f .
+(3.1)
+Remark 3.1. For any v ∈ U and vh ∈ Uh, we have
+∥[[vh]]∥2
+L2(f ) = ∥[[(v − vh)]]∥2
+L2(f ),
+∀f ∈ �h.
+In fact, v ∈ U implies that [[v]]|f = 0 (see Chapter 5 of [16]).
+We summarize our main result in this subsection as follows.
+Theorem 3.1. Let (u, p) ∈ U×Q and (uh, ph) ∈ Uh ×Qh be the solutions of (2.1)-(2.2) and
+(2.13)-(2.14), respectively. Let η(uh, ph;�h) be the residual error indicator of (2.17). Then
+we have the following estimate
+∥(u, p) − (uh, ph)∥2
+DG ≤ C1η2(uh, ph;�h),
+(3.2)
+where the constant C1 depending on the shape regularity of mesh.
+Let (uh, ph) ∈ Uh × Qh be the solution of (2.13)-(2.14), similarly to [4], we introduce
+the nonconformity of the MSIPDG method results in some consistency error:
+ζ := min
+˜vh∈U
+� �
+τ∈�h
+(∥uh − ˜vh∥2
+L2(τ) + ∥∇ × (uh − ˜vh)∥2
+L2(τ))
+�1/2.
+(3.3)
+We denote that ˜uh ∈ U is the unique minimizer of (3.3), namely
+˜ζ =
+� �
+τ∈�h
+(∥uh − ˜uh∥2
+L2(τ) + ∥∇ × (uh − ˜uh)∥2
+L2(τ))
+�1/2.
+(3.4)
+
+Convergence of AMIPDG methods for H(cur l)-elliptic problems
+9
+Lemma 3.1. Let (u, p) ∈ U × Q and (uh, ph) ∈ Uh × Qh be the solutions of (2.1)-(2.2) and
+(2.13)-(2.14), respectively, let ˜uh be the unique minimizer of (3.3), then
+∥(u − ˜uh, p − ph)∥U×Q = (∥u − ˜uh∥2
+curl,Ω + ∥p − ph∥2
+0)1/2 ≲ ∥˜ℓ1∥Q∗ + ∥˜ℓ2∥U∗,
+where the residuals ˜ℓ1 ∈ Q∗ and ˜ℓ2 ∈ U∗ defined by
+˜ℓ1(q) = ℓ1(q) − a(ph,q) + b(˜uh,q),
+∀q ∈ Q,
+(3.5)
+˜ℓ2(v) = ℓ2(v) − d(v, ph) − c(˜uh, v),
+∀v ∈ U.
+(3.6)
+Proof. For any q1,q2,q ∈ Q and any v1, v2, v ∈ U. we have the following property by
+(2.9)
+(� (v1 + v2,q1 + q2))(v,q)
+= a(q1 + q2,q) − b(v1 + v2,q) + d(v,q1 + q2) + c(v1 + v2, v)
+= a(q1,q) − b(v1,q) + d(v,q1) + c(v1, v)
++a(q2,q) − b(v2,q) + d(v,q2) + c(v2, v)
+= (� (v1,q1))(v,q) + (� (v2,q2))(v,q).
+Thus,
+(� (u − ˜uh, p − ph))(v,q)
+= (� (u, p))(v,q) − (� (˜uh, ph))(v,q)
+= (ℓ1(q) + ℓ2(v)) − (a(ph,q) − b(˜uh,q) + d(v, ph) + c(˜uh, v))
+= ˜ℓ1(q) + ˜ℓ2(v).
+In fact that (u − ˜uh, p −ph) ∈ U ×Q and combining the Lemma 2.1 can concludes the proof.
+Next, we will provide upper bounds for ∥˜ℓ1∥Q∗ and ∥˜ℓ2∥U∗ in Lemmas 3.2 and 3.4,
+respectively.
+Lemma 3.2. Let (uh, ph) ∈ Uh × Qh be the solutions of (2.13)-(2.14), and ˜uh be the unique
+minimizer of (3.3). Then we get the estimate of the linear functional ˜ℓ1 defined in (3.5) as
+following
+∥˜ℓ1∥Q∗ ≲
+� �
+τ∈�h
+∥R1(uh, ph)∥2
+L2(τ)
+�1/2 +
+� �
+τ∈�h
+∥∇ × (˜uh − uh)∥2
+L2(τ)
+�1/2.
+(3.7)
+Proof. For any q ∈ Q, by the definition of ˜ℓ1, we have
+˜ℓ1(q) =
+�
+τ∈�h
+�
+τ
+�
+(µ∇ × uh − ph) + µ∇ × (˜uh − uh)
+�
+· qdx.
+
+10
+K Liu et al.
+Then applying the Hölder inequality and the Cauchy-Schwarz inequality,
+|˜ℓ1(q)| ≤
+�
+τ∈�h
+∥µ∇ × uh − ph∥L2(τ)∥q∥L2(Ω) +
+�
+τ∈�h
+∥µ∇ × (˜uh − uh)∥L2(τ)∥q∥L2(Ω)
+≲
+�� �
+τ∈�h
+∥R1(uh, ph)∥2
+L2(τ)
+�1/2 +
+� �
+τ∈�h
+∥∇ × (˜uh − uh)∥2
+L2(τ)
+�1/2�
+∥q∥L2(Ω),
+conclude the proof.
+Before estimating the term ∥˜ℓ2∥U∗, we need to introduce the following interpolation
+operator with the corresponding approximations.
+Lemma 3.3. [ [19], Theorem 1] Let Nd1
+0(Ω;�h) be the lowest order edge elements of Nédélec
+first family. Then there exists an operator Πh : H0(curl;Ω) → Nd1
+0(Ω;�h) with the following
+properties: For every v ∈ H0(curl;Ω), there exist ϕ ∈ H1
+0(Ω) and z ∈ H1
+0(Ω), such that
+v − Πhv = ∇ϕ + z.
+And for any τ ∈ �h and f ∈ �h, we have
+h−1
+τ ∥ϕ∥L2(τ) + ∥∇ϕ∥L2(τ) ≲ hτ∥v∥L2(Ωτ),
+h−1
+τ ∥z∥L2(τ) + ∥∇z∥L2(τ) ≲ hτ∥∇ × v∥L2(Ωτ),
+where Ωτ =
+�
+f ∈τ
+Ωf , Ωf = {τ′ ∈ �h, f ∈ τ′}, and the constants depending on the shape
+regularity of the mesh.
+Lemma 3.4. Let (uh, ph) ∈ Uh × Qh be the solution of (2.13)-(2.14), and ˜uh be the unique
+solution of (3.3). Then the linear functional ˜ℓ2 defined in (3.6) satisfies the following estimate
+∥˜ℓ2∥U∗ ≲
+� �
+τ∈�
+h2
+τ(∥R2(uh, ph)∥2
+L2(τ) + ∥R2(uh)∥2
+L2(τ))
++
+�
+f ∈�
+hf (∥J1(ph)∥2
+L2(f ) + ∥J2(uh)∥2
+L2(f )) +
+�
+τ∈�
+∥uh − ˜uh∥2
+L2(τ)
+�1/2
+.
+(3.8)
+Proof. For any v ∈ U and Πh given by Lemma 3.3, we have
+v − Πhv = ∇ϕ + z,
+(3.9)
+where ϕ ∈ H1
+0(Ω) and z ∈ H1
+0(Ω). According to linearity of the operator ˜ℓ2 and (3.9), we
+have
+˜ℓ2(v) = ˜ℓ2(Πhv) + ˜ℓ2(v − Πhv) = ˜ℓ2(Πhv) + ˜ℓ2(∇ϕ) + ˜ℓ2(z).
+(3.10)
+We will next estimate the three terms on the right hand side of (3.10).
+
+Convergence of AMIPDG methods for H(cur l)-elliptic problems
+11
+For the first term ˜ℓ2(Πhv) of (3.10), using the definition of ˜ℓ2, we have
+˜ℓ2(Πhv)
+=
+ℓ2(Πhv) − d(Πhv, ph) − c(˜uh,Πhv)
+=
+ℓ2(Πhv) − d(Πhv, ph) − c(uh,Πhv) + c(uh − ˜uh,Πhv).
+Noting that Πhv ∈ Nd1
+0(Ω;�h) ⊆ Uh has zero jumps, and combining (2.14), we have
+ℓ2(Πhv) − d(Πhv, ph) − c(uh,Πhv) = ℓ2,h(Πhv) − dh(Πhv, ph) − ch(uh,Πhv) = 0.
+Thus, we have
+˜ℓ2(Πhv)
+=
+c(vh − ˜uh,Πhv)
+=
+c(vh − ˜uh, v) + c(vh − ˜uh,Πhv − v)
+≤
+∥κ∥0,∞∥vh − ˜uh∥0,�h(∥v∥0,�h + ∥Πhv − v∥0,�h).
+Then using (3.9), triangle inequality and Lemma 3.3, we get
+˜ℓ2(Πhv)
+≤
+∥κ∥0,∞∥vh − ˜uh∥0,�h(∥v∥0,�h + ∥∇ϕ + z∥0,�h)
+≤
+∥κ∥0,∞∥vh − ˜uh∥0,�h(∥v∥0,�h + ∥∇ϕ∥0,�h + ∥z∥0,�h)
+≤
+∥κ∥0,∞∥vh − ˜uh∥0,�h∥v∥curl,�h.
+(3.11)
+For the second term ˜ℓ2(∇ϕ) of (3.10), using the definition of ˜ℓ2, (2.8), (2.4), (2.6) and
+the fact ∇ × ∇ϕ = 0, which implies
+˜ℓ2(∇ϕ)
+=
+ℓ2(∇ϕ) − d(∇ϕ, ph) − c(˜uh,∇ϕ)
+=
+( f ,∇ϕ) − (∇ × ∇ϕ, ph) − (κ˜uh,∇ϕ)
+=
+( f ,∇ϕ) − (κ˜uh,∇ϕ).
+(3.12)
+By (3.12) and Green’s formula, we have
+˜ℓ2(∇ϕ)
+=
+( f ,∇ϕ) − (κuh,∇ϕ) + (κ(uh − ˜uh),∇ϕ)
+≤
+�
+τ∈�h
+(R3(uh),ϕ)0,τ +
+�
+f ∈�h
+< J2(uh),ϕ >0,f +(κ(uh − ˜uh),∇ϕ).
+Applying the Cauchy-Schwarz inequality, Lemma 3.3 and trace inequality, we have
+˜ℓ2(∇ϕ) ≤
+� �
+τ∈�h
+h2
+τ∥R3(uh)∥2
+0,τ +
+�
+f ∈�h
+hf ∥J2(uh)∥2
+0,f
++
+�
+τ∈�h
+∥κ∥0,∞∥uh − ˜uh∥2
+0,τ
+�1/2
+∥v∥curl,�h.
+(3.13)
+
+12
+K Liu et al.
+Similarly, for the third term ˜ℓ2(z) of (3.10), we have
+˜ℓ2(z)
+=
+( f , z) − (∇ × z, ph) − (κ˜uh, z)
+=
+( f , z) − (∇ × z, ph) − (κuh, z) + (κ(uh − ˜uh), z)
+≤
+� �
+τ∈�h
+h2
+τ∥R2(uh, ph)∥2
+0,τ +
+�
+f ∈�h
+hf ∥J1(ph)∥2
+0,f
++
+�
+τ∈�h
+∥κ∥0,∞∥uh − ˜uh∥2
+0,τ
+�1/2
+∥v∥curl,�h.
+(3.14)
+Substituting (3.11), (3.13) and (3.14) into (3.10), the proof is completed.
+Notice that both (3.7) and (3.8) are related to the terms
+�
+τ∈�h
+∥∇ × (˜uh − uh)∥2
+L2(τ) and
+�
+τ∈�
+∥uh − ˜uh∥2
+L2(τ), which are a part of ˜ζ. Therefore, we prove upper bounds for ˜ζ in the
+following Lemma.
+Lemma 3.5. Let (uh, ph) ∈ Uh × Qh be the solutions of (2.13)-(2.14) and ˜ζ be consistency
+error of (3.4), we have
+˜ζ2 ≲ η2(uh, ph;�h).
+(3.15)
+Proof. For any vh ∈ Uh, there exit an interpolation operator �h : H1(Ω;�h) → Uc
+h, such
+that(see Proposition 4.5 of [11])
+∥vh − �hvh∥2
+L2(Ω) ≲
+�
+f ∈�h
+hf ∥[[vh]]∥2
+L2(f ),
+(3.16)
+�
+τ∈�h
+∥∇ × (vh − �hvh)∥2
+L2(τ) ≲
+�
+f ∈�h
+h−1
+f ∥[[vh]]∥2
+L2(f ).
+(3.17)
+Then, combining (3.3), (3.4), (3.16), (3.17), and the fact hf < 1, we get
+˜ζ2
+=
+�
+τ∈�h
+(∥uh − ˜uh∥2
+L2(τ) + ∥∇ × (uh − ˜uh)∥2
+L2(τ))
+≤
+�
+τ∈�h
+(∥uh − �huh∥2
+L2(τ) + ∥∇ × (uh − �huh)∥2
+L2(τ))
+≲
+�
+f ∈�h
+hf ∥[[uh]]∥2
+L2(f ) +
+�
+f ∈�h
+h−1
+f ∥[[uh]]∥2
+L2(f )
+≲
+�
+f ∈�h
+h−1
+f ∥[[uh]]∥2
+L2(f ).
+(3.18)
+Noting that (uh, ph) ∈ Uh × Qh is the solution of discrete variational problem (2.13)-
+(2.14). Then by using Lemma 2.2, we know that uh is the solution of discrete variational
+problem (2.15). Hence, we have ( see Lemma 5 of [20])
+α∥h−1/2
+f
+[[uh]]∥L2(�h) ≲ η(uh, ph;�h).
+(3.19)
+
+Convergence of AMIPDG methods for H(cur l)-elliptic problems
+13
+At last, combining (3.18) and (3.19), we have
+˜ζ2
+≲
+η2(uh, ph;� ).
+Combining Lemmas 3.1, 3.2, 3.4 and 3.5, we will prove Theorem 3.1.
+Proof. [ Proof of Theorem 3.1:] By using (3.1), the triangle inequality, (3.4), Lemmas
+3.1, 3.2, 3.4, 3.5 and (3.19), we get
+∥(u, p) − (uh, ph)∥2
+DG
+≲
+∥p − ph∥2
+L2(Ω) + ∥κ(u − uh)∥2
+L2(Ω)
++
+�
+τ∈�h
+∥∇ × µ(u − uh)∥2
+L2(τ) +
+�
+f ∈�h
+αh−1
+f
+< [[uh]],[[uh]] >f
+≲
+∥p − ph∥2
+L2(Ω) + ∥u − ˜uh∥2
+cur l,Ω + ˜ζ2 +
+�
+f ∈�h
+αh−1
+f
+< [[uh]],[[uh]] >f
+=
+∥(u − ˜uh, p − ph)∥U×Q + ˜ζ2 +
+�
+f ∈�h
+αh−1
+f
+< [[uh]],[[uh]] >f
+≲
+∥˜ℓ1∥2
+Q∗ + ∥˜ℓ2∥2
+U∗ + ˜ζ2 +
+�
+f ∈�h
+αh−1
+f
+< [[uh]],[[uh]] >f
+≤
+C1η2(uh, ph;�h).
+3.2. The error reduces on two successive meshes
+For convenience, for any v ∈ U and vh ∈ Uh, we denote
+∥|v − vh|∥2
+h
+=
+∥κ(v − vh)∥2
+L2(Ω) +
+�
+τ∈�h
+∥∇ × µ(v − vh)∥2
+L2(τ)
++
+�
+f ∈�h
+αh−1
+f
+< [[vh]],[[vh]] >f .
+(3.20)
+Let Uc
+h be the H(cur l) conforming subspace of Uh given by
+Uc
+h := Uh ∩ H0(curl;Ω).
+Then, there is a subspace U⊥
+h which can orthogonally decompose Uh under L2 inner product
+such that Uh := Uc
+h ⊕ U⊥
+h . Especially, if (uh, ph) ∈ Uh × Qh is the solution of (2.13)-(2.14),
+then we have
+∥|u⊥
+h |∥2
+h ≲ α
+�
+f ∈∂ τ
+∥h−1/2
+f
+[[uh]]∥2
+L2(f ).
+(3.21)
+In fact, from the Lemma 2.2, notice that uh satisfies the IPDG scheme of (2.15), and ac-
+cording to Lemma 2 in [20], we can obtain (3.21).
+
+14
+K Liu et al.
+In order to easily estimate the jump term of face �h, we need to introduce the lifting
+operators and the corresponding stability estimates, more details are referenced to Propo-
+sition 12 in [18].
+Let �h : H1(Ω;�h) → Uh be the lifting operators, which satisfies the following equality
+�
+Ω
+�h(v) · wdx =< [[v]],{{w}} >�h,
+∀w ∈ Uh,
+(3.22)
+and
+∥�h(v)∥L2(Ω) ≤ C� ∥h−1/2[[v]]∥L2(�h),
+(3.23)
+where the constant C� depending on the shape regularity of mesh �h and the degree of
+polynomial l.
+Lemma 3.6. Let (u, p) ∈ U × Q and (uh, ph) ∈ Uh × Qh be the solutions of (2.1)-(2.2) and
+(2.13)-(2.14), respectively, we have
+∥p − ph∥L2(Ω)
+≲
+∥∇ × (u − uh)∥L2(Ω) + η(uh, ph;�h),
+(3.24)
+∥ph − pH∥L2(Ω)
+≲
+∥∇ × (uh − uH)∥L2(Ω)
++
+�
+η(uh, ph;�h) + η(uH, pH;�H)
+�
+.
+(3.25)
+Proof. Noting that Qh ⊆ Q, and using (2.1), the definition of R1(uh, ph) and (2.16), we
+have
+∥p − ph∥L2(�h)
+≤
+sup
+∀q∈Q
+(p − ph,q)�h
+∥q∥L2(�h)
+=
+sup
+∀q∈Q
+(µ∇ × u,q)�h −
+�
+R1(uh, ph) + µ∇ × uh,q
+�
+�h
+∥q∥L2(�h)
+≤
+sup
+∀q∈Q
+(µ∇ × (u − uh),q)�h −
+�
+R1(uh, ph),q
+�
+�h
+∥q∥L2(�h)
+≲
+∥∇ × (u − uh)∥L2(�h) + η(uh, ph;�h).
+Similarly, using the definition of R1(uh, ph), (2.13), (3.21)-(3.23), and the fact [[uh]] =
+
+Convergence of AMIPDG methods for H(cur l)-elliptic problems
+15
+[[uc
+h + u⊥
+h ]] = [[u⊥
+h ]], we have
+∥ph − pH∥L2(�h) ≤
+sup
+∀qh∈Qh
+(ph − pH,qh)�h
+∥qh∥L2(�h)
+≤
+sup
+∀qh∈Qh
+(ph,qh)�h −
+�
+R1(uH, pH) + µ∇ × uH,qh
+�
+�h
+∥qh∥L2(�h)
+≤
+sup
+∀qh∈Qh
+(µ∇ × uh,qh)�h+ < {{qh}},[[µuh]] >�h −
+�
+R1(uH, pH) + µ∇ × uH,qh
+�
+�h
+∥qh∥L2(�h)
+=
+sup
+∀qh∈Qh
+(µ∇ × (uh − uH),qh)�h+ < {{qh}},[[µuh]] >�h −
+�
+R1(uH, pH),qh
+�
+�h
+∥qh∥L2(�h)
+≲
+∥∇ × (uh − uH)∥L2(�h) + ∥h−1/2
+τ
+[[uh]]∥L2(�h) + η(uH, pH;�H)
+≲
+∥∇ × (uh − uH)∥L2(�h) + C� ∥h−1/2
+τ
+[[u⊥
+h ]]∥L2(�h) + η(uH, pH;�H)
+≲
+∥∇ × (uh − uH)∥L2(τ) +
+�
+η(uh, ph;�h) + η(uH, pH;�H)
+�
+.
+Remark 3.2. Noting that ∥(u, p)−(uh, ph)∥2
+DG+η2(uh, ph;�h) and ∥|u−uh|∥2
+h+η2(uh, ph;�h)
+are equivalent. In fact, by (3.24), we first know that
+∥(u, p) − (uh, ph)∥2
+DG + η2(uh, ph;�h)
+= ∥|u − uh|∥2
+h + ∥p − ph∥2
+L2(�h) + η2(uh, ph;�h)
+≲ ∥|u − uh|∥2
+h + η2(uh, ph;�h).
+Secondly, it is shown by the definition of ∥ · ∥DG
+∥|u − uh|∥2
+h + η2(uh, ph;�h) ≤ ∥(u, p) − (uh, ph)∥2
+DG + η2(uh, ph;�h).
+Thus, we next only need to consider the convergence of ∥|u − uh|∥2
+h + η2(uh, ph;�h).
+We first show that the error plus some quantity reduces with a fixed factor on two
+successive meshes.
+Lemma 3.7. Given f ∈ L2(Ω) and two tetrahedral mesh �h and �H, where �H ≤ �h. Let
+(u, p) ∈ U × Q be the solution of (2.1)-(2.2), and (uh, ph) ∈ Uh × Qh, (uH, pH) ∈ UH × QH
+be the solutions of (2.13)-(2.14), respectively. Then there exit two constants δ1,δ2 ∈ (0,1),
+such that
+∥|u − uh|∥2
+h
+≤
+(1 + δ1)∥|u − uH|∥2
+H − 1 − δ2
+2
+∥|uh − uH|∥2
+h
++
+C3
+δ1δ2α
+�
+η2(uh, ph;�h) + η2(uH, pH;�H)
+�
+.
+(3.26)
+where C3 depending on the C� .
+
+16
+K Liu et al.
+Proof. Choosing that q = ∇ × v, and subtracting (2.1) from (2.2), we obtain
+(κu, v) + (µ∇ × u,∇ × v) = ( f , v).
+(3.27)
+Subtracting (2.15) from (3.27) with v = vh = uc
+h − uc
+H, and using [[uc
+h − uc
+H]] = 0, we
+have
+(κ(u − uh), uc
+h − uc
+H)0,�h + (µ∇ × (u − uh),∇ × (uc
+h − uc
+H))0,�h
++ < [[uh]],{{µ∇ × (uc
+h − uc
+H)}} >�h= 0,
+which leads to
+(κ(u − uh), uc
+h − uc
+H)0,�h + (µ∇ × (u − uh),∇ × (uc
+h − uc
+H))0,�h
+= − < [[uh]],{{µuc
+h − uc
+H}} >�h .
+(3.28)
+Using (3.22) and (3.23), we have
+< [[uh]],{{∇ × (uc
+h − uc
+H)}} >�h
+=
+(�h(uh),∇ × (uc
+h − uc
+H))0,�h
+≤ C� ∥h−1/2[[uh]]∥0,�h∥∇ × (uc
+h − uc
+H)∥0,�h.
+(3.29)
+Let uh = uc
+h + u⊥
+h and uH = uc
+H + u⊥
+H, we have
+uh + uc
+H − uc
+h = uH − u⊥
+H + u⊥
+h ,
+(3.30)
+where uc
+H ∈ Uc
+H, uc
+h ∈ Uc
+h, u⊥
+H ∈ U⊥
+H, u⊥
+h ∈ U⊥
+h . By (3.30), (3.28), (3.29) and Young’s
+inequality, we get
+∥|u − uh|∥2
+h
+= ∥κ(u − uh)∥2
+L2(Ω) + ∥∇ × µ(u − uh)∥2
+L2(Ω)
++
+�
+f ∈�h
+αh−1
+f
+< [[(u − uh)]],[[u − uh]] >�h
+= ∥|u − uh − uc
+H + uc
+h|∥2
+h − ∥|uc
+h − uc
+H|∥2
+h − 2(κ(u − uh), uc
+h − uc
+H)0,�h
+−2(µ∇ × (u − uh),∇ × (uc
+h − uc
+H))0,�h
+−2
+�
+f ∈�h
+αh−1
+f
+< [[(u − uh)]],[[uc
+h − uc
+H]] >
+≲ ∥|u − uH|∥2
+H + 2∥|u − uH|∥H∥|u⊥
+h − u⊥
+H|∥h + ∥|u⊥
+h − u⊥
+H|∥2
+h − ∥|uc
+h − uc
+H|∥2
+h
++2∥h−1/2[[uh]]∥0,�h∥∇ × (uc
+h − uc
+H)∥0,�h
+≤ (1 + δ1)∥|u − uH|∥2
+H + (1 + 1
+δ1
+)∥|u⊥
+h − u⊥
+H|∥2
+h − (1 − ˆδ2C� )∥|uc
+h − uc
+H|∥2
+h
++C�
+ˆδ2
+∥h−1/2[[uh]]∥2
+0,�h
+= (1 + δ1)∥|u − uH|∥2
+H + (1 + 1
+δ1
+)∥|u⊥
+h − u⊥
+H|∥2
+h − (1 − δ2)∥|uc
+h − uc
+H|∥2
+h
++
+C2
+�
+δ2
+∥h−1/2[[uh]]∥2
+0,�h,
+
+Convergence of AMIPDG methods for H(cur l)-elliptic problems
+17
+where δ2 = ˆδ2C� . Using uc
+H = uH − u⊥
+H, uc
+h = uh − u⊥
+h , triangle inequality and average
+inequality, we have
+∥|uc
+h − uc
+H|∥2
+h ≥ 1
+2∥|uh − uH|∥2
+h − ∥|u⊥
+h − u⊥
+H|∥2
+h.
+By triangle inequality and (3.21), we obtain
+∥|u⊥
+h − u⊥
+H|∥2
+h
+≤
+2(∥|u⊥
+h |∥2
+h + ∥|u⊥
+H|∥2
+H)
+≤
+2α∥h−1/2[[u⊥
+h ]]∥2
+0,�h + 2α∥h−1/2[[u⊥
+H]]∥2
+0,�h.
+Combining [[uH]] = [[u⊥
+H + uc
+H]] = [[u⊥
+H]] and (3.19), we have
+∥|u − uh|∥2
+h
+≤
+(1 + δ1)∥|u − uH|∥2
+H − 1 − δ2
+2
+∥|uh − uH|∥2
+h
++
+C3
+δ1δ2α
+�
+η2(uh, ph;�h) + η2(uH, pH;�H)
+�
+.
+3.3. Contraction of the error estimator
+In this subsection, we prove the reduction of error indicators. Let us first consider the
+effect of changing the finite element function used in the estimator.
+Lemma 3.8. Given f ∈ L2(Ω) and two tetrahedral mesh �h, �H with �H ≤ �h. Let (vh,qh) ∈
+Uh × Qh and (v H,q H) ∈ UH × QH. For any ε > 0, we have
+η2(vh,qh;�h) ≤ (1 + ε)η2(v H,q H;�h) + Cε∥(vh,qh) − (v H,q H)∥2
+DG,
+(3.31)
+where Cε depending on the ε, and the mesh size h < 1.
+Proof.
+For any τ∗ ∈ �h, we will discuss each of the five components of the mark
+η2(vh,qh;�h).
+Firstly, using the definition of R1(vh,qh) and triangle inequality, we have
+∥R1(vh,qh)∥L2(τ∗)
+(3.32)
+= ∥qh − µ∇ × vh∥L2(τ∗)
+= ∥qh − q H + µ∇ × (v H − vh) + q H − µ∇ × v H∥L2(τ∗)
+≲ ∥q H − ∇ × v H∥L2(τ∗) + ∥qh − q H∥L2(τ∗) + ∥∇ × (vh − v H)∥L2(τ∗).
+Secondly, using the definition of R2(vh,qh), triangle inequality and inverse inequality,
+we get
+hτ∗∥R2(vh,qh)∥L2(τ∗)
+(3.33)
+= hτ∗(∥ f − ∇ × qh − κvh∥L2(τ∗))
+= hτ∗(∥ f − ∇ × (qh − q H) − κ(vh − v H) − ∇ × q H − κv H∥L2(τ∗))
+≤ hτ∗(∥ f − ∇ × q H − κv H∥L2(τ∗) + ∥∇ × (qh − q H)∥L2(τ∗) + ∥κ(vh − v H)∥L2(τ∗))
+≲ hτ∗(∥R2(v H,q H)∥L2(τ∗) + h−1
+τ∗ ∥(qh − q H)∥L2(τ∗) + ∥κ(vh − v H)∥L2(τ∗))
+≲ hτ∗∥R2(v H,q H)∥L2(τ∗) + ∥(qh − q H)∥L2(τ∗) + hτ∗∥κ(vh − v H)∥L2(τ∗).
+
+18
+K Liu et al.
+Similarly, using the definition of R3(vh), triangle inequality and inverse inequality, we
+get
+hτ∗∥R3(vh)∥L2(τ∗)
+(3.34)
+= hτ∗∥∇ · ( f − κvh)∥L2(τ∗)
+= hτ∗∥∇ · ( f − κv H + κv H − κvh)∥L2(τ∗)
+≤ hτ∗(∥∇ · ( f − κv H)∥L2(τ∗) + ∥∇ · κ(v H − vh)∥L2(τ∗))
+≲ hτ∗(∥R3(v H)∥L2(τ∗) + h−1
+τ∗ ∥κ(v H − vh)∥L2(τ∗))
+≲ hτ∗∥R3(v H)∥L2(τ∗) + ∥κ(v H − vh)∥L2(τ∗).
+Next, we discuss the jump J1(qh) and J2(vh). For any f ∈ �(�h), we let f = τ1
+∗
+�
+τ2
+∗
+with τ1
+∗,τ2
+∗ ∈ �h. Furthermore, using the definition of J1(qh), triangle inequality and trace
+inequality, we have
+h1/2
+f
+∥J1(qh)∥L2(f )
+(3.35)
+= h1/2
+f
+∥[[qh]]∥L2(f )
+= h1/2
+f
+∥[[q H + qh − q H]]∥L2(f )
+≤ h1/2
+f
+(∥[[q H]]∥L2(f ) + ∥[[qh − q H]]∥L2(f ))
+≤ h1/2
+f
+∥[[q H]]∥L2(f ) + h1/2
+f
+∥(qh − q H)|τ1
+∗∥L2(f ) + h1/2
+f
+∥(qh − q H)|τ2
+∗∥L2(f )
+≲ h1/2
+f
+∥J1(q H)∥L2(f ) + ∥(qh − q H)∥L2(τ1
+∗∪τ2
+∗).
+Similarly, using the definition of J2(vh), triangle inequality and trace inequality, we
+have
+h1/2
+f
+∥J2(vh)∥L2(f )
+(3.36)
+= h1/2
+f
+∥[[( f − κvh)]]∥L2(f )
+= h1/2
+f
+∥[[( f − κv H + κv H − κvh)]]∥L2(f )
+≤ h1/2
+f
+(∥[[(f − κv H)]]∥L2(f ) + ∥[[κ(v H − vh)]]∥L2(f ))
+≤ h1/2
+f
+∥J2(v H)∥L2(f ) + h1/2
+f
+(∥κ(v H − vh)|τ1
+∗∥L2(f ) + ∥κ(v H − vh)|τ2
+∗∥L2(f ))
+≲ h1/2
+f
+∥J2(v H)∥L2(f ) + ∥κv H − κvh∥L2(τ1
+∗∪τ2
+∗).
+Finally, the desired result (3.31) is obtained by combining (3.32)-(3.36), Young’s in-
+equality and the shape regularity of mesh �h.
+We then prove the contraction of the error estimator under the assumptions on the
+problem of (2.13)-(2.14).
+Lemma 3.9. Given constant θ ∈ (0,1) and two tetrahedral mesh �h, �H(�H ≤ �h). Let
+(uH, pH) ∈ UH × QH be the solution of (2.13)-(2.14), and ��H−→�h = �H \ (�h ∩ �H) be the
+
+Convergence of AMIPDG methods for H(cur l)-elliptic problems
+19
+set of all element refined into �h on �H. Then, there is a constant λ ∈ (0,1) independent of
+mesh size, such that
+η2(uH, pH;�h) ≤ η2(uH, pH;�H) − λη2(uH, pH;��H→�h).
+(3.37)
+Proof. Assume that the tetrahedral mesh τ ∈ �H is divided into two new tetrahedral
+mesh τ1
+∗ and τ2
+∗ with equal volumes, where τ1
+∗,τ2
+∗ ∈ �h. Thus, h3
+τ1
+∗ = |τ1
+∗| = |τ2
+∗| = h3
+τ2
+∗ =
+2−1h3
+τ by the shape regularity of mesh, which implies hτ1
+∗ = hτ2
+∗ = 2−1/3hτ. Then, we have
+∥R1(uH, pH)∥2
+L2(τ1
+∗) + ∥R1(uH, pH)∥2
+L2(τ2
+∗) ≤ ∥R1(uH, pH)∥2
+L2(τ),
+(3.38)
+and
+h2
+τ1
+∗(∥R2(uH, pH)∥2
+L2(τ1
+∗) + ∥R3(uH)∥2
+L2(τ1
+∗))
++ h2
+τ2
+∗(∥R2(uH, pH)∥2
+L2(τ2
+∗) + ∥R3(uH)∥2
+L2(τ2
+∗))
+≤ 2−2/3h2
+τ(∥R2(uH, pH)∥2
+L2(τ) + ∥R3(uH)∥2
+L2(τ)).
+(3.39)
+For any f ∈ ∂ (τ1
+∗ ∪ τ2
+∗), which can be divided into three parts;
+(1) For the first part, there are two of the faces are constant and belong to τ .
+(2) For the second part, there are two new faces that overlap and are used to divide the
+mesh τ. Since (uH, ph) ∈ UH × QH is a continuous polynomial in the region τ, it follows
+that the value of [[ph]] and [[( f − κuH)]] on this surface is equal to zero.
+(3) For the third part, there are four faces that are obtained by dividing the two faces
+in the τ into two.
+Furthermore, we obtain
+η2(uH, pH;τ1
+∗) + η2(uH, pH;τ2
+∗) ≤ γη2(uH, pH;τ).
+(3.40)
+where constant γ ∈ (0,1) independent of mesh τ.
+Next, since ��H→�h represents the part of the set in the tetrahedral set �H that will
+be used to be refined, it follows that ��H→�h ⊂ �H. Let ��H→�h denote the part of the
+cell set that has been refined in the tetrahedral set �H, we have ��h→�H ∈ �h. Obviously,
+�H \��H→�h = �h \��H→�h. Then combining the (3.40), and the marking strategy (2.18),
+we have
+η2(uH, pH;�h)
+=
+η2(uH, pH;�h \ ��H→�h) + η2(uH, pH;��H→�h)
+≤
+η2(uH, pH;�H \ ��H→�h) + γη2(uH, pH;��H→�h)
+≤
+η2(uH, pH;�H) + (γ − 1)η2(uH, pH;��H→�h)
+≤
+η2(uH, pH;�H) − λη2(uH, pH;��H→�h),
+where λ = 1 − γ ∈ (0,1) independent of mesh size.
+Now, we combine the Lemmas 3.6, 3.8 and 3.9 to prove the reduction of error indicators.
+
+20
+K Liu et al.
+Lemma 3.10. Given a constant θ ∈ (0,1) and two tetrahedral mesh �h, �H(�H ≤ �h). Let
+(uh, ph) ∈ Uh × Qh and (uH, pH) ∈ UH × QH be the solutions of (2.13)-(2.14), respectively.
+For any ε > 0 and λ ∈ (0,1), we have
+(1 − Cε
+α )η2(uh, ph;�h)
+≤
+(1 + ε + Cε
+α )η2(uH, pH;�H)
+− (1 + ε)λη2(uH, pH;��H→�h) + Cε∥|uh − uH|∥2
+h,
+(3.41)
+where constant Cε depending on the ε and mesh size.
+Proof. Using the Lemmas 3.6, 3.8 and 3.9, we have
+η2(uh, ph;�h)
+≤
+(1 + ε)
+�
+η2(uH, pH;�H) − λη2(uH, pH;��H→�h)
+�
++Cε∥(uh, ph) − (uH, pH)∥2
+DG
+≤
+(1 + ε)
+�
+η2(uH, pH;�H) − λη2(uH, pH;��H→�h)
+�
++Cε∥|uh − uH|∥2
+h + ∥ph − pH∥2
+L2(Ω)
+≤
+(1 + ε)
+�
+η2(uH, pH;�H) − λη2(uH, pH;��H→�h)
+�
++Cε∥|uh − uH|∥2
+h + Cε
+α
+�
+η2(uh, ph;�h) + η2(uH, pH;�H)
+�
+,
+which completes the proof.
+3.4. Convergence result
+Now, we proved that the sum of the norm of the error and the scaled error indicator is
+attenuated.
+Theorem 3.2. For a given θ ∈ (0,1),let {�k,Uk,Qk, uk, pk,η(uk, pk;�k)}k≥0 be the se-
+quence of meshes, Mixed discrete solution (defined by (2.13)-(2.14)), and the estimate in-
+dicator produced by the AMIPDG algorithm. Then there exist constants ρ > 0, δ ∈ (0,1),
+which depend on marking parameter θ and the shape regularity of the initial mesh �0, such
+that
+∥|u − uk+1|∥2
+k+1 + ρη2(uk+1, pk+1;�k+1) ≤ δ
+�
+∥|u − uk|∥2
+k + ρη2(uk, pk;�k)
+�
+.
+Proof. Setting �ρ = 1−δ2
+2Cε , then multiply the both sides of the (3.41) inequality by �ρ, we
+get
+�ρ(1 − Cε
+α )η2(uk+1, pk+1;�k+1)
+≤ �ρ(1 + ε + Cε
+α )η2(uk, pk;�k) − �ρ(1 + ε)λη2(uk, pk;��k→�k+1)
++1 − δ2
+2
+∥|uk+1 − uk|∥2
+h.
+(3.42)
+
+Convergence of AMIPDG methods for H(cur l)-elliptic problems
+21
+Next, by the (3.26) and (3.42), we have
+∥|u − uk+1|∥2
+k+1 + �ρ(1 − Cε
+α )η2(uk+1, pk+1;�k+1)
+≤ (1 + δ1)∥|u − uk|∥2
+k +
+C3
+δ1δ2α
+�
+η2(v k+1,q k+1;�k+1) + η2(v k,q k;�k)
+�
++�ρ(1 + ε + Cε
+α )η2(uk, pk;�k) − �ρ(1 + ε)λη2(uk, pk;��k→�k+1).
+(3.43)
+First move the term and then according to Dörfler marking strategy (2.18), the Theorem
+3.1 and ∥| · |∥h ≤ ∥ · ∥DG, we know −η2(v k,q k;��k→�k+1) ≤ −θη2(v k,q k;�k), then
+∥|u − uk+1|∥2
+k+1
++
+�ρ(1 − Cε
+α −
+C3
+�ρδ1δ2α)η2(uk+1, pk+1;�k+1)
+≤
+(1 + δ1)∥|u − uk|∥2
+k − �ρ(1 + ε)λθ
+2
+η2(uk, pk;�k)
++�ρ
+�
+1 + ε + Cε
+α +
+C3
+�ρδ1δ2α − (1 + ε)λθ
+2
+�
+η2(uk, pk;�k)
+≤
+(1 + δ1 −
+�ρ(1 + ε)λθC−1
+1
+2
+)∥|u − uk|∥2
+k
++�ρ
+�
+1 + ε + Cε
+α +
+C3
+�ρδ1δ2α − (1 + ε)λθ
+2
+�
+η2(uk, pk;�k).
+For convenience, denote
+β1
+=
+1 − Cε
+α −
+C3
+�ρδ1δ2α,
+β2
+=
+1 + δ1 −
+�ρ(1 + ε)λθC−1
+1
+2
+,
+β3
+=
+(1 + ε)(1 − λθ
+2 ) + Cε
+α +
+C3
+�ρδ1δ2α.
+Thus
+∥|u − uk+1|∥2
+k+1 + �ρβ1η2(uk+1, pk+1;�k+1) ≤ β2∥|u − uk|∥2
+k + �ρβ3η2(uk, pk;�k).
+Next, we firstly choose δ1 =
+�ρ(1+ε)λθC−1
+1
+4
+, then select the appropriate δ2 to make �ρ =
+1−δ2
+2Cε smaller to ensure 0 < δ1 < 1, Secondly, we let ε > 0 and (1 + ε)(1 − λθ
+2 ) = 1 − λθ
+4 (
+λθ ∈ (0,1)), therefore
+β2 = 1 − δ1 ∈ (0,1), (1 + ε)(1 − λθ
+2 ) < 1.
+Furthermore, we choose a sufficiently large penalty parameter α such that
+β1 > β3.
+
+22
+K Liu et al.
+Finally, there is a constant δ = max{β2, β1
+β3 }. Then, we let ρ = �ρβ1, and obtain
+∥|u − uk+1|∥2
+k+1 + ρη2(uk+1, pk+1;�k+1) ≤ δ
+�
+∥|u − uk|∥2
+k + ρη2(uk, pk;�k)
+�
+.
+Corollary 3.1. Under the conditions of Theorem 3.2, we have
+∥(u, p) − (uk, pk)∥2
+DG + ρη2(uk, pk;�k) ≤ δk �Cδ.
+where �Cδ = C
+�
+∥(u, p) − (u0, p0)∥2
+DG + ρη2(u0, p0;�0)
+�
+. Therefore, for a given precision,
+the AMIPDG method will terminate after a finite number of operations.
+Proof. Using the Remark 3.2 and Theorem 3.2, we have
+∥(u, p) − (uk, pk)∥2
+DG + ρη2(uk, pk;�k)
+≤
+C
+�
+∥|u − uk|∥2
+k + ρη2(uk, pk;�k)
+�
+≤
+δk �Cδ.
+4. Numerical experiments
+In this section, we test some numerical experiments to show the efficiency and the
+robustness of AMIPDG. We carry out these numerical experiments by using the MATLAB
+software package iFEM [6]. In Experiments 4.1 and 4.2, we take p = ∇ × u.
+In Example 4.1, we discuss the influence of the penalty parameter α on the error in
+∥ · ∥DG norm, and observe the dependency of the condition number of stiffness matrix on
+α.
+Example 4.1. Let Ω := [0,1] × [0,1] × [0,1], we construct the following analytical solution
+of the model (1.1)-(1.2):
+u =
+�
+�
+x(x − 1)y(y − 1)z(z − 1)
+sin(πx)sin(πy)sin(πz)
+(1 − ex)(1 − ex−1)(1 − e y)(1 − e y−1)(1 − ez)(1 − ez−1)
+�
+�.
+It is easy to see that the solution u satisfies the boundary condition u × n = 0 on ∂ Ω.
+In this example, we get a uniform mesh by partitioning the x−, y− and z−axes into
+equally distributed M(M ≥ 2) subintervals, and then dividing one cube into six tetrahe-
+drons. Let h = 1/M be mesh sizes for different tetrahedrons meshes. We fixed mesh with
+h = 1/4 and report the error estimates in ∥ · ∥DG norm and condition number of stiffness
+matrices for different penalty parameters α = 1,10,100,500 and 1000 in Table 1. We note
+that ∥u − uh∥0 increases at first and then decreases as the penalty parameter α increases.
+
+Convergence of AMIPDG methods for H(cur l)-elliptic problems
+23
+Table 1: The error in ∥ · ∥DG norms and condition number of stiffness matrices with h = 1/4.
+α
+1
+10
+100
+500
+1000
+∥
+�
+p − ph, u − uh
+�
+∥DG
+3.949e+00
+1.133e-00
+8.614e-01
+8.649e-01
+8.659e-01
+Cond
+3.235e+04
+7.021e+04
+5.959e+05
+2.995e+06
+6.150e+06
+The condition numbers of stiffness matrices increase with the increase of penalty parame-
+ters α.
+As a way to balance, in the following numerical tests, we always choose α = 100.
+Noting that we only consider uniform meshes in Example 4.1. Next we test adaptive
+meshes.
+Example 4.2. Let Ω := [0,1] × [0,1] × [0,1], we construct the following analytical solution
+of the model (1.1)-(1.2)
+u =
+�
+�
+�
+x(x−1)y(y−1)z(z−1)
+x2+y2+z2+0.001
+x(x−1)y(y−1)z(z−1)
+x2+y2+z2+0.001
+− x(x−1)y(y−1)z(z−1)
+x2+y2+z2+0.001
+�
+�
+�.
+Note that the solution u satisfies the condition u × n = 0 on ∂ Ω.
+The right of Figure 1 shows an adaptively refined mesh with marking parameter- θ =
+0.7 after k = 18. The grid is locally refined near the origin.
+Figure 1: Left: the initial mesh with 1152 DoFs. Right: the adaptive mesh(θ = 0.7) with 181104 DoFs
+after 18 refinements.
+The Figure 2 shows the curves of log N−logη
+�
+uk, pk;�k
+�
+for parameters θ = 0.3,0.5,0.7.
+The curves indicate the convergence and the quasi-optimality of the adaptive algorithm
+AMIPDG of η
+�
+uk, pk;�k
+�
+.
+Acknowledgment
+The first author is supported by the East China University of Technology (DHBK2019209)
+and Jiangxi Province Education Department (GJJ200755). The second, third and fourth
+authors are supported by the National Natural Science Foundation of China (Grant No.
+12071160). The third author is also supported by the National Natural Science Foundation
+of China (Grant No. 11901212).
+
+24
+K Liu et al.
+Figure 2: Quasi optimality of the AMIPDG of the error η
+�
+uk, pk;�k
+�
+with different marking parameters
+θ.
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+

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@@ -0,0 +1,742 @@
+arXiv:2301.00731v1  [math.DS]  2 Jan 2023
+Feuerbach’s and Poncelet’s theorems meet in space
+(On the occasion of their bicentennial)
+E. A. Avksentyev
+December 29, 2022
+Abstract
+Three-dimensional analogues of the Feuerbach theorem are proposed in this paper. One of them
+concerns some tetrahedron analogue of the Euler circle. Another one is pretty interesting «up-in-ex-
+touch» construction. And the third one, it turns out, is closely related to Poncelet’s theorem. This is
+very beautiful Grace’s theorem. It seems that this theorem is not widely known, and that no elementary
+proof has been given. Such an elementary proof of the Grace theorem is obtained in this paper by using
+properties of imaginary generators on a sphere and of isotropic tangents to a conic. An applying of the
+Grace theorem leads to several corollaries. One of them is Laguerre’s theorem, which generalizes the
+Euler-Chapple formulas. Further, we consider a spatial analog of Poncelet’s theorem. We prove that
+the Grace spheres touch some fixed sphere under the Poncelet rotation of bicentric tetrahedron. Finaly,
+going out from a plane into the third dimension, we obtain a new proof of Feuerbach’s theorem and
+perhaps the shortest proof of Euler-Chapple formulas.
+Введение
+Данная работа посвящена двум знаменитым геометрическим теоремам, кажется никак не связанным
+между собой, разве что они были опубликованны в один год двести лет назад [5, 14]. Приведем их
+формулировки
+Теорема (Feuerbach, 1822). Окружность девяти точек произвольного треугольника касается его
+вписанной и трех вневписанных окружностей.
+Теорема (Poncelet, 1822). Пусть для двух данных коник существует вписано-описанный в них
+многоугольник. Тогда этот многоугольник может динамически «вращаться» около данных коник,
+оставаясь вписано-описанным в них.
+У обеих теорем есть масса обобщений, но пространственные аналоги, насколько нам известно,
+имеются только у теоремы Понселе. Их довольно много (см., например, [6,8,9,15]) и среди них есть
+множество замечательных, но малоизвестных результатов.
+Задача трехмерного обобщения теоремы Фейербаха поставлена еще более ста лет назад в моно-
+графии Кулиджа [2]:
+«The geometry of the tetrahedron lags far behind that of the triangle... Is there an analogue
+to Feuerbach’s theorem? Above all what corresponds to the Hart systems? ...These difficult
+but important and interesting questions offer ample scope for serious work» (p. 247).
+Теорема Фейербаха содержит в себе два удивительных геометрических факта. Первый состоит в
+том, что четыре замечательные окружности треугольника – вписанная и три описанные – имеют об-
+щую касательную окружность. Второй же заключается в том, что эта общая касательная окружность
+является еще и окружностью девяти точек, которая и без того сама по себе замечательна.
+Первая попытка найти аналог теоремы Фейербаха в пространстве приводит к вопросу: существу-
+ет ли сфера, которая касалась бы вписанной и вневписанных сфер?
+Но здесь нас ожидает первый «сюрприз»: у произвольного тетраэдра кроме обычных четырех
+вневписанных сфер, аналогичных трем вневписанным сферам треугольника, существует еще три
+дважды-вневписанные сферы или чердачные (от англ. «roof»), как они названы в [20] (см. также [21]).
+Т.е., всего существует целых восемь сфер (см. рис. 1), касающихся граней тетраэдра! Назовем
+их касательными сферами. Было бы слишком оптимистично ожидать, что все восемь касательных
+1
+
+Рис. 1: Восемь касательных сфер тетраэдра
+сфер могли бы касаться ��дной сферы. И действительно, ответ на поставленный вопрос оказывается
+отрицательным: в общем случае произвольного тетраэдра такой сферы не существует.
+Проверить это очень легко: для этого достаточно рассмотреть лишь один пример подходящего
+тетраэдра. И нет сомнений, что такой знаток геометрии как Кулидж хорошо знал, что такой сферы
+в общем случае нет. Однако, он все-таки поставил вопрос поиска трехмерных аналогов теоремы
+Фейербаха, находя его важным, интересным и открывающим «широкие возможности для серьезной
+работы».
+В каком же направлении искать тогда аналоги теоремы Фейербаха в пространстве? Кажется,
+что осталась лишь задача описания частных случаев тетраэдров, у которых существует сфера, ка-
+сающаяся внутренним или внешним образом пяти, шести, семи или всех восьми касательных сфер.
+В работе [11] есть некоторое продвижение в этой задаче и для существования такой сферы получе-
+ны аналитические условия в специальных связанных с тетраэдром пентасферических координатах.
+К сожалению, эти условия весьма громоздкие и из них совершенно не ясно, существуют ли такие
+тетраэдры и как они устроены. Таким образом, задача в такой постановке остается незакрытой.
+Возникает еще идея поискать пространственный аналог теоремы Фейербаха в таком направле-
+нии: существует ли окружность, действительная или мнимая, которая касалась бы всех восьми
+касательных сфер? Кажется маловероятным, что ответ мог бы быть положительным, но задача пред-
+ставляется интересной.
+Оставив пока эти вопросы, мы приведем далее целых три трехмерных аналога теоремы Фейербаха.
+Первый аналог, которую мы хотим предложить в § 1 в качестве трехмерного обобщения теоремы
+Фейербаха, является довольно интересным фактом. У него очень простое доказательство, которое,
+2
+
+тем не менее, раскрывает связь этой конструкции с неевклидовой геометрией и приводит к трехмер-
+ному обобщению окружности Эйлера. Поэтому из трех аналогов этот наиболее аутентичен.
+Второй является очень красивой теоремой геометрии тетраэдра, открытой сто двадцать пять лет
+назад, но, кажется, до сих пор малоизвестной. Ее единственное оригинальное доказательство столь
+сложно, что есть целая статья с его реконструкцией. В § 2 мы получим элементарное доказательство
+этой теоремы, в котором обнаружится ее связь с теоремой Понселе. Второй аналог выглядит наименее
+аутентичным, но на наш взгляд, он ближе и роднее к теореме Фейербаха, чем другие два.
+Третий аналог представляет из себя довольно интересную конструкцию касающихся сфер, кото-
+рую мы назвали «up-in-ex-touch»-конструкция. Мы приведем ее в конце § 3, в котором мы также
+получим, возможно, самое короткое доказательство формул Эйлера-Чаппла.
+С помощью теоремы Грейса мы в §4 получим короткое и простое доказательство теоремы Лагерра,
+обобщающей формулы Эйлера-Чаппла. §5 посвящен трехмерному аналогу формул Эйлера-Чаппла.
+Далее в §6 мы рассмотрим пространственные аналоги теоремы Понселе. Мы покажем, что при
+вращении Понселе вписано-вневписанного тетраэдра его сферы Грейса касаются некоторой фикси-
+рованной сферы.
+В конце, совершая «выход в пространство», мы дадим новое доказательство теоремы Фейербаха.
+1 Первый аналог теоремы Фейербаха для тетраэдра
+Итак, рассмотрим произвольный тетраэдр общего вида, у которого имеется восемь касательных сфер.
+В качестве первого аналога теоремы Фейербаха для тетраэдра предлагаем следующую теорему.
+Теорема 1.1. Существует четыре круговых конуса, каждый из которых касается всех восьми его
+касательных сфер.
+Доказательство. Рассмотрим сферу ζD с центром в вершине D тетраэдра ABCD и спроектируем
+из центра D на сферу ζD все восемь касательных сфер. Их проекциями будут четыре окружности
+на сфере ζD, поскольку каждая пара гомотетичных относительно D сфер спроектируются в одну и
+ту же окружность. Эти четыре окружности касаются сторон сферического треугольника, стороны
+которого являются проекциями плоскостей трехгранного угла при вершине D. По теореме Фейербаха
+для сферического треугольника существует окружность, касающаяся этих четырех окружностей.
+Конус с вершиной D, содержащий эту окружность, очевидно удовлетворяет утверждению теоремы.
+Такой конус есть у каждой вершины.
+✷
+Теорема Фейербаха в сферической геометрии, в той облегченной форме, которую мы использо-
+вали в доказательстве, равносильна теореме Харта (см. [2]). Таким образом, в какой-то степени мы
+ответили на оба вопроса Кулиджа, которые мы цитировали во введении. На самом деле, можно про-
+двинуться еще дальше в этом направлении, если применить результат Акопяна [19], в котором он
+нашел такие свойства окружности Харта, которые во многом аналогичны свойствам окружности
+девяти точек. Хотя в [19] все утверждения формулируются для плоскости Лобачевского, но мы их
+естественным образом адаптируем применительно к трехгранным углам нашего тетраэдра.
+Избытком трехгранного угла называется величина, равная разнице между суммой его двух-
+гранных углов и 180◦. Медиатором трехгранного угла назовем плоскость, содержащую его ребро
+и делящую его на два трехгранных угла с равными избытками. При рассмотренной выше проекции
+трехгранного угла на сферу медиатор переходит в сферическую чевиану, делящую пополам пло-
+щадь соответственного треугольника (в [19] эта чевиана называется биссектором или биссекторным
+отрезком). Три медиатора пересекаются по прямой, которую можно назвать псевдоцентроидалью,
+поскольку ей соответствует псевдоцентроид сферического треугольника.
+Четыре прямые из одного пучка назовем вписанной четверкой, если все они являются образую-
+щими одного кругового конуса. Следующее утверждение является аналогом Леммы 5 из [19].
+3
+
+Предложение 1.2. Пусть a, b, c – ребра трехгранного угла с вершиной D. Тогда существует един-
+ственная тройка прямых ha, hb, hc, лежащих в плоскостях ⟨ab⟩, ⟨ac⟩, ⟨bc⟩ соответственно, таких
+что четверки {a, b, ha, hb}; {a, c, ha, hc}; {b, c, hb, hc} являются вписанными.
+Плоскости aha, bhb, chc являются аналогами так называемых псевдовысот, которым в [19] дается
+еще и другое определение через углы. Эти три плоскости пересекаются по общей прямой, назовем ее
+псевдоортоцентралью по аналогии с псевдоортоцентрами гиперболических треугольников.
+Круговой конус, содержащий все три ребра трехгранного угла в качестве своих образующих,
+назовем описанным.
+В [19, §§ 4-6] показано, что основания трех псевдовысот и трех биссекторных чевиан лежат на
+одной окружности. Центр этой окружности лежит на одной прямой с центром описанной, псевдоцен-
+троидом и всевдоортоцентром. Сформулируем аналогичные утверждение для тетраэдра.
+Теорема 1.3 (Конус Эйлера трехгранного угла). У любого трехгранного угла основания трех его
+медиаторов и трех его псевдовысот лежат на одном круговом конусе.
+Теорема 1.4 (Плоскость Эйлера трехгранного угла). У произвольного трехгранного угла четыре
+прямых – псевдоцентроидаль, псевдоортоцентраль, ось описанного конуса и ось конуса Эйлера –
+лежат в одной плоскости.
+Главным же результатом работы [19] является гиперболический аналог теоремы Фейербаха, со-
+гласно которому окружность Эйлера гиперболического треугольника касается его вписанной и трех
+вневписанных окружностей. Применительно к тетраэдру мы получаем следующее усиление Теоре-
+мы 1.1
+Теорема 1.5 (Аналог теоремы Фейербаха для тетраэдра). Четыре конуса Эйлера трехгранных углов
+тетраэдра касаются всех восьми его касательных сфер.
+Отметим несколько вопросов, которые возникают в связи с рассмотренными конструкциями.
+Вопрос 1.6. Инцидентны ли какие либо из следующих четверок замечательных прямых тетраэд-
+ра: псевдоцентроидали, псевдоортоцентрали, оси четырех описанных конусов, оси четырех конусов
+Эйлера?
+Вопрос 1.7. Существуют ли еще какие-либо квадрики, касающиеся всех касательных сфер, отлич-
+ные от четырех конусов Эйлера и четырех плоскостей граней?
+Вопрос 1.8. Любые три конуса общего положения пересекаются в восьми точках. Не окажется
+ли так, что четыре конуса Эйлера тетраэдра имеют восемь общих точек? Есть ли какие-то
+примечательные свойства биквадратических кривых, по которым пересекаются конусы Эйлера?
+2 Теорема Грейса как трехмерный аналог теоремы Фейербаха
+Более ста лет назад, британский математик Джон Хилтон Грейс в своей работе [7] открыл и доказал
+следующее замечательное свойство касательных сфер тетраэдра.
+Теорема 2.1 (Grace, 1897). Касательные сферы тетраэдра ABCD могут быть разбиты на че-
+тыре пары так, что парные сферы гомотетичны с центром D, и для каждой пары существует
+касающаяся их сфера, проходящая через вершины A, B, C.
+Замечание 2.2. Все касательные сферы можно разбить на две группы по четыре сферы. В одну
+входят вписанная и три дважды-вневписанные сферы, а в другую – четыре вневписанные. Любые
+две сферы из разных групп гомотетичны относительно одной из вершин тетраэдра. Для каждой
+4
+
+такой пары сфер существует единственная касающаяся их сфера Грейса, которая проходит через
+вершины грани, противоположной к той вершине, относительно которой данная пара касательных
+сфер гомотетична. Таким образом, всего получается шестнадцать сфер Грейса: для каждой из
+четырех граней тетраэдра через ее вершины проходит четыре различные сферы Грейса.
+Теорема Грейса связывает касательные сферы тетраэдра с замечательными точками, его верши-
+нами, с помощью общих касающихся их сфер. Это ее сближает с теоремой Фейербаха, с которой она,
+на наш взгляд, сравнима по красоте и имеет некоторое сходство. В этом смысле, можно было бы
+считать теорему Грейса неким трехмерным аналогом теоремы Фейербаха.
+Рис. 2: Сфера Грейса GD, касающаяся вписанной сферы σ, вневписанной сферы σD и проходящая через A, B, C.
+В недавней статье [13] Maehara и Martini замечают, что «по-видимому, эта теорема малоизвестна
+и до сих пор не имеет элементарного доказательства». В качестве результата они приводят такое
+доказательство, но лишь для частного случая триортогонального тетраэдра, пользуясь при этом
+аналитической техникой.
+Оригинальное же доказательство Грейса очень красивое и геометрическое, но довольно трудное.
+Поскольку Грейс дал лишь его набросок, Maehara и Tokushige в работе [12] подробно реконструиро-
+вали это доказательство.
+Мы получим элементарное и вполне короткое геометрическое доказательство теоремы Грейса,
+но сначала напомним некоторые определения и факты проективной геометрии. Пусть E3 – веще-
+ственное трехмерное евклидово пространство. Мы будет рассматривать его проективное пополнение
+«бесконечно удаленной» плоскостью. Эта модель проективного пространства получается переходом
+от декартовых координат (x, y, z) в E3 к однородным координатам (x : y : z : w), в которых бесконеч-
+но удаленной плоскости соответствуют точки с координатами (x : y : z : 0). Кроме того рассмотрим
+комплексификацию пространства, позволяя координатам принимать комплексные значения. Добав-
+ленные точки будем называть мнимыми.
+Записывая в однородных координатах (x : y : z : w) общее уравнение сферы
+x2 + y2 + z2 + 2axw + 2byw + 2czw + dw2 = 0,
+легко видеть, что она пересекает бесконечно-удаленную плоскость w = 0 по кривой
+x2 + y2 + z2 = 0, w = 0,
+5
+
+D
+GD
+A
+C
+B
+ODкоторая является общей для всех сфер. Она называется абсолютной окружностью.
+Всякая плоскость пересекает абсолютную окружность в двух сточках – круговых точках этой
+плоскости. В однородных координатах (x : y : z) на плоскости ее круговыми точками являются точки
+I = (1 : i : 0) и J = (1 : −i : 0). Все окружности плоскости проходят через ее круго��ые точки и каждая
+коника плоскости, проходящая через ее круговые точки, является окружностью (см. [16, § 4·8]).
+Прямая, пересекающая абсолютную окружность, называется изотропной. Каждая такая прямая
+является, естественно, мнимой.
+Предложение 2.3 ( [22, Гл. 12, § 2]). Касательные к невырожденной конике, проведенные из любого
+ее фокуса, являются изотропными.
+Таким образом, каждая прямая, проходящая через фокус коники и круговую точку ее плоскости,
+является изотропной. Для окружности это означает, что касательные из ее центра проходят через
+круговые точки.
+Образующей квадрики называется прямая, которая целиком принадлежит поверхности этой квад-
+рики. В комплексном проективном пространстве все невырожденные квадрики эквивалентны.
+Предложение 2.4 ( [9, § 2]).
+(i) Через каждую точку невырожденной квадрики проходят ровно две образующие, действительны
+или мнимые. Касательная плоскость пересекает квадрику по двум образующим, проходящим
+через точку касания.
+(ii) Все образующие квадрики распадаются на два семейства таким образом, что любые две обра-
+зующие из одного семейства не пересекаются, а любые две образующие из разных семейств
+пересекаются. Через любую точку образующей одного семейства проходит единственная об-
+разующая другого семейства.
+(iii) Любая плоскость, проходящая через образующую квадрики касается этой квадрики в некото-
+рой точке этой образующей.
+Пусть даны две сферы γ и η. Рассмотрим множество M(γ, η) сфер, которые касаются обеих сфер
+γ и η. Заметим что множество M(γ, η) распадается на два класса эквивалентности по типу касаний.
+Если сфера α касается γ и η одинаковым образом (обеих внутренним, или обеих внешним), то α
+принадлежит одному классу. Если же α касается γ и η различным образом (одной сферы внутренним,
+а другой внешним, или наоборот), то α принадлежит другому классу. Прямые, проходящие через
+точки касания γ и η со сферами одного класса, проходят через общую точку. Для сфер одного класса
+эта точка – один из двух центров инверсии, переводящей γ и η друг в друга, а для сфер другого
+класса – второй такой центр (эти точки – центры подобия сфер γ и η).
+Замечание 2.5. Все это имеет место быть и в случае, если, скажем, сфера η вырождается в
+плоскость π (сферу бесконечно большого радиуса). Тогда рассмотренные выше инверсные центры γ
+и π – это точки сферы γ, касательные плоскости в которых параллельны π.
+Следующая теорема является главным результатом этого параграфа. Она описывает семейство
+коник σ, которые вместе с данной окружностью Σ образуют 3-пару Понселе (Σ, σ), т.е. для них суще-
+ствует треугольник, вписанный в Σ и описанный около σ. Из этой теоремы практически мгновенно
+следует теорема Грейса, что мы сразу покажем после ее формулировки.
+Теорема 2.6 (О 3-парах Понселе). Пусть даны плоскость π и окружность Σ на ней. Фиксируем
+сферу γ, содержащую окружность Σ, и р��ссмотрим множество M(γ, π) сфер, касающихся сферы
+γ и плоскости π. Тогда если сферы α и β пробегают разные классы множества M(γ, π), то описан-
+ный около них конус K высекает на плоскости π семейство коник σ, образующих 3-пару Понселе с
+окружностью Σ.
+6
+
+Доказательство Теоремы Грейса. Пусть α и β – две касательные сферы тетраэдра ABCD, гомо-
+тетичные относительно вершины D. Рассмотрим сферу γ, касающуюся сфер α и β и проходящую
+через вершины A и B. Таких сфер, вообще говоря, целых четыре. Но две из них в данном случае
+вырождены в плоскости ⟨DAB⟩ и ⟨ABC⟩, которые принадлежат разным классам множества M(α, β).
+Тогда оставшиеся две сферы тоже принадлежат разным классам и в качестве γ выберем ту, которая
+принадлежит другому, нежели плоскость ⟨ABC⟩, классу. Пусть она пересекает плоскость ⟨ABC⟩ по
+окружности Σ. Описанный около α и β конус с вершиной D пересекает плоскость ⟨ABC⟩ по конике
+σ, касающейся сторон треугольника ABC. По Теореме о 3-парах Понселе вершина C также должна
+лежать на окружности Σ.
+✷
+Доказательство Теоремы 2.6 о 3-парах Понселе.
+Пусть Fα и Fβ – тоски касания сфер α и β с плоскостью π, которые по теореме Данделена (1822, [3])
+являются фокусами коники σ. Далее будем считать, что точки Fα и Fβ не совпадают друг с другом
+и с центром окружности Σ. Эти частные случаи сводятся к общему малым шевелением сфер α и β и
+утверждение теоремы для них получается предельным переходом. Если I – одна из круговых точек
+плоскости π, то I ∈ Σ. Обозначим через Pα и Pβ точки вторичного пересечения прямых IFα и IFβ с
+коникой Σ. Тогда треугольник IPαPβ вписан в окружность Σ, прямые IPα и IPβ касаются коники σ,
+и нам достаточно доказать, в силу теоремы Понселе, что прямая PαPβ тоже касается коники σ.
+Рис. 3: 3-пары Понселе (Σ, σ). Мнимые касательные представлены дугообразными розовыми отрезками.
+Пусть A и B – точки касания сферы γ со сферами α и β. Заметим, что прямая IFα является
+образующей сферы α. Обозначим через lA одну из двух образующих сферы α в точке A, которая
+пересекает образующую IFα (т.е. lA и IFα принадлежат разным семействам образующих сферы γ).
+Поскольку lA является также образующей и сферы γ, точка пересечения lA ∩ IFα – это одна из двух
+точек пересечения прямой IFα со сферой γ, т.е. это либо точка I, либо точка Pα.
+Заметим, что первый случай не возможен в силу нашей договоренности считать, что точка Fα
+отлична от центра окружности Σ. В самом деле, I лежала бы тогда в пересечении касательных плос-
+костей сферы α в точках A и Fα, т.е. полярно-сопряженная к AFα относительно α прямая содержала
+бы круговую точку I. А так как она вещественная и потому не может быть изотропной, она являлась
+7
+
+人
+T
+A
+P
+a
+P
+Fp
+D
+Fa
+Bбы бесконечно-удаленной, т.е. касательные плоскости сферы α в точках A и Fα были бы параллельны,
+а точка Fα совпадала бы с центром окружности Σ.
+Таким образом, прямая APα является общей образующей lA сфер α и γ в точке A, и а��алогично,
+прямая BPβ совпадает с lB – одной из двух общих образующих сфер β и γ в точке B. Покажем, что
+lA и lB компланарны.
+Для этого рассмотрим гомотетию с центром A, переводящую α в γ. Пусть gA – образующая
+сферы γ, в которую переходит образующая IFα сферы α. Заметим, что
+1) I ∈ gA, поскольку gA ∥ IFα,
+2) прямая gA инцидентна с прямой lA, т. к. прямая lA инвариантна при рассмотренной гомоте-
+тии и инцидентна с прямой IFα. Т. е. gA и lA – две образующие сферы γ, принадлежащие разным
+семействам.
+Аналогично, если gB – образующая сферы γ, в которую переходит образующая IFβ сферы β при
+гомотетии с центром B, переводящей β в γ, то
+3) I ∈ gB,
+4) gB и lB – тоже две образующие сферы γ, принадлежащие разным семействам.
+Из замечания 2.5 следует, что прямые gA и gB проходят через различные инверсные центры
+сферы γ и плоскости π, а потому различны. Тогда из 1) и 3) следует, что образующие gA и gB сферы
+α имеют общую точку и, значит, принадлежат разным семействам, откуда в силу 2) и 4) следует, что
+образующие lA и lB тоже из разных семейств, а потому компланарны.
+Теперь рассмотрим плоскость ⟨lA; lB⟩, которая в силу утверждения [iii] Предложения 2.4 касается
+обеих сфер α и β. Заметим, что вершина конуса K содержит прямую AB. Действительно, поскольку
+конус K пересекает π по невырожденной конике, его вершина не лежит на π. Так как α и β из
+разных классов множества M(γ, π), то γ и π из разных классов множества M(α, β). Значит, прямая
+AB проходит через инверсный центр сфер α и β, который не лежит на плоскости π.
+Т.о., ⟨lA; lB⟩ – касательная плоскость конуса K, а потому пересекает плоскость π по прямой,
+касающейся коники σ. Осталось заметить, что ⟨lA; lB⟩ пересекает π по прямой PαPβ, и таким образом,
+треугольник IPαPβ является вписано-описанным.
+✷
+3 Формулы Эйлера-Чаппла и up-in-ex-touch-аналог теоремы Фейер-
+баха
+Теорема 3.1 (Euler, Chapple). Пусть R, r и ra – радиусы описанной, вписанной и вневписанной
+окружностей произвольного треугольника, d и da – расстояния от центра описанной окружности
+до центров вписанной и вневписанной. Тогда выполняются следующие соотношения
+d2 = R2 − 2Rr
+(1)
+d2
+a = R2 + 2Rra
+(2)
+Мы приведем два, наверное, самых коротких доказательства этой теоремы. Для этого рассмотрим
+сферу ∆, построенную диаметрально на описанной окружности, наовем ее описанной сферой тре-
+угольника, сферу δ радиуса r, касающуюся плоскости треугольника в центре его вписанной окруж-
+ности, наовем ее вписано-поднятой, и сферу δa радиуса ra, касающуюся плоскости треугольника в
+центре соответствующей вневписанной окружности, наовем ее вневписано-поднятой.
+Заметим, что соотношения (1), (2) можно переписать в виде равенств
+d2 + r2 = (R − r)2,
+d2 + r2
+a = (R + ra)2,
+которые равносильны касанию сфер ∆ и δ, ∆ и δa.
+8
+
+Рис. 4: Сферы ∆ и δ касаются друг друга
+Доказательство 1. Касания ∆ и δ, ∆ и δa сразу следует
+из Теоремы Грейса. Действительно, рассмотрим тетраэдр с
+основанием ABC и вершиной D на бесконечно��ти в перпен-
+дикулярном к плоскости (ABC) направлении. Тогда сфера
+δ является его вписанной сферой, симметричная ей относи-
+тельно плоскости (ABC) – его вневписанной сферой, а сле-
+довательно, сфера ∆ – его сферой Грейса. Для пары ∆ и δa
+рассуждение аналогично.
+✷
+Это доказательство примечательно своей лаконичностью
+и красотой, но использование сложной Теоремы Грейса мо-
+жет выглядеть как «стрельба из пушки по воробьям». Поэто-
+му приводим другое
+Доказательство 2. Сделаем инверсию относительно сфе-
+ры, построенной диаметрально на вписанной окружности.
+Заметим, что сфера ∆ переходит в сферу ∆′, построенную
+диаметрально на окружности, проходящей через середины сторон треугольника Жергона (верши-
+нами которого являются точки касания вписанной окружности △ABC со сторонами). А сфера δ
+переходит в плоскость δ′, удаленную от плоскости (ABC) параллельно на расстояние r
+2 . Поскольку,
+радиус сферы ∆′, очевидно, тоже равен r
+2, сферы ∆′ и δ′, а следовательно, и сферы ∆ и δ касаются
+друг друга.
+✷
+Заметим, что доказанное свойство касания сферы ∆ с четырьмя сферами δ, δa, δb, δc является
+своего рода тоже неким аналогом теоремы Фейербаха в пространстве.
+Теорема 3.2 (Up-in-ex-touch). Описанная сфера треугольника касается его вписано-поднятой и че-
+тырех вневписано-поднятых сфер.
+Рис. 5: Up-in-ex-touch-аналог теоремы Фейербаха.
+9
+
+Заметим также, что сфера ∆ касается не только сфер δ, δa, δb, δc, но и еще четырех симметричных
+им относительно плоскости треугольника, т.е. целых восьми сфер.
+4 Теорема Лагерра и ее применение к тетраэдру
+Теорема 4.1 (Laguerre [10], 1879). Окружность Σ радиуса R с центром в точке O и коника σ с
+фокусами Fα, Fβ и малой полуосью b образуют 3-пару Понселе тогда и только тогда, когда выпол-
+няется соотношение
+(R2 − d2
+α)(R2 − d2
+β) = 4R2b2,
+(3)
+где dα = |OFα|, dβ = |OFβ|.
+Замечание 4.2. Малая полуось b может быть как действительной (у эллипсов), так и мнимой (у
+гипербол). В первом случае из формулы Лагерра видно, что фокусы эллипса должны лежать либо
+оба внутри окружности, либо оба вне. Во втором случае, у гиперболы, один фокус должен лежать
+внутри окружности, другой – снаружи.
+Замечание 4.3. Если коника σ является параболой, то условие существования вписано-описанных
+треугольников для пары (Σ, σ) становится совсем простым: d = R, где d = |OF|, т.е. фокус F
+параболы должен лежать на окружности. Это следует из известной теоремы Ламбера.
+Доказательство Теоремы Лагерра (⇒) Пусть γ – произвольная сфера, содержащая окружность Σ,
+а cфера α касается в точке Fα плоскости π, содержащей окружность Σ, а также касается сферы γ.
+Рассмотрим произвольный вписано-описанный треугольник ABC и проведем через его стороны ка-
+сательные плоскости к сфере α. Они пересекаются в некоторой точке D, образуя тетраэдр ABCD,
+у которого сфера α является одной из касательных сфер, а γ – сферой Грейса, которая касает-
+ся также другой касатеьной сферы β тетраэдра ABCD, гомотетичной α относительно вершины D.
+Как известно, сферы α и β касаются плоскости π в точках, изогонально сопряженных относительно
+△ABC. Кроме того, поскольку Fα и Fβ – фокусы вписанной в △ABC коники σ, они также изого-
+нально сопряжены. Отсюда заключаем, что сфера β касается плоскости π в точке Fβ.
+Нам понадобится одна очень простая лемма
+Лемма 4.4 (Thebault [17], 1922). Для малой полуоси b коники, высекаемой описанным около сфер α
+и β конусом на их общей касательной плоскости, выполняется соотношение
+|b2| = rαrβ
+(4)
+Пусть Sα и Sβ – две диаметрально противоположные точки на γ в перпендикулярном к плоскости
+π направлении, которые являются инверсными центрами сферы γ и плоскости π (см. замечание 2.5).
+Учитывая, что сферы α и β принадлежат разным классам множества M(γ, π) (см. доказательство
+теоремы Грейса), легко выразить радиусы сфер α и β:
+rα =
+����
+Σ(Fα)
+2π(Sα)
+���� ,
+rβ =
+����
+Σ(Fβ)
+2π(Sβ)
+���� ,
+(5)
+где Σ(Fα) = d2
+α − R2 и Σ(Fβ) = d2
+β − R2 – степени точек Fα и Fβ относительно окружности Σ,
+а π(Sα), π(Sβ) – расстояния от точек Sα и Sβ до плоскости π.
+Перемножим равенства (5) и учтем, что π(Sα)π(Sβ) = R2. Получим, что в равенстве (3) левая и
+правая части равны по модулю. Правая часть отрицательна только в случае, если коника σ является
+гиперболой. Такое происходит только тогда, когда сферы α и β касаются описанного около них
+конуса с вершиной D по разные стороны от D, а плоскости π – по одну сторону. Тогда сферы γ они
+должны касаться по разные стороны, а следовательно, точки Fα и Fβ их касания с π относительно
+10
+
+окружности Σ лежат тоже по разные стороны и левая часть (3) в этом случае также отрицательна.
+Таким образом, модули можно снять и равенство (3) считать доказанным.
+(⇐) Пусть выполняется (3). Если коники (Σ, σ) не образуют 3-пару Понселе, то можно изменить
+малую полуось b коники σ так, чтобы они образовали 3-пару Понселе. Тогда по уже доказанному
+тоже должно выполняться равенство (3), следовательно величина b не изменилась, т.е. (Σ, σ) как раз
+и образуют 3-пару Понселе
+✷
+Теорема Лаггера, примененная к тетраэдру, позволяет получить следующее интересное метриче-
+ское соотношение для касательных сфер тетраэдра.
+Теорема 4.5. Пусть ∆D – сфера, описанная около грани ABC тетраэдра ABCD, α и β – две
+касательные сферы, гомотетичные относительно D. Тогда произведение косинусов углов, которые
+сфера ∆D образует с α и β (среди них один угол мнимый), равно 1, если α и β касаются ⟨ABC⟩ с
+одной стороны, или −1, если с разных.
+cos(�
+∆D, α) cos(�
+∆D, β) = sign k,
+(6)
+где k – коэффициент упомянутой гомотетии с центром D.
+Доказательство Пусть OD и R – центр и радиус сферы ∆D; rα, rβ – радиусы сфер α и β; Dα, Dβ –
+расстояния между центрами ∆D и α, ∆D и α; dα, dβ – расстояния от OD до точек Fα и Fβ касания
+плоскости ⟨ABC⟩ со сферами α и β. Пусть Σ = ⊙(ABC), а конус K с вершиной D, описанный около
+α и β, пересекает плоскость ⟨ABC⟩ по конике σ.
+Воспол��зуемся леммой 4.4 и заметим, что в нашей конструкции с тетраэдром равенство (4) можно
+уточнить
+b2 = rαrβ sign k,
+(7)
+поскольку b2 может быть отрицательным, только если коника σ является гиперболой, что возможно
+лишь в том случае, если вершина конуса K является центром отрицательной гомотетии сфер α и β,
+т.е. они вписаны в K по разные стороны от его вершины.
+По теореме Лагерра для пары (Σ, σ) имеем
+(R2 − d2
+α)(R2 − d2
+β) = 4R2b2,
+(8)
+По теореме Пифагора
+d2
+α = D2
+α − r2
+α,
+d2
+β = D2
+β − r2
+β
+Подставляя эти равенства и (7) в соотношение (8), получаем требуемое соотношение
+R2 + r2
+α − D2
+α
+2R rα
+·
+R2 + r2
+β − D2
+β
+2R rβ
+= sign k
+✷
+5 Трехмерный аналог формулы Эйлера-Чаппла
+В связи с теоремой Эйлера-Чаппла возникает естественный вопрос о возможности ее трехмерного
+обобщения на случай тетраэдра. Этот вопрос был поставлен впервые Ж. Д. Жергонном в 1816 году
+в издаваемом им журнале1 в виде краткой сноски, относящейся к тетраэдру с радиусами описанной
+сферы R, вписанной – r и расстоянием d между их центрами:
+1Annales de math´ematiques pures et appliqu´ees, 6 (1815-1816), p. 228.
+11
+
+«Il
+serait
+sur-tout
+int´eressant
+de
+savoir
+si
+d
+peut
+ˆetre
+exprim´e
+uniquement
+en fonction de R et r.
+J. D. G.»
+Спустя восемь лет в том же журнале было опубликовано положительное решение этой задачи в
+работе Дюрранда [4], где он доказал следующее соотношение:
+d2 = (R + r)(R − 3r).
+(9)
+Этот результат получил широкое признание и в течение многих лет на него ссылались в литерату-
+ре, например, в таких почтенных изданиях как Математическая энциклопедия Клейна «Encyklop¨adie
+der mathematischen Wissenschaften» [18] (первая в мире математическая энциклопедия) и «Enciclopedia
+delle matematiche elementari» [1] (крупнейшая энциклопедия по математике, изданная в Италии). Од-
+нако, формула Дюрранда (9) оказалась неверной, а ответ на вопрос Жергонна – отрицательным: не
+существует общей для всех тетраэдров функциональной зависимости между R, r и d. Доказатель-
+ство Дюрранда было практически безупречным, но незаметная ошибка заключалась в его убежден-
+ности, что описанная и вписанная сфера непременно должны иметь некоторую зависимость. Вопрос
+Жергонна можно было бы сформулировать так: каковы условия существования вписано-описанного
+тетраэдра для двух данных сфер?
+Оказывается никаких необходимых условий для этого не требуется.
+Теорема 5.1. Для любых двух невырожденных квадрик общего положения существует бесконечное
+семейство вписано-описанных тетраэдров. Любая касательная плоскость ко вписанной квадрике
+может содержать грань такого тетраэдра, а его вершиной может быть произвольная точка
+описанной квадрики.
+В работе Фонтене [6] 1899 года эта теорема считается уже известной (см. также [8]).
+Итак, в отличие от плоского случая в пространстве для любых двух произвольных сфер всегда
+существует вписано-описанный в них тетраэдр, причем он может динамически вращаться около этих
+сфер, все время оставаясь вписано-описанным. При этом, любая точка описанной сферы может быть
+вершиной такого тетраэдра.
+Но оказывается, что не для любых двух вещественных сфер такой тетраэдр может быть веще-
+ственным. Критерием существования вещественного вписано-описанного тетраэдра является следу-
+ющее условие Грейса, исправляющее соотношение Дюрранда (9):
+Теорема 5.2 (Grace [8], 1917). Для данных двух сфер S и T необходимым и достаточным условием
+существования вписано-описанного вещественного тетраэдра, у которого вершины лежат на S, а
+плоскости граней касаются T, является следующее условие в зависимости от взаимного располо-
+жения S и T:
+(a) T вложена в S и
+d2 ⩽ (R + r)(R − 3r);
+(b) T и S расположены одна вне другой;
+(c) T и S пересекаются по действительной окружности и
+d2 ⩽ (R − r)(R + 3r).
+6 Вращение Понселе вписано-описанного тетраэдра
+Теорема 5.1 позволяет рассмотреть динамику «вращения» вписано-описанного тетраэдра. Эта дина-
+мика не столь однозначна, как в плоской теореме Понселе. Это показывает следующая теорема.
+12
+
+Теорема 6.1 ( [8]). Пусть вершины тетраэдра лежат на квадрике S, а грани касаются квадрики
+T. Тогда при фиксации плоскости π одной из его граней противоположная вершина P может при
+этом варьироваться, пробегая плоское сечение π′ квадрики S.
+Таким образом, тетраэдр вращается с намного большей свободой, чем вписано-описанный мно-
+гоугольник. Когда выбрана плоскость π, существует целая коника для выбора произвольной точки
+на ней в качестве вершины P, а для каждой такой пары P и π существует однопараметрическое
+семейство вписано-описанных треугольников, каждый из которых может быть противоположной к
+вершине P гранью вписано-описанного тетраэдра. Таким образом, в общем случае существует 4-
+параметрическое семейство тетраэдров.
+У плоской теоремы Понселе есть такой «эффект замыкания»: если начиная с некоторой начальной
+точки A1 строится последовательно вписано-описанная ломаная A1A2 . . . An и оказывается, что звено
+A1An тоже касается вписанной коники, замыкая ее, то такое замыкание будет происходить всегда.
+Если же, по аналогии, строить вписано-описанный тетраэдр для двух данных квадрик S и T,
+последовательно выбирая касательные плоскости его граней, то возникает следующий вопрос. Когда
+мы провели уже три плоскости, которые образовали вписано-описанный трехгранный угол, всегда
+ли можно его замкнуть четвертой плоскостью, чтобы образовался вписано-описанный тетраэдр?
+Ответ дает следующая теорема Фонтене.
+Теорема 6.2 (Fonten´e [6]). Последовательный процесс построения вписано-описанного тетраэдра
+всегда замыкается тогда и только тогда, когда квадрики S и T имеют четыре общих образующих.
+В этом случае, плоскость π и вершина P могут быть выбраны совсем произвольно и, таким
+образом, существует 5-параметрическое семейство вписано-описанных тетраэдров.
+Теорема 6.3. Пусть фиксированы описанная сфера S тетраэдра и одна из восьми его касательных
+сфер T, а тетраэдр динамически «вращается» около них, оставаясь вписано-описанным. Тогда все
+четыре касающиеся T сферы Грейса все время касаются некоторой фиксированной сферы, концен-
+тричной с описанной сферой S.
+Доказательство. Пусть сфера Грейса G проходит через вершины грани a и пусть вписанная
+сфера S касается сферы G в точке P, а плоскости ⟨a⟩ – в точке Q. Обозначим центры сфер S
+и T через OS и OT . Прямая PQ при вращении тетраэдра проходит через фиксированную точку –
+предельную точку K пучка сфер ⟨S, T⟩. Кроме того, на прямой PQ лежит инверсный центр E сферы
+G и плоскости ⟨a⟩, касательная в котором к G параллельна плоскости ⟨a⟩. Следовательно, OSE∥OT Q
+и △OSEK ∼ △OT QK, откуда получаем такое выражение
+OSE = OSK
+OT K · rT ,
+правая часть которого является величиной постоянной при вращении тетраэдра. Тогда, сфера с ради-
+усом, равным этой величине, и центром в точке OS касается сферы Грейса в любой момент вращения.
+✷
+7 Доказательство теоремы Фейербаха через выход в пространство
+Пусть δ – вписанная окружность треугольника ABC с центром в точке I и радиусом r, H – ор-
+тоцентр треугольника ABC, точки A1, B1, C1, I1 – середины отрезков AH, BH, CH, IH (I1 – инцентр
+△A1B1C1). Описанная около △A1B1C1 окружность ϑ – это окружность девяти точек △ABC. Пусть
+также ⊙a, ⊙b, ⊙c – окружности с диаметрами BC, CA, AB, ∆ и Θ – сферы, построенные диаметраль-
+но на окружностях δ и θ.
+13
+
+Доказательство Теоремы Фейербаха.
+Заметим, что касание окружностей δ и θ равносильно касанию сфер ∆ и Θ. По Теореме 3.2 для
+△A1B1C1 его описанная сфера Θ касается его вписано-поднятой сферы Υ. Поэтому касание Θ и ∆
+равносильно тому, что сфера Θ инвариантна при инверсии, переводящей сферы ∆ и Υ друг в друга.
+Заметим, что центр S этой инверсии расположен над точкой H на высоте r (т.е. SH⊥(ABC), |SH| =
+r), а коэффициент инверсии (квадрат радиуса сферы инверсии) равен |IH| · |I1H| = |IH|2
+2
+. Таким
+образом, достаточно доказать равенство Θ(S) = |IH|2
+2
+, которое в силу того, что Θ(S) = θ(H) + r2,
+равносильно соотношению
+|IH|2 − 2r2 = 2θ(H)
+(10)
+Заметим, что левая часть равенства (10) равна степени точки H относительно окружности ξ
+радиуса r
+√
+2 с центром I (ξ высекает на сторонах △ABC равные отрезки длины 2r). Осталось вос-
+пользоваться следующим замечательным свойством окружности ξ.
+Теорема 7.1. Окружности ξ, ⊙a, ⊙b, ⊙c имеют общий радикальный центр в точке H.
+Тогда заметим, что степень точки H относительно окружности θ в два раза меньше ее степени
+относительно окружностей ⊙a, ⊙b, ⊙c и равенство (10) равносильно утверждению ξ(H) = ⊙a(H) =
+⊙b(H) = ⊙c(H) Теоремы 7.1.
+✷
+Для доказательства Теоремы 7.1 рассмотрим окружность χa, диаметром которой является жер-
+гониана вершины A (т.е. отрезок, соединяющий A с точкой касания вписанной окружности δ со
+стороной BC) и воспользуемся следующим свойством окружности χa, возможно, имеющим и само-
+стоятельный интерес.
+Лемма 7.2 (χa-лемма). Окружности χa, ξ, ⊙a принадлежат одному пучку.
+Доказательство Теоремы 7.1. Достаточно проверить, что H ∈ rad(ξ, ⊙a).
+Заметим, что rad(χa, ⊙b) – это высота AH, rad(⊙a, ⊙b) – это высота CH, следовательно,
+H = rad(χa, ⊙a, ⊙b) ∈ rad(χa, ⊙a) = rad(ξ, ⊙a),
+где последнее равенство верно в силу χa-леммы.
+✷
+Доказательство χa-леммы.
+Воспользуемся следующим известным метрическим соотношением для пучков окружностей.
+Лемма 7.3 (О пучке). Если окружности α, β, γ лежат в одном пучке, то для любой точки P ∈ γ
+отношение ее степеней относительно α и β постоянно, причем
+α(P)
+β(P) = dαγ
+dβγ
+,
+(11)
+где dαγ и dβγ – расстояния между центрами α, γ и β, γ.
+Верно и обратное утверждение.
+Лемма 7.4 (Обратная лемма о пучке). Пусть центры окружностей α, β, γ коллинеарны, и на
+окружности γ имеется такая точка P, для которой выполняется соотношение (11). Тогда окруж-
+ности α, β, γ принадлежат одному пучку.
+14
+
+Рис. 6: Окружности ξ, χa, ⊙a принадлежат одному пучку
+В качестве окружностей α, β, γ из Обратной леммы о пучке возьмем окружности ⊙a, ξ, χa, центры
+M, I, L которых лежат на средней линии ML треугольника APQ. При этом,
+LM
+LI = AQ
+AN = ra
+r .
+Для точки P ∈ χa имеем
+α(P) = −(p − b)(p − c),
+β(P) = −r2.
+Тогда (11) запишется в виде соотношения
+(p − b)(p − c)
+r2
+= ra
+r ,
+которое равносильно легко проверяемому равенству
+(p − b)(p − c) = r ra.
+✷
+Доказательство χa-леммы выходом в пространство. Заметим, что окружности χa, δ и окруж-
+ность ⊙P Q с диаметром на отрезке PQ лежат в одном пучке. Поднимем их центры перпендикулярно
+плоскости ⟨ABC⟩, сохраняя коллинеарность: L → L, I → I, M → M, и пусть LL = r
+2, II = r. Тогда
+легко найти, что MM = r + ra
+2
+. При этом сферы S(L), S(J), S(M) с центрами L, I, M, содержащие
+окружности χa, δ, ⊙P Q соответственно, также принадлежат одному пучку. Рассмотрим плоскость
+π∥⟨ABC⟩, проходящую через I, и ортогональную проекцию △ABC → △A′B′C′ на плоскость π. Оста-
+лось заметить, что сечениями сфер S(L), S(J), S(M) плоскостью π являются окружности ξ′, χ′
+a, ⊙′
+a.
+Действительно, для сечений S(L), S(I) это очевидно, а для S(M) это легко проверить, поскольку
+квадрат радиуса окружности ее сечения плоскостью π равен
+|MP|2 +
+�ra + r
+2
+�2
+−
+�ra − r
+2
+�2
+= |MP|2 +rar = |MP|2 +(p−b)(p−c) = |MP|2 +|BP|·|CP| =
+� a
+2
+�2
+.
+Так как при пересечении сфер пучка плоскостью получается пучок окружностей, то ξ′, χ′
+a, ⊙′
+a
+принадлежат одному пучку.
+✷
+15
+
+3
+N
+L
+H
+B
+M
+P
+CСписок литературы
+[1]
+Biggiogero G.,
+La geometria del tetraedro. In: Enciclopedia delle Matematiche Elementari. A cura di
+L. Berzotari, G. Vivanti e D. Gigli. Volume II, parte I. Milano 1937. Ristampa anastatica, Maggio 1943, p. 237.
+[2] Coolidge J. L., A treatise on the circle and the sphere, Oxford: Clarendon Press, 1916.
+[3] Dandelin G., M´emoire sur quelques propri´et´es remarquables de la focale parabolique, Nouveaux m´emoires de
+l’Acad´emie rouale des sciences et belles-lettres de Bruxelles, T. 2 (1822) 171-200.
+[4] Durrande J. B., D´emonstrations ´el´ementaires des principales propriet´es des hexagones inscrits et circonscrits
+au cercle, suivies de la solution de divers probl`emes de la g´eometrie. Dissertation de la g´eometrie pure. Annales
+de math´ematiques pures et appliqu´ees, 14 (1823- 1824) 29-63.
+[5] Feuerbach K. W., Eigenschaften einiger merkw¨urdigen Punkte des geradlinigen Dreiecks, N¨urnberg, 1822.
+[6]
+Fonten´e G.,
+Sur des poly`edres mobiles comparables aux polygones de Poncelet,
+Nouvelles annales de
+math´ematiques 3e s´erie, tome 18 (1899) 67-74.
+[7] Grace J. H., Circles, spheres, and linear complexes, Trans. Cambridge Philosophical Soc. 14 (1898) 153–190.
+[8] Grace J. H., Tetrahedra in relation to spheres and quadrics, Proc. London Math. Soc. 17 (1918) 259-271.
+[9] Griffiths Ph., Harris J., A Poncelet theorem in space, Comm Math. Helv., 52 (1977) 145-160.
+[10] Laguerre E. N., Sur la relation qui existe entre un cercle circonscrit `а un triangle et les ´el´ements d’une conique
+inscrite dans ce triangle, Nouvelles annales de math´ematiques 2e s´erie, tome 18 (1879) 241-246.
+[11] Lewis T. C., Is there an analogue in solid geometry to Feuerbach’s theorem, Messenger of Mathematics, volume
+49 (1919) 187-192.
+[12]
+Hiroshi Maehara, Norihide Tokushige,
+Schl¨afli’s double six, Lie’s line-sphere transformation, and Grace’s
+theorem, European Journal of Combinatorics, 30 (2009) 1337–1351.
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+Hiroshi Maehara, Horst Martini, Tangent Spheres of Tetrahedra and a Theorem of Grace, The American
+Mathematical Monthly, 127:10 (2020) 897-910
+[14] Poncelet J. - V., Trait´e des propri´et´es projectives des figures, Gauthier-Villars, Paris, 1822.
+[15] Protasov V. Yu., Generalized closing theorems, Elem. Math., 66 (2011) 98-117.
+[16] Sommerville, D. M. Y., Analytical geometry of three dimensions, Cambridge University Press, 1943.
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+200-205.
+[18] Zacharias M. Elementargeometrie und elementare nicht-euklidische Geometrie in synthetischer Behandlung. In:
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+Матем. просв.,
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+78–92
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+16
+

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+A QUANTUM APPROACH FOR STOCHASTIC CONSTRAINED BINARY OPTIMIZATION
+Sarthak Gupta and Vassilis Kekatos
+Bradley Dept. of ECE, Virginia Tech, Blacksburg, VA 24061, USA; {gsarthak,kekatos}@vt.edu
+ABSTRACT
+Analytical and practical evidence indicates the advantage
+of quantum computing solutions over classical alternatives.
+Quantum-based heuristics relying on the variational quantum
+eigensolver (VQE) and the quantum approximate optimiza-
+tion algorithm (QAOA) have been shown numerically to
+generate high-quality solutions to hard combinatorial prob-
+lems, yet incorporating constraints to such problems has
+been elusive. To this end, this work puts forth a quantum
+heuristic to cope with stochastic binary quadratically con-
+strained quadratic programs (QCQP). Identifying the strength
+of quantum circuits to efficiently generate samples from prob-
+ability distributions that are otherwise hard to sample from,
+the variational quantum circuit is trained to generate binary-
+valued vectors to approximately solve the aforesaid stochastic
+program. The method builds upon dual decomposition and
+entails solving a sequence of judiciously modified standard
+VQE tasks. Tests on several synthetic problem instances us-
+ing a quantum simulator corroborate the near-optimality and
+feasibility of the method, and its potential to generate feasible
+solutions for the deterministic QCQP too.
+Index Terms— QAOA, VQE, dual decomposition, quan-
+tum unconstrained binary optimization (QUBO).
+1. INTRODUCTION
+Quantum computers exhibit an innate ability to handle ex-
+ponentially large computations in a parallel fashion yet with
+a strong probabilistic flavor.
+Quantum algorithms such as
+Shor’s integer factorization, Grover’s search, and the linear
+system solver of Harrow-Hassidim-Lloyd (HHL) can attain
+polynomial or even exponential speedups over the best known
+algorithms on classical computers [1]. Nonetheless, some of
+these quantum algorithms presume a large number of qubits
+on fault-tolerant quantum computers. In the near-term inter-
+mediate scale (NISQ) era, quantum computers are noisy and
+thus oftentimes limited in terms of number of gates and/or
+qubits. With such limitations in mind, variational quantum
+algorithms have been suggested as promising tools to practi-
+cally showcase quantum advantage in the NISQ setup [2].
+This work was supported by a seed funding grant from the Virginia Com-
+monwealth Cybersecurity Initiative (CCI) – Southwest Virginia node.
+Variational quantum computers involve a sequence of pa-
+rameterized gates. Their parameters are updated externally
+by a classical computer in a closed-loop fashion to steer the
+quantum state towards a desirable direction. The variational
+quantum eigensolver (VQE) used to provide high-quality
+solutions to combinatorial problems is a representative ex-
+ample. The Quantum Approximate Optimization Algorithm
+(QAOA) [3] is a special instance of VQE. In QAOA, not
+only the parameters but also the architecture of the quan-
+tum circuit become problem-dependent. The quantum circuit
+trained by QAOA operates as a sampler to efficiently gener-
+ate near-optimal solutions of binary quadratic problems (e.g.,
+MAXCUT); see [4] for a summary of claims on QAOA.
+While most VQE/QAOA schemes target unconstrained
+problems, dealing with constraints is crucial to several appli-
+cations in machine learning, wireless communications, and
+financial (stock trading) optimization.
+Adding constraints
+to QAOA or adiabetic quantum computing [5] (the QAOA
+counterpart for non-gate-based quantum computers) has been
+pursued in two ways.
+One approach has been to convert
+the constrained problem into an unconstrained minimization
+of a Lagrangian-like function [6, 7]. However, the weights
+for constraint penalties can be safely selected only if con-
+straints are expressed as Boolean functions or linear equal-
+ities. An alternative approach modifies the architecture of
+the quantum circuit (via the mixer Hamiltonian of QAOA)
+to confine quantum states on the subspace spanned by con-
+straints [8, 9, 4, 10]. Nonetheless, constructing such ‘driver’
+mixer Hamiltonians is again highly problem-dependent and
+often limited to equality constraints. Reference [11] devel-
+ops a quantum adiabetic approach to tackle binary linearly-
+constrained quadratic programs. It targets the dual problem
+and interfaces the quantum computer with a branch-and-
+bound scheme ran classically. Reference [12] treats mixed-
+binary quadratic-plus-convex problems using the alternating
+direction method of multipliers (ADMM) to split binary
+and continuous variables into separate minimizations, solved
+by QAOA and classical convex optimizers respectively per
+ADMM iteration.
+Relation to prior work.
+Addressing binary QCQPs by
+quantum heuristics has been largely unexplored to the au-
+thors’ knowledge. We put forth a quantum-based heuristic
+to solve a stochastic binary QCQP. Harnessing the power of
+quantum circuits to sample from probability mass functions
+arXiv:2301.01443v1  [quant-ph]  4 Jan 2023
+
+(PMF) that are hard to sample classically, we devise a dual
+decomposition technique that solves a sequence of standard
+VQE tasks to systematically adjust Lagrangian multipliers.
+Numerical tests using quantum computer simulators pro-
+vided by IBM evaluate this technique on randomly generated
+stochastic and deterministic binary QCQPs.
+2. QUANTUM COMPUTING PRELIMINARIES
+A quantum system consisting of n quantum bits (qubits) is de-
+scribed by an exponentially large state vector |x⟩ ∈ CN with
+N = 2n assuming the system is in a pure state. The Dirac no-
+tation |x⟩ named ket emphasizes that vector x is unit-norm or
+�N−1
+k=0 |xk|2 = 1. If ek is the k-th canonical vector of length
+N, we can write |x⟩ = �N−1
+k=0 xk |ek⟩. The vector ek is of-
+tentimes alternatively expressed as |ek⟩ = |k⟩, where k is the
+binary representation of index k. For example, a system with
+n = 2 qubits has a state in C4, which is spanned by canonical
+vectors {ek}3
+k=0 and e0 = [1 0 0 0]⊤ = |00⟩. Vector |x⟩
+provides a statistical characterization for the quantum state:
+If we measure the quantum system output, its qubits will be
+in configuration |k⟩ with probability |xk|2 for all k. Symbol
+⟨x| termed bra denotes the conjugate transpose of |x⟩, while
+the braket ⟨x|y⟩ denotes the inner product between states.
+The fundamental operations we can perform on a quan-
+tum system is evolution and measurement. The former can
+be described by the application of a unitary U on state |x⟩
+to produce state |y⟩ = U |x⟩. Although U is exponentially
+large, it is usually implemented efficiently using quantum
+gates. Among various types of measurements, we focus on
+projective measurements. A projective measurement is asso-
+ciated with a Hermitian matrix (named observable) and its
+eigenvalue decomposition H = �M
+m=1 λmvmvH
+m. If such
+measurement is performed on |x⟩, outcome m is observed
+with probability pm := | ⟨x|vm⟩ |2. Define a random variable
+taking value λm when outcome m is observed. The expected
+value of this variable is ⟨x|H|x⟩ = �M
+m=1 pmλm. If H is di-
+agonal, the measurement is on the computational basis. This
+is practically important because now vm = em, outcome m
+relates to |m⟩, and each qubit can be measured individually.
+If quantum system i has been prepared in state |xi⟩ for
+i = 1, 2, their joint state would be |x1⟩ ⊗ |x2⟩, where ⊗
+is the Kronecker product. This is oftentimes represented as
+|x1⟩ |x2⟩ or |x1, x2⟩. The Kronecker product rule generalizes
+to the composition of n systems. For example, |1⟩ |1⟩ |0⟩ =
+e1 ⊗ e1 ⊗ e0 = e6 = |110⟩, where the canonical vectors
+shown in the middle are in R2 and those at the end are in R8.
+3. VARIATIONAL QUANTUM EIGENSOLVER (VQE)
+VQE is a heuristic approach to find near-optimal solutions for
+combinatorial problems of the general form
+min
+b∈{0,1}n f(b).
+(1)
+A particular example of interest is the quadratic unconstrained
+binary optimization (QUBO) problem with
+f(b) = b⊤Ab + b⊤c + d
+(2)
+which is known to be NP-hard. For later developments, it is
+convenient to reformulate QUBO in terms of the spin {±1}
+variables through the transformation
+si = 1 − 2bi = (−1)bi for i = 0, . . . , n − 1.
+(3)
+Collecting the spin variables in vector s = 1 − 2b, the
+quadratic objective can be equivalently expressed as
+f(b) = ¯f(s) = s⊤ ¯As + s⊤¯c + ¯d
+(4)
+where ¯A := 1
+4A; ¯c := − 1
+2(A1 + c); and ¯d := 1
+41⊤A1 +
+1
+21⊤c + d. We next explain how VQE samples high-quality
+solutions of (1) by solving an eigenvalue minimization task.
+The VQE method falls under the family of variational
+quantum algorithms. The term variational pertains to the fact
+that the quantum circuit is not fixed, but parameterized by
+relatively few parameters collected in vector θ ∈ RP . These
+parameters are iteratively adjusted by classical computer in
+a closed-loop fashion so that the quantum system eventually
+reaches a desirable state. The process resembles the training
+of a neural network whose weights are updated by an opti-
+mization algorithm. Similarly to neural networks where the
+learner has to select an architecture (e.g., network depth/width
+and type of activations), the parameterized form (also termed
+ansatz) of the variational quantum circuit is specified a pri-
+ori. We will be using a 2-local ansatz where single-qubit RY
+gates are applied to all qubits, followed by a full entanglement
+circuit, all repeated for 3 layers (iterations) [2].
+Given θ and driven by input |0⟩n, the quantum circuit pro-
+duces at its output the quantum state |x(θ)⟩ = U(θ) |0⟩n for
+a unitary N × N matrix U(θ). To simplify notation, we will
+oftentimes write |x⟩ in lieu of |x(θ)⟩. Albeit |x⟩ ∈ CN is
+exponentially long, it can be easily generated by the quan-
+tum circuit though it cannot be read out of the circuit as a
+vector in a computationally efficient manner. Instead, it is rel-
+atively easy to sample from it. Every time we run the quan-
+tum circuit driven by |0⟩n, we will be observing one of the
+binary outputs |k⟩ = |ek⟩ with probability pk := |xk|2 for
+k = 0, . . . , N − 1. The quantum circuit thus serves as an ef-
+ficient sampler from the exponentially large probability mass
+function (PMF) {pk}N−1
+k=0 .
+To exploit this sampling property, we next relate the cost
+f(b) with a so-termed Hamiltonian matrix H so that
+H |ek⟩ = f(|k⟩) |ek⟩
+for all k.
+(5)
+Matrix H is apparently diagonal and carries all N function
+evaluations f(ek) on its diagonal. Moreover, the canonical
+vectors ek constitute the eigenvectors of H, each with cor-
+responding eigenvalue f(|k⟩). Therefore, the minimization
+
+in (1) can be reformulated as the problem of finding the eigen-
+vector corresponding to the minimum eigenvalue of H
+min
+|x⟩ ⟨x| H |x⟩ .
+(6)
+As long as |x⟩ is allowed to take any of the values {ek}N−1
+k=0 ,
+the minimizer of (6) corresponds to the minimizer of (1). For
+example, if a quantum system has n = 3 qubits, its state
+would be |x⟩ ∈ C8. Here ek’s are the columns of the identity
+matrix I8. If the minimizer of (6) is |e5⟩ = |b1b2b3⟩ = |101⟩,
+then the minimizer of (1) is b = [1 0 1]⊤; and vice versa.
+Although H is exponentially large, it can be implemented
+using only O(n2) quantum gates since it can be expressed as
+H =
+n−1
+�
+i=0
+n−1
+�
+j=0
+¯AijZiZj +
+n−1
+�
+i=0
+¯ciZi + ¯dIN
+(7)
+where the N × N Hermitian matrix Zi is defined as
+Zi = I2 ⊗ · · · ⊗ Z ⊗ · · · ⊗ I2 with Z =
+� 1
+0
+0
+−1
+�
+.
+This is a Kronecker product involving (n − 1) identity matri-
+ces I2 and one Pauli-Z operator Z applied to the i-th qubit.
+Matrix H as defined in (7) is obviously diagonal. To estab-
+lish (5), note first that Z |0⟩ = |0⟩ and Z |1⟩ = − |1⟩, or
+more compactly, Z |b⟩ = (−1)b |b⟩. Consequently, when Zi
+is applied to a state |b⟩ = |b1b2 · · · bn⟩, the effect is Zi |b⟩ =
+(−1)bi |b⟩ = si |b⟩ from (3). Similarly, it also holds that
+ZiZj |b⟩ = sisj |b⟩. Property (5) now follows immediately
+by postmultiplying (7) by any |ek⟩ and using f(b) = ¯f(s).
+If |x⟩ in (6) is restricted to set E := {ek}N−1
+k=0 , problem
+(6) is as hard as (1). VQE relaxes (6) to the set of all quantum
+states |x(θ)⟩ that can be parameterized by the chosen ansatz
+and via θ. Problem (6) is then solved over θ rather than |x⟩
+min
+θ
+F(θ) := ⟨x(θ)|H|x(θ)⟩ .
+(8)
+From the eigenvalue property (5), it follows ⟨en| H |ek⟩ =
+f(|k⟩) for all k. How about ⟨x| H |x⟩ for a general state |x⟩?
+Because |x⟩ = �N−1
+k=0 xk |ek⟩, it is easy to show that
+⟨x|H|x⟩ =
+N−1
+�
+k=0
+|xk|2f(|k⟩) =
+N−1
+�
+k=0
+pkf(|k⟩).
+(9)
+In other words, function F(θ) is the average of f under the
+PMF defined by |x⟩. For instance, the random outcome |k⟩ =
+|101⟩ occurring with probability |x5|2 is assigned to the ran-
+dom variable f taking the value f([1 0 1]⊤). Hence, func-
+tion F(θ) is really an expectation (an observable in the quan-
+tum computation parlance) of function f(b) when b is drawn
+from the PMF {|xk(θ)|2}N−1
+k=0 . Ideally, the global minimizer
+θ of (8) defines a PMF via |x(θ)⟩ that samples with non-zero
+probability only the canonical vectors |ek⟩ associated with the
+smallest eigenvalue of H.
+Problem (8) is solved in a hybrid fashion: The quantum
+computer samples from |x(θ)⟩ and estimates F(θ) and pos-
+sibly its gradient ∇θF. A classical computer uses the pre-
+vious information and iteratively updates θ based on a zero-
+or first-order optimization algorithm, such as gradient descent
+or Bayesian optimization. As with training neural networks,
+F(θ) is nonconvex due to the form of the ansatz. Moreover,
+the ensemble statistic F(θ) cannot be computed exactly, but
+estimated as the sample average ˆF(θ) := �R
+r=1 f(br)/R
+over R runs, where br is the quantum output after run r.
+4. CONSTRAINED VQE
+As discussed earlier, VQE provides a successful heuristic for
+solving QUBO through the variational formulation of (8).
+Can VQE be generalized to deal with a binary QCQP of the
+ensuing form?
+min
+b∈{0,1}n f0(b)
+(10)
+s.to fm(b) ≤ 0,
+m = 1 : M.
+Here fm(b) := b⊤Amb + b⊤cm + dm for m = 0, . . . , M.
+Solving such problems is also known to be NP-hard. Provid-
+ing a quantum heuristic to directly deal with (10) seems to
+be challenging. To this end, we relax expectations and aim
+at designing a quantum state |x⟩ from which we can draw
+binary-valued b that solve the stochastic binary QCQP:
+min
+|x⟩
+Ex[f0(b)]
+(11)
+s.to Ex[fm(b)] ≤ 0,
+m = 1 : M.
+As in the unconstrained setup, rather than minimizing over
+|x⟩, we propose optimizing over a PMF parameterized by θ
+and captured by quantum state |x(θ)⟩. Specifically, we sug-
+gest solving the constrained minimization
+min
+θ
+F0(θ)
+(12)
+s.to Fm(θ) ≤ 0 :
+λm,
+m = 1 : M
+where each observable Fm(θ) := ⟨x(θ)|Hm|x(θ)⟩ depends
+on the Hamiltonian Hm defined similar to H in (7) for all
+m. Heed that problem (12) can be reformulated and solved
+as a linear program (LP) over the PMF of b. Nonetheless,
+that requires evaluating {fm(b)}M
+m=0 for all 2n values of b.
+Moreover, the optimization variable of this LP is the vector
+of PMF values that is exponentially large too. That is also the
+case with standard VQE/QAOA.
+Contrary to (10), problem (12) is over the continuous vari-
+able θ, and thus, we can associate a dual variable λm for each
+constraint and define its Lagrangian function
+L(θ; λ) := F0(θ) +
+M
+�
+m=1
+λmFm(θ)
+(13)
+
+where λ ∈ RM collects all dual variables. Problem (12) could
+be solved via dual decomposition, according to which λ is
+updated iteratively via a subgradient ascent step on L as
+λt+1
+m
+:= max
+�
+λt
+m + µtFm(θt), 0
+�
+, m = 1 : M
+(14)
+for a positive step size µt = µ0/(t + α) with α > 0, and θt
+is a minimizer of the Lagrangian L(θ; λt) evaluated at λt:
+θt ∈ arg min
+θ ⟨x(θ)|H0 +
+M
+�
+m=1
+λt
+mHm|x(θ)⟩ .
+(15)
+Problem (15) takes the QUBO form of (8), and is therefore
+amenable to standard VQE or even the celebrated QAOA ap-
+proach. Under the latter, the ansatz takes a particular form that
+depends on the problem Hamiltonian H0 + �M
+m=1 λt
+mHm.
+Here, we used a problem-independent ansatz under the gen-
+eral VQE framework and leave QAOA for future work.
+5. NUMERICAL TESTS
+The novel solver for (12) was implemented in Python us-
+ing the Qiskit library [13].
+The VQE class in Qiskit was
+used to solve the minimization for the primal update (15).
+In addition to providing the ansatz described in Section 3,
+the VQE class was configured with the ‘SLSQP’ optimizer.
+The maximum number of iterations was set to 1, 000, and we
+used the aer simulator statevector quantum simu-
+lation backend. For the dual update in (14), constraint vi-
+olations were measured over the observables Hm using the
+minimum eigenstate returned by VQE. The stopping criteria
+∥λt −λt−1∥2 ≤ 1·10−5 was utilized to ascertain the conver-
+gence of the dual updates (14).
+To illustrate the application of the proposed strategy to
+solving the stochastic binary QCQP in (11), several 2-bit
+problem instances were sampled randomly by drawing the
+entries of {A0, c0, d0} and {A1, c1, d1} from the standard
+normal distribution, while ensuring the resulting problem was
+feasible. The VQE approach was compared against a linear
+program that finds a PMF solving (12); this was possible due
+to the small value of 2n. For the two approaches, the obtained
+PMFs along with the associated dual variables are reported in
+Table 1 for 4 randomly sampled problem instances.
+To study the scalability of the approach and to verify the
+compatibility of the solutions with the deterministic QCQP
+in (10), we also sampled 30 feasible 5-bit problem instances
+with three constraints each. The quadratic cost and constraint
+functions were generated as in the previous test. To avoid
+instances with non-binding constraints, the constants dm in
+the constraint functions were manually adjusted so that at
+least one of the constraints was active and yielded a non-zero
+dual variable. From the sampled problems, it was found that
+the dual decomposition involving VQE was able to produce
+the optimal solutions for 28 out of the 30 problem instances
+Table 1. Comparing the exact solution of (12) obtained via a
+linear program and the proposed quantum-based approach.
+#
+Found PMF
+Dual
+Quantum
+LP
+Quantum
+LP
+1
+[0.44, 0, 0.56, 0]
+[0.44, 0, 0.56, 0]
+0.854
+0.851
+2
+[0.71, 0, 0.29, 0]
+[0.70, 0, 0.30, 0]
+0.337
+0.337
+3
+[0, 0.80, 0, 0.20]
+[0, 0.80, 0, 0.20]
+0.459
+0.459
+4
+[0, 0, 0.61, 0.39]
+[0, 0, 0.60, 0.40]
+0.566
+0.566
+Fig. 1. Convergence of dual variables under dual updates (14)
+for a stochastic binary QCQP with M = 3 constraints.
+tested, whereas infeasible binary candidates were obtained for
+the remaining 2 instances. Figure 1 illustrates the conver-
+gence of the dual variables for one of the problem instances,
+where all three constraints were found to be active.
+6. CONCLUSIONS
+A novel generalization of VQE to address the need for dealing
+with stochastic binary QCQPs has been developed. Lever-
+aging dual decomposition, the approach entails solving a
+sequence of judiciously modified VQE tasks. Numerical tests
+demonstrate that upon convergence of the constrained VQE
+algorithm, the variational quantum circuit is able to sample
+from a stochastic policy to generate binary-valued vectors
+that minimize the binary QCQP and satisfy its constraints
+in expectation. Some of these samples seem to be feasible
+for the deterministic binary QCQP too. This novel heuristic
+sets the foundation for further developments towards con-
+strained discrete optimization.
+We are currently exploring
+several exciting directions: i) Coupling this approach with
+QAOA rather than VQE; ii) skipping the nested optimization
+in (15) through a primal-dual decomposition alternative as
+in [14, 15]; and iii) dealing with mixed-binary setups.
+
+Convergence of dual variables
+入1
+1.2
+入2
+入3
+1.0
+0.8
+0.6
+0.4
+0.2
+0.0
+0
+20
+40
+60
+80
+100
+120
+140
+Iterations7. REFERENCES
+[1] Michael A. Nielsen and Isaac L. Chuang,
+Quantum
+Computation and Quantum Information,
+Cambridge
+University Press, 2000.
+[2] Osvaldo Simeone,
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+in Signal Processing, vol. 16, no. 1–2, pp. 1–223, 2022.
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+2, pp. 1310–1321, Mar. 2022.
+

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+page_content='A QUANTUM APPROACH FOR STOCHASTIC CONSTRAINED BINARY OPTIMIZATION  Sarthak Gupta and Vassilis Kekatos  Bradley Dept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content=' of ECE, Virginia Tech, Blacksburg, VA 24061, USA;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content=' {gsarthak,kekatos}@vt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content='edu  ABSTRACT  Analytical and practical evidence indicates the advantage  of quantum computing solutions over classical alternatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content='  Quantum-based heuristics relying on the variational quantum  eigensolver (VQE) and the quantum approximate optimiza-  tion algorithm (QAOA) have been shown numerically to  generate high-quality solutions to hard combinatorial prob-  lems, yet incorporating constraints to such problems has  been elusive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content=' To this end, this work puts forth a quantum  heuristic to cope with stochastic binary quadratically con-  strained quadratic programs (QCQP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content=' Identifying the strength  of quantum circuits to efficiently generate samples from prob-  ability distributions that are otherwise hard to sample from,  the variational quantum circuit is trained to generate binary-  valued vectors to approximately solve the aforesaid stochastic  program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content=' The method builds upon dual decomposition and  entails solving a sequence of judiciously modified standard  VQE tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content=' Tests on several synthetic problem instances us-  ing a quantum simulator corroborate the near-optimality and  feasibility of the method, and its potential to generate feasible  solutions for the deterministic QCQP too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content='  Index Terms— QAOA, VQE, dual decomposition, quan-  tum unconstrained binary optimization (QUBO).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content=' INTRODUCTION  Quantum computers exhibit an innate ability to handle ex-  ponentially large computations in a parallel fashion yet with  a strong probabilistic flavor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content='  Quantum algorithms such as  Shor’s integer factorization, Grover’s search, and the linear  system solver of Harrow-Hassidim-Lloyd (HHL) can attain  polynomial or even exponential speedups over the best known  algorithms on classical computers [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content=' Nonetheless, some of  these quantum algorithms presume a large number of qubits  on fault-tolerant quantum computers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content=' In the near-term inter-  mediate scale (NISQ) era, quantum computers are noisy and  thus oftentimes limited in terms of number of gates and/or  qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content=' With such limitations in mind, variational quantum  algorithms have been suggested as promising tools to practi-  cally showcase quantum advantage in the NISQ setup [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content='  This work was supported by a seed funding grant from the Virginia Com-  monwealth Cybersecurity Initiative (CCI) – Southwest Virginia node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content='  Variational quantum computers involve a sequence of pa-  rameterized gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content=' Their parameters are updated externally  by a classical computer in a closed-loop fashion to steer the  quantum state towards a desirable direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content=' The variational  quantum eigensolver (VQE) used to provide high-quality  solutions to combinatorial problems is a representative ex-  ample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content=' The Quantum Approximate Optimization Algorithm  (QAOA) [3] is a special instance of VQE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content=' In QAOA, not  only the parameters but also the architecture of the quan-  tum circuit become problem-dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content=' The quantum circuit  trained by QAOA operates as a sampler to efficiently gener-  ate near-optimal solutions of binary quadratic problems (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content=',  MAXCUT);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content=' see [4] for a summary of claims on QAOA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content='  While most VQE/QAOA schemes target unconstrained  problems, dealing with constraints is crucial to several appli-  cations in machine learning, wireless communications, and  financial (stock trading) optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content='  Adding constraints  to QAOA or adiabetic quantum computing [5] (the QAOA  counterpart for non-gate-based quantum computers) has been  pursued in two ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content='  One approach has been to convert  the constrained problem into an unconstrained minimization  of a Lagrangian-like function [6, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content=' However, the weights  for constraint penalties can be safely selected only if con-  straints are expressed as Boolean functions or linear equal-  ities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content=' An alternative approach modifies the architecture of  the quantum circuit (via the mixer Hamiltonian of QAOA)  to confine quantum states on the subspace spanned by con-  straints [8, 9, 4, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content=' Nonetheless, constructing such ‘driver’  mixer Hamiltonians is again highly problem-dependent and  often limited to equality constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content=' Reference [11] devel-  ops a quantum adiabetic approach to tackle binary linearly-  constrained quadratic programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content=' It targets the dual problem  and interfaces the quantum computer with a branch-and-  bound scheme ran classically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content=' Reference [12] treats mixed-  binary quadratic-plus-convex problems using the alternating  direction method of multipliers (ADMM) to split binary  and continuous variables into separate minimizations, solved  by QAOA and classical convex optimizers respectively per  ADMM iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content='  Relation to prior work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content='  Addressing binary QCQPs by  quantum heuristics has been largely unexplored to the au-  thors’ knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content=' We put forth a quantum-based heuristic  to solve a stochastic binary QCQP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content=' Harnessing the power of  quantum circuits to sample from probability mass functions  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content='01443v1  [quant-ph]  4 Jan 2023  (PMF) that are hard to sample classically, we devise a dual  decomposition technique that solves a sequence of standard  VQE tasks to systematically adjust Lagrangian multipliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content='  Numerical tests using quantum computer simulators pro-  vided by IBM evaluate this technique on randomly generated  stochastic and deterministic binary QCQPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content=' QUANTUM COMPUTING PRELIMINARIES  A quantum system consisting of n quantum bits (qubits) is de-  scribed by an exponentially large state vector |x⟩ ∈ CN with  N = 2n assuming the system is in a pure state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content=' The Dirac no-  tation |x⟩ named ket emphasizes that vector x is unit-norm or  �N−1  k=0 |xk|2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content=' If ek is the k-th canonical vector of length  N, we can write |x⟩ = �N−1  k=0 xk |ek⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content=' The vector ek is of-  tentimes alternatively expressed as |ek⟩ = |k⟩, where k is the  binary representation of index k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content=' For example, a system with  n = 2 qubits has a state in C4, which is spanned by canonical  vectors {ek}3  k=0 and e0 = [1 0 0 0]⊤ = |00⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content=' Vector |x⟩  provides a statistical characterization for the quantum state:  If we measure the quantum system output, its qubits will be  in configuration |k⟩ with probability |xk|2 for all k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content=' Symbol  ⟨x| termed bra denotes the conjugate transpose of |x⟩, while  the braket ⟨x|y⟩ denotes the inner product between states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content='  The fundamental operations we can perform on a quan-  tum system is evolution and measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content=' The former can  be described by the application of a unitary U on state |x⟩  to produce state |y⟩ = U |x⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content=' Although U is exponentially  large, it is usually implemented efficiently using quantum  gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content=' Among various types of measurements, we focus on  projective measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content=' A projective measurement is asso-  ciated with a Hermitian matrix (named observable) and its  eigenvalue decomposition H = �M  m=1 λmvmvH  m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content=' If such  measurement is performed on |x⟩, outcome m is observed  with probability pm := | ⟨x|vm⟩ |2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content=' Define a random variable  taking value λm when outcome m is observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content=' The expected  value of this variable is ⟨x|H|x⟩ = �M  m=1 pmλm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content=' If H is di-  agonal, the measurement is on the computational basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content=' This  is practically important because now vm = em, outcome m  relates to |m⟩, and each qubit can be measured individually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content='  If quantum system i has been prepared in state |xi⟩ for  i = 1, 2, their joint state would be |x1⟩ ⊗ |x2⟩, where ⊗  is the Kronecker product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content=' This is oftentimes represented as  |x1⟩ |x2⟩ or |x1, x2⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content=' The Kronecker product rule generalizes  to the composition of n systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content=' For example, |1⟩ |1⟩ |0⟩ =  e1 ⊗ e1 ⊗ e0 = e6 = |110⟩, where the canonical vectors  shown in the middle are in R2 and those at the end are in R8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content=' VARIATIONAL QUANTUM EIGENSOLVER (VQE)  VQE is a heuristic approach to find near-optimal solutions for  combinatorial problems of the general form  min  b∈{0,1}n f(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content='  (1)  A particular example of interest is the quadratic unconstrained  binary optimization (QUBO) problem with  f(b) = b⊤Ab + b⊤c + d  (2)  which is known to be NP-hard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content=' For later developments, it is  convenient to reformulate QUBO in terms of the spin {±1}  variables through the transformation  si = 1 − 2bi = (−1)bi for i = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content=' , n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content='  (3)  Collecting the spin variables in vector s = 1 − 2b, the  quadratic objective can be equivalently expressed as  f(b) = ¯f(s) = s⊤ ¯As + s⊤¯c + ¯d  (4)  where ¯A := 1  4A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content=' ¯c := − 1  2(A1 + c);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content=' and ¯d := 1  41⊤A1 +  1  21⊤c + d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content=' We next explain how VQE samples high-quality  solutions of (1) by solving an eigenvalue minimization task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content='  The VQE method falls under the family of variational  quantum algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content=' The term variational pertains to the fact  that the quantum circuit is not fixed, but parameterized by  relatively few parameters collected in vector θ ∈ RP .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content=' These  parameters are iteratively adjusted by classical computer in  a closed-loop fashion so that the quantum system eventually  reaches a desirable state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content=' The process resembles the training  of a neural network whose weights are updated by an opti-  mization algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content=' Similarly to neural networks where the  learner has to select an architecture (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content=', network depth/width  and type of activations), the parameterized form (also termed  ansatz) of the variational quantum circuit is specified a pri-  ori.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content=' We will be using a 2-local ansatz where single-qubit RY  gates are applied to all qubits, followed by a full entanglement  circuit, all repeated for 3 layers (iterations) [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content='  Given θ and driven by input |0⟩n, the quantum circuit pro-  duces at its output the quantum state |x(θ)⟩ = U(θ) |0⟩n for  a unitary N × N matrix U(θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content=' To simplify notation, we will  oftentimes write |x⟩ in lieu of |x(θ)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content=' Albeit |x⟩ ∈ CN is  exponentially long, it can be easily generated by the quan-  tum circuit though it cannot be read out of the circuit as a  vector in a computationally efficient manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content=' Instead, it is rel-  atively easy to sample from it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content=' Every time we run the quan-  tum circuit driven by |0⟩n, we will be observing one of the  binary outputs |k⟩ = |ek⟩ with probability pk := |xk|2 for  k = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content=' , N − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content=' The quantum circuit thus serves as an ef-  ficient sampler from the exponentially large probability mass  function (PMF) {pk}N−1  k=0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content='  To exploit this sampling property, we next relate the cost  f(b) with a so-termed Hamiltonian matrix H so that  H |ek⟩ = f(|k⟩) |ek⟩  for all k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content='  (5)  Matrix H is apparently diagonal and carries all N function  evaluations f(ek) on its diagonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content=' Moreover, the canonical  vectors ek constitute the eigenvectors of H, each with cor-  responding eigenvalue f(|k⟩).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content=' Therefore, the minimization  in (1) can be reformulated as the problem of finding the eigen-  vector corresponding to the minimum eigenvalue of H  min  |x⟩ ⟨x| H |x⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content='  (6)  As long as |x⟩ is allowed to take any of the values {ek}N−1  k=0 ,  the minimizer of (6) corresponds to the minimizer of (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content=' For  example, if a quantum system has n = 3 qubits, its state  would be |x⟩ ∈ C8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content=' Here ek’s are the columns of the identity  matrix I8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content=' If the minimizer of (6) is |e5⟩ = |b1b2b3⟩ = |101⟩,  then the minimizer of (1) is b = [1 0 1]⊤;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content=' and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content='  Although H is exponentially large, it can be implemented  using only O(n2) quantum gates since it can be expressed as  H =  n−1  �  i=0  n−1  �  j=0  ¯AijZiZj +  n−1  �  i=0  ¯ciZi + ¯dIN  (7)  where the N × N Hermitian matrix Zi is defined as  Zi = I2 ⊗ · · · ⊗ Z ⊗ · · · ⊗ I2 with Z =  � 1  0  0  −1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content='  This is a Kronecker product involving (n − 1) identity matri-  ces I2 and one Pauli-Z operator Z applied to the i-th qubit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content='  Matrix H as defined in (7) is obviously diagonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content=' To estab-  lish (5), note first that Z |0⟩ = |0⟩ and Z |1⟩ = − |1⟩, or  more compactly, Z |b⟩ = (−1)b |b⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content=' Consequently, when Zi  is applied to a state |b⟩ = |b1b2 · · · bn⟩, the effect is Zi |b⟩ =  (−1)bi |b⟩ = si |b⟩ from (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content=' Similarly, it also holds that  ZiZj |b⟩ = sisj |b⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content=' Property (5) now follows immediately  by postmultiplying (7) by any |ek⟩ and using f(b) = ¯f(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content='  If |x⟩ in (6) is restricted to set E := {ek}N−1  k=0 , problem  (6) is as hard as (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content=' VQE relaxes (6) to the set of all quantum  states |x(θ)⟩ that can be parameterized by the chosen ansatz  and via θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content=' Problem (6) is then solved over θ rather than |x⟩  min  θ  F(θ) := ⟨x(θ)|H|x(θ)⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content='  (8)  From the eigenvalue property (5), it follows ⟨en| H |ek⟩ =  f(|k⟩) for all k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content=' How about ⟨x| H |x⟩ for a general state |x⟩?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content='  Because |x⟩ = �N−1  k=0 xk |ek⟩, it is easy to show that  ⟨x|H|x⟩ =  N−1  �  k=0  |xk|2f(|k⟩) =  N−1  �  k=0  pkf(|k⟩).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content='  (9)  In other words, function F(θ) is the average of f under the  PMF defined by |x⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content=' For instance, the random outcome |k⟩ =  |101⟩ occurring with probability |x5|2 is assigned to the ran-  dom variable f taking the value f([1 0 1]⊤).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content=' Hence, func-  tion F(θ) is really an expectation (an observable in the quan-  tum computation parlance) of function f(b) when b is drawn  from the PMF {|xk(θ)|2}N−1  k=0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content=' Ideally, the global minimizer  θ of (8) defines a PMF via |x(θ)⟩ that samples with non-zero  probability only the canonical vectors |ek⟩ associated with the  smallest eigenvalue of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content='  Problem (8) is solved in a hybrid fashion: The quantum  computer samples from |x(θ)⟩ and estimates F(θ) and pos-  sibly its gradient ∇θF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content=' A classical computer uses the pre-  vious information and iteratively updates θ based on a zero-  or first-order optimization algorithm, such as gradient descent  or Bayesian optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content=' As with training neural networks,  F(θ) is nonconvex due to the form of the ansatz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content=' Moreover,  the ensemble statistic F(θ) cannot be computed exactly, but  estimated as the sample average ˆF(θ) := �R  r=1 f(br)/R  over R runs, where br is the quantum output after run r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content=' CONSTRAINED VQE  As discussed earlier, VQE provides a successful heuristic for  solving QUBO through the variational formulation of (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content='  Can VQE be generalized to deal with a binary QCQP of the  ensuing form?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content='  min  b∈{0,1}n f0(b)  (10)  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content='to fm(b) ≤ 0,  m = 1 : M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content='  Here fm(b) := b⊤Amb + b⊤cm + dm for m = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content=' , M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content='  Solving such problems is also known to be NP-hard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content=' Provid-  ing a quantum heuristic to directly deal with (10) seems to  be challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content=' To this end, we relax expectations and aim  at designing a quantum state |x⟩ from which we can draw  binary-valued b that solve the stochastic binary QCQP:  min  |x⟩  Ex[f0(b)]  (11)  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content='to Ex[fm(b)] ≤ 0,  m = 1 : M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content='  As in the unconstrained setup, rather than minimizing over  |x⟩, we propose optimizing over a PMF parameterized by θ  and captured by quantum state |x(θ)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content=' Specifically, we sug-  gest solving the constrained minimization  min  θ  F0(θ)  (12)  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content='to Fm(θ) ≤ 0 :  λm,  m = 1 : M  where each observable Fm(θ) := ⟨x(θ)|Hm|x(θ)⟩ depends  on the Hamiltonian Hm defined similar to H in (7) for all  m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content=' Heed that problem (12) can be reformulated and solved  as a linear program (LP) over the PMF of b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content=' Nonetheless,  that requires evaluating {fm(b)}M  m=0 for all 2n values of b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content='  Moreover, the optimization variable of this LP is the vector  of PMF values that is exponentially large too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content=' That is also the  case with standard VQE/QAOA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content='  Contrary to (10), problem (12) is over the continuous vari-  able θ, and thus, we can associate a dual variable λm for each  constraint and define its Lagrangian function  L(θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content=' λ) := F0(θ) +  M  �  m=1  λmFm(θ)  (13)  where λ ∈ RM collects all dual variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content=' Problem (12) could  be solved via dual decomposition, according to which λ is  updated iteratively via a subgradient ascent step on L as  λt+1  m  := max  �  λt  m + µtFm(θt), 0  �  , m = 1 : M  (14)  for a positive step size µt = µ0/(t + α) with α > 0, and θt  is a minimizer of the Lagrangian L(θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content=' λt) evaluated at λt:  θt ∈ arg min  θ ⟨x(θ)|H0 +  M  �  m=1  λt  mHm|x(θ)⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content='  (15)  Problem (15) takes the QUBO form of (8), and is therefore  amenable to standard VQE or even the celebrated QAOA ap-  proach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content=' Under the latter, the ansatz takes a particular form that  depends on the problem Hamiltonian H0 + �M  m=1 λt  mHm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content='  Here, we used a problem-independent ansatz under the gen-  eral VQE framework and leave QAOA for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content=' NUMERICAL TESTS  The novel solver for (12) was implemented in Python us-  ing the Qiskit library [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content='  The VQE class in Qiskit was  used to solve the minimization for the primal update (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content='  In addition to providing the ansatz described in Section 3,  the VQE class was configured with the ‘SLSQP’ optimizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content='  The maximum number of iterations was set to 1, 000, and we  used the aer simulator statevector quantum simu-  lation backend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content=' For the dual update in (14), constraint vi-  olations were measured over the observables Hm using the  minimum eigenstate returned by VQE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content=' The stopping criteria  ∥λt −λt−1∥2 ≤ 1·10−5 was utilized to ascertain the conver-  gence of the dual updates (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content='  To illustrate the application of the proposed strategy to  solving the stochastic binary QCQP in (11), several 2-bit  problem instances were sampled randomly by drawing the  entries of {A0, c0, d0} and {A1, c1, d1} from the standard  normal distribution, while ensuring the resulting problem was  feasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content=' The VQE approach was compared against a linear  program that finds a PMF solving (12);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content=' this was possible due  to the small value of 2n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content=' For the two approaches, the obtained  PMFs along with the associated dual variables are reported in  Table 1 for 4 randomly sampled problem instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content='  To study the scalability of the approach and to verify the  compatibility of the solutions with the deterministic QCQP  in (10), we also sampled 30 feasible 5-bit problem instances  with three constraints each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content=' The quadratic cost and constraint  functions were generated as in the previous test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content=' To avoid  instances with non-binding constraints, the constants dm in  the constraint functions were manually adjusted so that at  least one of the constraints was active and yielded a non-zero  dual variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content=' From the sampled problems, it was found that  the dual decomposition involving VQE was able to produce  the optimal solutions for 28 out of the 30 problem instances  Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content=' Comparing the exact solution of (12) obtained via a  linear program and the proposed quantum-based approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content='  #  Found PMF  Dual  Quantum  LP  Quantum  LP  1  [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content='44, 0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content='56, 0]  [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content='44, 0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content='56, 0]  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content='854  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content='851  2  [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content='71, 0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content='29, 0]  [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content='70, 0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content='30, 0]  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content='337  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content='337  3  [0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content='80, 0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content='20]  [0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content='80, 0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content='20]  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content='459  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content='459  4  [0, 0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content='61, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content='39]  [0, 0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content='60, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content='40]  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content='566  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content='566  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content=' Convergence of dual variables under dual updates (14)  for a stochastic binary QCQP with M = 3 constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content='  tested, whereas infeasible binary candidates were obtained for  the remaining 2 instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content=' Figure 1 illustrates the conver-  gence of the dual variables for one of the problem instances,  where all three constraints were found to be active.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content=' CONCLUSIONS  A novel generalization of VQE to address the need for dealing  with stochastic binary QCQPs has been developed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content=' Lever-  aging dual decomposition, the approach entails solving a  sequence of judiciously modified VQE tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content=' Numerical tests  demonstrate that upon convergence of the constrained VQE  algorithm, the variational quantum circuit is able to sample  from a stochastic policy to generate binary-valued vectors  that minimize the binary QCQP and satisfy its constraints  in expectation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content=' Some of these samples seem to be feasible  for the deterministic binary QCQP too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content=' This novel heuristic  sets the foundation for further developments towards con-  strained discrete optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content='  We are currently exploring  several exciting directions: i) Coupling this approach with  QAOA rather than VQE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content=' ii) skipping the nested optimization  in (15) through a primal-dual decomposition alternative as  in [14, 15];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content=' and iii) dealing with mixed-binary setups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content='  Convergence of dual variables  入1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content='2  入2  入3  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
+page_content='0  0  20  40  60  80  100  120  140  Iterations7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'}
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+Printed 12 January 2023
+(MN LATEX style file v2.2)
+Comprehensive spectroscopic and photometric study of
+pulsating eclipsing binary star AI Hya
+F. Kahraman Ali¸cavu¸s1,2⋆, T. Pawar3†, K. G. He�lminiak3, G. Handler4, A. Moharana3,
+F. Ali¸cavu¸s1,2, P. De Cat5, F. Leone6,7, G. Catanzaro7, M. Giarrusso7,8, N. Ukita9,10,
+E. Kambe11
+1C¸anakkale Onsekiz Mart University, Faculty of Science, Physics Department, 17100, Canakkale, Turkey
+2C¸anakkale Onsekiz Mart University, Astrophysics Research Center and Ulupınar Observatory, TR-17100, anakkale, Turkey
+3Nicolaus Copernicus Astronomical Center, Department of Astrophysics, ul. Rabia´nska 8, PL-87-100 Toru´n, Poland
+4Nicolaus Copernicus Astronomical Center, Polish Academy of Sciences, Bartycka 18, PL-00-716 Warsaw, Poland
+5Royal Observatory of Belgium, Ringlaan 3, B-1180 Brussel, Belgium
+6Dipartimento di Fisica e Astronomia, Sezione Astrofisica, Universit?a di Catania, Via S. Sofia 78, I-95123 Catania, Italy
+7INAF, Osservatorio Astrofisico di Catania, Via S. Sofia 78, I-95123 Catania, Italy
+8University of Florence, Department of Physics and Astronomy, Via Giovanni Sansone 1, I-50019 Sesto Fiorentino, Italy
+9Okayama Astrophysical Observatory, National Astronomical Observatory of Japan, 3037-5 Honjo, Kamogata, Asakuchi, Okayama 719-0232, Japan
+10The Graduate University for Advanced Studies, 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan
+11Subaru Telescope, National Astronomical Observatory of Japan, 650 North Aohoku Place, Hilo, HI 96720, USA
+Accepted ... Received ...; in original form ...
+ABSTRACT
+The pulsating eclipsing binaries are remarkable systems that provide an opportu-
+nity to probe the stellar interior and to determine the fundamental stellar parameters
+precisely. Especially the detached eclipsing binary systems with (a) pulsating compo-
+nent(s) are significant objects to understand the nature of the oscillations since the
+binary effects in these systems are negligible. Recent studies based on space data have
+shown that the pulsation mechanisms of some oscillating stars are not completely
+understood. Hence, comprehensive studies of a number of pulsating stars within de-
+tached eclipsing binaries are important. In this study, we present a detailed analysis
+of the pulsating detached eclipsing binary system AI Hya which was studied by two
+independent groups with different methods. We carried out a spectroscopic survey to
+estimate the orbital parameters via radial velocity measurements and the atmospheric
+parameters of each binary component using the composite and/or disentangled spec-
+tra. We found that the more luminous component of the system is a massive, cool
+and chemically normal star while the hotter binary component is a slightly metal-rich
+object. The fundamental parameters of AI Hya were determined by the analysis of
+binary variations and subsequently used in the evolutionary modelling. Consequently,
+we obtained the age of the system as 850 ± 20 Myr and found that both binary com-
+ponents are situated in the δ Scuti instability strip. The frequency analysis revealed
+pulsation frequencies between the 5.5 – 13.0 d−1 and we tried to estimate which binary
+component is the pulsating one. However, it turned out that those frequencies could
+originate from both binary components.
+Key words:
+stars: binaries: eclipsing – stars: atmospheres – stars: fundamental
+parameters – stars: variables: δ Scuti – stars: individual: AI Hya
+⋆ E-mail: filizkahraman01@gmail.com
+† E-mail: pawar@ncac.torun.pl
+1
+INTRODUCTION
+To understand the universe, it is necessary to comprehend
+stars which are its building blocks. For a deep investigation
+of stars, we should know their basic stellar parameters such
+© 2021 RAS
+arXiv:2301.04409v1  [astro-ph.SR]  11 Jan 2023
+
+2
+F. Kahraman Ali¸cavu¸s et. al.
+as mass (M), radius (R) and chemical composition. Binary
+stars, in particular the eclipsing ones, are the most suitable
+objects to derive these parameters as M and R can be de-
+rived with an accuracy better than 1% (Torres, Andersen, &
+Gim´enez 2010; Southworth 2013). Therefore, these systems
+are substantial for a better understanding of the universe,
+our Galaxy, and, most directly, stellar evolution. However,
+eclipsing binary systems as such do not provide information
+about the stellar interior. This is where the pulsating stars
+come in. The oscillation frequencies of pulsating stars can be
+used to probe the stellar interior by applying asteroseismic
+methods, making eclipsing binary systems with (a) pulsat-
+ing component(s) one of the most valuable tools to improve
+our knowledge of stellar evolution.
+Various types of pulsating stars in different evolution-
+ary states exist. Some of them, such as β Cephei, δ Scuti, and
+γ Doradus stars (Lampens 2021; Southworth 2021), are also
+found in eclipsing binary systems. The δ Scuti variables are
+the most common pulsating stars found in eclipsing bina-
+ries because of their relatively short pulsation periods. The
+δ Scuti stars are A to F-type dwarf or giant stars generally
+exhibiting pressure mode oscillations with periods between
+18 min and 8 h and amplitudes below 0m.1 in the V-band
+(Aerts, Christensen-Dalsgaard, & Kurtz 2010). Their the-
+oretical instability strip (e.g. Dupret et al. 2005) indicates
+the location of objects in the Hertzsprung-Russell (H-R) di-
+agram that are expected to show δ Scuti-type oscillations.
+Thanks to space missions such as Kepler (Borucki et al.
+2010) and the Transiting Exoplanet Survey Satellite (TESS,
+Ricker et al. 2014), we learned that δ Scuti stars are also ob-
+served beyond the borders of the theoretical instability strip,
+showing the necessity to revise them (Uytterhoeven et al.
+2011; Antoci et al. 2014; Bowman & Kurtz 2018). Accord-
+ing to the latest catalog of δ Scuti stars in eclipsing binaries,
+there are around 90 such objects (Kahraman Ali¸cavu¸s et al.
+2017). This number is now increasing especially by the dis-
+coveries of new systems from the investigation of the space
+data (e.g. Kahraman Ali¸cavu¸s et al. 2022; Gaulme & Guzik
+2019). The pulsations of the δ Scuti stars in eclipsing bina-
+ries are affected by the other binary component (Kahraman
+Ali¸cavu¸s et al. 2017; Liakos & Niarchos 2017). Indeed, their
+pulsation period (Ppuls) decreases when the orbital period
+(Porb) becomes shorter and, hence, the other component ap-
+proaches the pulsating component. It was also thought that
+the tidal forces between the binary components can alter the
+pulsation axis (Kurtz et al. 2020). The first observational
+proof of this was presented by Handler et al. (2020) thanks
+to the high-quality data of TESS. These authors showed that
+in some binary systems the pulsation axis can align with the
+orbital axis because of the tidal forces. This type of object
+is now known as tidally tilted pulsators and they are a clear
+proof of binary effects on pulsations.
+For a deep understanding of the effects of binarity on
+pulsations in eclipsing binary systems and on stellar evolu-
+tion and structure, comprehensive investigations of such sys-
+tems are necessary. AI Hya (V = 9m.35) is an eclipsing binary
+system with a δ Scuti component consisting of a F2m and
+F0V star (Stancliffe et al. 2015). It has an eccentric orbit and
+an orbital period of 8.289649(2) days (Kreiner 2004). Spec-
+troscopic observations revealed that AI Hya is a double-lined
+binary system (Popper 1988). In a recent study, an updated
+photometric analysis based on the TESS data of AI Hya was
+given which shows that the secondary component exhibits
+multiperiodic oscillations (Lee, Hong, & Kristiansen 2020).
+However, no detailed spectral analysis with high-resolution
+spectra has been carried out for the system so far. There-
+fore, we provide a detailed photometric and spectral analysis
+of AI Hya in this study to reveal the true character of this
+interesting object.
+Two teams were working on this system independently.
+One group was led by TP (group-P with KH, AM, NU, and
+EK) and the second group by FKA (group-K with GH, FA,
+PDC, FL, GC, and MG). We used the same photometric but
+different spectroscopic data. We compared our partial re-
+sults as the work progressed. However, the overall approach
+used by each group was different. In the end, we combined
+our results to obtain the final parameters of the system.
+The paper is organized as follows. In Sect. 2 the observa-
+tional data are introduced. The radial velocity and spectral
+analyses are given in Sect. 3 and Sect. 4, respectively. The
+binary modelling and the pulsation frequency analysis are
+presented in Sect. 5 and Sect. 6. In Sect. 7, discussions and
+conclusions are given.
+2
+OBSERVATIONAL DATA
+In the photometric analysis of AI Hya, TESS data was used
+by both groups. TESS was launched in April 2018 mainly to
+detect new exoplanets (Ricker et al. 2014). TESS has moni-
+tored almost the entire sky which has been subdivided into
+sectors that are observed for about 27 days each. The TESS
+observations were taken in 2-min. short (SC) and 30-min
+long (LC) cadence in the nominal phase of the mission (first
+two years). For the extended mission, the LC was reduced
+to 10-min. The data are available in the Barbara A. Mikul-
+ski Archive for Telescopes (MAST)1 where they are released
+in different versions: simple aperture photometry (SAP) and
+pre-search data conditioning SAP fluxes (PDCSAP). AI Hya
+was observed in one sector only (sector 7). The 2-min SAP
+fluxes were used in our analysis since SAP fluxes have lower
+flux uncertainty and 2-min data are more suitable for the
+analysis of AI Hya (see Sect. 6). They were converted into
+magnitude by using the same method as Kahraman Ali¸cavu¸s
+et al. (2022).
+Photometric data from ground-based surveys also exist,
+e.g. from ASAS 3 (Pojma´nski 2002) and ASAS-SN (Jayas-
+inghe et al. 2018), but they are of inferior quality and do not
+allow for proper analysis of pulsations. The TESS sector 7
+data are the best ones available so far, although AI Hya will
+again be visible in the satellite’s field of view in sector 61.
+The spectroscopic data of the system were taken from
+four different instruments. The list of the instruments and
+the basic information about them are given in Table 1. One
+spectrum was taken with Catania Astrophysical Observatory
+Spectropolarimeter (CAOS, Leone, et al. 2016). The CAOS
+is a high-resolution, fibre-fed, cross-dispersed ´echelle spec-
+trograph installed to the 91-cm telescope at the Catania
+Astrophysical Observatory (Mt. Etna, Italy). Three spectra
+of AI Hya were collected from the CORALIE ´echelle spec-
+trograph which is mounted on the 1.2-m Leonhard Euler
+1 https://mast.stsci.edu
+© 2021 RAS, MNRAS 000, 1–13
+
+Comprehensive study of AI Hya
+3
+Table 1. Information about the spectroscopic observations. N,
+R and SNR represent the number of the spectra, resolving power
+and the signal-to-noise ratio, respectively.
+Spectrometer
+N
+Observations
+R
+SNR
+Spectral
+years
+range [˚A]
+CAOS
+1
+2021
+38000
+50
+415 − 670
+CORALIE
+3
+2015
+60000
+20 − 34
+390 − 680
+HERMES
+15
+2020
+85000
+50 − 70
+377 − 900
+HIDES
+13
+2014 − 2017
+50000
+40 − 88
+408 − 752
+telescope at La Silla Observatory (Chile) (Pepe et al. 2018).
+The High Efficiency and Resolution Mercator ´echelle spec-
+trograph (HERMES) was also used to obtain high-resolution
+spectra of AI Hya. HERMES is mounted on the 1.2-m Mer-
+cator telescope at the Roque de Los Muchchos observa-
+tory on the Canary Island La Palma in Spain (Raskin et
+al. 2011). The last instrument used in this study is the
+HIgh-Dispersion ´Echelle spectrograph (HIDES). HIDES is
+attached to the 1.88-m telescope of Okayama astrophysical
+observatory in Japan (Kambe et al. 2013). The spectra of
+CORALIE and HIDES were taken by group-P, while the
+spectra of CAOS and HERMES were gathered by group-K.
+In total 32 spectra of AI Hya were gathered and these spec-
+tra are well distributed in orbital phases of AI Hya. Each
+group used the obtained spectra to measure the radial ve-
+locity (vr ) changes. Additionally, these data were taken into
+account to derive the atmospheric parameters (e.g. effective
+temperature Teff, surface gravity log g, metallicity) and the
+projected rotational velocity (v sin i) of the components of
+AI Hya.
+3
+RADIAL VELOCITY ANALYSIS
+The vr values of the AI Hya system were measured with
+different approaches by both group-P and group-K using
+different spectra taken from the distinct instruments.
+3.1
+vr measurements
+Group-P
+calculated
+the
+vr
+values
+from
+HIDES
+and
+CORALIE
+spectra,
+using
+the
+two-dimensional
+cross-
+correlation todcor program (Zucker & Mazeh 1994). In the
+analysis, a synthetic spectrum was used as a template and
+this spectrum was generated using an ATLAS9 model at-
+mosphere (Kurucz 1993) having Teff, metallicity [M/H] and
+v sin i parameters of 6800 K, 0.0 and 30 km s−1, respectively.
+When a template with 60 km s−1(the v sin i value found in
+further analysis) was used, the results did not improve in
+terms of rms of the orbital fit, nor did the uncertainties of
+orbital elements. Moreover, some points, with the smallest
+difference in vr measurements, seemed to suffer from sys-
+tematic effects, and had to be rejected. We therefore believe
+the use of 30 km s−1templates was justified. The calculated
+vr values for each binary component are given in Table A1.
+Group-K used the RaVeSpAn code (Pilecki et al. 2017)
+to determine the vr values of the binary components using
+the broadening function formalism. In the analysis, local
+thermodynamic equilibrium (LTE) synthetic spectra with
+atmospheric parameters similar to that of group-P were used
+as templates (Coelho et al. 2005). The spectra of CAOS and
+HERMES were used in the vr measurements. The resulting
+vr measurements are given in Table A1.
+3.2
+vr curve modelling
+For the spectroscopic orbital fitting, group-P used all the
+available vr measurements, including those made by group-
+K and from Popper (1988). Group-P used the v2fit code
+(Konacki et al. 2010) which adjusts a double-Keplerian with
+a Levenberg-Marquardt algorithm. In this analysis, the am-
+plitude of vr curves (K), Porb, the time of phase zero (T0),
+mass centre’s velocity (γ), eccentricity (e) and argument of
+the periastron (ω) were set as free parameters. Thanks to
+the long time span of the data (>51 years), it was possi-
+ble to detect the apsidal motion ( ˙ω) of the binary’s orbit:
+0.186(56) deg/yr. This is in reasonable agreement (1.75σ)
+with the value given by Lee, Hong, & Kristiansen (2020):
+0.075(31) deg/yr. The results of the analysis are given in
+Table 2 and the theoretical vr curve fits to the measured vr
+data are illustrated in Fig.1.
+Group-K used the rvfit code2 for the radial velocity
+analysis. The rvfit program can analyse single and double-
+lined binary systems by using the adaptive simulated an-
+nealing method (Iglesias-Marzoa et al. 2015). In the analy-
+sis, the Porb taken from Kreiner (2004) was considered as a
+fixed parameter. Other orbital parameters such as T0, K, γ,
+ω and e were taken as free parameters during the analysis.
+Both groups vr measurements were used in the analysis and
+as a result, the orbital parameters of the system were ob-
+tained. The resulting parameters of the current vr analysis
+are given in Table 2. The consistency between the theoretical
+vr curve and measurements is shown in Fig. 2.
+Both groups found the resulting mass ratio (q
+=
+M2/M1
+=
+K1/K2)3 larger than 1 (1.075 ± 0.011 and
+1.080 ± 0.007 for groups -K and -P, respectively). Accord-
+ing to this q value, the vr curve and the results, the star
+(generally called secondary) covered by the hotter binary
+component at orbital phase 0.5 is more massive than the
+hotter binary component (primary). To test these findings,
+binary modelling is necessary. Therefore, these results will
+be tested in the binary modelling sections.
+4
+SPECTRAL ANALYSIS
+4.1
+Group-K
+4.1.1
+Spectral disentangling
+To obtain the atmospheric parameters (Teff, log g), v sin i
+and the chemical composition of each binary component of
+AI Hya, a detailed spectral analysis is necessary. As AI Hya
+is a double-lined binary system, its spectrum consists of the
+spectral lines of both binary components. Therefore, group-
+K carried out a spectral disentangling analysis to extract the
+individual spectra of each binary component from the com-
+posite spectra of AI Hya. In the analysis, the code fdbinary
+2 http://www.cefca.es/people/riglesias/rvfit html
+3 The subscripts 1 and 2 refer to hotter primary and cooler sec-
+ondary components, respectively.
+© 2021 RAS, MNRAS 000, 1–13
+
+4
+F. Kahraman Ali¸cavu¸s et. al.
+Table 2. The results of the radial velocity analysis. The sub-
+scripts 1 and 2 refer to hotter primary and cooler secondary com-
+ponents, respectively. a shows the fixed parameters.
+Parameter
+Group-P
+Group-K
+T0 (HJD)
+2458491.570 ± 0.028
+2452506.383 ± 0.032
+Porb(d)
+8.289761 ± 0.000027
+8.2896490a
+γ (km/s)
+45.90 ± 0.24
+45.70 ± 0.35
+K1 (km/s)
+90.42 ± 0.37
+89.52 ± 0.65
+K2 (km/s)
+83.71 ± 0.46
+83.29 ± 0.63
+e
+0.2419 ± 0.0036
+0.2432 ± 0.0050
+ω (deg)
+254.03 ± 1.30
+250.92 ± 1.63
+˙ω (deg/yr)
+0.186 ± 0.056
+a1 sin i (R⊙)
+14.380 ± 0.061
+14.222 ± 0.105
+a2 sin i (R⊙)
+13.312 ± 0.072
+13.233 ± 0.101
+a sin i (R⊙)
+27.692 ± 0.094
+27.454 ± 0.145
+M1 sin3 i (M⊙)
+1.992 ± 0.023
+1.950 ± 0.033
+M2 sin3 i (M⊙)
+2.151 ± 0.022
+2.095 ± 0.035
+q = M2/M1
+1.080 ± 0.007
+1.075 ± 0.011
+50
+25
+0
+25
+50
+75
+100
+125
+RV (Km/s)
+=245.283
+=253.526
+=254.424
+Popper_rv1
+Popper_rv2
+Group-P_rv1
+Group-P_rv2
+Group-K_rv1
+Group-K_rv2
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Phase
+10
+0
+10
+O-C
+Figure 1. Upper panel: The model vr fit to the combined vr mea-
+surements from Popper (1988), Group-P (HIDES+CORALIE)
+and Group-K (HERMES+CAOS). Lower panel: residuals. Model
+made by Group-P.
+was used (Ilijic et al. 2004). fdbinary is capable of disen-
+tangling a composite spectrum, which includes flux contri-
+butions from two or three components, in Fourier space.
+Before the analysis with fdbinary, one should know the
+light contributions of the binary components at the orbital
+phases corresponding to the times the spectra were taken.
+These values should be fixed during the analysis. Hence,
+to determine the light contributions of both binary compo-
+nents at the different orbital phases, we carried out a pre-
+liminary binary modelling of AI Hya by taking Teff of the
+TESS Input Catalog (TIC; Stassun et al. 2019) as the Teff
+of the hotter component. The analysis was performed utiliz-
+ing the Wilson-Devinney code (Wilson & Devinney 1971).
+As a result of this preliminary analysis, it was found that
+the hotter and cooler binary components contribute around
+38% and 62% to the total, respectively. However, one should
+keep in mind that these light contributions change accord-
+ing to the orbital phases. For example, the primary eclipse
+Figure 2. Upper panel: The model vr fit to the vr measure-
+ments of Groups-K and -P. Lower panel: residuals. Model made
+by Group-K.
+is a total eclipse where the light contribution of the hotter
+components is negligible.
+In the analysis, we used the HERMES spectra as they
+are well distributed over the orbital phases and have a higher
+resolving power. Taking into account the observation time
+of each HERMES spectrum, the light contributions at these
+times were first determined using the fluxes measured from
+the photometric solution and subsequently fixed during the
+analysis. In addition to this, we also fixed all results de-
+rived in the vr analysis during the spectral disentangling.
+For the disentangling progress, we used the spectral inter-
+val of ∼4200 − 6400 ˚A by ignoring the parts polluted by tel-
+luric lines. For the analysis, this spectral window was di-
+vided into 15 spectral parts with steps of ∼ 100 − 150 ˚A.
+Each small spectral part was then analysed separately. As
+a result, we obtained the individual spectra of each binary
+component. The separated spectra derived with fdbinary
+were re-normalised by taking into account the light ratio of
+the binary components, as described by Ilijic et al. (2004).
+4.1.2
+Determination of the atmospheric parameters and
+chemical compositions
+After the individual spectra of the components of AI Hya
+were obtained, we were able to determine the atmospheric
+parameters, v sin i, and the chemical composition. To de-
+rive these parameters, we used the plane-parallel and line-
+blanketed local thermodynamic equilibrium (LTE) ATLAS9
+model atmospheres (Kurucz 1993) and the synthe code
+(Kurucz & Avrett 1981) to generate theoretical spectra.
+First, the hydrogen lines of the binary components were used
+to obtain initial Teff values.
+In this analysis, the Hβ lines of the components were
+compared with many theoretical Hβ lines which were de-
+rived for a wide range of Teff (5000 − 9000 K) with a step
+size of 100 K, where log g and metallicity were fixed to 4.0
+and solar, respectively. During the analysis, we took into
+account the minimization method described by Catanzaro,
+Leone, & Dall (2004) and successfully applied in a series
+© 2021 RAS, MNRAS 000, 1–13
+
+150
+100
+米
+米
+xnl↓
+50
+Normalized
+采
+0
+米
+米
+米
+米
+CAOS
+50
+△ CORALIE
+ HERMES
+ HIDES
+-100
+15
+10
+A
+1
+s
+5
+米
+中
+uy)
+米
+中
+采米
+0
+米
+米
+5
+米
+O-C.
+15
+15
+10
+5
+uy)
+中谷
+米
+日米日
+米
+-5E
+-10
+-
+0
+15
+0.0
+0.2
+0.6
+0.8
+1.0
+0.4
+PhaseComprehensive study of AI Hya
+5
+Figure 3. Theoretical hydrogen line fits (red dashed lines) to
+the Hβ lines (solid black line) of the hotter and cooler binary
+components (Group-K).
+of papers (i.e., Catanzaro et al. 2022, 2019). Consequently,
+the Teff of the hotter and cooler components were found to
+be 7500 ± 200 K and 7000 ± 150 K, respectively. We did not
+attempt to optimize log g because the hydrogen lines are
+not sensitive to this parameter for stars cooler than 8000 K
+(Smalley et al. 2002). The best theoretical Hβ line fits to the
+separated spectra of the components are shown in Fig. 3.
+We also determined values for log g, the microturbulent
+velocity ξ, and v sin i by improving the initially determined
+Teff value using the excitation potential−abundance rela-
+tionship. For the correct atmospheric parameters, different
+excitation potentials of the same element should give the
+same abundances. Therefore, by using this relation for iron
+(Fe), we determined the atmospheric parameters. Detailed
+information about this analysis method is given by Kahra-
+man Ali¸cavu¸s et al. (2016). The results of this analysis are
+listed in Table 3. To determine the errors on the atmospheric
+parameters, we checked how their values change for differ-
+ences in the excitation potential−abundance correlation of
+about 5%.
+In the next step, the chemical composition of the binary
+components was derived after fixing the atmospheric param-
+eters to their final values. For the chemical abundance deter-
+mination, we first identified the lines based on the Kurucz
+line list4. The spectral synthesizing method and the identi-
+fied lines were used in this examination. Consequently, the
+chemical compositions of both binary components were ob-
+tained and the results are listed in Table 4. The consistency
+between the synthetic and observed spectra of both binary
+4 http://kurucz.harvard.edu/linelists.html
+Figure 4. Consistency between the synthetic (dashed-lines) and
+disentangled spectra of the components of AI Hya (Group-K).
+Figure 5. Abundance distribution of the components of AI Hya
+relative to solar values (Asplund et al. 2009) (Group-K).
+components is illustrated in Fig. 4. The abundance distribu-
+tions relative to solar abundance (Asplund et al. 2009) are
+shown in Fig. 5, indicating that the hotter binary component
+has an overabundance compared to the Sun for some ele-
+ments. The errors of the chemical compositions were deter-
+mined including the uncertainties in the derived atmospheric
+parameters and the effects of the resolving power and the
+SNR of the spectra, as described by Kahraman Ali¸cavu¸s et
+al. (2016).
+4.2
+Group-P
+For the spectral decomposition and analysis, group-P used
+the HIDES data only. Spectral analysis was performed on
+both the observed composite spectra and the disentangled
+spectra of the individual components. For the spectral disen-
+tangling, we used a python wrapper5 made for using version
+3 of fdbinary (FD3; Ilijic et al. 2004). A particular por-
+tion of the total spectra was taken to ensure good quality
+in terms of SNR and spectral features. The light fractions
+used for the disentangling procedure were obtained from the
+light curve analysis as 38% and 62% for the primary and sec-
+ondary respectively.
+4.2.1
+gssp
+On the other hand, we also modelled the composite spec-
+trum using the gssp composite module of the Grid Search
+5 https://github.com/ayushmoharana/fd3 initiator
+© 2021 RAS, MNRAS 000, 1–13
+
+1.0
+0.8
+0.6
+0.4
+xn
+T
+0.2
+otter
+Normalized
+0.0
+1.0
+0.8
+0.6
+0.4
+0.2
+Al
+cooler
+0.0
+4800
+4820
+4840
+4860
+4880
+4900
+4920
+Wavelength (A)Al HyaHot
+xnl
+1.0
+Normalized
+0.9
+Fel
+Fel
+0.8
+Fel
+Fel
+0.7
+5439
+5448
+5427
+5430
+5433
+5436
+5444
+Wavelength (A)
+xn
+00
+.0
+Normalized
+Fel
+Fel
+0.9
+Fel
+Fel
+Fel
+0.8
+5382
+5385
+5400
+5403
+5379
+5391
+5394
+5397
+5388
+Wavelength (A)3
+Hotter star
+Cooler star
+loge(El)-
+O
+Mg
+Si
+Ca
+Sc
+Cr
+Fe
+Ti
+Mn
+Ni
+Element6
+F. Kahraman Ali¸cavu¸s et. al.
+Table 3. The final atmospheric parameters and v sin i value of the hot (primary) and cool binary components of AI Hya. log ϵ (Fe)
+represent the relative abundance with respect to hydrogen (H=12.0)
+.
+Group-K
+Teff (K)
+log g (cgs)
+ξ (km s−1)
+v sin i (km s−1)
+log ϵ (Fe)
+Primary
+7700 ± 100
+3.8 ± 0.1
+3.4 ± 0.3
+57 ± 6
+8.25 ± 0.54
+Secondary
+7200 ± 100
+3.6 ± 0.2
+1.9 ± 0.3
+64 ± 4
+7.64 ± 0.20
+Group-P (gssp)
+Teff (K)
+log g (cgs)
+ξ (km s−1)
+v sin i (km s−1)
+[M/H]
+Primary
+7350 ± 300
+3.8 (fixed)
+4.83 ± 1.15
+50 (fixed)
+0.14 ± 0.14
+Secondary
+7150 ± 250
+3.6 (fixed)
+3.07 ± 0.52
+62 (fixed)
+0.06 ± 0.10
+Group-P (iSpec)
+Teff (K)
+log g (cgs)
+ξ (km s−1)
+v sin i (km s−1)
+[M/H]
+Primary
+7300 ± 170
+3.83 (fixed)
+5.33 ± 0.86
+50 (fixed)
+0.15 (fixed)
+Secondary
+7260 ± 175
+3.58 (fixed)
+3.98 ± 0.70
+62 (fixed)
+0.01 (fixed)
+Table 4. Abundances of individual elements of the binary com-
+ponents and Sun (Asplund et al. 2009).
+Group-K
+Elements
+Hotter
+Cooler
+Solar
+component
+component
+abundance
+12Mg
+7.96 ± 0.16
+8.01 ± 0.63
+7.60 ± 0.04
+14Si
+8.03 ± 0.36
+7.12 ± 0.51
+7.51 ± 0.03
+20Ca
+6.93 ± 0.27
+6.69 ± 0.27
+6.34 ± 0.04
+21Sc
+3.11 ± 0.32
+3.15 ± 0.04
+22Ti
+5.71 ± 0.49
+5.17 ± 0.30
+4.95 ± 0.05
+24Cr
+6.63 ± 0.42
+5.80 ± 0.30
+5.64 ± 0.04
+25Mn
+6.84 ± 0.82
+6.06 ± 0.45
+5.43 ± 0.05
+26Fe
+8.25 ± 0.23
+7.64 ± 0.24
+7.50 ± 0.04
+28Ni
+7.44 ± 0.38
+6.73 ± 0.33
+6.22 ± 0.04
+Group-P (iSpec)
+Elements
+Hotter
+Cooler
+Solar
+component
+component
+abundance
+24Cr
+5.95 ± 0.19
+5.63 ± 0.23
+5.64 ± 0.04
+26Fe
+7.83 ± 0.16
+7.48 ± 0.17
+7.50 ± 0.04
+28Ni
+6.76 ± 0.18
+6.53 ± 0.22
+6.22 ± 0.04
+in Stellar Parameter (gssp) software package (Tkachenko
+2015). As its name implies, gssp is based on a grid search
+in the fundamental atmospheric parameters. It uses the
+method of atmosphere models and spectrum synthesis,
+which performs a comparison of the observations with the-
+oretical spectra from the grid. These synthetic spectra are
+calculated using the synthV LTE-based radiative transfer
+code (Tsymbal 1996) and a grid of atmospheric models pre-
+computed using llmodels (Shulyak et al. 2004). Specifi-
+cally, in the composite module, the user can set the radial
+velocity of the components as a free parameter so that all
+the possible combinations of the synthetic spectra of primary
+and secondary from the computed grid are used to build the
+composite theoretical spectra of the binary. This synthetic
+spectrum is then compared against the a-priori normalized
+observed spectrum and a χ2 merit function is used to judge
+the goodness of the fit.
+The broadening function (BF) is a representation of
+spectral profiles in velocity space. The BF contains signa-
+tures of the vr shifts of different lines and also intrinsic stel-
+lar effects like rotational broadening, spots, pulsations, etc.
+(Rucinski 1999). We calculated the BF for one of the com-
+posite spectra of AI Hya to estimate v sin i values for the
+primary and secondary components, respectively. This pro-
+cess serves to remove the degeneracy between v sin i and
+other atmospheric parameters like T eff and [M/H]. A mod-
+ified version of the treatment described in Rucinski (1999)
+was adopted and a multi Gaussian fit was implemented. The
+BF was calculated in a wavelength range of 4080-5000 ˚A. A
+synthetic solar-type spectrum with zero projected rotational
+velocity v sin i was used as our template. To deal with the
+noise in the data, a Gaussian smoother of 3 km s−1 rolling
+window was applied to the BF. Two clear peaks were visible
+in the velocity space, as shown in Figure 6, corresponding
+to the primary and secondary components. The peaks were
+fitted with the rotational profile,
+G(v) = A
+�
+�c1
+�
+1 −
+�
+v
+vmax
+�2
++ c2
+�
+1 −
+�
+v
+vmax
+�2��
+�+lv+k
+(1)
+where A is the area under the profile, vmax is the maximum
+velocity shift which occurs at the equator (Gray 2005), c1
+and c2 are constants which are a function of limb darkening
+themselves, while l and k are correction factors to the BF
+continuum. The BF fit was calculated for the spectra with
+the highest SNR and good separation between the compo-
+nents in velocity space. The best BF fit to the line profile of
+the primary and secondary binary components are shown in
+Fig.6. Fixing the obtained values of v sin i from this analysis
+and log g from the light curve solution, the gssp composite
+fitting routine was applied to obtain stellar temperatures
+Teff (1,2), microturbulent velocities ξ and global metallicities
+[M/H].
+The step size of the grid gives us a rough idea of the
+errors involved. However, to obtain more robust error esti-
+mates we plotted the χ2 data for each parameter and fitted
+a parabola to obtain the minimum; its distance to the in-
+tercepts on the abscissa are taken as the errors. These pa-
+rameters are obtained for a total of four spectra and then
+averaged out. The remaining spectra were not suitable for
+© 2021 RAS, MNRAS 000, 1–13
+
+Comprehensive study of AI Hya
+7
+100
+50
+0
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+Primary
+50
+100
+150
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1.0
+Secondary
+Relative Flux
+2457109.96513 BJD
+Radial Velocities (km/s)
+Figure 6. Broadening functions for the primary and secondary
+components of AI Hydrae calculated using HIDES spectra (epoch:
+2457109.96513 HJD), which provided a good SNR and velocity
+separation between the two components. The blue, dashed line
+represents best-fit rotational function (Group-P).
+5320
+5330
+5340
+5350
+5360
+5370
+5380
+5390
+5400
+Wavelength (A)
+0.75
+0.80
+0.85
+0.90
+0.95
+1.00
+1.05
+Normalized Flux
+Data
+Model
+Figure 7. A snippet of the best-fit model generated by gssp for
+the given set of parameters (Group-P).
+the analysis in gssp due to lower SNR. The results of the
+analysis are compiled in Table 3 and a sample of the fit to
+one of the spectra is shown in Figure 7.
+4.2.2
+iSpec
+A complimentary spectroscopic analysis was performed on
+the disentangled spectra of the primary and secondary stars
+using iSpec (Blanco-Cuaresma et al. 2014). Before the anal-
+ysis, the spectra are treated for vr offset and continuum cor-
+rection. Estimates of flux errors were introduced as a sum of
+errors calculated from SNR, and flux-scaled residuals from
+the disentangled routine. For the spectroscopic analysis we
+fixed the log g parameter with values obtained from the light
+curve solution and limb darkening parameters with values
+adopted from Claret & Bloemen (2011).
+We fit the model using the spectral synthesis approach.
+This is done by implementing the use of the spectrum code
+(Gray & Corbally 1994), a marcs (Gustafsson et al. 2008)
+grid of model atmospheres, and solar abundances taken from
+Asplund et al. (2009). We adopt a two-step process. The ini-
+tial run is aimed at estimating the global metallicity ([M/H])
+by keeping it as a free parameter. The macroturbulent ve-
+locity (vmac) and alpha enhancement parameters were set
+to zero as vmac has a negligible contribution for stars in the
+concerned temperature range and alpha enhancement, when
+set as a free parameter, produced implausible values. v sin i
+was set to the values obtained by the BF analysis. We com-
+pared the obtained value for [M/H] with results from the
+gssp analysis and found it to be consistent with the errors.
+The average value of [M/H] was calculated and fixed for the
+next step where we fit for temperature Teff, microturbulent
+velocity ξ, and abundances of Iron (Fe), Nickel (Ni) and
+Chromium (Cr), as these were the prominent lines in the
+chosen spectral range.
+The output parameters obtained from iSpec are given
+in Table 3 and Table 4. It is to be noted that Fe, Ni, and Cr
+are more abundant in the primary compared to solar values
+and those of the secondary star. This trend in the abun-
+dances is in agreement with the values obtained by group-K.
+The output parameters for the secondary star agree fairly
+well with those from the gssp analysis and from the group-
+K. The best fit solution for the primary component, as in
+the case of gssp analysis, also hinted towards a lower Teff
+compared to the group-K solution.
+5
+BINARY MODELLING
+5.1
+Group-K
+To update the fundamental stellar parameters (M, R) of
+AI Hya, we performed binary modelling with the help of the
+determined atmospheric parameters and the results of the
+vr investigation.
+In binary modelling, the TESS data were used. How-
+ever, the shapes of the eclipses of AI Hya are distorted due
+to the pulsations. Thus we first cleaned the pulsations and
+only then carried out the binary modelling. Therefore, the
+Period04 program (Lenz & Breger 2005) was used to detect
+the variations caused by oscillations. The derived pulsation
+frequencies6 were cleaned from the light curve and the resid-
+uals were used in the binary modelling.
+In this analysis, we used the Wilson-Devinney code
+(Wilson & Devinney 1971) combined with Monte-Carlo sim-
+ulations (Zola et al. 2004, 2010). The pulsation removed data
+were binned to around 4000 points to be used in the binary
+modelling code. AI Hya is classified as a detached binary
+system in the literature (Lee, Hong, & Kristiansen 2020).
+According to their results (e.g., for Ω, q, a), both compo-
+nents do not seem to fill their Roche lobe, hence the sys-
+tem is defined as a detached binary. Also, the morphology
+of the light curve, i.e. very small ellipsoidal variations and
+eclipses spanning a small fraction of the orbital period, con-
+firm this classification. Therefore, a detached binary config-
+uration was considered our analysis. In the modelling, we
+took some parameters fixed, such as the Teff of the hotter
+component, Porb, q taken from our results and bolometric
+albedos (Ruci´nski 1969), bolometric gravity-darkening coef-
+ficient (von Zeipel 1924), and the logarithmic limb darken-
+ing coefficient (van Hamme 1993) taken the same as given
+Kahraman Ali¸cavu¸s & Ali¸cavu¸s (2019). The orbital inclina-
+tion (i), Teff of the cooler component, phase shift (φ), e, a,
+ω, and dimensionless potential (Ω) of the components were
+set free.
+6 The frequencies given in Sect. 6.
+© 2021 RAS, MNRAS 000, 1–13
+
+8
+F. Kahraman Ali¸cavu¸s et. al.
+Table 5. Results of the light curve analysis and the fundamental
+stellar parameters. The Subscripts 1, 2 and 3 represent the hotter,
+the cooler, and third binary components, respectively. a Shows
+the Fixed Parameters.
+Parameter
+Value
+Value
+Group-K
+Group-P
+i (o)
+89.866 ± 0.015
+89.837 ± 0.136
+T 1a (K)
+7700 ± 100
+7330 ± 170
+T 2 (K)
+7180 ± 230
+7210 ± 150
+Ω1
+11.412 ± 0.046
+-
+Ω2
+8.961 ± 0.035
+-
+Phase shift
+-0.0310 ± 0.0001
+-
+q
+1.074a
+1.075
+r1∗ (mean)
+0.1001 ± 0.0036
+0.1015 ± 0.0005
+r2∗ (mean)
+0.1412 ± 0.0026
+0.1412 ± 0.0006
+l1 / (l1+l2)
+0.381 ± 0.016
+0.374 ±0.02
+l2 / (l1+l2)
+0.619 ± 0.016
+0.616 ± 0.02
+l3
+0.0
+0.0
+Derived Quantities
+M1 (M⊙)
+1.950 ± 0.033
+1.950 ± 0.033
+M2 (M⊙)
+2.096 ± 0.035
+2.096 ± 0.035
+R1 (R⊙)
+2.754 ± 0.015
+2.787 ± 0.020
+R2 (R⊙)
+3.863 ± 0.021
+3.877 ± 0.026
+log (L1/L⊙)
+1.381 ± 0.034
+1.311 ± 0.081
+log (L2/L⊙)
+1.554 ± 0.035
+1.549 ± 0.097
+log g1 (cgs)
+3.848 ± 0.003
+3.838 ± 0.005
+log g2 (cgs)
+3.586 ± 0.003
+3.582 ± 0.005
+Mbol1 (mag)
+1.30 ± 0.08
+1.474 ± 0.202
+Mbol2 (mag)
+0.87 ± 0.08
+0.877 ± 0.243
+MV 1 (mag)
+1.25 ± 0.08
+1.424 ± 0.208
+MV 2 (mag)
+0.79 ± 0.08
+0.822 ± 0.258
+Distance (pc)
+659 ± 30
+642 ± 36
+As a result of this analysis, the fundamental parameters
+of both components of AI Hya were calculated. Additionally,
+the bolometric (Mbol) and absolute (MV ) magnitudes were
+estimated. The jktabsdim code (Southworth, Maxted, &
+Smalley 2004b) and the bolometric correction (Eker et al.
+2020) are used in the calculations of these parameters. The
+outcome of the binary modelling is given in Table 5 and the
+consistency of the theoretical light curve with the observa-
+tion is shown in Fig. 8.
+When the results of this analysis were examined, one
+can notice that the more luminous star is the more massive
+and also the cooler component. This result is consistent with
+the results found in the vr analysis by group-K.
+5.2
+Group-P
+Aiming to determine precise physical and orbital parame-
+ters of AI Hya, we performed its modelling in version 40 of
+the jktebop (Southworth, Maxted, & Smalley 2004b). This
+program is written by J. Southworth and aimed at modelling
+light curves of detached eclipsing binaries and is based on
+the ebop program (Popper & Etzel 1981). The code treats
+stars as spheres to calculate the eclipse shapes, and biaxial
+ellipsoids to calculate proximity effects. The light curves are
+calculated by numerical integration of concentric circles over
+each stellar surface. It can deal with stellar oblateness of up
+Figure 8. Theoretical binary modelling fit without spot assump-
+tion (solid-line) (Group-K).
+to 4% making it a good choice for AI Hya. The photometric
+data remain the same as used by Group-K.
+The parameters set as free are Porb, time of minima
+of the primary eclipse To, inclination i, eccentricity e, ar-
+gument of periastron ω, surface brightness ratio J (sec-
+ondary/primary), ratio of radii ( rA
+rB ), and the sum of radii
+(rA+rB). These radii are relative to the semi-major axis. For
+the limb darkening coefficients, we use a logarithmic law and
+set their initial values according to Claret (2017). The coef-
+ficients were fixed for the initial fit and were perturbed at
+the error estimation step.
+The code gives an option to include multiple sine and
+polynomial functions during the light curve modelling to ac-
+count for periodic and long-term trends. We use this func-
+tionality to our advantage to pseudo-model the observed
+pulsations so that their effect on the binary model is mini-
+mal, giving us an improved precision. We analyse the out-of-
+eclipse portions of the light curve using pyriod7, and use the
+frequencies to initialise the sinusoids in the jktebop input
+files. This is done in an iterative way where we add one sine
+with a constant period and fit for its epoch and amplitude.
+The frequency is kept if the model is improved significantly;
+otherwise the next most prominent frequency is taken. In
+this analysis, we used a total of 9 sines, which is the limit
+for jktebop. The number of independent frequencies of AI
+Hya is higher than this maximum limit, hence we are left
+with some residual pulsation signals as seen in Figure 9 and
+Figure 10.
+Once the sines are fixed to the best fit values of epoch,
+period and amplitudes, we make the Monte Carlo runs for
+error estimation. The results of this analysis are mentioned
+in Table 5, in comparison to the values obtained by group-K.
+Similarly to the other group, we used the results of vr, and
+jktebop solutions to calculate a set of absolute parameters,
+including masses, radii, luminosities, and distance. The ef-
+fective temperatures mentioned in the table are an average
+over the sum of Teff obtained from gssp and iSpec analysis.
+7 https://github.com/keatonb/Pyriod
+© 2021 RAS, MNRAS 000, 1–13
+
+1.0
+0.9
+xnl
+Normalised f
+0.8
+0.7
+0.6
+Data
+Model
+0.03
+Res.
+0.00
+0.03
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1.0
+1.1
+1.2
+PhaseComprehensive study of AI Hya
+9
+Figure 9. jktebop model with 9 sines used to model the pulsa-
+tions (Group-P).
+Figure 10. Zoomed-in view of the model over an orbit (Group-
+P).
+6
+FREQUENCY ANALYSIS OF THE
+PULSATIONS
+AI Hya was observed by TESS during observation sector 7
+in January/February 2019. We used the Simple Aperture
+Photometry data from the 2-min cadence light curves avail-
+able at the Mikulski Archive for Space Telescopes8 (MAST).
+This time series spans 24.45 d and contains 16362 measure-
+ments. To determine the pulsation frequencies, we used only
+the data that were taken out of eclipse, which reduced the
+data set to 14019 measurements (time span 24.07 d).
+This time series was analysed using the Period04 soft-
+ware (Lenz & Breger 2005) by group-K. This package applies
+single-frequency power spectrum analysis and simultaneous
+multi-frequency sine-wave fitting. These sine-wave fits are
+subtracted from the data and the residuals examined for
+the presence of further periodicities. The application of this
+procedure to AI Hya is illustrated in Fig. 11.
+During such a process, it is important to decide where
+to stop. Often this is facilitated via the application of SNR
+criteria. In this work, we have adopted the strategy proposed
+by Breger et al. (1993) which is to compute the ratio of the
+signal amplitude relative to the local noise level to deter-
+mine whether the frequency under consideration represents
+a significant detection. Whereas Breger et al. (1993) propose
+SNR > 4 for a detection, recent findings for space-based data
+8 https://mast.stsci.edu/portal/Mashup/Clients/Mast/Portal.html
+Figure 11. The Fourier Transform of the out-of-eclipse TESS
+light curve of AI Hya (top) and subsequent prewhitening steps.
+The blue arrows denote the signals detected. Outside of the fre-
+quency range shown no significant signal is present.
+Table 6. A least squares fit of the pulsation frequencies of AI
+Hya. Formal error estimates for the independent frequencies and
+phases (Montgomery & O’Donoghue 1999) are given in braces in
+units of the last digits after the comma.
+Frequency
+Amplitude
+SNR
+d−1
+mmag
+±0.02
+ν1
+6.2412(1)
+4.75
+54.2
+ν2
+9.2654(4)
+1.18
+9.7
+ν3
+9.9065(4)
+1.20
+9.4
+ν4
+12.715(1)
+0.48
+4.5
+ν5
+12.928(1)
+0.54
+5.4
+ν6
+9.3689(4)
+1.42
+11.5
+3νorb
+0.3619
+1.76
+7.5
+4νorb
+0.4825
+1.32
+5.9
+ν7
+5.5599(7)
+0.78
+8.3
+ν8
+5.7804(1)
+0.69
+7.5
+2νorb
+0.2413
+1.75
+7.3
+ν9
+5.6375(7)
+0.73
+7.7
+ν10
+7.136(1)
+0.37
+6.0
+ν11
+7.751(1)
+0.39
+5.2
+ν12
+9.3051(6)
+0.82
+6.7
+ν13
+9.8432(8)
+0.69
+5.3
+ν3 + ν7
+15.464(1)
+0.43
+5.6
+(e.g., Baran & Koen 2021) suggest that a more conservative
+limit must be chosen. Given the restricted frequency range
+in which we search for periodicities, our requirement was
+SNR > 4.5. Furthermore, in unresolved frequency spectra,
+the periodic content present in the time series can easily
+be overinterpreted (Balona 2014) which suggests caution re-
+garding the present data set. Consequently, we stopped the
+frequency search after the detection of 17 signals (lowest
+panel of Fig. 11). More periodicities are certainly present,
+but these need to await a longer data set for reliable detec-
+tion. We list the frequency solution so derived in Table 6.
+© 2021 RAS, MNRAS 000, 1–13
+
+8.5
+8.6
+8.7
+08.8
+a
+M
+8.9
+9.0
+Data
+9.1
+Model
+Residuals
+0.01
+Resi.
+0.00
+0.01
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+Phase8.48
+8.49
+0
+8.50
+8.51
+Data
+Model
+1494
+1496
+1495
+1498
+1497
+1499
+1500
+1501
+1502
+Time (ID-2457000) days10
+F. Kahraman Ali¸cavu¸s et. al.
+This table also contains three harmonics of the orbital
+period. These are not pulsation frequencies, but a conse-
+quence of residual binary-induced variability (see Section on
+binary modeling for a discussion). The pulsation frequencies
+themselves were found in an interval between 5.5 – 13.0 d−1,
+with one possible combination frequency. It is however not
+clear whether this is a real combination or just a numeri-
+cal coincidence keeping in mind the short data set, hence
+poor frequency resolution. Our frequency solution is similar
+to that reported by Lee, Hong, & Kristiansen (2020) apart
+from their identification of possible combination frequencies
+that are partly implausible.
+To use the pulsations to learn more about the indi-
+vidual components by applying asteroseismic methods, it
+is essential to know from which star the pulsations orig-
+inate. A quick look at the TESS light curve reveals that
+pulsations are clearly visible during the total part of the
+primary eclipse, meaning that the secondary is the source
+of the highest amplitude oscillations. However, both com-
+ponents of AI Hya are located within the pulsational in-
+stability strip of the δ Scuti stars (Murphy et al. 2019, see
+Fig. 12), thus the primary may pulsate as well. δ Scuti stars
+generally pulsate in pressure and mixed modes of low ra-
+dial order (e.g., Breger 2000). Using the stellar parameters
+from Table 5, we can compute the expected frequency of the
+radial fundamental mode of both pulsators from the pulsa-
+tion constant Q = P
+�
+ρ/ρ⊙ = PM 1/2R−3/2, assuming Q to
+be 0.033 d for this mode (Fitch 1981). We thus expect the
+radial fundamental mode frequency of the primary compo-
+nent to be around 9.3 d−1, and around 5.8 d−1 for the sec-
+ondary component, respectively. In Table 6 oscillation fre-
+quencies around both these values are seen, which allows
+no more than the educated guess that the pulsations below
+∼ 8 d−1 would arise from the secondary component, whereas
+the higher frequency modes could originate from either star.
+A determination of the origin of the pulsations from the
+orbital light time effect is unfortunately out of reach. The
+expected light time effect would be about 30 s (cf. Table 2).
+An attempt to measure the effect for the strongest pulsa-
+tion frequency yielded 35 ± 111 s, a null result. To conclude,
+because it is impossible to say with confidence which pulsa-
+tion frequencies arise from which component of AI Hya, an
+asteroseismic analysis cannot be carried out.
+7
+EVOLUTIONARY MODELS
+The evolutionary status of the binary components was ex-
+amined by utilizing the Modules for Experiments in Stel-
+lar Astrophysics (mesa) evolution code (Paxton et al. 2011,
+2013) which includes a binary module (Paxton et al. 2015) to
+examine the binary orbital evolution and to determine the
+initial parameters of binary systems. In this examination,
+various evolutionary models were generated considering dif-
+ferent metallicity (Z). In the models, MESA equation-of-
+state (EOS) were used. The EOS tables are based on the
+OPAL EOS tables (Rogers & Nayfonov 2002). The OPAL
+opacity tables and the default solar mixtures were adopted
+as Z initial fraction from Asplund et al. (2009). Helium
+mass fraction were taken Y=0.28, for Z=0.02. Convective
+core overshoot was described by the exponentially decaying
+prescription of Herwig (2000) and overshooting parameter
+adopted 0.20 for both components (Claret & Torres (2016)
+find 0.208 for both components). A mixing length αMLT
+value of 1.8 was used as the theoretical δ Scuti instability
+strip (Dupret et al. 2004, 2005) was obtained with this αMLT
+value.
+Taking into account the calculated parameters in the
+binary modelling for both groups, the evolutionary status
+of the binary components was investigated. As a result, we
+found that the secondary (more luminous) binary compo-
+nent can be represented with the same evolutionary tracks
+according to both groups’ results. However, the less lumi-
+nous primary component’s position was determined with
+different Z parameters as the parameters of this star were
+found to be slightly different in the study of the two groups.
+According to the evolutionary models, the Z parameters of
+both binary components were found similar to solar (As-
+plund et al. 2009) within the errors which differs from the
+results of the groups as we determined that the less luminous
+component’s atmosphere is somewhat enhanced in metals.
+The results of this analysis are given in Table 7 and a H-R
+diagram is shown in Fig. 12. The observational borders of
+the δ Scuti instability strip were taken from Murphy et al.
+(2019). As can be seen from the H-R diagram, both binary
+components are placed inside the δ Scuti instability strip.
+8
+DISCUSSION AND CONCLUSIONS
+In this analysis, we present the results of the detailed anal-
+ysis of AI Hya carried out by two independent groups. The
+system was observed with different high-resolution spectro-
+graphs (R≳38000). The radial velocity variations of AI Hya
+were modelled using the vr measurements of both groups
+and the orbital parameters such as T0, Porb, e and q were
+updated. The resulting parameters of the analysis of both
+groups are consistent with each other within the errors and
+they slightly differ from the results of Popper (1988). Espe-
+cially the e value shows a discrepancy. Popper (1988) found
+e to be 0.2301 ± 0.0015 while in our study it was determined
+as 0.2419 ± 0.0036 and 0.2432 ± 0.0050 by group-P and -K,
+respectively.
+Since our high-resolution spectra are spread over all or-
+bital phases, we were able to derive the atmospheric param-
+eters of both binary components by modelling either the
+composite spectra or the spectra of the individual compo-
+nents after applying spectral disentangling. To derive the
+atmospheric parameters, v sin i and the chemical composi-
+tion of the binary components, group-K analysed disentan-
+gled spectra of the components, while group-P performed
+their analysis using both the composite and disentangled
+spectra. As a result, group-K found that the more lumi-
+nous star is cooler than the less luminous component. They
+found the Teff
+values from the Hβ line fit and Fe lines
+to be 7500 ± 200 K and 7700 ± 100 K for the primary and
+7000 ± 150 K and 7200 ± 100 K for the secondary compo-
+nent, respectively. Group-P used two different codes in their
+analysis. With the gssp code analysis they found a similar
+result with group-K even though the resulting Teff values
+differ from each other, they determined that the more lu-
+minous star is cooler (7150 ± 250 K) and less luminous one
+is hotter (7350 ± 300 K). In the iSpec analysis of group-P,
+Teff values of both components were found similar to the
+© 2021 RAS, MNRAS 000, 1–13
+
+Comprehensive study of AI Hya
+11
+Table 7. Results obtained from the best-fit evolutionary models.
+Parameter
+Group-K
+Group-P
+P initial (days)
+8.34 (1)
+8.34 (1)
+einitial
+0.242 (2)
+0.243 (2)
+Z1
+0.013 (2)
+0.016 (2)
+Z2
+0.018 (2)
+0.018 (2)
+Age (Myr)
+850 (20)
+860 (20)
+results of the gssp analysis within error bars. The primary’s
+temperature is the most significant discrepancy between the
+values derived by the two groups. The exact reason for this
+temperature inconsistency is not fully understood, although
+it is still only at a level of ∼1.1σ.
+In the chemical abundance analysis, both groups found
+the less luminous but hotter binary component to show
+overabundance while the other component has chemical
+abundance similar to solar. Both groups determined the
+abundances of some individual elements such as iron (Fe).
+They derived Fe abundances as 8.25 ± 0.23 (group-K) and
+7.83 ± 0.16 (group-P). These values are consistent with each
+other within their 1σ errors, and both demonstrate that the
+hotter component has a slightly metal-rich chemical abun-
+dance compared to solar values (see Table 4). This comes
+somewhat to a surprise, as this binary system should have
+been formed in the same interstellar environment and hence
+its components should have the same chemical composition.
+The difference could be due to the consequences of the evolu-
+tion of the system. If AI Hya had a very eccentric orbit when
+the system was formed, there could be some material flows
+from one component to another that could have changed the
+diffusion in one component. Another explanation was given
+by Yushchenko et al. (2015) and they pointed out that pos-
+sible gas and dust accretion from the circumstellar envelope
+could alter the atmospheric composition of one component.
+After the determination of the atmospheric parameters,
+they were used as input in the binary modelling. Overall,
+even though both working groups used different approaches
+to estimate the parameters of the binary component of
+AI Hya, the values determined by both groups are found to
+be consistent with each other within the error bars. The two
+groups obtained very similar M and R values with a ⩽1.7%
+and ∼0.5% accuracy, respectively. When we compare these
+values with the ones found by Lee, Hong, & Kristiansen
+(2020), we notice that there are slight differences, especially
+in the R parameters, and there is significant diversity in the
+calculated distance. These differences could be caused by
+the different assumptions of the atmospheric parameters.
+The evolutionary status of the system was examined
+and it was found that both binary components are inside the
+δ Scuti instability strip. The age of the system is determined
+as well. According to the determined ages, we could say that
+AI Hya is in an important evolutionary phase in terms of
+binary evolution. The rapidly evolving massive component
+will begin the mass transfer process to the less massive one
+approximately 20 Myr from now. This situation could cause
+significant variations in the oscillation properties. Increas-
+ing the number of such bodies is important in terms of ex-
+amining the pulsating structures before the mass transfer
+processes.
+The pulsation properties of AI Hya were examined us-
+Figure 12. The positions of the binary components in the H-R
+diagram according the results of both group-K (g-K) and group-P
+(g-P). The instability strip (IS) borders of the δ Scuti stars were
+taken from Murphy et al. (2019).
+ing the TESS data. However, the system has only one sector
+of SC data, which offers us a poor frequency resolution. In
+the analysis, pulsation frequencies were found between 5.5
+and 13 d−1. As both binary components are placed in the
+δ Scuti instability strip, we were unable to say whether one
+or both pulsate. Apart from that, we could not find pulsa-
+tions related to the orbital frequency.
+As a result of this study, we thoroughly examined
+a detached binary system showing oscillations. This kind
+of objects is particularly important to examine the insta-
+bility strip of δ Scuti stars since they allow us to deter-
+mine fundamental astrophysical, atmospheric parameters
+and the chemical abundances of individual binary compo-
+nents. Hence an increasing number of analyses of such sys-
+tems is expected to be essential to deeply understand the
+nature of pulsations.
+ACKNOWLEDGMENTS
+The authors would like to thank the reviewer for useful
+comments and suggestions that helped to improve the
+publication. This study has been supported by the Sci-
+entific and Technological Research Council (TUBITAK)
+project 120F330. GH thanks the Polish National Center
+for Science (NCN) for supporting the study through
+grants 2015/18/A/ST9/00578 and 2021/43/B/ST9/02972.
+TP’s research is supported through NCN OPUS project
+number 2017/27/B/ST9/02727. AM’s acknowledges the
+support provided by the Polish National Science Centre
+(NCN) OPUS project number 2017/27/B/ST9/02727 and
+2021/41/N/ST9/02746. Based on observations made with
+the Mercator Telescope, operated on the island of La Palma
+by the Flemish Community, at the Spanish Observatorio
+del Roque de los Muchachos of the Instituto de Astrof`ısica
+de Canarias. The TESS data presented in this paper were
+obtained from the Mikulski Archive for Space Telescopes
+(MAST). Funding for the TESS mission is provided by
+© 2021 RAS, MNRAS 000, 1–13
+
+2.0
+tAl Hya
+1.8
+1.6
+(L/ Lo)
+1.4
+1.2
+Primary (g-K), ☆ Primary (g-P)
+Secondary (g-K), ☆ Secondary (g-P)
+1.950 MO track (Z=0.013)
+1.950 MO track (Z=0.016)
+1.0
+ 2.096 MO track (Z=0.018)
+IS
+0.8
+4.00
+3.95
+3.90
+3.85
+3.80
+3.75
+3.70
+3.65
+3.60
+log T (K)12
+F. Kahraman Ali¸cavu¸s et. al.
+the NASA Explorer Program. This work has made use
+of data from the European Space Agency (ESA) mission
+Gaia (http://www.cosmos.esa.int/gaia), processed by the
+Gaia Data Processing and Analysis Consortium (DPAC,
+http://www.cosmos.esa.int/web/gaia/dpac/consortium).
+Funding for the DPAC has been provided by national
+institutions, in particular the institutions participating
+in the Gaia Multilateral Agreement. This research has
+made use of the SIMBAD data base, operated at CDS,
+Strasbourq, France.
+DATA AVAILABILITY
+The data underlying this work will be shared at reasonable
+request to the corresponding author.
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+Table A1. The vr measurements. The subscripts “1” and “2”
+represent the more and the less luminous components, respec-
+tively.
+HJD
+vr,1
+vr,2
+Instrument
++2450000
+(km s−1)
+(km s−1)
+9263.45270
+-12.6 ± 2.8
+109.3 ± 2.7
+CAOS
+9161.65803
+132.8 ± 1.8
+-48.3 ± 1.7
+HERMES
+9162.64306
+109.8 ± 2.0
+-21.7 ± 1.5
+HERMES
+9230.65226
+121.5 ± 1.6
+-24.7 ± 1.8
+HERMES
+9231.66393
+124.1 ± 1.7
+-24.9 ± 1.7
+HERMES
+9233.62648
+59.8 ± 5.7
+36.1 ± 3.4
+HERMES
+9234.55784
+20.6 ± 2.1
+72.1 ± 2.5
+HERMES
+9237.61273
+15.0 ± 1.6
+77.3 ± 1.5
+HERMES
+9235.43315
+-46.3 ± 1.5
+130.3 ± 1.8
+HERMES
+9257.49195
+98.8 ± 1.8
+-2.9 ± 2.0
+HERMES
+9260.61123
+-33.9 ± 1.7
+117.9 ± 1.9
+HERMES
+9276.55613
+96.6 ± 2.0
+-8.4 ± 1.2
+HERMES
+9296.42427
+88.7 ± 1.7
+-3.8 ± 2.0
+HERMES
+9297.44747
+126.7 ± 1.8
+-37.4 ± 2.2
+HERMES
+9298.45846
+113.5 ± 1.7
+-16.0 ± 1.8
+HERMES
+9299.46357
+78.1 ± 2.1
+12.5 ± 2.3
+HERMES
+7075.62231
+-39.7 ± 1.5
+129.9 ± 0.5
+CORALIE
+7076.63954
+-20.4 ± 1.2
+120.0 ± 1.3
+CORALIE
+7109.63123
+-17.9 ± 2.4
+126.0 ± 1.3
+CORALIE
+7022.31643
+118.5 ± 1.9
+-31.4 ± 0.5
+HIDES
+7060.09414
+-13.6 ± 0.7
+118.2 ± 0.6
+HIDES
+7109.96513
+-18.6 ± 1.1
+114.6 ± 0.6
+HIDES
+7114.92732
+120.1 ± 1.2
+-34.0 ± 0.7
+HIDES
+7146.98403
+118.5 ± 1.3
+-42.9 ± 0.8
+HIDES
+7147.96084
+131.3 ± 1.2
+-42.3 ± 0.6
+HIDES
+7363.28986
+135.7 ± 0.6
+-48.4 ± 0.7
+HIDES
+7755.22744
+-34.0 ± 0.7
+126.5 ± 0.7
+HIDES
+7813.13416
+-25.2 ± 0.8
+124.3 ± 0.9
+HIDES
+7814.08321
+-28.3 ± 0.9
+126.6 ± 0.9
+HIDES
+7846.01822
+-10.8 ± 1.7
+113.4 ± 1.0
+HIDES
+8035.34461
+101.6 ± 0.7
+-13.9 ± 0.8
+HIDES
+8066.24908
+85.2 ± 0.8
+-4.1 ± 1.3
+HIDES
+This paper has been typeset from a TEX/ LATEX file prepared
+by the author.
+© 2021 RAS, MNRAS 000, 1–13
+

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+arXiv:2301.03075v1  [math.DS]  8 Jan 2023
+CONSTRUCTION OF FRACTAL FUNCTIONS USING KANNAN
+MAPPINGS AND SMOOTHNESS ANALYSIS
+SUBHASH CHANDRA, SAURABH VERMA, AND SYED ABBAS
+Abstract. Let T be a self-map on a metric space (X, d). Then T is called
+Kannan map if there exists α, 0 < α < 1
+2, such that
+d(T(x), T(y)) ≤ α[d(x, T(x)) + d(y, T(y))], for all x, y ∈ X.
+This paper aims to introduce a new method to construct fractal functions
+using Kannan mappings. First, we give the rigorous construction of fractal
+functions with the help of the Kannan iterated function system (IFS). We also
+show the existence of a Borel probability measure supported on the attractor
+of the Kannan IFS satisfying the strong separation condition. Moreover, we
+study the smoothness of the constructed fractal functions. We end the paper
+with some examples and graphical illustrations.
+1. INTRODUCTION
+The concept of fractal interpolation function (FIF) was introduced by Barnsley
+[2, 3] through iterated function system (IFS), and their construction is rooted in the
+theory of IFS [9]. The FIF is an interpolation function whose graph is an invariant
+set of an IFS. The pioneering research on fractal interpolation has gotten much at-
+tention in the literature, and it continues to flourish. The concept of FIF has been
+extended and generalized in several ways given in the literature. Wang and Yu [27]
+gave the construction of new class IFSs with variable parameters and generated as-
+sociated FIFs. Also, they studied the smoothness and stability of FIFs under some
+conditions on data points. The construction of nonlinear FIF using Matkowski and
+the Rakotch fixed point theorems is given in [20]. In this order, Songli [21] gave
+the construction of nonlinear FIF on Sierpi´nski gasket. The reader may refer to
+books [3, 16] for the details on fractal functions. The fractal dimension is one of
+the major themes in fractal geometry. Many works on the fractal dimensions of
+fractals functions are in the literature. There are various approaches, such as the
+mass-distribution principle, potential theory, Fourier transform, positive operators,
+etc., to compute or estimate the Hausdorff dimension of a set [11, 26]. Using the
+potential theoretic approach, Barnsley gave results on the Hausdorff dimension of
+an affine FIF in [3]. Falconer [11] also gave the estimate of the Hausdorff dimension
+of an affine FIF. The results on the Hausdorff dimension using the positive oper-
+ators approach are given in [26]. Priyadarshi [19] gave an algorithm to determine
+lower bounds for the Hausdorff dimension of a set of complex continued fractions
+and estimated the best lower bound. Jha and Verma [12] established very inter-
+esting results for fractal dimensions of fractal functions and some invariant sets.
+2020 Mathematics Subject Classification.
+28A80, 47H10, 28A33, 28A78.
+Key words and phrases. Kannan IFS, Fractal Functions, Borel Probability Measure, Fractal
+Dimension.
+1
+
+2
+SUBHASH CHANDRA, SAURABH VERMA, AND SYED ABBAS
+They estimated fractal dimensions for a class of FIFs, widely known as α-fractal
+functions, by using function spaces such as H¨older space, oscillation space, and
+space of bounded variation. Ruan et al. [22] estimated the box dimension of the
+new class of linear FIFs by using the δ-covering method. Additionally, they have
+established a relationship between the order of fractional integral and box dimen-
+sions of two linear FIFs. As we know, recurrent FIF is the generalization of linear
+FIF, and the graph of recurrent FIF is the invariant set of recurrent IFS. Barnsley
+and Massopust [4] gave results on the bilinear FIFs and their box dimension. Few
+recent developments on fractal dimensions can be seen in [6, 24, 25]. Cheng et
+al. [6] introduced the notion of upper metric mean dimension with potential on
+any subset via Carath´eodory-Pesin structures. Selmi [24] studied the multifractal
+Hausdorff and packing dimensions of Borel probability measures and studied their
+behaviors under orthogonal projections. In this order, Selimi estimated the multi-
+fractal Hausdorff and the packing dimensions of product measures in [25].
+Barnsley [2, 3] considered the collection of self-contraction mappings and used the
+Hutchinson operator and Banach fixed point principle to construct fractal functions.
+Kannan [13, 14] introduced a new fixed point theorem widely known as Kannan
+fixed point theorem.
+Other related results on Kannan mapping can be seen in
+[8, 10]. By using the concept of Kannan mapping, Sahu et al. [23] introduced the
+notion of the Kannan iterated function system.
+Theorem 1.1. [14] Let T is a map of the complete metric space X into itself and
+if
+d(T (x), T (y)) ≤ α[d(x, T (x)) + d(y, T (y))], ∀ x, y ∈ X, 0 < α < 1
+2.
+Then T has the unique fixed point in X.
+A natural question arises can we construct fractal functions using the concept
+of Kannan fixed point theory? This question motivates us to conduct the current
+study. In this study, we use the concept of Kannan IFS and Kannan fixed point
+theorem and derive very interesting results.
+This paper is organized as follows: Section 2 is devoted to preliminaries and required
+terminologies related to this article. Section 3 presents the construction of fractal
+functions and the existence of self-similar measures. In Section 4, the smoothness
+result of the Kannan fractal function is given. The graphical illustration of the
+Kannan fractal functions is given in Section 5.
+2. Background and preliminaries
+This section aims to provide some basic definitions and results that act as prelude
+to this article. Let F ̸= ∅ be a subset of Rn. The diameter of F is given by
+diamd(F) = sup {d(x, y) : x, y ∈ F} .
+If {Fi} is a countable (or finite) collection of sets having a diameter at most δ
+which cover set E ⊆ Rn, then we say that {Fi} is a δ-cover of E. For δ > 0 and a
+non-negative real number s, we define
+Hs
+δ,d(E) = inf
+� ∞
+�
+i=1
+diamd(Fi)s : {Fi} is a δ − cover of E
+�
+.
+Definition 2.1. The s-dimensional Hausdorff measure of set E is given by Hs(E) =
+limδ→0 Hs
+δ (E).
+
+CONSTRUCTION OF FRACTAL FUNCTIONS USING KANNAN MAPPINGS
+3
+Definition 2.2. ( Hausdorff dimension) Let s ≥ 0 and E ⊆ Rn. The Hausdorff
+dimension of E is defined as
+dimH(E) = inf{s : Hs(E) = 0} = sup{s : Hs(E) = ∞}.
+Definition 2.3. (Box Dimension) Let E ⊆ Rn be bounded and non-empty and let
+Nδ(E) be the smallest number of sets of diameter at most δ which cover E. The
+lower box dimension of E is
+dimB(E) = lim
+δ→0
+log Nδ(E)
+− log δ
+,
+and the upper box dimension of E is
+dimB(E) = lim
+δ→0
+log Nδ(E)
+− log δ
+,
+If both lower and upper box dimensions are the same, then that quantity is called
+the box dimension of E and it is given by
+dimB(E) = lim
+δ→0
+log Nδ(E)
+− log δ
+.
+For the details on the Hausdorff and box dimensions, the reader may be referred
+to [11].
+Definition 2.4. Let d1 and d2 are two matrices on X, then d1 and d2 are topo-
+logically equivalent if and only if
+d1(xn, x) → 0 ⇐⇒ d2(xn, x) → 0,
+for {xn} ⊂ X and x ∈ X.
+Let d1 and d2 are two matrices on X, then d1 and d2 are metrically equivalent if
+and only if there exists c1, c2 > 0 and x, y ∈ X such that
+c1d1(x, y) ≤ d2(x, y) ≤ c2d1(x, y).
+Fractal Interpolation Function. Now, we introduce FIF in brief.
+Here, we
+consider a set for interpolation as {(xn, yn) : n = 1, 2, . . . , N}.
+We set J =
+{1, 2, ..., N − 1}, I = [x1, xN] and for j ∈ J, let Ij = [xj, xj+1]. For j ∈ J, let
+Lj : I → Ij be a contractive homomorphism such that
+Lj(x1) = xj, Lj(xN) = xj+1, j ∈ J.
+Now, define Fj : K = I × R → R, j ∈ J, which is a contraction in the second
+variable, that is, |Fj(x, y) − Fj(x, y′)|≤ rj|y − y′|, for all x ∈ I, rj ∈ [0, 1) and
+y, y′ ∈ R and satisfying Fj(x1, y1) = yj, Fj(xN, yN) = yj+1, j ∈ J. We shall take
+(2.1)
+Lj(x) = ajx + bj Fj(x, y) = αjy + qj(x),
+In the above expression aj and bj are determined by using conditions Lj(x1) =
+xj, Lj(xN) = xj+1. Here, αj is the scaling factor with |αj|< 1 and continuous
+functions qj : I → R, j ∈ J satisfy “join-up conditions” imposed for the bivariate
+maps Fj. That is, qj(x1) = yj − αjy1 and qj(xN) = yj+1 − αjyN for all j ∈ J. Now
+define functions Wj : I × R → I × R for j ∈ J by
+Wj(x, y) = (Lj(x), Fj(x, y)).
+
+4
+SUBHASH CHANDRA, SAURABH VERMA, AND SYED ABBAS
+Theorem 1 in [3] says that the IFS I := {I × R; W1, W2, . . . , WN−1} defined above
+has a unique attractor which is the graph of a function f which satisfies the following
+functional equation reflects self-referentiality:
+f(x) = αjf(L−1
+j (x)) + qj(L−1
+j (x)), x ∈ Ij, j ∈ J.
+The above function f is known as the fractal interpolation function.
+Kannan mapping. In 1969, Kannan [13] introduced a mapping, which was an
+improvement over the contraction mapping, known as Kannan mapping, defined as
+follows:
+If there exists a number α, 0 < α < 1
+2, such that, for all x, y ∈ X,
+d(T (x), T (y)) ≤ α[d(x, T (x)) + d(y, T (y))].
+Then T is called a Kannan mapping and α is called Kannan-contractivity factor
+of T . Let Tn : X → X are Kannan mappings having contractivity factor αn, for
+n = 1, 2, . . ., N and (X, d) be a complete metric space. Then, the set {X; Tn, n =
+1, 2, . . ., N} is said to be Kannan IFS.
+Remark 2.5. Let f : [0, 1] → [0, 1] be defined by f(x) = x
+3. Then this function f is
+a contraction mapping with contraction factor 1
+3, but it is not a Kannan mapping.
+On the other hand, the function g : [0, 1] → [0, 1] defined by
+g(x) =
+�
+x
+4, if 0 ≤ x < 1
+2
+x
+5,
+if 1
+2 ≤ x ≤ 1.
+is a Kannan mapping with β = 4
+9 but it is not a contraction mapping. The concepts
+of the Kannan operator and contraction are independent. The self-map T given in
+the previous example is Kannan, but it is not a contraction due to its discontinuity.
+The following simple note can be seen in [10]. However, we include its details
+for the reader’s convenience.
+Note 2.6. Let (X, d) be a metric space and T : X → X is contraction with constant
+c < 1
+3. Then T is Kannan contractive with respect to metric d.
+Because of the contractivity of T , we have
+d(T (x1), T (x2)) ≤ cd(x1, x2) ≤ cd(x1, T x1)+cd(T x1, T x2)+c(T x2, x2), ∀ x1, x2 ∈ X.
+This turns
+d(T (x1), T (x2)) ≤ α[d(x1, T (x1)) + d(x2, T (x2))], ∀ x1, x2 ∈ X.
+Since 0 < α :=
+c
+1−c < 1
+2, T is a Kannan mapping.
+Proposition 2.7. Let X be a complete metric space and d1 and d2 are equivalent
+metrics on X, i.e., there exist positive constants c1, c2 such that
+c1d1(x1, x2) ≤ d2(x1, x2) ≤ c2d1(x1, x2), x1, x2 ∈ X.
+If T is a contraction on X with respect to the metric d1 then there exists an m ∈ N
+such that T m is a Kannan contraction with respect to the metric d2.
+Proof. Since T is contraction on (X, d1), there exits 0 ≤ k < 1 such that
+d1(T x1, T x2) ≤ kd1(x1, x2), x1, x2 ∈ X.
+
+CONSTRUCTION OF FRACTAL FUNCTIONS USING KANNAN MAPPINGS
+5
+Whence d2(T x1, T x2) ≤ c2d1(T x1, T x2) ≤ c2kd1(x1, x2) ≤
+�
+c2
+c1 k
+�
+d2(x1, x2). Take
+m ∈ N such that c2
+c1 km < 1
+3. Then
+d2(T mx1, T mx2) ≤ c2d1(T mx1, T mx2) ≤ c2kmd1(x1, x2) ≤
+�
+c2
+c1
+km
+�
+d2(x1, x2).
+Hence, T m is a contraction with respect to the metric d2.
+From Note 2.6, we
+conclude that T m is a Kannan contraction with respect to the metric d2. Thus, the
+proof is completed.
+□
+Theorem 2.8. [14] Let T is a map of the complete metric space X into itself and
+if
+d(T (x), T (y)) ≤ α[d(x, T (x)) + d(y, T (y))], ∀ x, y ∈ X, 0 < α < 1
+2.
+Then T has the unique fixed point in X.
+The Hausdorff distance from the set A to the set B is defined as
+h(A, B) = max{sup
+a∈A
+inf
+b∈B d(a, b), sup
+b∈B
+inf
+a∈A d(a, b)}.
+Note 2.9. In [23, Lemma 3.5] the authors claimed that for all B, C ∈ H(X),
+h(T (B), T (C)) ≤ β[h(B, T (B)) + h(C, T (C))].
+The above claim is not true, for instance, see the following example, which is
+borrowed from [8].
+Example 2.10. Let X = {0, 1, 2}, and the function d : X × X → R and the map
+f : X → X be given by
+d(0, 0) = d(1, 1) = d(2, 2) = 0
+d(0, 1) = d(1, 0) = 5, d(1, 2) = d(2, 1) = 2, d(0, 2) = d(2, 0) = 3
+f(1) = f(2) = 2, f(0) = 1.
+Then the map f : X → X is Kannan on (X, d) with contractivity factor α ∈ [ 2
+5, 1
+2)
+but the map T : H(X) → H(X) given by T (B) = ∪x∈Bf(x) for all B ∈ H(X) is
+not a Kannan map on (H(X), h(d)) for any contractivity factor α ∈ [0, 1
+2).
+Remark 2.11. The above example does not work for the following lemma due to
+different contractive factors. Here, we correct Lemma 3.5 of [23], and we show that
+it holds under certain additional conditions.
+Lemma 2.12. Suppose T : X → X be a continuous Kannan mapping on the metric
+space (X, d) with contractivity factor 0 < β < 1
+6. Then T : H(X) → H(X) given by
+T (B) = {T (x) : x ∈ B} for every B ∈ H(X) is Kannan mapping on (H(X), h(d))
+with contractivity factor 0 < γ =
+β
+(1−4β) < 1
+2.
+Proof. Let us first recall a basic real-analysis result that the image of a compact
+set under a continuous map is compact. Since T is continuous, it maps H(X) into
+itself. Now, since T is Kannan mapping on (X, d), for x, y ∈ X, we have
+d(T (x), T (y)) ≤ β[d(x, T (x)) + d(y, T (y))]
+≤ β[d(x, T (y)) + d(T (y), T (x)) + d(y, T (x)) + d(T (x), T (y))]
+= β[d(x, T (y)) + d(y, T (x))] + 2βd(T (x), T (y)).
+
+6
+SUBHASH CHANDRA, SAURABH VERMA, AND SYED ABBAS
+So, we obtain
+d(T (x), T (y)) ≤
+β
+(1 − 2β)[d(x, T (y)) + d(y, T (x))].
+Now, for B, C ∈ H(X)
+sup
+x∈B
+inf
+y∈C d(T (x), T (y)) ≤
+β
+(1 − 2β)[sup
+x∈B
+inf
+y∈C d(x, T (y)) + sup
+x∈B
+inf
+y∈C d(y, T (x))].
+That is,
+sup
+x∈B
+inf
+y∈C d(T (x), T (y)) ≤
+β
+(1 − 2β)[h(B, T (C)) + h(C, T (B))].
+Thanks to the triangle inequality,
+h(T (B), T (C)) ≤
+β
+(1 − 2β)[h(B, T (B)) + h(T (B), T (C))
++h(C, T (C)) + h(T (C), T (B))].
+Consequently,
+�
+1 −
+2β
+(1 − 2β)
+�
+h(T (B), T (C)) ≤
+β
+(1 − 2β)[h(B, T (B)) + h(C, T (C))].
+Therefore,
+h(T (B), T (C)) ≤ γ[h(B, T (B)) + h(C, T (C))],
+where γ =
+β
+(1−4β) < 1
+2 for 0 < β < 1
+6. This completes the proof.
+□
+Theorem 2.13. For a complete metric space (X, d), let Tn : n = 1, 2, ..., N are
+continuous Kannan mappings on (H(X), h) with contractivity factor 0 < βn <
+1
+6, for all n. Define T : H(X) → H(X) by T (B) = ∪N
+n=1Tn(B) for each B ∈
+H(X). Then T is a Kannan mapping with contractivity factor γ = max{γn : n =
+1, 2, ..., N}.
+Proof. For B, C ∈ H(X), we have
+h(T (B), T (C)) = h(T1(B) ∪ T2(B) ∪ . . . ∪ Tn(B), T1(C) ∪ T2(C) ∪ . . . ∪ Tn(C))
+≤ max{h(T1(B), T1(C)), h(T2(B), T2(C)), . . . , h(Tn(B), Tn(C))}.
+By using the above lemma, we obtain
+h(T (B), T (C))
+= max
+�
+β1
+1 − 4β1
+[h(B, T1(B)) + h(C, T1(C))],
+β2
+1 − 4β2
+[h(B, T2(B))
++ h(C, T2(C))], . . . ,
+βn
+1 − 4βn
+[h(B, Tn(B)) + h(C, Tn(C))]
+�
+≤ max
+1≤i≤n
+�
+βi
+1 − 4βi
+�
+[max{h(B, T1(B)), h(B, T2(B)), ..., h(B, Tn(B))}
++ max{h(C, T1(C)), h(C, T2(C)), ..., h(C, Tn(C))}]
+≤ max
+1≤i≤n{γi}[h(B, T1(B) ∪ T2(B)) . . . ∪ Tn(B) + h(C, T1(C) ∪ T2(C)) . . . ∪ Tn(C)]
+≤ γ[h(B, T (B)) + h(C, T (C))],
+
+CONSTRUCTION OF FRACTAL FUNCTIONS USING KANNAN MAPPINGS
+7
+where γ = max1≤i≤n{γi} = max1≤i≤n
+�
+βi
+1−4βi
+�
+< 1
+2. Hence, T is Kannan with
+contractivity factor γ.
+□
+Remark 2.14. In [8] Dung et al. proposed a question, whether their results are true
+or not for n ≥ 3. In this order, in the above theorem, we show that T is Kannan
+for all n. Moreover, from Theorem 2.8, T has a unique fixed point.
+We use the following notations throughout the article: C(I) denotes the set of
+continuous functions f : I = [x0, xN] → [a, b]. Let C∗(I) ⊂ C(I) and given by
+C∗ = {f ∈ C(I) : f(x0) = y0, f(xN) = yn}. K = I × R.
+Let us define a metric dθ on K as follows
+dθ((x, y), (z, w)) = |x − z|+θ|y − w|, θ > 0.
+Note that (C∗(I), Hθ) is a complete metric space with respect to Hausdorff metric
+Hθ, where
+Hθ(f, g) = Hθ(Gf, Gg) = max{ sup
+x∈Gf
+inf
+y∈Gg dθ(x, y), sup
+y∈Gg
+inf
+x∈Gf dθ(x, y)}.
+In the following section, we give the construction of fractal functions using Kannan
+IFS and the existence of self-similar measures.
+3. Construction of fractal functions via Kannan Iterated function
+systems
+Let Fi : K → [a, b] be continuous mappings and satisfying for some k ≥ 0, and
+0 ≤ βi < 1
+2
+|Fi(x, y)−Fi(w, y)|≤ k|x−w|, |Fi(x, y)−Fi(x, z)|≤ βi
+�
+|y −Fi(., y)|+|z −Fi(., z)|
+�
+for all x, w ∈ I, y, z ∈ [a, b], and i = 1, 2, . . . , N. Now, let {K; Wi, i = 1, 2, . . ., N}
+be an IFS with
+Wi(x, y) = (Li(x), Fi(x, y)) = (aix + bi, Fi(x, y)),
+where transformations are constrained by the data according to
+Wi(x0, y0) = (xi−1, yi−1), Wi(xN, yN) = (xi, yi)
+for i = 1, 2, . . . , N. For all i = 1, 2, ..., N, Wi : K → K are Kannan mappings. Then
+{K; Wi : i = 1, 2, ..., N} is the Kannan IFS.
+Theorem 3.1. Let N > 1, and {K; Wi, i = 1, 2, . . . , N} denote the IFS defined as
+above, associated with the set of data
+{(xi, yi) : i = 1, 2, ..., N} such that amax = max
+i (xi+1 − xi) < 1
+3.
+Then, there is a metric dθ on K = I × R, equivalent to the Euclidean metric such
+that for all i = 1, 2, ..., N, Wi are Kannan maps with respect to dθ.
+
+8
+SUBHASH CHANDRA, SAURABH VERMA, AND SYED ABBAS
+Proof. For all (x, y), (w, z) ∈ K, we have
+dθ(Wi(x, y), Wi(w, z)) = dθ
+�
+(Li(x), Fi(x, y)), (Li(w), Fi(w, z))
+�
+= |Li(x) − Li(w)|+θ|Fi(x, y) − Fi(w, z)|
+≤ |ai||x − w|+θ|Fi(x, y) − Fi(w, z)|
+≤ amax|x − w|+θ|Fi(x, y) − Fi(w, z)|.
+Now, thanks to the triangle inequality, we have
+|x − w|≤ |x − Li(x)|+|Li(x) − Li(w)|+|Li(w) − w|,
+this further yields
+|x − w|≤
+1
+1 − amax
+(|x − Li(x)|+|Li(w) − w|).
+We now estimate
+|Fi(x, y) − Fi(w, z)|≤ |Fi(x, y) − Fi(w, y)|+|Fi(w, y) − Fi(w, z)|
+≤ k|x − w|+βi
+�
+|y − Fi(w, y)|+|z − Fi(w, z)|
+�
+≤ k|x − w|+βi
+�
+|y − Fi(x, y)|+|Fi(x, y) − Fi(w, y)|+|z − Fi(w, z)|
+�
+≤ k|x − w|+βi
+�
+|y − Fi(x, y)|+k|x − w|+|z − Fi(w, z)|
+�
+≤ (k + kβmax)|x − w|+βmax
+�
+|y − Fi(x, y)|+|z − Fi(w, z)|
+�
+.
+With the help of the above estimates, we obtain
+dθ(Wi(x, y), Wi(w, z))
+= amax + (k + kβmax)θ
+1 − amax
+(|x
+− Li(x)|+|Li(w) − w|) + βmaxθ
+�
+|y − Fi(x, y)|+|z − Fi(w, z)|
+�
+≤ γ
+�
+dθ((x, y), Wi(x, y)) + dθ((w, z), Wi(w, z))
+�
+,
+where γ = max
+�
+amax+(k+kβmax)θ
+1−amax
+, βmaxθ
+�
+. Using the condition amax < 1
+3, we may
+choose a suitable (sufficiently small) θ > 0 such that γ < 1
+2. For this value of θ, the
+mapping Wi is a Kannan mapping, completing the proof.
+□
+Remark 3.2. From the above proof, if amax < 1
+5 then we may choose a suitable
+(sufficiently small) θ > 0 such that the Kannan contractivity factor γ < 1
+4.
+Theorem 3.3. Let N > 1 and {K; Wi, i = 1, 2, . . ., N} denote the IFS defined as
+above, associated with the set of data
+{(xi, yi) : i = 1, 2, ..., N} such that amax = max
+i (xi+1 − xi) < 1
+7.
+Then, there exists a unique non empty compact set G ⊂ K = I × [a, b] such that
+G = ∪N
+i=1Wi(G).
+
+CONSTRUCTION OF FRACTAL FUNCTIONS USING KANNAN MAPPINGS
+9
+Proof. On similar lines of the proof of Theorem 3.1, we have
+dθ(Wi(x, y), Wi(w, z)) ≤ γ
+�
+dθ((x, y), Wi(x, y)) + dθ((w, z), Wi(w, z))
+�
+,
+where γ = max
+�
+amax+(k+kβmax)θ
+1−amax
+, βmaxθ
+�
+. Using the condition amax < 1
+7, we may
+choose a suitable (sufficiently small) θ > 0 such that γ < 1
+6. For this value of θ,
+the mapping Wi is a Kannan mapping with contractivity factor γ < 1
+6. Now, using
+Theorem 2.13 and Remark 2.14, we obtain a unique compact set G satisfying
+G = ∪N
+i=1Wi(G).
+This completes the proof.
+□
+Theorem 3.4. Let the IFS {K; Wi, i = 1, 2, ..., N} defined as above associated with
+the set of data
+{(xi, yi) : i = 1, 2, ..., N} such that amax = max
+i (xi+1 − xi) < 1
+5.
+Let G denote the attractor of the IFS. Then, G is the graph Gf of continuous
+function f : [x0, xN] → [a, b] satisfying f(xi) = yi for all i = 0, 1, ..., N. That is,
+Gf = {(x, f(x)) : x ∈ [x0, xN]},
+where f(xi) = yi for all i = 0, 1, ..., N.
+Proof. Let C∗(I) = {f ∈ C(I) : f(x1) = y1, f(xN) = yN}. Here, C∗(I) is a
+closed subset of C(I) and (C∗(I), Hθ) is a complete metric space. Define Read-
+Bajraktarevi´c (RB) operator T : C∗(I) → C∗(I) by
+(3.1)
+(T g)(x) = Fi(L−1
+i (x), g(L−1
+i (x))).
+Now, we show that T is a Kannan mapping w.r.t. Hθ. We will proceed as follows.
+Here, graphs of T g and T h are given by
+GT g = {(x, T g(x)) : x ∈ I}
+and
+GT h = {(y, T h(y)) : y ∈ I}.
+Let (x, T g(x)) ∈ GT g and (y, T h(y)) ∈ GT h. Since Wi is Kannan with contractivity
+factor γ, from Theorem 3.1, we get
+dθ
+�
+(x, T g(x)), (y, T h(y))
+�
+= dθ
+�
+(x, Fi(L−1
+i (x), g(L−1
+i
+(x))), (y, Fi(L−1
+i (y), h(L−1
+i (y))))
+�
+= dθ
+�
+(Li(w), Fi(w, g(w)), (Li(z), Fi(z, g(z))
+�
+= dθ
+�
+Wi(w, g(w)), Wi(z, h(z))
+�
+≤ γ
+�
+dθ
+�
+(w, g(w)), Wi(w, g(w))
+�
++ dθ
+�
+(z, h(z)), Wi(z, h(z))
+��
+= γ
+�
+dθ
+�
+L−1
+i (x), g(L−1
+i (x)), (x, T g(x))
+�
++ dθ
+�
+L−1
+i (y), h(L−1
+i (y)), (y, T h(y))
+��
+,
+
+10
+SUBHASH CHANDRA, SAURABH VERMA, AND SYED ABBAS
+where w = L−1
+i (x) and z = L−1
+i (y). Thanks to the triangle inequality,
+dθ
+�
+(x, T g(x)), (y, T h(y))
+�
+≤ γ
+�
+dθ
+�
+(L−1
+i (x), g(L−1
+i
+(x))), (y, T g(y))
+�
++ dθ
+�
+(x, T g(x)), (y, T h(y))
+�
++ dθ
+�
+(L−1
+i (y), h(L−1
+i (y))), (x, T h(x))
+�
++ dθ
+�
+(x, T g(x)), (y, T h(y))
+��
+.
+On taking infimum both sides, we have
+inf
+y ∈I dθ
+�
+(x, T g(x)), (y, T h(y))
+�
+≤ γ
+�
+inf
+y∈I dθ
+�
+(L−1
+i (x), g(L−1
+i
+(x))), (y, T g(y))
+�
++ inf
+y∈I dθ
+�
+(L−1
+i (y), h(L−1
+i (y))), (x, T h(x))
+�
++ 2 inf
+y∈I dθ
+�
+(x, T g(x)), (y, T h(y))
+��
+.
+That is,
+(1 − 2γ) inf
+y ∈I dθ
+�
+(x, T g(x)), (y, T h(y))
+�
+≤ γ
+�
+inf
+y∈I dθ
+�
+(L−1
+i (x), g(L−1
+i (x))), (y, T g(y))
+�
++ inf
+y∈I dθ
+�
+(L−1
+i (y), h(L−1
+i (y))), (x, T h(x))
+��
+.
+By taking supremum over x ∈ I, we have
+Hθ(GT g, GT h) ≤
+γ
+1 − 2γ [Hθ(Gg, GT g) + Hθ(Gh, GT h)].
+That is,
+Hθ(T g, T h) ≤
+γ
+1 − 2γ
+�
+Hθ(g, T g) + Hθ(h, T h)
+�
+.
+Since amax < 1
+5, Remark 3.2 yields that β :=
+γ
+1−2γ < 1
+2. Therefore, T is Kannan
+w.r.t. Hθ. By Theorem 2.8 , T has a unique fixed point f ∈ C∗(I). Further, it is
+easy to check that f interpolates the data set. Now, we show that the graph Gf of
+f is an attractor of the IFS. Since Wi(x, y) = (Li(x), Fi(x, y)) for all i = 1, 2, ..., N,
+I = ∪j∈JLj(I), and from the functional Equation 2.1, we get that
+Wi(Gf) = Wi({(x, f(x)) : x ∈ [x0, xN]})
+= {(Li(x), Fi(x, f(x))) : x ∈ [x0, xN]}
+= {(Li(x), f(Li(x))) : x ∈ [x0, xN]}
+= {(x, f(x)) : x ∈ [xi−1, xi]}.
+Hence
+Gf = {(x, f(x)) : x ∈ [x0, xN]}
+= ∪N
+i=1{(x, f(x)) : x ∈ [xi−1, xi]}
+= ∪N
+i=1Wi(Gf).
+By Theorem 3.3, G is the unique attractor of the IFS {K; Wi, i = 1, 2, ..., N}. Thus,
+G = Gf. This completes the proof.
+□
+
+CONSTRUCTION OF FRACTAL FUNCTIONS USING KANNAN MAPPINGS
+11
+Example 3.5. The continuous function h : [−1, 21
+10] → [−1, 21
+10] defined by
+h(x) =
+�
+x2
+4 − x
+8 , if − 1 ≤ x < 1
+2
+x2
+5 − x
+10,
+if 1
+2 ≤ x ≤ 21
+10.
+is Kannan mapping with β = 10
+21 but it is not a contraction mapping.
+First, we show that T is Kannan contraction, and we choose β = 10
+21 < 1
+2.
+(i) For the range −1 ≤ x, y < 1
+2, we have
+|T (x) − T (y)|= 1
+8|x(2x − 1) − y(2y − 1)|
+and
+|x − T (x)|+|y − T (y)|= 1
+8
+�
+|x|9 − 2x|+|y||9 − 2y|
+�
+.
+For β = 10
+21, we can see that
+|T (x) − T (y)|≤ β
+�
+|x − T (x)|+|y − T (y)|
+�
+.
+(ii) For 1
+2 ≤ x, y < 21
+10, we have
+|T (x) − T (y)|= 1
+10|x(2x − 1) − y(2y − 1)|
+and
+|x − T (x)|+|y − T (y)|= 1
+10
+�
+|x|11 − 2x|+|y||11 − 2y|
+�
+.
+For β = 10
+21, we can see that
+|T (x) − T (y)|≤ β
+�
+|x − T (x)|+|y − T (y)|
+�
+.
+(iii) For −1 ≤ x < 1
+2 and 1
+2 ≤ y < 21
+10, we have
+|T (x) − T (y)|= |x(2x − 1)
+8
+− y(2y − 1)
+10
+|
+and
+|x − T (x)|+|y − T (y)|=
+�1
+8|x|9 − 2x|+ 1
+10|y||11 − 2y|
+�
+.
+For β = 10
+21, we can see that
+|T (x) − T (y)|≤ β
+�
+|x − T (x)|+|y − T (y)|
+�
+.
+Now, one can see that T is not a contraction because for x = −1, y = −0.99, we
+have
+|T (x) − T (y)|= 0.2537 > |x − y|= 0.01.
+Remark 3.6. In this order, we can construct many different Kannan mappings with
+the help of the functional Equation 2.1
+Fj(x, y) = αjy + qj(x), j ∈ J.
+For, instance
+Fj(x, y) = T (y) + qj(x),
+
+12
+SUBHASH CHANDRA, SAURABH VERMA, AND SYED ABBAS
+where
+T (y) =
+�
+y2
+4 − y
+8, if − 1 ≤ y < 1
+2
+y2
+5 − y
+10,
+if 1
+2 ≤ y ≤ 21
+10,
+and qj : I → R, j ∈ J are suitable continuous functions satisfying qj(x1) = yj −αjy1
+and qj(xN) = yj+1 − αjyN for all j ∈ J.
+3.1. Existence of Self-Similar Measures.
+Definition 3.7. Let I = {K; Wi : i = 1, 2, ..., N} be an IFS and A be the attarctor
+of the IFS I. Then, we say that I satisfies strong separation condition (SSC) if
+Wi(A) ∩ Wj(A) = ∅ whenever i ̸= j.
+Note that there are several separation conditions are available for any IFS, for
+instance, [9, 11].
+Hutchinson [9] computed the Hausdorff dimension of self-similar sets under the
+open set condition. Assuming the SSC, Priyadarshi and his collaborators [26] gave
+a formula for the Hausdorff dimension of the invariant set of generalized graph-
+directed systems.
+Theorem 3.8. Let I = {K; Ti : i = 1, 2, ..., N} be an IFS consisting of Kannan
+mappings satisfies the SSC. Let (p1, p2, · · · , pN) be a probability vector. Then, there
+exists a Borel probability measure µ∗ supported on the attractor A of the IFS such
+that
+µ∗ =
+N
+�
+i=1
+piµ∗ ◦ T −1
+i
+.
+Proof. Since the Kannan IFS {K; Ti : i = 1, 2, ..., N} satisfies the SSC. That is,
+Ti(A) ∩ Tj(A) = φ ∀ i ̸= j. Let E0 = A. We have
+A =
+N
+�
+i=1
+Ti(A) =
+N
+�
+i,j=1
+Tij(A) =
+N
+�
+i1,i2,...,in
+Ti1,i2,...,in(A).
+For k = 1, 2, ..., we define Ek as follows:
+Ek =
+�
+Ti1i2···ik(A) : ij ∈ {1, 2, ..., N}, j = 1, 2, ..., k
+�
+,
+where Ek denotes the collection of disjoint Borel subsets of A. Let B ∈ Ek. Note
+that each B is contained in one of the sets in Ek−1 and contains a finite number
+of the sets in Ek+1. We can see that |Ti1i2···ik(A)|→ 0 as k → ∞ as follows. Let
+Ti : A → A and |A|= supx,y∈A d(x, y) = d(x0, y0), x0, y0 ∈ A. Since Ti is Kannan
+contraction, we have
+d(Ti(x), Ti(y)) ≤ βi[d(x, Ti(x)) + d(y, Ti(y))]
+≤ βi[d(x0, y0) + d(x0, y0)]
+≤ 2βid(x0, y0).
+By taking supremum of both side, we have
+sup
+x,y∈A
+d(Ti(x), Ti(y)) ≤ 2βid(x0, y0),
+and 2βi < 1. Now,
+|Ti(A)|≤ 2βid(x0, y0) = ci|A|, ci = 2βi.
+
+CONSTRUCTION OF FRACTAL FUNCTIONS USING KANNAN MAPPINGS
+13
+In a similar way, we get
+|Ti1i2···ik(A)|≤ ck
+max|A|→ 0 when k → ∞,
+where cmax = 2βmax = 2 max{β1, β2, ..., βk}.
+Let a probability vector p = (p1, p2, ..., pN) satisfying pi > 0 for all i and �N
+i=1 pi =
+1. We assign µ(A) with µ(A) = 1 = �N
+i=1 pi. Let B ∈ Ek such that B = Ti1i2···ik(A).
+Let Ek = �
+B∈Ek B = �
+ij,j=1,2,...,k Ti1i2···ik(A) = A. Hence, we have µ(C) = 0 ∀ C
+with C ∩ A = ∅ and E = � Ek
+� Rn \Ek and µ(Rn) = 1. It follows that (Cf. [11,
+Proposition 1.7]) the definition of µ may be extended to all subsets of Rn so that
+µ becomes a measure. Now, we show that µ = �N
+i=1 piµ ◦ T −1
+i
+.
+Let Tj(A) be an arbitrary cylinder in the first stage. Then
+N
+�
+i=1
+piµ ◦ T −1
+i
+(Tj(A)) = pjµ(A) = pj,
+and from the construction of measure µ, we have µ(Tj(A)) = pj.
+Therefore,
+µ(Tj(A)) = �N
+i=1 piµ ◦ T −1
+i
+(Tj(A)) = pj for all j.
+Similarly, we have µ(B) =
+�N
+i=1 piµ ◦ T −1
+i
+(B) for all cylinders B ∈ Ek at any stage k. Thus, the proof is
+complete.
+□
+Remark 3.9. Recall that the collection of all Borel probability measures on Rn, de-
+noted by P(Rn), is a complete metric space with respect to the Monge-Kantorovich
+metric dH defined as
+dH(µ, ν) = sup
+�����
+�
+fdµ(x) −
+�
+fdν(x)
+���� : where f : Rn → R, Lip(f) ≤ 1
+�
+.
+Define a mapping M : P(Rn) → P(Rn) by M(µ) = �N
+i=1 piµ ◦ f −1
+i
+. Now, we have
+dH(M(µ), M(ν)) = sup
+����
+�
+fdM(µ)(x) −
+�
+fdM(ν)(x)
+��� : Lip(f) ≤ 1
+�
+= sup
+����
+N
+�
+i=1
+pi
+�
+fdµ ◦ f −1
+i
+(x) −
+N
+�
+i=1
+pi
+�
+fdν ◦ f −1
+i
+(x)
+��� : Lip(f) ≤ 1
+�
+.
+Hutchinson [9] showed that if all fi are contractions, then M is the contraction
+with respect to the Monge-Kantorovich metric.
+Here, a natural question arises
+whether M is Kannan with respect to the Monge-Kantorovich metric when all fi
+are Kannan. It is open for further investigation.
+4. Smooth fractal functions
+Let us denote the space of m-times continuously differentiable functions by
+Cm(I).
+Now, we define a new metric with the help of Hausdorff distance such
+as
+D(g, h) := max
+0≤k≤m Hθ(Gg(k), Gh(k)),
+where Hθ(Gg, Gh) denotes the Hausdorff distance induced from the metric dθ be-
+tween the graphs of f and g.
+Since Hθ(Gg, Gh) and ∥g − h∥∞ are equivalent,
+(Cm(I), D) will be a complete metric space.
+
+14
+SUBHASH CHANDRA, SAURABH VERMA, AND SYED ABBAS
+Theorem 4.1. Let g ∈ Cm(I), where m ∈ N. Suppose that Lj : I → Ij is affine
+map defined by Lj(x) = ajx + bj satisfying Lj(x1) = xj, Lj(xN) = xj+1, j ∈ J and
+Fj(x, y) = αjy + qj(x). Let q(k)
+j
+(x1) = g(k)(x1), q(k)
+j
+(xN) = g(k)(xN), j ∈ J, 0 ≤
+k ≤ m, and scaling factor αj satisfying αj < ak
+5 , where ak = min{ak
+j : j ∈ J}.
+Then T has a unique fractal function f ∗
+∆ ∈ Cm
+∗ (I). Moreover, dimH(Gr(f ∗
+∆)) =
+dimB(Gr(f ∗
+∆)) = 1.
+Proof. Let Cm
+∗ (I) = {g ∈ Cm(I) : h(k)(x1) = g(k)(x1), h(k)(xN) = g(k)(xN), 0 ≤
+k ≤ m}. Here, Cm
+∗ (I) is a closed subset of Cm(I). It can be seen that (Cm
+∗ (I), D) is
+a complete metric space. Define the RB operator T : Cm
+∗ (I) → Cm
+∗ (I) by
+(T g)(x) = αjf(L−1
+j (x)) + qj(L−1
+j (x)), x ∈ Ij, j ∈ J.
+It can be observed that T is well-defined. Let g, h ∈ Cm
+∗ (I),
+D(g, h) := max
+0≤k≤m Hθ(Gg(k), Gh(k))
+for each 0 ≤ k ≤ m, we have
+(T g)(k)(x) = a−k
+j [αjg(k)(L−1
+j (x)) + q(k)
+j
+(L−1
+j (x))].
+ˆF k
+j (x, y) = a−k
+j αjy + a−k
+j q(k)
+j
+(x) and ˆW k
+j (x, y) = (Lj(x), ˆF k
+j (x, y)). It can be seen
+that ˆ
+W k
+j are Kannan contractions as follows.
+D( ˆW k
+j (x1, y1), ˆW k
+j (x2, y2))
+≤ αj
+ak D((x1, y1), (x2, y2))
+≤ αj
+ak D((x1, y1), ˆ
+W k
+j (x1, y1)) + αj
+ak D( ˆ
+W k
+j (x1, y1), ˆ
+W k
+j (x2, y2)) + αj
+ak D( ˆ
+W k
+j (x2, y2), (x2, y2)).
+Hence, we obtain
+D( ˆW k
+j (x1, y1), ˆW k
+j (x2, y2)) ≤
+αj
+ak − αj
+[D((x1, y1), ˆW k
+j (x1, y1))+D((x2, y2), ˆ
+W k
+j (x2, y2))].
+Hence, ˆ
+W k
+j are Kannan contractions with contractivity factor
+αj
+ak−αj .
+Now, we show that T is Kannan w.r.t. metric D. That is
+D(T g, T h) ≤ γ′[D(g, T g) + D(h, T h)].
+Let (x, (T g)(k)(x)) ∈ G(T g)(k) and (y, (T h)(k)(y)) ∈ G(T h)(k). We have
+dθ
+�
+(x, (T g)(k)(x)), (y, (T h)(k)(y))
+�
+= dθ
+��
+x, ˆF k
+j (L−1
+j (x), g(k)(L−1
+j (x)))
+�
+,
+�
+y, ˆF k
+j (L−1
+j (y), h(k)(L−1
+j (y)))
+��
+= dθ
+��
+Lj(w), ˆF k
+j (w, g(k)(w))
+�
+,
+�
+Lj(z), ˆF k
+j (z, h(k)(z))
+��
+= dθ
+�
+ˆW k
+j (w, g(k)(w)), ˆ
+W k
+j (z, h(k)(z))
+�
+.
+
+CONSTRUCTION OF FRACTAL FUNCTIONS USING KANNAN MAPPINGS
+15
+Since ˆ
+W k
+j
+are Kannan contraction and by substituting again w = L−1
+j (x) and
+z = L−1
+j (y), we have
+dθ
+�
+(x, (T g)(k)(x)), (y, (T h)(k)(y))
+�
+≤
+αj
+ak − αj
+�
+dθ
+�
+(L−1
+j (x), g(k)(L−1
+j (x))), ˆ
+W k
+j (L−1
+j (x), g(k)(L−1
+j (x)))
+�
++ dθ
+�
+(L−1
+j (y), h(k)(L−1
+j (y))), ˆW k
+j (L−1
+j (y), h(k)(L−1
+j (y)))
+��
+=
+αj
+ak − αj
+�
+dθ
+�
+L−1
+j (x), g(k)(L−1
+j (x)), (x, (T g)(k)(x))
+�
++ dθ
+�
+L−1
+j (y), h(k)(L−1
+j (y)), (y, (T h)(k)(y))
+��
+≤
+αj
+ak − αj
+�
+dθ
+�
+(L−1
+j (x), g(k)(L−1
+j (x))), (y, (T g)(k)(y))
+�
++ dθ
+�
+(x, (T g)(k)(x)), (y, (T h)(k)(y))
+�
++ dθ
+�
+(L−1
+j (y), h(k)(L−1
+j (y))), (x, (T h)(k)(x))
+�
++ dθ
+�
+(x, (T g)(k)(x)), (y, (T h)(k)(y))
+��
+.
+On taking infimum both sides, we have
+inf
+y∈I dθ
+�
+(x, (T g)(k)(x)), (y, (T h)(k)(y))
+�
+≤
+αj
+ak − αj
+�
+inf
+y∈I dθ
+�
+(L−1
+j (x), g(k)(L−1
+j (x))), (y, (T g)(k)(y))
+�
++ inf
+y∈I dθ
+�
+(L−1
+j (y), (h)(k)(L−1
+j (y))), (x, (T h)(k)(x))
+�
++ 2 inf
+y∈I dθ
+�
+(x, (T g)(k)(x)), (y, (T h)(k)(y))
+��
+.
+That is,
+(1 −
+2αj
+ak − αj
+) inf
+y∈I dθ
+�
+(x, (T g)(k)(x)), (y, (T h)(k)(y))
+�
+≤
+αj
+ak − αj
+�
+inf
+y∈I dθ
+�
+(L−1
+j (x), g(k)(L−1
+j (x))), (y, (T g)(k)(y))
+�
++ inf
+y∈I dθ
+�
+(L−1
+j (y), h(k)(L−1
+j (y))), (x, (T h)(k)(x))
+��
+.
+By taking supremum over x ∈ I, we have
+max
+0≤k≤m Hθ(G(T g)(k), G(T h)(k)) ≤
+αj
+ak − 3αj
+[ max
+0≤k≤m Hθ(Gg(k), G(T g)(k))
++ max
+0≤k≤m Hθ(Gh(k), G(T h)(k))]
+That is
+D(T g, T h) ≤ γ′[D(g, T g) + D(h, T h)],
+where γ′ =
+αj
+ak−3αj < 1
+2. Hence, T is Kannan contrcation with contractivity factor
+γ′ =
+αj
+ak−3αj < 1
+2. By Theorem 2.8 , T has a unique fractal function f ∗
+∆ ∈ Cm
+∗ (I) and
+obeys the equation 3.1. We know that any continuous function with the bounded
+derivative is of bounded variation. This result with [15, Theorem 1.3] yields
+dimH(Gr(f ∗
+∆)) = dimB(Gr(f ∗
+∆)) = 1.
+
+16
+SUBHASH CHANDRA, SAURABH VERMA, AND SYED ABBAS
+Hence, the proof is complete.
+□
+Remark 4.2. [5] The relationship between the Heisenberg and Euclidean geometry
+on H = R3 is rather intricate.
+The Heisenberg-Hausdorff dimension is always
+greater than or equal to its Euclidean counterpart. The Hausdorff dimension of
+(R3, dH) is equal to 4 (in fact, balls in the metric dH have a measure proportional
+to the fourth power of their radius). This implies, for instance, that the Heisenberg
+metric dH cannot be locally bi-Lipschitz equivalent with any Riemannian metric,
+particularly with the Euclidean metric dE.
+Remark 4.3. From the above remark, we can conclude that if two metrics are
+topologically equivalent, that does not imply that the dimension of the graph of any
+function can be equal, but if metrics are metrically equivalent, then the Hausdorff
+dimension will be equal, but the Hausdorff measure need not be equal; see the
+following
+Let δ > 0. Then
+Hs
+δ,d2(E) = inf
+� ∞
+�
+i=1
+diamd2(Fi)s : {Fi} is a δ − cover of E
+�
+≤ inf
+� ∞
+�
+i=1
+cs
+2diamd1(Fi)s : {Fi} is a δ − cover of E
+�
+= cs
+2Hs
+δ,d1(E).
+Similarly, cs
+1Hs
+δ,d1(E) ≤ Hs
+δ,d2(E). Therefore,
+cs
+1Hs
+δ,d1(E) ≤ Hs
+δ,d2(E) ≤ cs
+2Hs
+δ,d1(E) holds for all δ > 0.
+As δ → 0+, we have
+cs
+1Hs
+d1(E) ≤ Hs
+d2(E) ≤ cs
+2Hs
+d1(E).
+Now, using the definition of the Hausdorff dimension and the above inequality, we
+get
+dimH,d1(E) = inf{s : Hs
+δ,d1(E) = 0} = inf{s : Hs
+δ,d2(E) = 0} = dimH,d2(E),
+completing the reamark.
+5. Graph of Kannan fractal functions
+Here, we have the functional equation
+Fj(x, y) = T (y) + qj(x), j ∈ J,
+and the self-referential equation is
+f(Lj(x)) = T (f(x)) + qj(x), x ∈ Ij, j ∈ J.
+We choose qj(x) = cjx + dj satisfying the join-up conditions such as
+T (y1) + cjx1 + dj = yj and T (yN) + cjxN + dj = yj+1, j ∈ J.
+So, we have
+cj = (yj − yj+1) − (T (y1) − T (yN))
+(x1 − xN)
+
+CONSTRUCTION OF FRACTAL FUNCTIONS USING KANNAN MAPPINGS
+17
+0
+0.2
+0.4
+0.6
+0.8
+1
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1
+Figure 1.
+0
+0.2
+0.4
+0.6
+0.8
+1
+0
+0.2
+0.4
+0.6
+0.8
+1
+1.2
+1.4
+1.6
+1.8
+Figure 2.
+
+18
+SUBHASH CHANDRA, SAURABH VERMA, AND SYED ABBAS
+0
+0.2
+0.4
+0.6
+0.8
+1
+-1
+-0.5
+0
+0.5
+1
+1.5
+Figure 3.
+0
+0.2
+0.4
+0.6
+0.8
+1
+-1
+-0.5
+0
+0.5
+1
+1.5
+Figure 4.
+
+CONSTRUCTION OF FRACTAL FUNCTIONS USING KANNAN MAPPINGS
+19
+and
+dj = yj − T (y1) − (yj − yj+1) − (T (y1) − T (yN))
+(x1 − xN)
+x1.
+The initial data is taken as follows for Figure 1 and Figure 2, respectively.
+{(0, 0.9), (0.1, 0.5), (0.2, 0.75), (0.3, 0.95), (0.4, 1.25), (0.5, 1.5), (0.6, 1.6), (0.7, 1.2),
+(0.8, 1.8), (0.9, 1.9), (1.0, 2.0)} and {(0, 0.6), (0.1, 1.4), (0.2, 0.9), (0.3, 1.2), (0.4, 1.8),
+(0.5, 1.3), (0.6, 0.9), (0.7, 1.75), (0.8, 0.85), (0.9, 1.75), (1.0, 2.0)}.
+For Figure 3 and Figure 4, we consider qj(x) = x2 + cjx + dj satisfying the join-up
+conditions such as
+T (y1) + x2
+1 + cjx1 + dj = yj and T (yN) + x2
+n + cjxN + dj = yj+1, j ∈ J.
+So, we have
+cj = (yj − yj+1) − (T (y1) − T (yN)) − (x2
+1 − x2
+N)
+(x1 − xN)
+and
+dj = yj − T (y1) − (yj − yj+1) − (T (y1) − T (yN)) − (x2
+1 − x2
+N)
+(x1 − xN)
+x1.
+The initial data is taken as follows for Figure 3 and Figure 4, respectively.
+{(0, 0.75), (0.1, 1.4), (0.2, 0.65), (0.3, 1.55), (0.4, 1.25), (0.5, 1.0), (0.6, 1.75), (0.7, 1.3),
+(0.8, 2.0), (0.9, 1.15), (1.0, 0.95)} and {(0, 0.5), (0.1, 1.5), (0.2, 1.75), (0.3, 0.95), (0.4, 1.0),
+(0.5, 1.8), (0.6, 1.2), (0.7, 1.6), (0.8, 1.4), (0.9, 0.85), (1.0, 1.4)}.
+Acknowledgements. The first author’s work is financially supported by the CSIR,
+India, with grant number 09/1058(0012)/2018-EMR-I.
+References
+1. G. Beer, Metric spaces on which continuous functions are uniformly continuous and Hausdorff
+distance, Proc. Amer. Math. Soc. 95 (1985) 653-658.
+2. M. F. Barnsley, Fractal functions and interpolation, Constr. Approx. 2 (1986) 303-329.
+3. M. F. Barnsley, Fractal Everywhere, Academic Press, Orlando, Florida, 1988. SIAM J. Math.
+Anal. 20(5) (1989) 1218–1242.
+4. M. F. Barnsley and P. R. Massopust, Bilinear fractal interpolation and box dimension, J.
+Approx. Theory 192 (2015) 362–378.
+5. Z. M. Balogh and J. T. Tyson, Hausdorff dimension of self-similar and self-affine fractals the
+Heisenberg group, Proc. London Math. Soc. (3) 91 (2005) 153-183.
+6. D. Cheng, Z. Liand and B. Selmi, Upper metric mean dimensions with potential on subsets.
+Nonlinearity. 34 (2021) 852–867.
+7. S. Chandra and S. Abbas, On fractal dimensions of fractal functions using functions spaces,
+Bull. Aust. Math. Soc. (2022) 1-11.
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+maps, Reich maps, Chatterjea type maps, and related results, Journal of Fixed Point Theory
+and Applications 19, no. 4 (2017), 2271-2285.
+9. J. E. Hutchinson, Fractals and self-similarity, Indian Univ. Math. J. 30 (1981) 713-747.
+10. Janos Ludvik, On Mappings Contractive in the Sense of Kannan, Proceedings of the American
+Mathematical Society 61, no. 1 (1976): 171-75.
+11. K. J. Falconer, Fractal Geometry: Mathematical Foundations and Applications, John Wiley
+Sons Inc., New York, 1999.
+12. S. Jha and S. Verma, Dimensional analysis of α-fractal functions, Results in Mathematics 76
+(4) (2021) 1-24.
+13. R. Kannan, Some results on fixed points, Bull. Calcutta Math. Soc. 60 (1968) 71-76.
+14. R. Kannan, Some results on fixed points II, Amer. Math. Monthly 76 (1969) 405-408.
+
+20
+SUBHASH CHANDRA, SAURABH VERMA, AND SYED ABBAS
+15. Y. S. Liang, Box dimensions of Riemann-Liouville fractional integrals of continuous functions
+of bounded variation, Nonlin. Anal. 72(11) (2010) 4304-4306.
+16. P. R. Massopust, Fractal Functions, Fractal Surfaces, and Wavelets. 2nd ed., Academic Press,
+San Diego, 2016.
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+18. M. A. Navascu´es, Fractal approximation, Complex Anal. Oper. Theory 4(4) (2010) 953-974.
+19. A. Priyadarshi, Lower bound on the Hausdorf dimension of a set of complex continued frac-
+tions. J. Math. Anal. Appl. 449(1), (2017) 91–95.
+20. S.I. Ri, A new idea to construct the fractal interpolation function, Indag. Math. 29(3) (2018)
+962-971.
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+22. H.J. Ruan, W.-Y. Sub and K. Yao, Box dimension and fractional integral of linear fractal
+interpolation functions, J. Approx. Theory 161 (2009) 187-197.
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+139–144.
+24. B. Selmi, Multifractal dimensions for projections of measures. Bol. Soc. Paran. Mat. 40 (2022)
+1-15.
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+of invariant sets, Trans. Amer. Math. Soc. 364(2) (2012) 1029-1066.
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+analytical properties, J. Approx. Theory 175 (2013) 1-18.
+School of Mathematical and Statistical Sciences, Indian Institute of Technology
+Mandi, Kamand (H.P.) - 175005, India
+Email address: sahusubhash77@gmail.com
+Department of Applied Sciences, IIIT Allahabad, Prayagraj-211015, India
+Email address: saurabh331146@gmail.com
+School of Mathematical and Statistical Sciences, Indian Institute of Technology
+Mandi, Kamand (H.P.)- 175005, India
+Email address: sabbas.iitk@gmail.com
+

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+High-fidelity imaging of a band insulator in a three-dimensional optical lattice clock
+William R. Milner,1, ∗ Lingfeng Yan,1 Ross B. Hutson,1 Christian Sanner,1 and Jun Ye1, †
+1JILA, NIST and University of Colorado, 440 UCB, Boulder, Colorado 80309, USA
+We report on the observation of a high-density, band insulating state in a three-dimensional optical lattice
+clock. Filled with a nuclear-spin polarized degenerate Fermi gas of 87Sr, the 3D lattice has one atom per site in
+the ground motional state, thus guarding against frequency shifts due to contact interactions. At this high density
+where the average distance between atoms is comparable to the probe wavelength, standard imaging techniques
+suffer from large systematic errors. To spatially probe frequency shifts in the clock and measure thermodynamic
+properties of this system, accurate imaging techniques at high optical depths are required. Using a combination
+of highly saturated fluorescence and absorption imaging, we confirm the density distribution in our 3D optical
+lattice in agreement with a single spin band insulating state. Combining our clock platform with this high filling
+fraction opens the door to studying new classes of long-lived, many-body states arising from dipolar interactions.
+Optical
+lattice
+clocks
+integrate
+quantum
+many-body
+physics and precision metrology to achieve state-of-the-art
+measurement precision [1–5]. To advance clock performance,
+one wishes to probe as many atoms as feasible for the longest
+possible coherence time. Improvements in both precision and
+accuracy of optical lattice clocks, with increased atom num-
+bers, have been enabled by the development of high-fidelity,
+microscopic imaging of the atomic cloud to spatially resolve
+clock shifts [6, 7]. The combination of high density and long
+coherence time will allow characterization of novel systematic
+effects such as that arising from dipolar interactions between
+atoms on neighboring lattice sites [8–11]. Lattice thermom-
+etry [12] and studies of novel physics such as SU(N) mag-
+netism [13, 14] will also benefit from accurate imaging at high
+density where these phenomena emerge.
+To optimize atom number while minimizing interaction-
+related dephasing, a clock platform based on a 3D lattice
+geometry and Fermi-degenerate matter has been developed
+[7, 15]. Following nuclear spin polarization [16, 17], the Pauli
+exclusion principle mandates there is at most one atom per lat-
+tice site in the ground motional state. To ensure this ground
+state motional occupation during lattice loading we operate
+with kBT < kBTF < ℏωbg, where T, TF , ℏωbg refers to the
+atomic temperature, Fermi temperature and lattice bandgap
+respectively [18]. At the highest density affordable with one
+fermion per lattice site, this system realizes an insulating state
+of matter where tunnelling is suppressed [15, 19]. Combining
+this high-density system with spin-orbit coupling generated
+from clock addressing will enable exploring cluster state gen-
+eration and tunable spin models [20, 21].
+Differential frequency shifts across the optical lattice en-
+coding potential systematic effects can be spatially resolved
+by combining in situ imaging and narrow-line clock spec-
+troscopy [6]. To extract these frequency shifts, two subse-
+quent images of the ground and excited state density distribu-
+tions are required. Thus for our clock platform, accurate in
+situ imaging at high density is imperative. In our lattice where
+the average distance between atoms (407 nm) is comparable
+to the probe wavelength (461 nm), imaging with a weak, reso-
+nant probe is strongly perturbed. Both collective effects medi-
+ated by dipolar interactions [22] and systematic defects such
+as lensing of the probe beam [23, 24] introduce errors to the
+reconstructed density distribution at high density.
+To mitigate these systematic effects, different techniques
+can be used to reduce the absorption cross section and make
+the cloud ”optically thin”. These techniques can be broadly
+divided into two categories: dispersive imaging at large de-
+tuning from resonance [25–27] and saturated imaging at high
+intensity [28–31]. For dispersive imaging extracting informa-
+tion about the atomic density often requires spatially filtering
+the scattered and unscattered light in the Fourier plane of the
+imaging system, demanding precise fabrication and alignment
+of custom optics. Additionally, careful studies of dispersive
+imaging show that residual systematic effects at finite detun-
+ing are non-negligible and can be addressed using differential
+measurement schemes at opposite detuning [32]. To address
+these imaging errors in this work, we use both highly saturated
+fluorescence and absorption imaging.
+In this Letter, we report on the observation of a band in-
+sulating state in our 3D optical lattice clock. Using highly
+saturated imaging to mitigate imaging errors, with a satura-
+tion parameter far greater than the optical depth, we accu-
+rately confirm the density distribution in our 3D optical lattice
+in good agreement with thermodynamic calculation. We ex-
+tend previous work using high intensity fluorescence imaging
+[28], confirming the accuracy of this imaging technique in a
+new high density regime with a degenerate Fermi gas of 87Sr
+[16, 33]. With atomic densities as high as 6×1014 atoms/cm3,
+we observe systematic agreement with atom counts obtained
+via time-of-flight absorption imaging and identify the range
+where the extracted atomic density distribution is not blurred
+by our imaging pulse.
+Our high intensity imaging scheme is outlined in Fig. 1.
+The combination of atomic level structure and relatively large
+mass of 87Sr is particularly well suited for our imaging tech-
+nique, providing a cycling transition with a large scattering
+rate while avoiding significant motional effects from the imag-
+ing pulse.
+The transition from 1S0 to 1P1 with linewidth
+Γ = 2π × 30.5 MHz provides a large photon scattering rate
+with minimal depumping to dark states during the imaging
+time [34]. During a 1 µs pulse at full saturation about 100
+photons per atom are scattered and the atoms accelerate at
+a =
+ℏkΓ
+2m where k is the imaging light wavenumber and m
+is the atomic mass. The net momentum transfer amounts to
+arXiv:2301.03343v1  [physics.atom-ph]  9 Jan 2023
+
+2
+FIG. 1. Schematic of our clock platform. Vertical and horizontal
+imaging systems with numerical apertures of 0.2 and 0.1 respectively
+provide measurements of the 2D density distribution ˜n. Accounting
+for the lattice spacing a = 407 nm, ˜na2 is determined from highly
+saturated absorption imaging. To mitigate imaging errors, the atoms
+are highly saturated and each scatters photons with a maximum rate
+of Γ/2. Measurements from our high resolution imaging system in-
+tegrated along gravity are presented in panel (a), where the density
+distribution is extracted for thermodynamic modeling. Images from
+the horizontal imaging system in panel (b) are just used to determine
+our atom cloud aspect ratio for our inverse Abel transform.
+a Doppler shift of kaτ = 2.8 MHz which is much less than
+the transition linewidth Γ/2π.
+Finally, the linear displace-
+ment for a 1 µs pulse at full saturation is just aτ 2
+2
+= 0.6 µm.
+This linear displacement and corresponding Doppler shift can
+be largely cancelled in fluorescence imaging by retroreflect-
+ing the incident beam. The spread in transverse position due
+to random momentum transfer from spontaneous emission
+is
+ℏk
+6mt3/2�
+Γ/2 < 0.1 µm over a 1 µs pulse duration and
+small compared to our 1.3 µm imaging resolution [35]. Using
+highly saturated absorption imaging, we measure the column
+density distribution ˜n in our optical lattice in Fig. 1(a). Ac-
+counting for the lattice spacing a = 407 nm corresponding
+to the 87Sr magic wavelength at 813 nm, the scaled column
+density ˜na2 is plotted.
+Saturated absorption and fluorescence imaging are benefi-
+cial in comparison to standard imaging techniques in a num-
+ber of ways. In this highly saturated regime the scattering rate
+is largely immune to beam intensity, frequency, and pointing
+fluctuations. Given the saturation intensity Isat = 40 mW/cm2
+for the imaging transition, a Gaussian probe beam with 20
+mW of optical power and a 100 µm waist corresponds to a
+peak intensity of I ∼ 3000 Isat, within the typical constraints
+of a standard imaging laser system. Given that the probe beam
+is attenuated through the atom cloud, a saturation parameter
+I/Isat much greater than the optical depth is required to fully
+saturate the imaging transition. We note parallels between flu-
+orescence and absorption imaging at high saturation. In both
+cases, the extracted atom number is determined by a single
+variable. For fluorescence imaging, this corresponds to the
+number of collected photons per atom and for saturated ab-
+sorption imaging the number of missing photons per atom in
+the probe beam. Thus, both fluorescence and saturated ab-
+sorption imaging can be calibrated via a single absolute atom
+number measurement. For images in our 3D lattice, we de-
+termine our atom number via clock excitation fraction fluctu-
+ations arising from quantum projection noise (QPN) [36, 37].
+For fluorescence imaging, only a single image of collected
+fluorescence in an arbitrary direction is required, minimizing
+fringing and simplifying image processing substantially. Flu-
+orescence imaging also avoids limited dynamic range issues
+suffered from high intensity absorption imaging. Strategies
+such as multiple measurements at varying intensity to deter-
+mine the atomic density in different regions of the cloud may
+be taken to confront this issue [30, 31]. The primary disad-
+vantage of fluorescence imaging in comparison to absorption
+imaging is that the signal-to-noise is generally worse [37]. To
+optimize signal-to-noise ratio (SNR) in fluorescence imaging,
+the photon collection efficiency and therefore the numerical
+aperture (NA) of the imaging system, must be maximized. In
+our experiment, the vertical and horizontal imaging systems
+have numerical apertures of 0.2 and 0.1, corresponding to col-
+lection efficiencies of approximately 1 and 0.2 percent. Al-
+ternatively, if spatial resolution is not required then the pulse
+duration can be extended enhancing the number of detected
+photons.
+To motivate the development of our high intensity imaging
+technique, systematic errors associated with standard in situ
+imaging techniques at high density are presented in Fig. 2.
+Absorption imaging at I ∼ Isat and high intensity fluorescence
+imaging are presented side-by-side for comparison. To study
+these systematic errors at high density, we prepare a sample
+with optical depth > 200 by producing a degenerate Fermi
+gas with 10 nuclear spin components, ≈ 2 × 105 atoms and a
+T/TF of approximately 0.1 in a crossed dipole trap. The errors
+associated with low intensity absorption imaging can be seen
+twofold. First, the reconstructed optical depth from absorp-
+tion detection in the upper left panel is far too low, two orders
+of magnitude less than the expected value of ∼ 200. This
+erroneously low optical depth is attributed to effects such as
+enhanced forward emission and lensing of probe light [23].
+Secondly, the reconstructed optical depth in the upper right
+panel increases after a 500 µs time-of-flight expansion con-
+clusively demonstrating the density dependence of these ob-
+served systematic errors.
+In comparison, saturated fluorescence imaging yields a far
+larger reconstructed optical depth and diffuses following ex-
+pansion as expected. We compare this reconstructed 2D den-
+sity distribution with the expected distribution corresponding
+to a Fermi gas. Using independently measured experimental
+values, we calculate this distribution with no free parameters
+[38]. The total atom number and reduced temperature T/TF
+are determined from time-of-flight absorption imaging at low
+density with an optical density ∼ 1. The trapping frequencies
+are extracted from parametric confinement modulation. Using
+
+皖A·23
+FIG. 2. A comparison of high intensity fluorescence and standard ab-
+sorption imaging (I ∼ Isat) at optical depths exceeding 200 in our
+highly degenerate Fermi gas is shown. In situ absorption imaging at
+low intensity yields strikingly erroneous measurements at high den-
+sity. The calculated 2D Fermi gas distribution according to our ex-
+perimental parameters is shared for comparison in qualitative agree-
+ment.
+these parameters, we calculate both an in situ and 500 µs time-
+of-flight Fermi gas profile for comparison with our measure-
+ments. We observe qualitative agreement between measure-
+ment and calculation at these extremely high optical depths.
+Intrigued by the measurements presented in Fig. 2, we un-
+dertake a quantitative study on the fidelity of our saturated
+imaging technique. We present a calibration method for flu-
+orescence detection, using the total number of collected fluo-
+rescence photons for comparison with an accurate atom num-
+ber reference. Absorption imaging at low density following
+time-of-flight expansion serves as an appropriate calibration.
+Following expansion for 7 ms, the optical depth is ∼ 1 and
+systematic imaging errors can be safely ignored. To inde-
+pendently calibrate the atom number in our 10 spin Fermi
+gas, we prepare a thermal sample and use measured density
+fluctuations to determine the effective absorption cross sec-
+tion [39–41]. In Fig. 3(a) we ensure this calibration shows
+systematic agreement with atom numbers between approxi-
+mately 1 × 105 and 4 × 105, varied by increasing our final
+evaporation trap depth. For the 3 µs pulse duration used, the
+fitted calibration is in reasonable agreement with calculation
+using the measured quantum efficiency and imaging system
+numerical aperture [37]. To ensure that the imaging transition
+is fully saturated, the laser intensity at 1 µs pulse duration is
+increased until the collected photon number plateaus, as seen
+in the figure inset.
+To perform accurate spatially resolved measurements, we
+must also determine the blurring induced by our imaging
+pulse.
+Just as collective effects introduce errors to the re-
+FIG. 3. (a) Calibration method for in situ fluorescence detection us-
+ing atom counts from time-of-flight absorption imaging. Collected
+photon counts from both the vertical and horizontal imaging systems
+are plotted, with solid and dashed lines representing fits to the hori-
+zontal and vertical measurements respectively. Inset: Collected pho-
+ton count with vertical imaging system as a function of I/Isat at 1
+µs pulse duration. (b) Peak column density as a function of fluo-
+rescence pulse duration. Measurements are normalized by 1.9×1011
+atoms/cm2, the column density at the shortest pulse duration of 500
+ns. Images at 500 ns and 2 µs in inset are plotted for comparison.
+The error bars denote the standard error of the mean.
+constructed density distribution, any systematic changes to ˜n
+introduced by our imaging pulse must be determined. To cal-
+ibrate this blurring in Fig. 3(b), we extend the fluorescence
+pulse duration and examine the peak column density as atoms
+diffuse. The inset shows averaged images from 500 ns and
+2 µs pulse durations. We note that we observe no atom loss
+or molecular formation over the full 2 µs range, confirmed by
+the detected photon count increasing linearly with pulse dura-
+tion. To minimize blurring, we carefully retroreflect our probe
+beam by optimizing the backcoupled light through the probe
+optical fiber. At pulse durations up to 1 µs, we confirm that
+the peak column density decreases by < 5%. [37].
+Motivated by the calibration reported in Fig. 3, we directly
+determine the 3D density distribution in a deep optical lattice
+via saturated in situ absorption imaging. We form a cubic lat-
+tice with trap depths of approximately 60, 70, and 50 Er in
+three orthogonal directions, where Er is the lattice photon re-
+coil energy ≈ h × 3.5 kHz. Following forced evaporation
+with 10 nuclear spin states we spin polarize using a focused
+beam detuned from the 3P1 intercombination line to form a
+state-dependent potential, removing nearly all but the mF = -
+9/2 atoms [16, 17]. Clock spectroscopy confirms ≈ 90% spin
+purity. An additional step of spin purification is applied by
+
+4
+FIG. 4. (a) The three-dimensional density distribution and the corre-
+sponding lattice filling fraction are determined from in situ absorp-
+tion image in Fig. 1(a) and the use of an inverse Abel transformation.
+(b) A linecut along z = 0 provides the data points in circle. Errorbars
+are both the statistical uncertainty of the Abel transformation and
+atom number uncertainty added in quadrature. We start with a pre-
+diction based on HTSE calculation, using independently measured
+values for the temperature, atom number, and harmonic confinement.
+The best fit to the data results in a 10% reduction of the measured as-
+pect ratio ωy/ωx and 5% reduction of the measured T/TF . The red
+line captures this fit, with temperature uncertainty in the shaded band.
+The blue dashed line is a fit to Gaussian in qualitative disagreement
+with na3.
+coherently driving the mF = -9/2 atoms into the excited clock
+state and removing any residual spins with a resonant imaging
+pulse. Absorption imaging directly provides us with the col-
+umn density distribution ˜n, integrated through the vertical axis
+along gravity as depicted in Fig. 1(a). Based on our Fig. 3(b)
+analysis, we choose a pulse duration of 1 µs to minimize blur-
+ring and a saturation intensity of 54(4), substantially larger
+than peak optical density of ∼ 15. To spatially probe the band
+insulator plateau we use an imaging magnification of 38.8 to
+achieve an effective pixel size of 412 nm, roughly equal to the
+lattice constant a = 407 nm. We note that our effective pixel
+size is smaller than our optical resolution of 1.3 µm, thus our
+imaging system is optically oversampled. To extract the 3D
+density distribution, we use an inverse Abel transform [42].
+Given our vertical imaging is not along an axis of cylindri-
+cal symmetry, n must be appropriately scaled by the aspect
+ratio of the spatial density distribution [37]. The aspect ra-
+tio is independently calibrated using the absorption imaging
+measurement in Fig. 1(b).
+At this high magnification, the SNR in fluorescence imag-
+ing for a 1 µs pulse duration is limited by a combination of
+read noise and photon shot noise. We found that even after
+extensive averaging the extracted 3D density distribution us-
+ing an inverse Abel transform was sensitive to small fluctua-
+tions in ˜n. Thus, saturated absorption imaging with a superior
+SNR provides a more robust technique to characterize the 3D
+density distribution. This extracted 3D density distribution is
+plotted in Fig. 4(a).
+To judge the fidelity of our measured 3D density distri-
+bution, we compare the line cut at z = 0 with calculation
+in Fig. 4(b). To estimate the density distribution, we use a
+High Temperature Series Expansion (HTSE) calculation in
+the atomic limit [12, 14, 37, 43]. The ingredients of this cal-
+culation include values for the atomic temperature, harmonic
+confinement, and total atom number. Given the density dis-
+tribution only depends on the ratio of the respective harmonic
+confinements, the measured aspect ratios from Fig. 1 are used
+for our HTSE calculation. The total atom number N is de-
+termined from quantum projection noise measurements [37].
+To estimate the temperature including heating during lattice
+loading, we measure the reduced temperature T/TF in time-
+of-flight after a round-trip from the lattice back to the dipole
+trap and determine an entropy-per-particle increase of 0.25(6)
+kB. Inferring an entropy increase of 0.13(3) kB in a single
+lattice loading sequence, we estimate a T/TF of 0.165(7).
+Although we did not perform a cross-dimensional thermaliza-
+tion measurement to directly verify thermal equilibrium, the
+uncertainty in our temperature is included in the shaded band
+of the HTSE calculation in Fig. 4(b) [44, 45]. We note that the
+extended plateau region is larger than our 1.3 µm imaging res-
+olution. To further quantify the imaging fidelity, we compare
+na3 to a Gaussian fit in clear disagreement with data.
+In conclusion, we report on the observation of a spin-
+polarized, band insulating state in our 3D optical lattice clock.
+This has been enabled by characterizing saturated in situ
+imaging techniques to accurately determine our density dis-
+tribution. Broadly, the saturated imaging techniques in this
+work will be applicable for studies of SU(N) magnetism and
+thermodynamics in the Mott-insulating regime [46, 47]. With
+the high filling fraction demonstrated in this work, many-body
+states arising from dipolar interactions can be generated be-
+tween atoms on neighboring lattice sites [8, 9].
+Acknowledgement. We thank D. Kedar for maintain-
+ing the ultrastable clock laser used in this work and A. Aep-
+pli, K. Kim, J. M. Robinson, M. Miklos, and Y. M. Tso for
+useful discussions. We thank K. Kim, N. D. Oppong, and
+L. Sonderhouse for careful reading of the manuscript and for
+providing insightful comments. Funding for this work is pro-
+vided by NSF QLCI OMA-2016244, DOE Center of Quan-
+tum System Accelerator, DARPA, AFOSR, V. Bush Fellow-
+ship, NIST, and NSF Phys-1734006.
+
+5
+∗ william.milner@colorado.edu
+† ye@jila.colorado.edu
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+Supplemental material to
+High-fidelity imaging of a band insulator in a three-dimensional optical lattice clock
+Density diffusion
+Here we provide supplemental analysis to the data pre-
+sented in Fig. 3(b). In panel A of Fig. S1, we plot the in-
+tegrated counts along the x axis of each image. We see an
+asymmetry emerge along the direction of the probe beam as
+the pulse duration is extended. This asymmetry suggests that
+the observed density diffusion may arise from inhomogeni-
+ety between the incident and retroreflected beams. While the
+power is certainly mismatched, this could also be due to ei-
+ther imperfect spatial alignment or mode mismatch given the
+divergence of the probe beam.
+We also plot the total counts in each image as a function of
+pulse duration in panel B. The linear character of the counts
+over the full pulse duration range suggests that we do not ob-
+serve appreciable atom loss or pumping to dark states. The
+counts at each pulse duration are normalized to the counts at
+500 ns. The inset shows the Gaussian RMS width of the cloud
+as a function of pulse duration.
+Signal-to-noise comparison
+In the main text of the paper we refer to both saturated ab-
+sorption and fluorescence imaging. We provide a quantita-
+tive comparison of the signal-to-noise ratio (SNR) between
+the two techniques here. We express our signal-to-noise for a
+detection pixel in terms of the normalized variance V(N)/N,
+where N denotes the number of atoms within the respective
+detection region. For fluorescence imaging the SNR is simply
+determined by the shot noise associated with the number of
+detected photons. To calculate the total atom number, we first
+convert the fluorescence counts detected on our camera to the
+number of collected photons. Then, using the collection effi-
+ciency of our imaging system and scattering rate of our atomic
+transition we determine the conversion of detected photons
+per atom. On our CCD camera, we measure na counts in a
+given pixel. Using the quantum efficiency q of the imaging
+system, and the camera conversion gain g in units of counts
+per photo electron, we infer na
+qg photons. At full saturation,
+the atomic scattering rate is Γ
+2 and the number of photons scat-
+tered per atom is Psc = Γ
+2 × τ, where τ is the pulse duration.
+Finally, we denote the collection efficiency as Y , determined
+by the numerical aperture of our imaging system and by ra-
+diation pattern anisotropies. Combining terms, the total atom
+number is N =
+na
+gqY Psc . Using error propagation, we deter-
+mine the variance VF l(N).
+VF l(N) =
+� ∂N
+∂na
+�2
+V(na) =
+�
+1
+gqY Psc
+�2
+gna
+(1)
+FIG. S1. Panel (a) shows the integrated counts from the images in
+Fig. 3(b) of the main text along the x axis as a function of pulse du-
+ration. The total counts at each pulse duration is plotted in panel (b),
+normalized by the counts at 500 ns. Given the detected photon count
+increases linearly with pulse duration, we observe minimal atom loss
+or molecular formation over the full 2 µs range. The inset shows the
+Gaussian RMS width of the cloud as a function of pulse duration.
+Here, we have used the fact that the distribution of gener-
+ated photo electrons ne is binomial. Thus, V(na) = V(g ×
+ne) = g2V(ne) = g2ne = gna. Combining terms:
+VF l(N)/N =
+1
+qY Psc
+(2)
+The SNR associated with absorption imaging is more com-
+plicated given the formula for the atom number in Eq. 3 has
+both logarithmic and linear terms and involves two images na
+and nb with and without atoms present. Here, A and σ0 refer
+to the effective pixel size accounting for the imaging system
+magnification and effective atomic absorption cross section,
+respectively. Similar to fluorescence imaging, an appropriate
+error propagation of the na and nb terms determines Eq. 4 and
+Eq. 5. We summarize the formulas here and point a reader to
+reference [1] for a full derivation.
+arXiv:2301.03343v1  [physics.atom-ph]  9 Jan 2023
+
+2
+N = A
+σ0
+log( nb
+na
+) +
+2
+Γτgq (nb − na)
+(3)
+VAbs(N) = g ˜A2( 1
+na
++ 1
+nb
+) + g ˜B2(na + nb) + 4g ˜A ˜B (4)
+˜A = A
+σ0
+, ˜B =
+2
+qgτΓ
+(5)
+We compare the different techniques in Fig. S2 using the
+experimentally relevant parameters for our imaging system.
+In both cases, a 1 µs resonant pulse is used with a numerical
+aperture of 0.2 and a quantum efficiency of 85%. For the fluo-
+rescence SNR in blue, the transition is assumed to be fully sat-
+urated and scatters photons with a rate of Γ/2. For the I/Isat
+= ∼ 55 we use for our inverse Abel measurements, the SNR
+in absorption imaging is superior to fluorescence imaging in
+regions where the column density is higher than 2 atoms/a2.
+Particularly given our peak density of ˜na2 = ∼ 20 in Fig. 1(a),
+absorption imaging provides a better SNR in the regions of
+high density where we extract our peak filling fraction. At a
+critical OD of 0.17, fluorescence detection under our experi-
+mental parameters provides a superior SNR at all imaging in-
+tensities. We note these calculations neglect technical noise,
+in particular camera readout noise, which can be accounted
+for by offsetting V(na) accordingly. This contribution will
+disproportionately reduce the SNR of fluorescence imaging,
+as the fluorescence counts are substantially lower than the ab-
+sorption counts.
+To probe fine spatial details in our atomic cloud, an imag-
+ing resolution smaller than the length scale of these spatial
+features is required. To achieve this condition, a sufficiently
+large numerical aperture imaging system must be utilized and
+aberrations must be minimized. In this case, the imaging res-
+olution is fundamentally limited by diffraction. We verified
+the diffraction-limited performance of our NA = 0.2 objec-
+tive lens by propagating a point source at 461 nm through
+a test setup (including all imaging path optics and vacuum
+viewports) and measuring the point-spread function.
+While absorption and fluorescence imaging rely on the
+same light scattering process (they only collect different parts
+of the scattered EM field [2]), the signal amplitudes for these
+two methods scale differently with the NA. When collect-
+ing fluorescence, the solid angle coverage of the imaging sys-
+tem proportionally affects the signal down to the lowest spa-
+tial frequencies. This is not the case for absorption imaging,
+where the amplitude of spatial frequency components below
+the NA-dependent bandwidth is constant as the NA is further
+increased (assuming the lens fully covers the probe beam). In
+other words, for fluorescence imaging, most of the signal light
+gets collected in the outer ring fraction of the lens aperture,
+which renders it particularly susceptible to lens imperfections.
+0
+20
+40
+60
+80
+100
+I/Isat
+0.0
+0.5
+1.0
+1.5
+2.0
+Var(N)/N
+Absorption beam intensity
+1 s pulse duration, NA = 0.2
+Saturated fluorescence
+OD = 0.17, 0.28 atoms/a2
+OD = 1.2, 2 atoms/a2
+OD = 3.1, 5 atoms/a2
+OD = 6.1, 10 atoms/a2
+OD = 12.2, 20 atoms/a2
+FIG. S2.
+SNR comparison between absorption and fluorescence
+imaging. The relevant imaging parameters from the main figures of
+the paper are used for this calculation. For absorption imaging the
+atom count variance scales inversely proportional with intensity in
+the non-saturated limit I ≪ Isat, and proportional with intensity in
+the high saturation limit. The variance is for both imaging methods
+proportional to 1/τ. In the fully saturated regime (and assuming no
+technical noise) the normalized variance for fluorescence imaging is
+independent of atomic column density. To avoid imaging defects at
+the high densities used in clock operation, an I/Isat > 50 was used
+in all imaging measurements. The black dashed line indicates the
+intensity used for our inverse Abel measurements.
+HTSE calculation
+To accurately model the density distribution in our 3D lat-
+tice, we use a High Temperature Series Expansion (HTSE)
+calculation in the atomic limit. The general Hamiltonian for
+SU(N) symmetric fermions in a 3D lattice in the atomic limit
+takes the following form:
+HAL = U
+2
+�
+i,σ̸=σ′
+ˆni,σˆni,σ′ +
+�
+i,σ
+Viˆni,σ
+(6)
+On a lattice site i, there are just two competing energy
+scales: an interaction energy U between particles and a po-
+sition dependent energy offset Vi according to the harmonic
+confinement. By using the local density approximation µ =
+V (x, y, z)−µ0, where V (x, y, z) = 1
+2m(ω2
+xx2+ω2
+yy2+ω2
+zz2)
+and µ0 corresponds to the peak chemical potential in the lat-
+tice. For the spin-polarized system in this work, U = 0 and the
+calculations are substantially simplified.
+Ultimately, we want to express the density distribution
+n(µ, T, r) in terms of the chemical potential, atomic tempera-
+ture, and position in the lattice. On a lattice site i, we express
+the Grand partition function Z and Grand potential Ω :
+Z(µ, T, r) =
+N=1
+�
+σ=0
+�N
+σ
+�
+e−βµσ
+(7)
+
+3
+Ω = −kBTln(Z)
+From here, we determine the entropy and occupancy per
+lattice site i:
+s(µ, T, r) = −∂Ω
+∂T = kB ln(Z) + ∆s
+∆s = kB
+Z βµe−βµ
+(8)
+n(µ, T, r) = −∂Ω
+∂µ =
+1
+Z(µ, T)e−βµ
+(9)
+We accurately determine the total atom number Nlat from
+in situ absorption imaging and total entropy Slat via time-of-
+flight fitting to a non-interacting Fermi-Dirac profile. Simi-
+larly, we express the entropy s and occupation n on a given
+lattice site using Eq. 8 and Eq. 9 expressed in terms of T
+and µ. We then determine global fitting parameters T and µ0
+to ensure the integrated entropy and occupancy over all lat-
+tice sites equals our experimentally measured values. After
+determining µ0 and T to realize the equality in Eq. 9, we cal-
+culate n(µ, T, r). A linecut of n(µ, T, r) at z = 0 is plotted in
+Fig. 4(b).
+Inverse Abel transform
+We outline our reconstruction procedure here using mea-
+surements of the atomic cloud aspect ratios and an inverse
+Abel transform: First, we use saturated absorption images
+along a vertical axis aligned with z and a horizontal axis
+aligned with x corresponding to Fig. 1(a) and Fig. 1(b) to
+determine the aspect ratios ωx/ωy and ωx/ωz respectively.
+Next, we perform an inverse Abel transform on the Fig. 1(a)
+image to reconstruct an initial three-dimensional density dis-
+tribution. Given there is no axis of cylindrical symmetry in
+our system geometry, the reconstructed density from the in-
+verse Abel transform must be appropriately re-scaled.
+Treating our system as an ellipsoid with radii rx, ry, rz
+and N atoms the density is nlat = N/Vlat where Vlat =
+4
+3πrxryrz.
+We extract the inverse Abel transform for the
+Fig. 1(a) image along the x axis, given the largest Band in-
+sulator plateau will occur along the axis with the weakest har-
+monic confinement. The density distribution from this pro-
+cedure assumes a volume of VAbel =
+4
+3πrxrxry. Thus we
+scale the extracted density by nAbel/nlat = rz
+rx = ωz/ωx us-
+ing the measured aspect ratio from Fig. 1(b). Given excess
+noise around the origin, the x = 0 point is interpolated with
+the neighboring point in Fig. 4(a). This reconstruction proce-
+dure was cross-checked with simulated density distributions
+to ensure its fidelity. The three-point Abel transform method
+was used for this work, which has been independently studied
+to verify its fidelity [3].
+QPN calculation
+To calibrate our atom number, we analyze quantum projec-
+tion fluctuations using the narrow-linewidth clock transition
+between the 1S0 and 3P0 states in 87Sr. Using a clock laser
+stabilized to our 8 mHz linewidth silicon reference cavity, ro-
+tation noise due to laser instability can be neglected in these
+measurements [4]. Additionally, fluctuations in total counts
+are < 2% and not a limiting systematic for determining the
+atom number calibration. Referenced in many texts [5], by
+preparing atoms in a superposition of 1S0 to 3P0 the variance
+V of the measured excitation fraction is related to the mean
+atom number ¯N and mean excitation ¯pe by:
+VQP N = ¯pe(1 − ¯pe)
+¯N
+(10)
+To determine this variance, we do many subsequent mea-
+surements of pe under identical operating conditions. For a
+measurement i to determine pi
+e, two fluorescence counts ˜Ci
+g
+and ˜Ci
+e are read off a region of interest of our camera includ-
+ing our atoms. These counts are subtracted by two averaged
+dark frames ¯Bg and ¯Be to yield Ci
+g = ˜Ci
+g− ¯Bg, Ci
+e = ˜Ci
+e− ¯Be.
+We would like to determine the coefficient a that satisfies
+N i
+e = aCi
+e/τ, N i
+g = aCi
+g/τ. We can immediately see that
+the excitation fraction has no dependence on this coefficient:
+pi
+e =
+�aCi
+e
+�aCie + �aCig
+(11)
+However, the total atom number N i = a(Ci
+e + Ci
+g)/τ =
+aCi
+t/τ does. Rewriting Eq. 10, we see a measurement of the
+variance VQP N, the mean excitation ¯pe, and the mean total
+counts ¯Ct can determine a.
+VQP N = ¯pe(1 − ¯pe)
+a ¯Ct/τ
+(12)
+The coefficient a can be interpreted as the ”atoms per count
+per pulse duration”. In principle, with knowledge of the quan-
+tum efficiency, gain, scattering rate, numerical aperture, and
+radiation pattern one could calculate this value. Practically,
+assumptions about the radiation pattern based on the quanti-
+zation axis and probe light polarization make this calculation
+more difficult. In practice, it is much more straightforward to
+directly measure a than to individually measure each of these
+values with high accuracy.
+The observed variance of the excitation fraction Vpe has
+contributions from quantum projection noise (QPN), photon
+shot noise (PSN), and camera readout noise (RN):
+Vpe = VQP N + VP SN + VRN
+(13)
+Here g is the detector gain in units of counts per electron.
+
+4
+VP SN = ¯pe(1 − ¯pe)
+¯Ct
+× g
+(14)
+VRN = R2
+¯Ct
+2 (2¯p2
+e − 2¯pe + 1)
+(15)
+VP SN can be understood intuitively considering the ratio
+VQP N/VP SN. The number of signal electrons (equivalently
+the number of collected photons multiplied by the camera
+quantum efficiency) per atom determines the relative scaling
+of VQP N and VP SN.
+VQP N
+VP SN
+=
+1
+g × a
+(16)
+105
+3 × 104
+4 × 104
+6 × 104
+Ct (counts)
+10
+5
+Var(pe)
+FIG. S3. Readout noise calibration. A π pulse on our optical clock
+transition is used so pe ≈ 1 and Vpe =
+R2
+¯
+Ct2 + C. We use 4 pulse
+durations between 5 and 20 µs to vary Ct. We fit R = 100.2 ± 24.6
+and C = 2.73 × 10−6 ± 1.02 × 10−6.
+To determine a we need to accurately calibrate VRN and
+VP SN. We see at pe = 1, VP SN, VQP N = 0. Thus, measur-
+ing Vpe at pe = 1 will independently determine VRN.
+We wish to fit R and ensure it is consistent with the cameras
+specified readout noise. To extract this value, we use 4 pulse
+durations between 5 and 20 µs to vary Ct. This is illustrated
+in Fig. S3. In practice, we fit
+Vpe = R2
+¯Ct
+2 + C
+(17)
+We fit R = 100.2 ± 24.6 and C = 2.73 × 10−6 ± 1.02 ×
+10−6. For our circular ROI there are X = 889 pixels in the
+masked radius. For the calibrated gain g = 1.59 counts/e- and
+readout noise r = 2.4 e- respectively , Rcalc = √Xgr =
+94.7 in agreement with R = 100.2 ± 24.6. We note that the
+gain and readout noise of the camera are close to specifica-
+tion. Dark counts over our 30 ms exposure are < .1 e- and
+considered negligible.
+Next, we wish to determine aQP N. To do so, we perform a
+second measurement at pe = 0.5. The variance of this dataset
+contains contributions from VQP N, VP SN, and VRN. Us-
+ing the measured R value, we subtract the VRN contribution.
+Next, we fit the data in Fig. S4 to:
+Vpe = 0.5(1 − 0.5)
+a ¯Ct/τ
++ 0.5(1 − 0.5)
+¯Ct
+× g
+(18)
+We fit aQP N = 1.72 ± 0.16. This is in reasonable agree-
+ment with the calculated value of 1.43 assuming Γ/2 scatter-
+ing into 4 π while also accounting for the measured quantum
+efficiency.
+105
+4 × 104
+6 × 104
+Ct (counts)
+2 × 10
+5
+3 × 10
+5
+4 × 10
+5
+Var(pe)
+FIG. S4. aQP N calibration. The atoms in our optical lattice are
+placed in a superposition of the ground and clock states with a π/2
+pulse so pe ≈ 0.5 for these measurements and Vpe is fit to Eq. 18.
+We determine aQP N = 1.72 ± 0.16.
+READOUT NOISE
+Here, we derive the readout noise term used in our variance
+measurements. The expressions used are somewhat different
+than other literature, given that we use averaged dark frames
+¯Be and ¯Bg. Recall, pe =
+Ce
+Ce+Cg . To determine the readnoise
+contribution to the excitation fraction, we perform standard
+error propagation:
+VRN =
+� ∂pe
+∂Ce
+�2
+V(Ce) +
+� ∂pe
+∂Cg
+�2
+V(Cg)
+(19)
+Here,
+∂pe
+∂Cg
+=
+Ce
+(Ce + Cg)2 =
+pe
+(Ce + Cg)
+(20)
+
+5
+∂pe
+∂Ce
+=
+Cg
+(Ce + Cg)2 =
+1 − pe
+(Ce + Cg)
+(21)
+To determine V(Ce) consider an X pixel region-of-interest
+for which we extract Cg, Ce in two separate measurements.
+Each pixel contains r read noise in electrons. The single pixel
+read noise in units of counts is thus g × ri. The total noise in
+this region of interest is summed in quadrature pixel-by-pixel
+V(Cg), V(Ce) = �
+X (ri × g)2 = Xr2g2 = R2. Plugging
+terms in Eq. 19:
+VRN = R2
+¯Ct
+2 (2¯p2
+e − 2¯pe + 1)
+(22)
+Imaging system parameters for Fig. 3(a)
+In Table 1 is a summary of the imaging parameters for the
+measurements in Fig. 3(a). For Fig. 1 and Fig. 4, a 1 µs pulse
+duration was used. In Fig. 3(b), we vary the pulse length be-
+tween 500 ns and 2 µs. Atom number fluctuations in time-
+of-flight absorption imaging for these measurements have a
+standard deviation less than 2 %.
+Table 1
+Vertical imaging system
+Numerical aperture
+0.23
+Pulse duration
+3 µs
+Total photons scattered per atom at full
+saturation
+287
+Collection efficiency
+1.3 %
+Camera quantum efficiency
+0.85
+Imaging system quantum efficiency
+0.65
+Calculated photon count per atom
+2.06
+Measured photon count per atom
+1.91(1)
+Horizontal imaging system
+Numerical aperture
+0.10
+Pulse duration
+3 µs
+Total photons scattered per atom at full
+saturation
+287
+Collection efficiency
+0.25 %
+Camera quantum efficiency
+0.78
+Imaging system quantum efficiency
+0.72
+Calculated photon count per atom
+0.402
+Measured photon count per atom
+0.445(3)
+[1] G. E. Marti, PhD Thesis (University of California, Berkeley,
+2014).
+[2] W. Ketterle, D. S. Durfee, and D. Stamper-Kurn, arXiv (1999).
+[3] D. D. Hickstein, S. T. Gibson, R. Yurchak, D. D. Das,
+and
+M. Ryazanov, Review of Scientific Instruments 90, 065115
+(2019).
+[4] D. Matei, T. Legero, S. H¨afner, C. Grebing, R. Weyrich,
+W. Zhang, L. Sonderhouse, J. Robinson, J. Ye, F. Riehle, et al.,
+Phys. Rev. Lett. 118, 263202 (2017).
+[5] W. M. Itano, J. C. Bergquist, J. J. Bollinger, J. M. Gilligan, D. J.
+Heinzen, F. L. Moore, M. G. Raizen, and D. J. Wineland, Phys.
+Rev. A 47, 3554 (1993).
+

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+arXiv:2301.02204v1  [math.CO]  5 Jan 2023
+ASSOCIATION SCHEMES ON TRIPLES FROM AFFINE SPECIAL
+SEMILINEAR GROUPS
+DOM VITO A. BRIONES
+Abstract. Association schemes on triples (ASTs) are 3-dimensional analogues of classical
+association schemes.
+If a group acts two-transitively on a set, the orbits of the action
+induced on the triple Cartesian product of that set yields an AST. By considering the
+actions of semidirect products of the affine special linear group ASL(k, n) with subgroups of
+the Galois group Gal(GF(n)), we obtain the sizes, third valencies, and intersection numbers
+of the ASTs obtained from subgroups of the affine special semilinear group.
+1. Introduction
+A classical association is an algebraic-combinatorial object with certain symmetry prop-
+erties. These properties suffice to afford classical association with desirable structural char-
+acteristics and are pliant enough to allow classical association schemes to be applicable to
+several areas of mathematics. For example, the adjacency algebra of a classical association
+scheme is semisimple and, when the adjacency matrices define a distance-regular graph, the
+structure constants of this algebra can be expressed in terms of certain families of orthogonal
+polynomials. [4]
+Mesner and Bhattacharya introduced the notion of association schemes on triples (or
+ASTs), a ternary analogue for classical association schemes [5].
+An AST on a set Ω is
+a partition of the triple Cartesian product Ω × Ω × Ω subject to regularity requirements
+paralleling the symmetry conditions for classical association schemes. In ASTs, the resulting
+adjacency hypermatrices produce a ternary algebra under a ternary product that extends
+the usual matrix multiplication.
+However, the structural properties of ASTs remain unclear, partly due to the ternary
+adjacency algebra not being associative nor commutative. As first steps in the investigation
+of ASTs, studies were conducted regarding analogues of identity and inverse elements [6],
+enumerations of ASTs over the smallest number of vertices [1], possible sources of ASTs such
+as those from group actions, two-graphs, designs, and other ASTs [5, 7, 3], as well as the
+intersection numbers of known families of ASTs [5, 2, 3].
+In particular, the actions of two-transitive groups yield ASTs [5]. The orbits of these
+actions are closely related to the parameters of the AST, providing their sizes, third valencies,
+and intersection numbers [5, 2]. In fact, [5] provides the sizes, third valencies, and intersection
+numbers of the ASTs obtained from the affine general linear group AGL(1, n) where n is a
+prime power. This was extended in [2], wherein these parameters were obtained for the ASTs
+(D.V.A. Briones, Corresponding author) Institute of Mathematics, University of the Philippines
+Diliman, 1101 Quezon City, Philippines
+E-mail address: dabriones@up.edu.ph.
+Date: January 6, 2023.
+Key words and phrases. algebraic combinatorics, ternary algebra, association scheme on triples
+MSC Classification: 05E30.
+1
+
+2
+D.V.A. BRIONES
+obtained from subgroups of the affine semilinear group AΓL(k, n) of the form AGL(k, n)⋊H),
+where k ≥ 1 and H ≤ Gal(GF(q). Further work was done in [3], where these parameters
+were obtained from ASTs obtained from the affine special linear group ASL(2, n).
+We extend this last result by determining the sizes, third valencies, and intersection num-
+bers of ASTs obtained from subgroups of the affine special semilinear group ASL(k, n) ⋊
+Gal(GF(n)) of the form ASL(k, n) ⋊ H, where k ≥ 2, n is a prime power, and H is a
+subgroup of Gal(GF(n)). In particular, we show that the ASTs obtained from ASLH(k, n)
+are the same as the ASTs obtained from AGLH(k, n) = AGL(k, n) ⋊ H for k ≥ 3.
+2. Preliminaries
+We define association schemes on triples, remark how ASTs arise from two-transitive
+groups, and review the actions of the affine special linear and affine special semilinear groups.
+2.1. Association schemes on triples. We define association schemes on triples and men-
+tion how the parameters of an AST obtained from a two-transitive group are related to the
+group action.
+Definition 2.1. [5, 7] Let Ω be a finite set with at least 3 elements. An association scheme
+on triples (AST) on Ω is a partition X = {Ri}m
+i=0 of Ω × Ω × Ω with m ≥ 4 such that the
+following hold.
+(1) For each i ∈ {0, . . . , m}, there exists an integer n(3)
+i
+such that for each pair of distinct
+x, y ∈ Ω, the number of z ∈ Ω with (x, y, z) ∈ Ri is n(3)
+i .
+(2) (Principal Regularity Condition.) For any i, j, k, l ∈ {0, . . . , m}, there exists a con-
+stant pl
+ijk such that for any (x, y, z) ∈ Rl, the number of w such that (w, y, z) ∈ Ri,
+(x, w, z) ∈ Rj, and (x, y, w) ∈ Rk is pl
+ijk.
+(3) For any i ∈ {0, . . . , m} and any σ ∈ S3, there exists a j ∈ {0, . . . , m} such that
+Rj = {(xσ(1), xσ(2), xσ(3)) : (x1, x2, x3) ∈ Ri}.
+(4) The first four relations are R0 = {(x, x, x) : x ∈ Ω}, R1 = {(x, y, y) : x, y ∈ Ω, x ̸= y},
+R2 = {(y, x, y) : x, y ∈ Ω, x ̸= y}, and R3 = {(y, y, x) : x, y ∈ Ω, x ̸= y}.
+The integer n(3)
+i
+is the third valency of Ri, and is the analogue of valency from classical
+association schemes. By Conditions 1 and 3 of Definition 2.1 there are for each i the constants
+n(1)
+i
+= |{z ∈ Ω : (z, x, y) ∈ Ri}| and n(2)
+i
+= |{z ∈ Ω : (x, z, y) ∈ Ri}| independent of any pair
+of distinct x, y ∈ Ω. Similarly, n(1)
+i
+is the first valency of Ri and n(2)
+i
+is the second valency of
+Ri. The trivial relations are R0, R1, R2 and R3 while the other relations are the nontrivial
+relations. Further, the numbers pl
+ijk are called the intersection numbers.
+ASTs arise naturally from the actions of two-transitive groups [5], mirroring how Schurian
+classical association schemes are induced by the actions of transitive groups [4]. Indeed,
+when a two-transitive group G acts on a set Ω, the orbits of the induced action on Ω×Ω×Ω
+is an AST [5]. Correspondences between the action and the parameters of the induced AST
+are summarized in the following remark.
+Remark 1 ([5, 2]). Let G be a group acting two-transitively on a set Ω and let X be the AST
+induced by this action. For any pair of distinct elements a, b ∈ Ω, the orbits of the two-point
+stabilizer Ga,b on Ω \ {a, b} are in bijection with the nontrivial relations of the AST. As a
+consequence of this bijection, the sizes of these orbits are also the third valencies.
+
+ASSOCIATION SCHEMES ON TRIPLES FROM AFFINE SPECIAL SEMILINEAR GROUPS
+3
+2.2. Affine special groups. Given a prime power n and k ≥ 1, the affine special linear
+group ASL(k, n) is the semidirect product GF(n) ⋊ SL(k, n), where SL(k, n) is the group
+of invertible linear transformations on the k-dimensional vector space V over GF(n) of
+determinant 1. Explicitly, the affine special linear group is the following group of maps from
+V to itself.
+ASL(k, n) = {x �→ Ax + b : A ∈ SL(k, n), b ∈ V } .
+Similarly, the affine special semilinear group ASL(k, n) ⋊ Gal(GF(n)) is the semidirect
+product of the affine special linear group ASL(k, n) with the Galois group Gal(GF(n)).
+Explicitly, the affine special semilinear group is the following group of maps from V to itself.
+ASL(k, n) ⋊ Gal(GF(n)) = {x �→ Aφ(x) + b : A ∈ SL(k, n), b ∈ GF(n), φ ∈ Gal(GF(n))} .
+3. ASTs from subgroups of the affine special semilinear group
+In this section we generalize work done in [3] by obtaining the sizes, third valencies, and
+intersection numbers of ASTs obtained from the actions of subgroups of the affine special
+semilinear group of the form
+ASLH(k, n) = ASL(k, n) ⋊ H,
+where k ≥ 2, n = pα a power of a prime number p, and H a subgroup of Gal(GF(n)).
+We obtain the sizes and third valencies of these ASTs by obtaining a two-point stabilizer of
+ASLH(k, n) and then determining its orbits. Finally, we obtain the intersection numbers of
+these ASTs through explicit orbit computations.
+For ease of discussion, we fix the following notations. Let n = pα be a power of a prime p,
+k ≥ 2, V be the k-dimensional vector space over GF(n), H be a subgroup of Gal(GF(n)),
+and X be the AST obtained from ASLH(k, n). For a ∈ GF(n), let ⃗a = (a, 0, . . . , 0)T ∈ V .
+Further, for (u, v, w) ∈ V × V × V , let [(u, v, w)] ∈ X denote the orbit of (u, v, w) under
+ASLH(k, n).
+We begin with the case where k = 2. To determine the size and third valencies of X, we
+exploit the relationships between these parameters and the orbits of a two-point stabilizer
+of ASLH(k, n).
+Theorem 3.1. Let n = pα be a power of a prime p, q = pω with ω|α, H = GalGF (q)(GF(n))
+and X be the AST obtained from the action of ASLH(2, n) on the 2-dimensional vector space
+V over GF(n). The two-point stabilizer ASLH(2, n)⃗0,⃗1 has the following orbits on V \{⃗0,⃗1}.
+(1) There are −2 + ω
+α
+� α
+ω
+β=1 qgcd ( α
+ω ,β) orbits of the form
+� ⃗
+φ(a) : φ ∈ H
+�
+, a ̸= 0, 1
+each of size degGF (q)(a).
+(2) There are −1 + ω
+α
+� α
+ω
+β=1 qgcd ( α
+ω ,β) orbits of the form
+�
+(c, φ(a))T : c ∈ GF(n), φ ∈ H
+�
+, a ̸= 0
+each of size n degGF (q)(a).
+As a consequence of Theorem 3.1, we obtain the sizes and third valencies of the ASTs
+obtained from ASLH(2, n).
+Theorem 3.2. Let n = pα be a power of a prime p, q = pω with ω|α, H = GalGF (q)(GF(n))
+and X be the AST obtained from the action of ASLH(2, n) on the 2-dimensional vector
+
+4
+D.V.A. BRIONES
+space V over GF(n). Then X has −3 + 2
+�
+ω
+α
+� α
+ω
+β=1 qgcd ( α
+ω ,β)�
+nontrivial relations. There are
+−2 + ω
+α
+� α
+ω
+β=1 qgcd ( α
+ω ,β) nontrivial relations of the form
+Ra = {[(⃗0,⃗1,⃗a)]}, a ̸= 0, 1,
+with corresponding third valency degGF (q)(a). The remaining −1+ ω
+α
+� α
+ω
+β=1 qgcd ( α
+ω ,β) nontrivial
+relations of X are of the form
+aR = {[(⃗0,⃗1, (0, a)T)]}, a ̸= 0,
+with corresponding third valency n degGF (q)(a).
+Proof. The two-point stabilizer is
+ASLH(2, n)⃗0,⃗1 = {(x, y)T �→
+�
+1
+c
+0
+1
+�
+(φ(x), φ(y))T : c ∈ GF(n), φ ∈ H}.
+Direct computation shows that the orbits of ASLH(2, n)⃗0,⃗1 have the following forms.
+(1) The first type of orbit has the form
+{(φ(a), 0)T : φ ∈ H},
+which consists of those vectors whose second coordinate is 0 and whose first coordinate
+is a Galois conjugate of an element a ∈ GF(n) with a ̸= 0, 1.
+(2) The remaining orbits are of the form
+{(x, φ(a))T : x ∈ GF(n), φ ∈ H},
+which consists of those vectors whose second coordinate is a Galois conjugate of an
+element a ∈ GF(n) with a ̸= 0.
+The sizes of these orbits follow directly from the Fundamental Theorem of Galois Theory.
+The number of orbits of each type are then obtained through the Fundamental Theorem of
+Galois Theory and a straightforward application of Burnside’s Orbit Counting Theorem to
+the action of Gal(GF(n)) on GF(n).
+□
+For notational convenience, let Aa denote the adjacency hypermatrix corresponding to
+the relation Ra whenever a ̸= 0, 1.
+Similarly, let aA denote the adjacency hypermatrix
+corresponding to the relation aR whenever a ̸= 0. Further, let T be a transversal of the
+orbits of H on GF(n) \ {0}. The intersection numbers of the subalgebra generated by the
+adjacency hypermatrices of the nontrivial relations of X are given in the next theorem.
+Theorem 3.3. Let n = pα be a power of a prime p, q = pω with ω|α, H = GalGF (q)(GF(n))
+and X be the AST obtained from the action of ASLH(2, n). The following equations hold
+for any a, b, c ̸= 0, 1 and a, b, c ̸= 0.
+(1) AaAbAc = �
+ℓ∈T\{1} pℓAℓ, where
+pℓ = |{φ(c) : φ ∈ H and (∃ψ, τ ∈ H) [(1 − φ(c))τ(a) + φ(c) = ℓ = φ(c)ψ(b)]}| .
+(2) AaAb cA = Aa cA Ab = cA AaAb = 0.
+(3) aA bA Ac = �
+ℓ∈T pℓ ℓA, where
+pℓ =
+����
+�
+φ(c) : φ ∈ H and (∃ψ, τ ∈ H)
+�
+τ(a)
+1 − φ(c) = ℓ = ψ(b)
+φ(c)
+������ .
+
+ASSOCIATION SCHEMES ON TRIPLES FROM AFFINE SPECIAL SEMILINEAR GROUPS
+5
+(4) aA Ac bA = �
+ℓ∈T pℓ ℓA, where
+pℓ = |{ψ(b) : ψ ∈ H and (∃φ, τ ∈ H) [ψ(b)φ(c) = ℓ = τ(a) + ψ(b)]}| .
+(5) Ac aA bA = �
+ℓ∈T pℓ ℓA, where
+pℓ =
+����
+�
+ψ(b) : ψ ∈ H and (∃φ, τ ∈ H)
+�
+ψ(b)(1 − φ(c)) = ℓ = τ(a)(φ(c) − 1)
+φ(c)
+������ .
+(6) aA bA cA = �
+ℓ∈T\{1} pℓAℓ + �
+∈T p A, where
+pℓ = q
+����
+�
+φ(c) : (∃ψ, τ ∈ H)
+�τ(a) + φ(c)
+φ(c)
+= d = −ψ(b)
+φ(c)
+������ ,
+p = |{ψ(b) : (∃φ, τ ∈ H) [τ(a) + ψ(b) + φ(c) = ]}| .
+Proof. We prove only the third statement, as the other statements are shown similarly. With
+Ri = aR, Rj = bR, and Rk = Rc, we determine the Rℓ such that the intersection number
+pℓ
+ijk is nonzero. If Rℓ =d R for some d ̸= 0, considering the viable w as in the the Principal
+Regularity Condition from Definition 2.1 necessitates that φ(c)ψ(b) = 0 for some φ, ψ ∈ H,
+which is impossible. If Rℓ = Rd for some d ̸= 0, 1, the Principal Regularity Conditions says
+that the number of viable w, pℓ
+ijk, is the number of vectors of the form (φ(c), 0)T with φ ∈ H
+such that there are ψ and τ in H that satisfy
+τ(a)
+1 − φ(c) = ℓ = ψ(b)
+φ(c)
+□
+The succeeding theorem gives the intersection numbers pl
+ijk of the ASTs obtained from
+ASLH(2, q) whenever exactly one of Ri, Rj, and Rk is trivial. Here I1, I2, and I3 denote the
+respective adjacency hypermatrices of the trivial relations R1, R2, and R3 of X. The proof,
+similar to that of the proof of Theorem 3.3, is omitted.
+Theorem 3.4. Let n = pα be a power of a prime p, q = pω with ω|α, H = GalGF (q)(GF(n))
+and X be the AST obtained from the action of ASLH(2, q). The following equations hold for
+any a, b ̸= 0, 1 and a, b ̸= 0.
+(1) I1AaAb = pI1, where
+p1 = |{ψ(b) : ψ ∈ H and (∃τ ∈ H)[τ(a)ψ(b) = 1]}|.
+(2) AaI2Ab = p2I2, where
+p2 = |{ψ(b) : ψ ∈ H and (∃τ ∈ H)[τ(a)ψ(b) = τ(a) + ψ(b)]}|.
+(3) AaAbI3 = p3I3, where
+p3 = |{ψ(b) : ψ ∈ H and (∃τ ∈ H)[τ(a) + ψ(b) = 1]}|.
+(4) I1Aa aA = I1 aA Aa = AaI2 aA = aA I2Aa = Aa aA I3 = aA AaI3 = 0.
+(5) I1 aA bA = p∗I1, aA I2 bA = p∗I2, aA bA I3 = p∗I3, where
+p∗ = q |{ψ(b) : ψ ∈ H and (∃τ ∈ H)[τ(a) = −ψ(b)]}| .
+Here we consider the AST obtained from ASLH(k, n) for k ≥ 3, n a prime power, and
+H a subgroup of Gal(GF(n)). The following theorem tells us that the AST obtained from
+ASLH(k, n) is the same as the AST obtained from the subgroup AGLH(k, n) = AGL(k, n)⋊
+H of the affine semilinear group AΓL(k, n) whenever k ≥ 3. In particular, the parameters
+of these ASTs have already been obtained in [3].
+
+6
+D.V.A. BRIONES
+Theorem 3.5. Let n = pα be a power of a prime p, q = pω with ω|α, and H = GalGF (q)(GF(n)).
+Then the AST obtained from the action of ASLH(k, n) is equal to the AST obtained from
+the action of AGLH(k, n).
+Proof. Notice that if a group G and a subgroup K of G both act two-transitively on a set,
+the orbits of G are unions of orbits of K. In particular, if G and K have the same number
+of orbits, then these orbits are exactly the same. Thus, to prove the theorem, it suffices to
+show that the ASTs obtained from AGLH(k, n) and ASLH(k, n) have the same size. By
+Remark 1, it suffices to show that the two-point stabilizer ASLH(k, n)⃗0,⃗1 has the same orbits
+as AGLH(k, n)⃗0,⃗1 on GF(n) \ {⃗0,⃗1}.
+Indeed, the two-point stabilizers above are given by
+ASLH(k, n)⃗0,⃗1 = {v �→ Aφ(v) : A ∈ SL(k, n), φ ∈ H},
+and
+AGLH(k, n)⃗0,⃗1 = {v �→ Aφ(v) : A ∈ GL(k, n), φ ∈ H}.
+Direct computation shows that the orbits of ASLH(k, n)⃗0,⃗1 have the following forms.
+(1) One type of orbit has the form
+{(φ(a), 0, . . . , 0)T : φ ∈ H},
+which consists of those vectors whose first coordinate is a Galois conjugate of an
+element a ∈ GF(n) with a ̸= 0, 1. The other coordinates are 0.
+(2) The remaining orbit is
+(GF(n))k \ Span(⃗1),
+consisting of the vectors linearly independent from ⃗1.
+These are also the orbits of AGLH(k, n)⃗0,⃗1, completing the proof.
+□
+References
+1. J.M.P.
+Balmaceda
+and
+D.V.A.
+Briones,
+Association
+schemes
+on
+triples
+over
+few
+vertices,
+Matimyas
+Matematika
+45
+(2022),
+13–26,
+http://mathsociety.ph/matimyas/images/vol45/BalmacedaMatimyas.pdf.
+2.
+, Families of association schemes on triples from two-transitive groups (preprint), arXiv (2022),
+https://arxiv.org/abs/2107.07753.
+3.
+,
+A
+survey
+on
+association
+schemes
+on
+triples
+(preprint),
+arXiv
+(2022),
+https://arxiv.org/abs/2206.10500.
+4. E. Bannai and T. Ito, Algebraic combinatorics I. Association schemes, Mathematics lecture note series,
+no. 58, Benjamin/Cummings Pub. Co, San Francisco, 1984.
+5. D.M.
+Mesner
+and
+P.
+Bhattacharya,
+Association
+schemes
+on
+triples
+and
+a
+ternary
+algebra,
+Journal
+of
+Combinatorial
+Theory,
+Series
+A
+55
+(1990),
+no.
+2,
+204–234,
+https://www.sciencedirect.com/science/article/pii/0097316590900688.
+6.
+, A ternary algebra arising from association schemes on triples, Journal of Algebra 164 (1994),
+no. 3, 595–613, https://www.sciencedirect.com/science/article/pii/S0021869384710817.
+7. C.E. Praeger and P. Bhattacharya, Circulant association schemes on triples, New Zealand Journal of
+Mathematics 52 (2021), 153–165, https://nzjmath.org/index.php/NZJMATH/article/view/106.
+

GdE0T4oBgHgl3EQfRQBK/content/tmp_files/load_file.txt

@@ -0,0 +1,223 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf,len=222
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'}
+page_content='02204v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'}
+page_content='CO]  5 Jan 2023  ASSOCIATION SCHEMES ON TRIPLES FROM AFFINE SPECIAL  SEMILINEAR GROUPS  DOM VITO A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'}
+page_content=' BRIONES  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'}
+page_content=' Association schemes on triples (ASTs) are 3-dimensional analogues of classical  association schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'}
+page_content='  If a group acts two-transitively on a set, the orbits of the action  induced on the triple Cartesian product of that set yields an AST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'}
+page_content=' By considering the  actions of semidirect products of the affine special linear group ASL(k, n) with subgroups of  the Galois group Gal(GF(n)), we obtain the sizes, third valencies, and intersection numbers  of the ASTs obtained from subgroups of the affine special semilinear group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'}
+page_content=' Introduction  A classical association is an algebraic-combinatorial object with certain symmetry prop-  erties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'}
+page_content=' These properties suffice to afford classical association with desirable structural char-  acteristics and are pliant enough to allow classical association schemes to be applicable to  several areas of mathematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'}
+page_content=' For example, the adjacency algebra of a classical association  scheme is semisimple and, when the adjacency matrices define a distance-regular graph, the  structure constants of this algebra can be expressed in terms of certain families of orthogonal  polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'}
+page_content=' [4]  Mesner and Bhattacharya introduced the notion of association schemes on triples (or  ASTs), a ternary analogue for classical association schemes [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'}
+page_content='  An AST on a set Ω is  a partition of the triple Cartesian product Ω × Ω × Ω subject to regularity requirements  paralleling the symmetry conditions for classical association schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'}
+page_content=' In ASTs, the resulting  adjacency hypermatrices produce a ternary algebra under a ternary product that extends  the usual matrix multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'}
+page_content='  However, the structural properties of ASTs remain unclear, partly due to the ternary  adjacency algebra not being associative nor commutative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'}
+page_content=' As first steps in the investigation  of ASTs, studies were conducted regarding analogues of identity and inverse elements [6],  enumerations of ASTs over the smallest number of vertices [1], possible sources of ASTs such  as those from group actions, two-graphs, designs, and other ASTs [5, 7, 3], as well as the  intersection numbers of known families of ASTs [5, 2, 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'}
+page_content='  In particular, the actions of two-transitive groups yield ASTs [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'}
+page_content=' The orbits of these  actions are closely related to the parameters of the AST, providing their sizes, third valencies,  and intersection numbers [5, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'}
+page_content=' In fact, [5] provides the sizes, third valencies, and intersection  numbers of the ASTs obtained from the affine general linear group AGL(1, n) where n is a  prime power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'}
+page_content=' This was extended in [2], wherein these parameters were obtained for the ASTs  (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'}
+page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'}
+page_content=' Briones, Corresponding author) Institute of Mathematics, University of the Philippines  Diliman, 1101 Quezon City, Philippines  E-mail address: dabriones@up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'}
+page_content='ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'}
+page_content='  Date: January 6, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'}
+page_content='  Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'}
+page_content=' algebraic combinatorics, ternary algebra, association scheme on triples  MSC Classification: 05E30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'}
+page_content='  1  2  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'}
+page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'}
+page_content=' BRIONES  obtained from subgroups of the affine semilinear group AΓL(k, n) of the form AGL(k, n)⋊H),  where k ≥ 1 and H ≤ Gal(GF(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'}
+page_content=' Further work was done in [3], where these parameters  were obtained from ASTs obtained from the affine special linear group ASL(2, n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'}
+page_content='  We extend this last result by determining the sizes, third valencies, and intersection num-  bers of ASTs obtained from subgroups of the affine special semilinear group ASL(k, n) ⋊  Gal(GF(n)) of the form ASL(k, n) ⋊ H, where k ≥ 2, n is a prime power, and H is a  subgroup of Gal(GF(n)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'}
+page_content=' In particular, we show that the ASTs obtained from ASLH(k, n)  are the same as the ASTs obtained from AGLH(k, n) = AGL(k, n) ⋊ H for k ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'}
+page_content=' Preliminaries  We define association schemes on triples, remark how ASTs arise from two-transitive  groups, and review the actions of the affine special linear and affine special semilinear groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'}
+page_content=' Association schemes on triples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'}
+page_content=' We define association schemes on triples and men-  tion how the parameters of an AST obtained from a two-transitive group are related to the  group action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'}
+page_content=' [5, 7] Let Ω be a finite set with at least 3 elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'}
+page_content=' An association scheme  on triples (AST) on Ω is a partition X = {Ri}m  i=0 of Ω × Ω × Ω with m ≥ 4 such that the  following hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'}
+page_content='  (1) For each i ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'}
+page_content=' , m}, there exists an integer n(3)  i  such that for each pair of distinct  x, y ∈ Ω, the number of z ∈ Ω with (x, y, z) ∈ Ri is n(3)  i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'}
+page_content='  (2) (Principal Regularity Condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'}
+page_content=') For any i, j, k, l ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'}
+page_content=' , m}, there exists a con-  stant pl  ijk such that for any (x, y, z) ∈ Rl, the number of w such that (w, y, z) ∈ Ri,  (x, w, z) ∈ Rj, and (x, y, w) ∈ Rk is pl  ijk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'}
+page_content='  (3) For any i ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'}
+page_content=' , m} and any σ ∈ S3, there exists a j ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'}
+page_content=' , m} such that  Rj = {(xσ(1), xσ(2), xσ(3)) : (x1, x2, x3) ∈ Ri}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'}
+page_content='  (4) The first four relations are R0 = {(x, x, x) : x ∈ Ω}, R1 = {(x, y, y) : x, y ∈ Ω, x ̸= y},  R2 = {(y, x, y) : x, y ∈ Ω, x ̸= y}, and R3 = {(y, y, x) : x, y ∈ Ω, x ̸= y}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'}
+page_content='  The integer n(3)  i  is the third valency of Ri, and is the analogue of valency from classical  association schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'}
+page_content=' By Conditions 1 and 3 of Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'}
+page_content='1 there are for each i the constants  n(1)  i  = |{z ∈ Ω : (z, x, y) ∈ Ri}| and n(2)  i  = |{z ∈ Ω : (x, z, y) ∈ Ri}| independent of any pair  of distinct x, y ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'}
+page_content=' Similarly, n(1)  i  is the first valency of Ri and n(2)  i  is the second valency of  Ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'}
+page_content=' The trivial relations are R0, R1, R2 and R3 while the other relations are the nontrivial  relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'}
+page_content=' Further, the numbers pl  ijk are called the intersection numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'}
+page_content='  ASTs arise naturally from the actions of two-transitive groups [5], mirroring how Schurian  classical association schemes are induced by the actions of transitive groups [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'}
+page_content=' Indeed,  when a two-transitive group G acts on a set Ω, the orbits of the induced action on Ω×Ω×Ω  is an AST [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'}
+page_content=' Correspondences between the action and the parameters of the induced AST  are summarized in the following remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'}
+page_content='  Remark 1 ([5, 2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'}
+page_content=' Let G be a group acting two-transitively on a set Ω and let X be the AST  induced by this action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'}
+page_content=' For any pair of distinct elements a, b ∈ Ω, the orbits of the two-point  stabilizer Ga,b on Ω \\ {a, b} are in bijection with the nontrivial relations of the AST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'}
+page_content=' As a  consequence of this bijection, the sizes of these orbits are also the third valencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'}
+page_content='  ASSOCIATION SCHEMES ON TRIPLES FROM AFFINE SPECIAL SEMILINEAR GROUPS  3  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'}
+page_content=' Affine special groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'}
+page_content=' Given a prime power n and k ≥ 1, the affine special linear  group ASL(k, n) is the semidirect product GF(n) ⋊ SL(k, n), where SL(k, n) is the group  of invertible linear transformations on the k-dimensional vector space V over GF(n) of  determinant 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'}
+page_content=' Explicitly, the affine special linear group is the following group of maps from  V to itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'}
+page_content='  ASL(k, n) = {x �→ Ax + b : A ∈ SL(k, n), b ∈ V } .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'}
+page_content='  Similarly, the affine special semilinear group ASL(k, n) ⋊ Gal(GF(n)) is the semidirect  product of the affine special linear group ASL(k, n) with the Galois group Gal(GF(n)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'}
+page_content='  Explicitly, the affine special semilinear group is the following group of maps from V to itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'}
+page_content='  ASL(k, n) ⋊ Gal(GF(n)) = {x �→ Aφ(x) + b : A ∈ SL(k, n), b ∈ GF(n), φ ∈ Gal(GF(n))} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'}
+page_content=' ASTs from subgroups of the affine special semilinear group  In this section we generalize work done in [3] by obtaining the sizes, third valencies, and  intersection numbers of ASTs obtained from the actions of subgroups of the affine special  semilinear group of the form  ASLH(k, n) = ASL(k, n) ⋊ H,  where k ≥ 2, n = pα a power of a prime number p, and H a subgroup of Gal(GF(n)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'}
+page_content='  We obtain the sizes and third valencies of these ASTs by obtaining a two-point stabilizer of  ASLH(k, n) and then determining its orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'}
+page_content=' Finally, we obtain the intersection numbers of  these ASTs through explicit orbit computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'}
+page_content='  For ease of discussion, we fix the following notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'}
+page_content=' Let n = pα be a power of a prime p,  k ≥ 2, V be the k-dimensional vector space over GF(n), H be a subgroup of Gal(GF(n)),  and X be the AST obtained from ASLH(k, n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'}
+page_content=' For a ∈ GF(n), let ⃗a = (a, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'}
+page_content=' , 0)T ∈ V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'}
+page_content='  Further, for (u, v, w) ∈ V × V × V , let [(u, v, w)] ∈ X denote the orbit of (u, v, w) under  ASLH(k, n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'}
+page_content='  We begin with the case where k = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'}
+page_content=' To determine the size and third valencies of X, we  exploit the relationships between these parameters and the orbits of a two-point stabilizer  of ASLH(k, n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'}
+page_content=' Let n = pα be a power of a prime p, q = pω with ω|α, H = GalGF (q)(GF(n))  and X be the AST obtained from the action of ASLH(2, n) on the 2-dimensional vector space  V over GF(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'}
+page_content=' The two-point stabilizer ASLH(2, n)⃗0,⃗1 has the following orbits on V \\{⃗0,⃗1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'}
+page_content='  (1) There are −2 + ω  α  � α  ω  β=1 qgcd ( α  ω ,β) orbits of the form  � ⃗  φ(a) : φ ∈ H  �  , a ̸= 0, 1  each of size degGF (q)(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'}
+page_content='  (2) There are −1 + ω  α  � α  ω  β=1 qgcd ( α  ω ,β) orbits of the form  �  (c, φ(a))T : c ∈ GF(n), φ ∈ H  �  , a ̸= 0  each of size n degGF (q)(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'}
+page_content='  As a consequence of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'}
+page_content='1, we obtain the sizes and third valencies of the ASTs  obtained from ASLH(2, n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'}
+page_content=' Let n = pα be a power of a prime p, q = pω with ω|α, H = GalGF (q)(GF(n))  and X be the AST obtained from the action of ASLH(2, n) on the 2-dimensional vector  4  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'}
+page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'}
+page_content=' BRIONES  space V over GF(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'}
+page_content=' Then X has −3 + 2  �  ω  α  � α  ω  β=1 qgcd ( α  ω ,β)�  nontrivial relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'}
+page_content=' There are  −2 + ω  α  � α  ω  β=1 qgcd ( α  ω ,β) nontrivial relations of the form  Ra = {[(⃗0,⃗1,⃗a)]}, a ̸= 0, 1,  with corresponding third valency degGF (q)(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'}
+page_content=' The remaining −1+ ω  α  � α  ω  β=1 qgcd ( α  ω ,β) nontrivial  relations of X are of the form  aR = {[(⃗0,⃗1, (0, a)T)]}, a ̸= 0,  with corresponding third valency n degGF (q)(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'}
+page_content=' The two-point stabilizer is  ASLH(2, n)⃗0,⃗1 = {(x, y)T �→  �  1  c  0  1  �  (φ(x), φ(y))T : c ∈ GF(n), φ ∈ H}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'}
+page_content='  Direct computation shows that the orbits of ASLH(2, n)⃗0,⃗1 have the following forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'}
+page_content='  (1) The first type of orbit has the form  {(φ(a), 0)T : φ ∈ H},  which consists of those vectors whose second coordinate is 0 and whose first coordinate  is a Galois conjugate of an element a ∈ GF(n) with a ̸= 0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'}
+page_content='  (2) The remaining orbits are of the form  {(x, φ(a))T : x ∈ GF(n), φ ∈ H},  which consists of those vectors whose second coordinate is a Galois conjugate of an  element a ∈ GF(n) with a ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'}
+page_content='  The sizes of these orbits follow directly from the Fundamental Theorem of Galois Theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'}
+page_content='  The number of orbits of each type are then obtained through the Fundamental Theorem of  Galois Theory and a straightforward application of Burnside’s Orbit Counting Theorem to  the action of Gal(GF(n)) on GF(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'}
+page_content='  □  For notational convenience, let Aa denote the adjacency hypermatrix corresponding to  the relation Ra whenever a ̸= 0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'}
+page_content='  Similarly, let aA denote the adjacency hypermatrix  corresponding to the relation aR whenever a ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'}
+page_content=' Further, let T be a transversal of the  orbits of H on GF(n) \\ {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'}
+page_content=' The intersection numbers of the subalgebra generated by the  adjacency hypermatrices of the nontrivial relations of X are given in the next theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'}
+page_content=' Let n = pα be a power of a prime p, q = pω with ω|α, H = GalGF (q)(GF(n))  and X be the AST obtained from the action of ASLH(2, n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'}
+page_content=' The following equations hold  for any a, b, c ̸= 0, 1 and a, b, c ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'}
+page_content='  (1) AaAbAc = �  ℓ∈T\\{1} pℓAℓ, where  pℓ = |{φ(c) : φ ∈ H and (∃ψ, τ ∈ H) [(1 − φ(c))τ(a) + φ(c) = ℓ = φ(c)ψ(b)]}| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'}
+page_content='  (2) AaAb cA = Aa cA Ab = cA AaAb = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'}
+page_content='  (3) aA bA Ac = �  ℓ∈T pℓ ℓA, where  pℓ =  ����  �  φ(c) : φ ∈ H and (∃ψ, τ ∈ H)  �  τ(a)  1 − φ(c) = ℓ = ψ(b)  φ(c)  ������ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'}
+page_content='  ASSOCIATION SCHEMES ON TRIPLES FROM AFFINE SPECIAL SEMILINEAR GROUPS  5  (4) aA Ac bA = �  ℓ∈T pℓ ℓA, where  pℓ = |{ψ(b) : ψ ∈ H and (∃φ, τ ∈ H) [ψ(b)φ(c) = ℓ = τ(a) + ψ(b)]}| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'}
+page_content='  (5) Ac aA bA = �  ℓ∈T pℓ ℓA, where  pℓ =  ����  �  ψ(b) : ψ ∈ H and (∃φ, τ ∈ H)  �  ψ(b)(1 − φ(c)) = ℓ = τ(a)(φ(c) − 1)  φ(c)  ������ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'}
+page_content='  (6) aA bA cA = �  ℓ∈T\\{1} pℓAℓ + �  \uf6be∈T p\uf6be \uf6beA, where  pℓ = q  ����  �  φ(c) : (∃ψ, τ ∈ H)  �τ(a) + φ(c)  φ(c)  = d = −ψ(b)  φ(c)  ������ ,  p\uf6be = |{ψ(b) : (∃φ, τ ∈ H) [τ(a) + ψ(b) + φ(c) = \uf6be]}| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'}
+page_content=' We prove only the third statement, as the other statements are shown similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'}
+page_content=' With  Ri = aR, Rj = bR, and Rk = Rc, we determine the Rℓ such that the intersection number  pℓ  ijk is nonzero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'}
+page_content=' If Rℓ =d R for some d ̸= 0, considering the viable w as in the the Principal  Regularity Condition from Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'}
+page_content='1 necessitates that φ(c)ψ(b) = 0 for some φ, ψ ∈ H,  which is impossible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'}
+page_content=' If Rℓ = Rd for some d ̸= 0, 1, the Principal Regularity Conditions says  that the number of viable w, pℓ  ijk, is the number of vectors of the form (φ(c), 0)T with φ ∈ H  such that there are ψ and τ in H that satisfy  τ(a)  1 − φ(c) = ℓ = ψ(b)  φ(c)  □  The succeeding theorem gives the intersection numbers pl  ijk of the ASTs obtained from  ASLH(2, q) whenever exactly one of Ri, Rj, and Rk is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'}
+page_content=' Here I1, I2, and I3 denote the  respective adjacency hypermatrices of the trivial relations R1, R2, and R3 of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'}
+page_content=' The proof,  similar to that of the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'}
+page_content='3, is omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'}
+page_content=' Let n = pα be a power of a prime p, q = pω with ω|α, H = GalGF (q)(GF(n))  and X be the AST obtained from the action of ASLH(2, q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'}
+page_content=' The following equations hold for  any a, b ̸= 0, 1 and a, b ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'}
+page_content='  (1) I1AaAb = pI1, where  p1 = |{ψ(b) : ψ ∈ H and (∃τ ∈ H)[τ(a)ψ(b) = 1]}|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'}
+page_content='  (2) AaI2Ab = p2I2, where  p2 = |{ψ(b) : ψ ∈ H and (∃τ ∈ H)[τ(a)ψ(b) = τ(a) + ψ(b)]}|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'}
+page_content='  (3) AaAbI3 = p3I3, where  p3 = |{ψ(b) : ψ ∈ H and (∃τ ∈ H)[τ(a) + ψ(b) = 1]}|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'}
+page_content='  (4) I1Aa aA = I1 aA Aa = AaI2 aA = aA I2Aa = Aa aA I3 = aA AaI3 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'}
+page_content='  (5) I1 aA bA = p∗I1, aA I2 bA = p∗I2, aA bA I3 = p∗I3, where  p∗ = q |{ψ(b) : ψ ∈ H and (∃τ ∈ H)[τ(a) = −ψ(b)]}| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'}
+page_content='  Here we consider the AST obtained from ASLH(k, n) for k ≥ 3, n a prime power, and  H a subgroup of Gal(GF(n)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'}
+page_content=' The following theorem tells us that the AST obtained from  ASLH(k, n) is the same as the AST obtained from the subgroup AGLH(k, n) = AGL(k, n)⋊  H of the affine semilinear group AΓL(k, n) whenever k ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'}
+page_content=' In particular, the parameters  of these ASTs have already been obtained in [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'}
+page_content='  6  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'}
+page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'}
+page_content=' BRIONES  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'}
+page_content=' Let n = pα be a power of a prime p, q = pω with ω|α, and H = GalGF (q)(GF(n)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'}
+page_content='  Then the AST obtained from the action of ASLH(k, n) is equal to the AST obtained from  the action of AGLH(k, n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'}
+page_content=' Notice that if a group G and a subgroup K of G both act two-transitively on a set,  the orbits of G are unions of orbits of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'}
+page_content=' In particular, if G and K have the same number  of orbits, then these orbits are exactly the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'}
+page_content=' Thus, to prove the theorem, it suffices to  show that the ASTs obtained from AGLH(k, n) and ASLH(k, n) have the same size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'}
+page_content=' By  Remark 1, it suffices to show that the two-point stabilizer ASLH(k, n)⃗0,⃗1 has the same orbits  as AGLH(k, n)⃗0,⃗1 on GF(n) \\ {⃗0,⃗1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'}
+page_content='  Indeed, the two-point stabilizers above are given by  ASLH(k, n)⃗0,⃗1 = {v �→ Aφ(v) : A ∈ SL(k, n), φ ∈ H},  and  AGLH(k, n)⃗0,⃗1 = {v �→ Aφ(v) : A ∈ GL(k, n), φ ∈ H}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'}
+page_content='  Direct computation shows that the orbits of ASLH(k, n)⃗0,⃗1 have the following forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'}
+page_content='  (1) One type of orbit has the form  {(φ(a), 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'}
+page_content=' , 0)T : φ ∈ H},  which consists of those vectors whose first coordinate is a Galois conjugate of an  element a ∈ GF(n) with a ̸= 0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'}
+page_content=' The other coordinates are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'}
+page_content='  (2) The remaining orbit is  (GF(n))k \\ Span(⃗1),  consisting of the vectors linearly independent from ⃗1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'}
+page_content='  These are also the orbits of AGLH(k, n)⃗0,⃗1, completing the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'}
+page_content='  □  References  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'}
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+Risk-Averse MDPs under Reward Ambiguity
+Haolin Ruan
+School of Data Science, City University of Hong Kong, Kowloon Tong, Hong Kong
+haolin.ruan@my.cityu.edu.hk
+Zhi Chen
+Department of Management Sciences, College of Business, City University of Hong Kong, Kowloon Tong, Hong Kong
+zhi.chen@cityu.edu.hk
+Chin Pang Ho
+School of Data Science, City University of Hong Kong, Kowloon Tong, Hong Kong
+clint.ho@cityu.edu.hk
+We propose a distributionally robust return-risk model for Markov decision processes (MDPs) under risk and
+reward ambiguity. The proposed model optimizes the weighted average of mean and percentile performances,
+and it covers the distributionally robust MDPs and the distributionally robust chance-constrained MDPs
+(both under reward ambiguity) as special cases. By considering that the unknown reward distribution lies
+in a Wasserstein ambiguity set, we derive the tractable reformulation for our model. In particular, we show
+that that the return-risk model can also account for risk from uncertain transition kernel when one only
+seeks deterministic policies, and that a distributionally robust MDP under the percentile criterion can be
+reformulated as its nominal counterpart at an adjusted risk level. A scalable first-order algorithm is designed
+to solve large-scale problems, and we demonstrate the advantages of our proposed model and algorithm
+through numerical experiments.
+1.
+Introduction
+Markov decision processes (MDPs) provide a powerful modeling framework for sequential decision-
+making problems and reinforcement learning in stochastic dynamic environments (Puterman 2014).
+Obtaining the model parameters of MDPs that perfectly reflect the environments, however, has
+always been a challenge in practice, as these parameters are estimated from limited data that are
+potentially contaminated (Mannor et al. 2007). Moreover, these parameters, such as transition
+kernel and reward function, are often time-dependent or even uncertain, but they are approximated
+as fixed values in an overly simplified setting (Mannor et al. 2016). Therefore, the output policies
+of MDPs are often disappointing in practice.
+Robust MDPs address the aforementioned issues of parameter ambiguity, by allowing the
+unknown values of transition kernels and reward functions to lie in a given ambiguity set (Behza-
+dian et al. 2021, Chen et al. 2019, Clement and Kroer 2021a, Delgado et al. 2016). Then, robust
+MDPs seek for policies that maximize the worst-case expected return over all transition kernels
+1
+arXiv:2301.01045v1  [cs.LG]  3 Jan 2023
+
+Ruan, Chen, Ho: Risk-Averse MDPs under Reward Ambiguity
+2
+and reward functions in the ambiguity sets. By specifying ambiguity sets that contain the unknown
+transition kernels with high confidence, the optimal policies of robust MDPs are robust to param-
+eter ambiguity (Iyengar 2005).
+In this paper, we focus on the case where the reward function is ambiguous, which sometimes
+is referred to as imprecise-reward MDPs (Alizadeh et al. 2015, Regan and Boutilier 2010, 2011a,b,
+2012). This particular setting is also closely related to imitation learning, which trains an agent to
+learn a certain behavior of an expert, while only some demonstrated trajectories of her is available
+(Chen et al. 2020, Ho and Ermon 2016, Osa et al. 2018, Rashidinejad et al. 2021). When applying
+inverse reinforcement learning approach to learn the reward function that completely represents
+the expert’s preference (Brown et al. 2020, Choi and Kim 2012, Ng et al. 2000), the yielded policies,
+which suffer from reward ambiguity, may perform poorly in practice.
+To handle reward ambiguity, we utilize techniques from distributionally robust optimization
+(DRO) (Derman and Mannor 2020) and distributionally robust chance-constrained program (Chen
+et al. 2007, Postek et al. 2018), assuming that the true reward distribution resides in an ambiguity
+set. This approach does not require the reward function to be precisely specified. Instead, only
+the descriptions of common distribution information such as support, moments and shape in the
+ambiguity set are needed, which are often much easier to be obtained/estimated (Hanasusanto
+et al. 2015, 2017, Zymler et al. 2013). In this paper, we consider a Wasserstein ambiguity set for our
+distributionally robust models as in Abdullah et al. (2019), Calafiore and Ghaoui (2006), Xie (2021).
+Unlike phi-divergence ambiguity sets which may contain too extreme member distributions, the
+closeness between points in the support set is incorporated in Wasserstein sets, thus their member
+distributions may be more reasonable (Gao and Kleywegt 2022); on the other hand, Wasserstein
+sets are often a better choice than moment-based ambiguity sets when the number of samples is
+too small to obtain a reliable estimation on moments (Yang 2020). We choose Wasserstein sets
+for these reasons, although other types of ambiguity sets such as nested ambiguity sets (Xu and
+Mannor 2010, 2012) and the ambiguity sets based on Prohorov metric (Erdo˘gan and Iyengar 2006)
+are also considered in literature. For our distributionally robust chance-constrained MDPs, we will
+furthermore show its equivalence with the nominal counterparts with an adjusted risk level. To the
+best of our knowledge, this is the first result in MDPs that establishes the mutual transformation
+between distributional ambiguity and risk.
+Our return-risk model (RR) is a risk-averse MDP model that not only takes into account reward
+ambiguity, but also considers both the average and risk of the return. MDPs that minimize the risk
+of the return instead of the expected cost are called risk-aware MDPs (also called risk-sensitive or
+risk-averse MDPs) (Ahmadi et al. 2021, B¨aauerle and Rieder 2017, Carpin et al. 2016, Haskell and
+Jain 2015, Huang and Haskell 2017). In risk-aware optimization, the objective function is taken as
+
+Ruan, Chen, Ho: Risk-Averse MDPs under Reward Ambiguity
+3
+a risk measure, such as value-at-risk (VaR) (Delage and Mannor 2007, 2010, Gilbert et al. 2017),
+conditional value-at-risk (CVaR) (B¨auerle and Ott 2011, Chow et al. 2017, Huang and Guo 2016)
+and other spectral risk measures (B¨auerle and Glauner 2021), and variants of expected utility
+(Bernard et al. 2022, Jaimungal et al. 2022, Pflug and Wozabal 2007).
+Among these risk measures, VaR and CVaR are arguably the most popular ones and have
+attracted the attention of many researchers (B¨auerle and Ott 2011, Chow et al. 2017, Delage and
+Mannor 2007, 2010, Gilbert et al. 2017, Huang and Guo 2016). By using CVaR, one aims to give a
+precise depiction of the extreme tail of the distribution (of the uncertain rewards), while VaR does
+not reflect the extreme scenerios exceeding VaR. It is well-known that CVaR is a coherent risk
+measure, which can be efficiently optimized by convex optimization tools (Chen and Xie 2021); in
+contrast, VaR is a more challenging risk measure because it is not a coherent one.
+One remarkable advantage of VaR is its stability of estimation (especially under fat-tailed reward
+distribution (Sarykalin et al. 2008)), which is particularly important under data-driven settings
+where the number of samples are limited and decision makers evaluate models based on their
+out-of-sample performances (Bertsimas and Thiele 2006, van de Berg et al. 2022, Zheng et al.
+2016). To demonstrate, we provide an example where we consider a one-step MDP with only 1
+state s and 2 actions a1 and a2 (Sutton and Barto 2018). In this one-step MDP, the decision
+maker only makes one decision in each episode, and she aims to maximize her VaR/CVaR of
+rewards for these episodes. We consider uncertain rewards ˜rs,a1 ∼ Pt-dist and ˜rs,a2 = ˜rs,a1 + ρ|s|
+where Pt-dist is a Student’s t-distribution and we vary its degree of freedom δ ∈ {2,3,4}. We set
+the shift ratios ρ = {0.05i}i∈[5], and for testing the estimation accuracy w.r.t. VaR (resp., CVaR)
+(where we choose the risk threshold 10%), we set the shift quantity s as Pt-dist-VaR0.1[˜rs,a1] (resp.,
+Pt-dist-CVaR0.1[˜rs,a1]), where both risk measures can be efficiently calculated (see Appendix B for
+more details). We evaluate the decision maker’s accuracy rate as the proportion of testing samples
+where she has chosen the action with a higher VaR/CVaR of rewards (i.e., action a2); for each
+pair of accuracy rate and shift ratio, following Yamai et al. (2002), 1000 random reward samples
+for each state-action pair are available for the decision maker, and we test her accuracy rate based
+on 10000 testing samples.
+As illustrated in Figure 1, the accuracy rate increases with the shift ratio ρ. As δ decreases, F
+becomes more fat-tailed, and the accuracy rate of VaR is remarkably higher than that of CVaR,
+which indicates that the statistical inference on VaR would be more accurate than on CVaR.
+Therefore, VaR may be a more preferable choice when only small sample sets are available.
+Our return-risk model is motivated by the soft-robust criterion/model, which optimizes a convex
+combination of the mean and a robust performance in the optimization literature (Ben-Tal et al.
+2010). MDPs with soft-robustness are also popular in recent years, where decision makers aim to
+
+Ruan, Chen, Ho: Risk-Averse MDPs under Reward Ambiguity
+4
+0.05
+0.10
+0.15
+0.20
+0.25
+Shift ratio
+0.6
+0.7
+0.8
+0.9
+1.0
+Accuracy rate
+VaR
+CVaR
+0.05
+0.10
+0.15
+0.20
+0.25
+Shift ratio
+0.6
+0.7
+0.8
+0.9
+1.0
+VaR
+CVaR
+0.05
+0.10
+0.15
+0.20
+0.25
+Shift ratio
+0.6
+0.7
+0.8
+0.9
+1.0
+VaR
+CVaR
+Figure 1
+The accuracy rates of the decision maker choosing the correct action (so that the VaR/CVaR of her
+rewards is maximized): δ = 4 (left), δ = 3 (middle) and δ = 2 (right).
+maximize a weighted average of the mean and percentile performances (Brown et al. 2020, Lobo
+et al. 2020). Unlike these existing soft-robust MDPs, however, the proposed return-risk model is
+fundamentally different in two aspects: first, these existing soft-robust models have no consideration
+for reward ambiguity, while we utilize distributionally robustness to account for reward ambiguity,
+by which we can hedge against the most adversarial realization of the distribution of rewards
+(within the ambiguity set), thus our model is more robust to reward uncertainty (Chen et al. 2019,
+Xu and Mannor 2010); second, we choose VaR as the risk measure which has a direct interpretation
+to percentile performances, and, as illustrated above, tends to be more advantageous in data-driven
+optimization.
+Our work concentrates on model-based setting, where our proposed models are motivated by
+the classical (dual formulation of) nominal MDPs (Puterman 2014) and the chance-constrained
+MDPs (Delage and Mannor 2010). It is worth noting that, beyond model-based setting, there are
+other inspiring and innovative researches on robust reinforcement learning, such as robust TDC
+algorithms and robust Q-learning (Roy et al. 2017, Wang and Zou 2021), robust policy gradient
+(Wang and Zou 2022), least squares policy iteration (Lagoudakis and Parr 2003) and sample
+complexity analysis (Panaganti and Kalathil 2022). Note that, though model-free reinforcement
+learning can be used to learn satisfactory policies for complex environment, the requirement of
+large amounts of interaction (with environment) may render the learning process slow (Kaiser et al.
+2019), while high sample efficiency is one strong advantage of model-based learning (Sutton and
+Barto 2018). We also note that MDPs with transition kernel ambiguity is another active research
+line where distributionally robustness is widely employed (Clement and Kroer 2021b, Shapiro 2016,
+2021, Xu and Mannor 2012).
+We may summarize our contributions as follows (and we also compare our contributions to those
+of related works in Table 2 in Appendix I).
+
+Ruan, Chen, Ho: Risk-Averse MDPs under Reward Ambiguity
+5
+(i) We show that the distributionally robust model of optimizing expected rewards can be
+reformulated as a convex conic program, which is equivalent to the nominal MDP with a convex
+regularization in the objective function.
+(ii) For distributionally robust chance-constrained MDPs (DCC), we show that it can be refor-
+mulated as nominal chance-constrained MDPs at adjusted risk levels. This observation bridges the
+gap between risk and parameter ambiguity.
+(iii) Combining the proposed models in (i) and (ii), we propose the return-risk MDP that
+maximizes the weighted average of the expectation and VaR of reward (both under distributionally
+robustness to reward uncertainty), which is flexible and can perform well under the criteria of mean
+and percentile returns.
+(iv) When only considering deterministic policies, we show that our return-risk model can also
+account for risk from uncertain transition kernel, and we derive its equivalent reformulation as a
+mixed-integer second-order cone program (MISOCP).
+(v) To solve the proposed return-risk model, we design a first-order method that is more scalable
+than the MOSEK solver, thus is faster with large-size problems.
+(vi) In the simulation and empirical experiments, we adopt a data-driven setting, where the
+decision maker aims at maximizing the expectation and VaR of the random reward. We compare
+the performances of distributionally robust MDPs (DRMDPs), DCC, RR, robust MDPs (RMDPs)
+(Delage and Mannor 2010) and BROIL (Brown et al. 2020), and results show that the third
+one performs the best under both expectation and different VaR’s (with risk thresholds 5%, 10%
+and 15%), which showcases its advantages and adjustability to the decision makers’ changeable
+preferences between return and risk.
+The remainder of this paper is organized as follows. We introduce the background in Section 2.
+In Sections 3 and 4, we study DRMDPs as well as the DCC model, respectively, and we derive
+their tractable reformulations. Combining these proposed models, we propose the RR model in
+Section 5. The designed first-order algorithm for the RR model is detailed in Section 6. We compare
+the performances of DRMDP, DCC, RR, RMDP and BROIL, and demonstrate the advantage of
+our proposed algorithm in Section 7. Conclusion is drawn in Section 8.
+2.
+Background
+We consider an infinite-horizon MDP with a finite state space S = {1,··· ,S} and a finite action
+space A = {1,··· ,A}. Let P ∈ RS×A×S be the transition probability kernel such that ps,a,s′ is
+denoted to be the transition probability of transiting to state s′ ∈ S when action a ∈ A is chosen
+in state s ∈ S; thus, ps,a ∈ ∆S is the transition probability distribution for every (s,a) ∈ S × A.
+Given the state-action pair (s,a), an agent will receive an expected reward rs,a ∈ R. To simplify
+our notation, we denote the reward function as a vector r = {rs,a}(s,a)∈S×A.
+
+Ruan, Chen, Ho: Risk-Averse MDPs under Reward Ambiguity
+6
+We seek for the optimal stationary randomized policy π = {πs}s∈S with πs ∈ ∆A for all s ∈ S,
+where an action a ∈ A will be taken in state s ∈ S with probability πs,a. A nominal MDP that
+maximizes the expected reward can be formulated (Puterman 2014) as
+ℓN = max
+x∈X r⊤x,
+(1)
+where the feasible set X is given by X =
+�
+x ∈ RSA
++
+�� (E − γ · ¯P )x = p0
+�
+. Here the coefficient
+matrices E = diag(e⊤,··· ,e⊤) ∈ RS×SA with S all-ones vectors e ∈ RA and ¯P = (¯p1,··· , ¯pS)⊤ ∈
+RS×SA with ¯ps = {ps′,a,s}(s′,a)∈S×A for all s ∈ S. For each (s,a) ∈ S × A, we denote the sth sub-
+vector of x as xs = {xi}i∈{(s−1)A+1,··· ,sA}; its ath component xs,a can be interpreted as the total
+discounted probability one occupying state s and choosing action a when applying the policy
+π⋆
+s,a = x⋆
+s,a/(�
+a∈A x⋆
+s,a) ∀(s,a) ∈ S × A (Puterman 2014)1. We have a discount factor γ ∈ (0,1) and
+the initial distribution p0 ∈ RS
+++ of the initial states. Problem (1) is a linear program that can be
+efficiently solved by simplex method and interior-point method (Nocedal and Wright 2006). One
+can also compute the optimal policy efficiently by applying value iteration or policy iteration to
+solve the associated Bellman equation of this problem (Bertsekas and Tsitsiklis 1995, Puterman
+2014).
+The nominal MDP (1) does not account for uncertainty in either rewards or transition kernel.
+To account for reward uncertainty, Delage and Mannor (2010) assume that the random reward
+vector ˜r follows a known Gaussian distribution P and propose a chance-constrained MDP model
+as follows:
+ℓCC(ε) =
+�
+�
+�
+�
+�
+�
+�
+max y
+s.t. P[˜r⊤x ≥ y] ≥ 1 − ε
+x ∈ X, y ∈ R.
+(2)
+In fact, the above chance-constrained model maximizes the VaR (at the risk level 1 − ε) of the
+reward with respect to the distribution P. Since P is assumed Gaussian, by theorem 10.4.1 in
+Pr´ekopa (2013), one can reformulate problem (2) as a second-order cone program as follows:
+ℓCC(ε) = max
+x∈X EP[˜r⊤x] − ∥F−1(1 − ε)Σ1/2x∥2,
+where F−1(·) is the inverse of the cumulative density function of the Gaussian distribution P
+and Σ is the covariance matrix of P. Second-order cone programs allow efficient solutions by
+state-of-the-art commercial solvers such as CPLEX, Gurobi and MOSEK (see, e.g., Ben-Tal and
+Nemirovski (2001)). Despite its tractability, the chance-constrained MDP (2) requires the precise
+underlying reward distribution as input. Moreover, the above reformulation does not hold for
+generic distribution P.
+1 By Puterman (2014), any x ∈ X admits such interpretation, thus we can retrieve our policies of all the proposed
+models in this paper in this way.
+
+Ruan, Chen, Ho: Risk-Averse MDPs under Reward Ambiguity
+7
+3.
+Distributionally Robust MDPs
+In many real-world situations, the true distribution of the uncertain reward is hard (if not impossi-
+ble) to obtain. Instead, we may have some firm knowledge, such as moments and shape about it. As
+one of the most efficacious treatments for such situations, the DRO approach models uncertainty
+as a random variable governed by an unknown probability distribution residing in an ambiguity
+set. Facing distributional ambiguity, a decision maker seeks for solutions that hedge against the
+most adversarial distribution from within the ambiguity set. To be specific, in our context, we
+assume that the true distribution of the uncertain reward resides in a Wasserstein ball of radius
+θ ≥ 0 around some reference distribution ˆP:
+F(θ) = {P ∈ P(RSA) | dW
+�
+P, ˆP
+�
+≤ θ}.
+(3)
+Here P(RSA) is the set of all probability distributions on RSA, and the Wasserstein distance
+between two distributions P1 and P2, equipped with a general norm ∥ · ∥ in RSA, is given by
+dW (P1,P2) = infP∈Q(P1,P2) EP[∥˜r1 − ˜r2∥], where Q(P1,P2) is the set of all joint distributions with
+marginal distributions P1 and P2 that govern ˜r1 and ˜r2, respectively.
+The random parameter in the nominal MDP (1) is the expectation of reward, which in practice,
+is often estimated by the average of historical samples. However, when the sample size is small,
+such a sample average is not close to the expectation but rather, is known to be optimistically
+biased (see, e.g., Smith and Winkler (2006)). Hence, the nominal MDP (1) based on samples
+may yield an unsatisfactory policy that does not perform well out-of-sample. For this reason, a
+possible alternative is to maximize instead the worst-case expected reward as in the following
+distributionally robust MDP:
+ℓDRMDP(θ) = max
+x∈X
+inf
+P∈F(θ)EP[˜r⊤x].
+(4)
+The following proposition offers an equivalent conic program for (4).
+Proposition 1. The distributionally robust MDP (4) can be reformulated a conic program
+ℓDRMDP(θ) = max
+x∈X EˆP[˜r⊤x] − θ · ∥x∥∗.
+It is not hard to observe that the distributionally robust MDPs can be viewed as a convex reg-
+ularization of the nominal MDP (4) under the reference distribution ˆP. In particular, the convex
+regularizing term in the distributionally robust MDP is θ∥x∥∗, which is sized by the Wasserstein
+radius θ. Interestingly, we have also found that an (distributionally) optimistic MDP can be refor-
+mulated as a reverse conic program with a (concave) regularization term −θ∥x∥∗. We relegate this
+result to Appendix D.
+
+Ruan, Chen, Ho: Risk-Averse MDPs under Reward Ambiguity
+8
+Figure 2
+Values of ε with respect to different θ’s: ε = 0.05 (left), ε = 0.1 (middle), and ε = 0.15 (right).
+We remark that, problem (4) is indeed a special case of the robust optimization problem consid-
+ered in Jaimungal et al. (2022), where we consider the expected utility framework. Compared to the
+policy gradient methods provided in Jaimungal et al. (2022) where convergence is not established,
+we have derived its equivalent reformulation as a tractable conic program which can be efficiently
+solved by state-of-the-art commercial solvers such as Gurobi, Mosek and CPLEX, and can also
+be seamlessly incorporated in the tractable reformulation of our proposed return-risk model in
+Section 5.
+4.
+Distributionally Robust Chance-Constrained MDPs
+In this section, we turn from optimizing the expectation of reward to its tailed performance, by
+exploring chance-constrained MDPs. In particular, we still consider Wasserstein ambiguity sets (3)
+to account for distributional ambiguity, meanwhile specifying the reference distribution ˆP and the
+norm ∥ · ∥ in the definition of the Wasserstein distance.
+For the former, we focus on an elliptical reference distribution ˆP = P(µ,Σ,g)
+2 throughout this
+section, whose probability density distribution is given by f(r) = k · g
+� 1
+2(r − µ)⊤Σ−1(r − µ)
+�
+,
+where k is a positive normalization scalar, µ is a mean vector, Σ is a positive definite matrix and g
+is a generating function. We emphasize that this assumption on ˆP is mild as this is only the center
+of the ambiguity set. In particular, our proposed distributionally robust chance-constrained MDPs
+can account for all types of distributions (as long as they are inside the ambiguity set) and they are
+not restricted to be all elliptical. As we shall see, such specifications lead to tractable reformulation
+of our proposed models. Preliminaries on elliptical distributions are relegated to Appendix C.
+For the latter, we adopt the Mahalanobis norm associated with the positive definite matrix Σ,
+captured by ∥x∥Σ =
+√
+x⊤Σ−1x. Note that the dual norm of a Mahalanobis norm ∥ · ∥Σ is another
+Mahalanobis norm ∥ · ∥Σ−1 that is defined by the inverse matrix Σ−1.
+2 Note that results in Section 3 hold for a general reference distribution.
+
+1e-03
+8e-04
+6e-04
+4e-04
+2e-04
+0e+00
+0.040
+0.045
+0.0501e-03
+8e-04
+6e-04
+4e-04
+2e-04
+0e+00
+0.085
+0.090
+0.095
+0.1001e-03
+8e-04
+6e-04
+4e-04
+2e-04
+0e+00
+0.130
+0.135
+0.140
+0.145
+0.150Ruan, Chen, Ho: Risk-Averse MDPs under Reward Ambiguity
+9
+In a distributionally robust chance-constrained MDP, we hope that even in the worst-case, with
+a high confidence the reward is no less than a lower bound, and we aim at maximizing such a lower
+bound by solving
+ℓDCC(θ,ε) =
+�
+�
+�
+�
+�
+�
+�
+�
+�
+max y
+s.t.
+inf
+P∈F(θ)P[˜r⊤x ≥ y] ≥ 1 − ε
+x ∈ X, y ∈ R.
+(5)
+Quite notably, the worst-case chance constraint in the pessimistic chance-constrained MDP (5) is
+equivalent to a nominal chance constraint in (2) with a higher risky level.
+Lemma 1. Suppose in the Wasserstein ambiguity set (3), the reference distribution is an ellip-
+tical distribution ˆP = P(µ,Σ,g) and the Wasserstein distance is equipped with a Mahalanobis norm
+associated with the positive definite matrix Σ. The distributionally robust chance constraint
+∀ P ∈ F(θ) : P[˜r⊤x ≥ y] ≥ 1 − ε
+(6)
+is satisfiable if and only if P(µ,Σ,g)[˜r⊤x ≥ y] ≥ 1 − ε, where ε = 1 − Φ(¯η⋆) ≤ ε with ¯η⋆ that can
+be computed via bisection method which searches for the smallest η ≥ Φ−1(1 − ε) that satisfies
+η(Φ(η) − (1 − ε)) −
+� η2/2
+(Φ−1(1−ε))
+2/2 kg(z)dz ≥ θ.
+Equipped with Lemma 1, it then turns out that the distributionally robust chance-constrained
+MDP (5) is equivalent to a nominal chance-constrained MDP (2) at a higher risky level. Conse-
+quently, the distributionally robust chance-constrained MDP (5) can be reformulated into a conic
+program, or more precisely, a second-order cone program owing to our choice of the Mahalanobis
+norm.
+Proposition 2. Suppose in the Wasserstein ambiguity set (3), the reference distribution is an
+elliptical distribution ˆP = P(µ,Σ,g) and the Wasserstein distance is equipped with a Mahalanobis
+norm associated with the positive definite matrix Σ. If the risk threshold satisfies ε < 0.5, then the
+distributionally robust chance-constrained MDP (5) is equivalent to the second-order cone program
+ℓDCC(θ,ε) = max
+x∈X µ⊤x − ∥Φ−1(1 − ε)Σ1/2x∥2,
+where ε = 1 − Φ(¯η⋆) ≤ ε with ¯η⋆ being the smallest η ≥ Φ−1(1 − ε) that satisfies η(Φ(η) − (1 − ε)) −
+� η2/2
+(Φ−1(1−ε))
+2/2 kg(z)dz ≥ θ.
+Similar to the distributionally robust MDPs in Section 3, the distributionally robust chance-
+constrained MDPs also admit an optimistic counterpart, which is equivalent to the nominal chance-
+constrained MDPs with a larger risk threshold. We relegate this result to Appendix E.
+To conclude this section, we present in Figure 2 the relations between ε and ε. Indeed, for any
+fixed ε, there is a one-to-one correspondence between the risk threshold ε and the Wasserstein
+radius θ. Following from this fact, for the chance-constrained model in our numerical experiments
+(Section 7), we only calibrate the risk threshold rather than the Wasserstein radius.
+
+Ruan, Chen, Ho: Risk-Averse MDPs under Reward Ambiguity
+10
+5.
+Return-Risk MDP
+For rational decision makers, two types of rewards are their chief concerns: the average and the
+worst-case rewards. However, the risk-averse models often can not achieve decent average return
+on which the model put no emphasis (Carpin et al. 2016, Delage and Mannor 2010, Jiang and
+Powell 2018). To take both concerns into considerations, we leverage the established DRMDPs and
+DCC model in Sections 3 and 4 as ingredients and propose the return-risk MDP that maximizes
+the weighted average of the worst-case expectation and VaR of reward as follows:
+ℓRR(α,θ,ε) = max
+x∈X α inf
+P∈F(θ)EP[˜r⊤x] + (1 − α)
+inf
+P∈F′(θ)P-VaRε[˜r⊤x].
+(7)
+Here the Wasserstein ball F(θ) is assumed equipped with a general reference distribution and an
+L2-norm in the definition of the Wasserstein distance, while an elliptical reference distribution
+ˆP = P(µ,Σ,g) and a Mahalanobis norm associated with the positive definite matrix Σ are assumed for
+F ′(θ). It is not hard to see that the return-risk MDP (7) takes the distributionally robust MDP (4)
+and the distributionally robust chance-constrained MDP (5) in as special cases by varying ε, θ and
+α ∈ {0,1}. Furthermore, by choosing a fractional α, the return-risk model enables one to tailor a
+balance between risk and return. Proposition 3 below provides an equivalent second-order cone
+program for the return-risk MDP (7) under these assumptions.
+Proposition 3. Suppose in (7) the Wasserstein ball F(θ) (resp., F ′(θ)) is equipped with a
+general distribution (resp., an elliptical reference distribution ˆP = P(µ,Σ,g)) and the norms in the
+definitions of the Wasserstein distances of F(θ) and F ′(θ) are an L2-norm and the Mahalanobis
+norm associated with Σ ≻ 0, respectively. Assume that the risk threshold satisfies ε < 0.5, then the
+return-risk MDP (7) is equivalent to a second-order cone program
+ℓRR(α,θ,ε) = max
+x∈X µ⊤x − αθ · ∥x∥2 − (1 − α) · ∥Φ−1(1 − ε)Σ1/2x∥2,
+(8)
+where ε = 1 − Φ(¯η⋆) ≤ ε with ¯η⋆ being the smallest η ≥ Φ−1(1 − ε) that satisfies η(Φ(η) − (1 − ε)) −
+� η2/2
+(Φ−1(1−ε))
+2/2 kg(z)dz ≥ θ, and it could be computed via bisection method.
+5.1.
+Risk-Awareness for Uncertain Transition Kernel
+By adopting the static soft-robust framework in Lobo et al. (2020), one can indeed also account
+for the uncertainty in transition kernel in our return-risk model. As in Lobo et al. (2020), suppose
+we have finite samples of transition kernel { ˆP i}i∈[N] with weights w ∈ ∆N := {w ∈ RN
++ | e⊤w = 1}
+that are generated by MCMC (see, e.g., Kruschke (2010)). Our proposed model is then as follows:
+max
+π∈(∆A)S ψ · EˆP[g(π, ˜P )] + (1 − ψ) · ˆP-CVaRι[g(π, ˜P )].
+(9)
+
+Ruan, Chen, Ho: Risk-Averse MDPs under Reward Ambiguity
+11
+max (1 − ψ)(η −
+1
+1 − ι
+�
+i∈[N]
+yi) + ψ ·
+�
+i∈[N]
+(µ⊤xi − αθ · ∥xi∥2 − (1 − α)∥Φ−1(1 − ε)Σ1/2xi∥2)
+s.t. yi − wiη ≥ αθ · ∥xi∥2 + (1 − α) · ∥Φ−1(1 − ε)Σ1/2xi∥2 − µ⊤xi
+∀i ∈ [N]
+(E − γ · ¯P i)xi = wi · p0
+∀i ∈ [N]
+xi ≤
+wi
+1−γπ
+∀i ∈ [N]
+xi
+s,a ≥
+wi
+1 − γ (πs,a − 1) +
+�
+a′∈A
+xi
+s,a′
+∀(i,s,a) ∈ N × S × A
+π ∈ (∆A)S ∩ {0,1}SA,η ∈ R,xi ∈ RSA
++ ,y ∈ RN
++
+∀i ∈ [N].
+Figure 3
+Reformulation of (9) as an MISOCP.
+Here the objective function in (9) is again soft-robust against the uncertainty (in transition kernel),
+with the weight ψ ∈ [0,1] as the controller for the robustness and ι ∈ [0,1] is the risk threshold (w.r.t.
+the uncertain transition kernel). The weighted empirical distribution ˆP[ ˜P = ˆP i] = wi ∀i ∈ [N] and
+the function
+g(π,P ) = max µ⊤x − αθ · ∥x∥2 − (1 − α) · ∥Φ−1(1 − ε)Σ1/2x∥2
+s.t. xs,a = πs,a ·
+�
+a′∈A
+xs,a′
+∀(s,a) ∈ S × A
+(E − γ · ¯P )x = p0
+x ∈ RSA
++
+represents the optimal value of the return-risk model with the additional constraint that the optimal
+policy should be the input π ∈ (∆A)S and with ¯P as the coefficient matrix corresponding to the
+input transition kernel P .
+Quite notably, when focusing on deterministic policies, one can reformulate (9) as an MISOCP.
+Proposition 4. If π is restricted to be a deterministic policy (i.e., π ∈ (∆A)S ∩{0,1}SA), prob-
+lem (9) has an equivalent MISOCP reformulation as in Figure 3.
+We remark that, though deterministic policies seem to be restricted compared to the randomized
+ones, they actually are more favored under some situations; for example, they may be a more
+suitable choice in some medical domains where randomized policies are unworkable for practical
+and philosophical reasons (Rosen et al. 2006). Also, randomized policies may be difficult to be
+evaluated after they have been deployed and may have poor reproducibility (Lobo et al. 2020).
+6.
+First-Order Method
+In this section, we introduce an efficient first-order algorithm to solve the equivalent formulation (8)
+of our return-risk model. Our algorithm is based on an alternating direction linearized proximal
+
+Ruan, Chen, Ho: Risk-Averse MDPs under Reward Ambiguity
+12
+method of multipliers (AD-LPMM) algorithm (Beck 2017, Shefi and Teboulle 2014), which is a
+variant of the alternating direction method of multiplier (ADMM) algorithm and also has a con-
+vergence rate of O(1/N) (here N is the number of iterations) proved by Beck (2017). The proposed
+splitting allows efficient update of variables in AD-LPMM (where the solutions are analytical or
+can be retrieved by an efficient bisection method).
+For the primal update of the ADMM algorithm, one needs to solve minimization problems with
+a quadratic term involved (in its objective function); in AD-LPMM, this quadratic term can be
+linearized by adding a proximity term to the objective function, which could render the primal
+update much easier. To implement our AD-LPMM algorithm, first we will introduce auxiliary
+variables and rewrite (8) (as a minimization problem) as follows:
+min αθ · ∥x∥2 + (1 − α) · ∥Φ−1(1 − ε)Σ1/2y∥2 − µ⊤z
+s.t. (E − γ · ¯P )x = p0
+x = y
+x = z
+x ∈ RSA,y ∈ RSA,z ∈ RSA
++ ,
+(10)
+where, in the spirit of AD-LPMM, we can split the decision variables into two groups and update
+them separately. The augmented Lagrangian function of (10) is:
+L(x,y,z;λ,ξ,η)
+= αθ · ∥x∥2 + (1 − α)Φ−1(1 − ε) · ∥Σ1/2y∥2 − µ⊤z + λ⊤((E − γ · ¯P )x − p0) + ξ⊤(x − y)
++η⊤(x − z) + c
+2 ·
+��������
+(E − γ · ¯P )x − p0
+x − y
+x − z
+��������
+2
+2
+.
+Based on our splitting method, we will update the two groups of variables (y,z) and x separately.
+For the update of (y,z), we define two primal update operators
+Py(x,ξ;c) = arg min
+y
+(1 − α)Φ−1(1 − ε) · ∥Σ1/2y∥2 − ξ⊤y + c
+2 · ∥x − y∥2
+2
+and Pz(x,η;c) = arg min
+z≥0
+−z⊤(µ + η) + c
+2 · ∥x − z∥2
+2; while for the update of x (i.e., the second
+group of variables), we define
+Px(y,z,λ,ξ,η;c,ν, ˆx) = arg min
+x
+αθ · ∥x∥2 + x⊤((E − γ · ¯P )⊤λ + ξ + η)
++ c
+2 ·
+��������
+(E − γ · ¯P )x − p0
+x − y
+x − z
+��������
+2
+2
++ 1
+2 · ℓ2
+Q(c,ν)(x − ˆx),
+
+Ruan, Chen, Ho: Risk-Averse MDPs under Reward Ambiguity
+13
+Algorithm 1: AD-LPMM for Problem (10)
+Input: Frobenius norm ν = ∥(E − γ · ¯P )⊤(E − γ · ¯P ) + 2 · I∥F, initial stepsize c0 > 0,
+stepsize growth rate β > 0, desired precision δ, x0, y0, z0, λ0, ξ0, η0, k ← 0
+while
+��������
+(E − γ · ¯P )xk − p0
+xk − yk
+xk − zk
+��������
+∞
+≥ δ do
+// Primal update
+step 1: yk+1 ← Py(xk,ξk;ck);
+step 2: zk+1 ← Pz(xk,ηk;ck);
+step 3: xk+1 ← Px(yk+1,zk+1,λk,ξk,ηk;ck,ν,xk);
+// Dual update
+step 4: λk+1 ← λk + ck · ((E − γ · ¯P )xk+1 − p0);
+step 5: ξk+1 ← ξk + ck · (xk+1 − yk+1);
+step 6: ηk+1 ← ηk + ck · (xk+1 − zk+1);
+// Increase stepsize
+step 7: ck+1 ← ck + βc0;
+step 8: k ← k + 1;
+end
+Output: Solution xk
+where Q(c,ν) = c · ((ν − 2) · I − (E − γ · ¯P )⊤(E − γ · ¯P )) and ℓQ(·) (equipped with a positive
+semi-definite matrix Q) is a weighted vector norm such that ℓQ(x) =
+�
+x⊤Qx. As we shall see in
+Section 6.3, the update of x is fast (where an analytical solution is available) with the proximity
+term (1/2)·ℓ2
+Q(c,ν)(x− ˆx) added. Note that when Q(c,ν) ≡ 0, the update in AD-LPMM degenerates
+to an ADMM’s one.
+We now introduce our AD-LPMM in Algorithm 1. Basically, the most time-consuming computa-
+tions lie in the primal update phase, where the updates are carried out by solving a minimization
+problem with other variables fixed at values after their last updates. As shall be detailed soon,
+owing to our variable splitting method, the primal updates are also quite fast, where analytical solu-
+tions or solutions obtained by bisection are available. Here we choose a stepsize that is increasing
+in every iteration (with a growth rate β > 0), which in practice accelerates the convergence.
+
+Ruan, Chen, Ho: Risk-Averse MDPs under Reward Ambiguity
+14
+6.1.
+Subproblem in Step 1: Proximal Mapping and Projection
+To solve Py(x,ξ;c), first we would utilize the technique of proximal mapping and establish the
+following equivalences:
+Py(x,ξ;c) = Prox (1−α)Φ−1(1−ε)
+c
+·���·∥Σ(x + 1
+c · ξ)
+= x + 1
+c · ξ − (1−α)Φ−1(1−ε)
+c
+· ProjBℓΣ−1 (·)
+�
+1
+(1−α)Φ−1(1−ε) · (c · x + ξ)
+�
+,
+(11)
+where Proxf(·)(x) = arg minv f(v) + 1
+2 · ∥v − x∥2
+2 is the proximal mapping operator and
+ProjBℓΣ(·)(x) = arg min
+v:ℓΣ(v)≤1
+1
+2 · ∥v − x∥2
+2
+(12)
+is the operator of projection on the unit ball BℓΣ(·) = {x ∈ RSA | ℓΣ(x) ≤ 1}. Here, the first equality
+in (11) holds by the definition of the proximal mapping operator, and the second equality follows
+from,e.g., example 6.4.7 in Beck (2017). Indeed, problem (12) allows an efficient solution obtained
+by a bisection method to locate its optimal dual solution λ⋆ ≥ 0 (after which the optimal primal
+solution can be retrieved immediately), where the upper bound of the bisection is provided in
+Lemma 2 relegated to Appendix A.4. The time complexity of the solution process (11), as well as
+the pseudocode for the bisection method, are provided in the following proposition.
+Proposition 5. Problem Py(x,ξ;c) can be solved in time O(SAlog(1/δ′)), where δ′ is the
+desired precision of the bisection method.
+6.2.
+Subproblem is Step 2: Componentwise Update
+Problem Pz(x,η;c) can be decomposed into SA single-variable quadratic programming problems,
+each allowing an analytical solution. We summarize the time complexity and details in the following
+proposition.
+Proposition 6. Problem Pz(x,η;c) can be solved in time O(SA).
+6.3.
+Subproblem in Step 3: Linearization and Proximal Mapping
+Compared to the update in ADMM, in our AD-LPMM, a proximity term (1/2) · ℓ2
+Q(c,ν)(x − ˆx)
+is added to the objective function of the update in step 3. By choosing Q(·,·) as mentioned in
+Section 6, we can linearize all the quadratic terms in Px(y,z,λ,ξ,η;c,ν, ˆx), thus the solution can
+be obtained analytically by the technique of proximal mapping (meanwhile assuring the positive
+semi-definiteness of Q(ck,ν) in every iteration of Algorithm 1). This solution process, as well as its
+time complexity, is provided in the following proposition.
+Proposition 7. Problem Px(y,z,λ,ξ,η;c,ν, ˆx) can be solved in time O(SA).
+
+Ruan, Chen, Ho: Risk-Averse MDPs under Reward Ambiguity
+15
+100
+200
+300
+400
+500
+Sample size
+15
+14
+13
+VaR ( '=0.15)
+ 
+DRMDP
+CC
+RR
+BROIL
+RMDP
+100
+200
+300
+400
+500
+Sample size
+14
+12
+10
+Mean
+ 
+DRMDP
+CC
+RR
+BROIL
+RMDP
+Figure 4
+Empirical study. Models DRMDP (4), CC (2), RR (7), RMDP and BROIL evaluated by VaR (risk
+threshold ε′ = 15%) and mean of reward. The upper and lower edges of the shaded areas are respectively
+the 95% and 5% percentiles of the 100 performances, while the solid lines are the medians.
+7.
+Numerical Experiments
+In this section, we conduct two numerical experiments to compare the performances of
+DRMDPs (4), CC (2)3, RR (7), RMDPs (Delage and Mannor 2010) and BROIL (Brown et al.
+2020) (please see Appendices F and G for more details for the last two models). In both experi-
+ments, we train our reward functions with different sample sizes (100,200,300,400,500). For each
+sample size, performance of each model is evaluated for 100 times. The performance of each model
+is evaluated by expectation and VaR with risk thresholds ε′ ∈ {5%,10%,15%}. Cross validations
+are conducted for parameter selection (please see Appendix H.1 for details).
+In Section 7.1, we conduct a simulation study where MDPs are generated randomly as in Regan
+and Boutilier (2012); In Section 7.2, we study a machine replacement problem introduced in
+Delage and Mannor (2010). As implied in our proofs, in this section, the Wasserstein ambiguity
+set of DRMDPs (4) will be equipped with a general reference distribution and an L2-norm for the
+Wasserstein distance; as for RR (7), we use a general reference distribution and an L2-norm in the
+definition of the Wasserstein distance for the Wasserstein ambiguity set F(θ), while for F ′(θ), we
+use an elliptical reference distribution ˆP = P(µ,Σ,g) and the Mahalanobis norm associated with the
+positive definite matrix Σ for the Wasserstein distance. All optimization problems are solved by
+MOSEK on a 2.3GHz processor with 32GB memory.
+
+Ruan, Chen, Ho: Risk-Averse MDPs under Reward Ambiguity
+16
+100
+200
+300
+400
+500
+Sample size
+1600
+1650
+1700
+1750
+VaR ( '=0.15)
+ 
+DRMDP
+CC
+RR
+BROIL
+RMDP
+100
+200
+300
+400
+500
+Sample size
+1750
+1800
+1850
+Mean
+ 
+DRMDP
+CC
+RR
+BROIL
+RMDP
+Figure 5
+Simulation. Models DRMDP (4), CC (2), RR (7), RMDP and BROIL evaluated by VaR (risk threshold
+ε′ = 15%) and mean of reward. The upper and lower edges of the shaded areas are respectively the 95%
+and 5% percentiles of the 100 performances, while the solid lines are the medians.
+7.1.
+Simulation Study
+In this experiment, we follow the experiment setup in Regan and Boutilier (2012) where the number
+of reachable next-states and the transition kernel are randomly generated (both of which are known
+to decision makers). More details of the experiment setting are relegated to Appendix H.2.
+As illustrated in Figures 5 and 7 (where the latter for VaR with ε′ ∈ {5%,10%} is relegated
+to Appendix H.4), when the decision maker aims to optimize her tailed performances, CC is a
+preferable choice compared to DRMDPs; on the contrary, when pursuing optimizing the average
+return, DRMDPs perform much better than CC. Observe that the RR model, which includes both
+DRMDPs and the DCC model as special cases, remains as the best model under all criteria. In
+particular, one can observe that, RR achieves higher percentile returns than BROIL (that is a
+model without robustness), which demonstrates the benefits of distributionally robustness and the
+advantage of the risk measure VaR for percentile performance optimization. As expected, RMDPs
+end up yielding over-conservative policies; as a result, it performs poorly in most instances under
+all criteria.
+7.2.
+Machine Replacement Problem
+In this experiment, we follow the experiment setup in Delage and Mannor (2010) and consider
+the case where a factory holds an extensive amount of machines, each of which is subject to the
+same underlying MDP (more details of the experiment setting can be found in Appendix H.2). Our
+setting is similar to Delage and Mannor (2010) except for the follows: we use a data-driven setting
+3 As we demonstrated in Section 4, a DCC is equivalent to a nominal chance-constrained one with an adjusted risk
+level, thus here we simply choose the latter as the benchmark.
+
+Ruan, Chen, Ho: Risk-Averse MDPs under Reward Ambiguity
+17
+as described above, and we evaluate our (policies of) models by looking at the various performance
+measures as in Section 7.1.
+We report the overall performances of the five models in Figures 4 and 8 (where the latter for
+VaR with ε′ ∈ {5%,10%} is relegated to Appendix H.5). Similar to the previous experiment, RR
+always performs better than or equal to the best model between CC and DRMDPs, and it provides
+the best performance under all criteria, which again manifest the merit of taking both the expected
+and worst-case performances into consideration and distributionally robustness.
+7.3.
+Computation Times of Different Algorithms
+Table 1
+The average of the runtimes of the MOSEK solver and the AD-LPMM algorithm in seconds and the
+relative gaps (%) to the optimal values computed by MOSEK.
+S=A
+Runtimes
+Relative gaps
+MOSEK AD-LPMM
+40
+0.60
+2.79
+< 0.1 %
+70
+5.58
+4.81
+< 0.1 %
+100
+25.50
+19.98
+0.2 %
+130
+93.54
+66.17
+< 0.1 %
+160
+444.06
+168.34
+0.4 %
+In this section, we compare the computation times of our AD-LPMM algorithm with the state-
+of-the-art solver MOSEK. Table 1 reports the runtimes of the the AD-LPMM and MOSEK when
+solving problem (8) at different problem sizes. Results indicate that, though our AD-LPMM is
+slower than the MOSEK solver when problem size is small, it showcases its strong scalability and
+become much faster than MOSEK with large-size problems (while always maintaining high solution
+quality), where the advantage is more notable when the problem scales up.
+8.
+Conclusion
+We consider risk-aware MDPs with ambiguous reward functions and propose the return-risk model,
+which is versatile and can optimize any weighted combination of the average and quantile perfor-
+mances of a policy. This model generalizes and combines the advantage of distributionally robust
+MDPs and distributionally robust chance-constrained MDPs, thus is powerful in both average
+and percentile performances optimization. In particular, risk from uncertain transition kernel can
+also be captured by the return-risk model when output policies are deterministic. Tractable refor-
+mulations are provided for all our proposed models, and we design an AD-LPMM algorithm for
+
+Ruan, Chen, Ho: Risk-Averse MDPs under Reward Ambiguity
+18
+the return-risk model, which is well scalable and faster than the MOSEK solver with large-scale
+problems. Experimental results showcase the versatility of the return-risk model as well as the
+scalability of the algorithm.
+In the future, we believe that it would be important to explore more efficient methods for
+obtaining solution of RR, where function approximation and policy gradient (Sutton and Barto
+2018) are possible choices to achieve this.
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+Ruan, Chen, Ho: Risk-Averse MDPs under Reward Ambiguity
+24
+A.
+Proof of Results
+A.1.
+Proofs of Results in Section 3
+Proof of Proposition 1.
+It is sufficient to rewrite the objective of (4) as follows:
+inf
+P∈F(θ)EP[˜r⊤x] = − sup
+P∈F(θ)
+EP[−˜r⊤x]
+= −min
+λ≥0
+�
+λθ −
+�
+RSA inf
+ξ∈RSA(λ∥ξ − r∥ + ξ⊤x) dˆPr
+�
+= − min
+λ≥∥x∥∗
+�
+λθ −
+�
+RSA r⊤x dˆPr
+�
+= EˆP[˜r⊤x] − θ∥x∥∗,
+where the second identity follows from theorem 1 in Gao and Kleywegt (2016) and the third
+identity follows from strong conic duality
+inf
+ξ∈RK(λ∥ξ − r∥ + ξ⊤x) =
+�
+�
+�
+r⊤x
+λ ≥ ∥x∥∗
+−∞
+λ ∈ [0,∥x∥∗).
+Substituting the above reexpression then concludes the proof.
+Q.E.D.
+A.2.
+Proofs of Results in Section 4
+Proof of Lemma 1.
+Notice that (6) is equivalent to
+sup
+P∈F(θ)
+P
+�˜r⊤x < y
+�
+≤ ε ⇐⇒ sup
+P∈F(θ)
+P
+�˜r⊤x ≤ y
+�
+≤ ε,
+where it is equivalent if we replace the strict inequality on the left-hand side with a weak one on
+the right-hand side; see proposition 3 in Gao and Kleywegt (2016). Exploring the definition of VaR,
+we note that
+sup
+P∈F(θ)
+P
+�˜r⊤x ≤ y
+�
+≤ ε ⇐⇒ sup
+P∈F(θ)
+P-VaR1−ε
+�
+y − ˜r⊤x
+�
+≤ 0.
+By corollary 4.9 in Chen and Xie (2021) and the assumption of Mahalanobis norm, it holds that
+sup
+P∈F(θ)
+P-VaR1−ε
+�
+y − ˜r⊤x
+�
+= P(µ,Σ,g)-VaR1−ε
+�
+y − ˜r⊤x
+�
+.
+In other words, the worst-case VaR around the elliptical distribution P(µ,Σ,g) with the risk threshold
+ε is equal to the nominal elliptical VaR with a small risk threshold ε ≤ ε (which, would correspond
+to a higher risk level). We thus obtain
+sup
+P∈F(θ)
+P-VaR1−ε
+�
+y − ˜r⊤x
+�
+≤ 0 ⇐⇒ P(µ,Σ,g)-VaR1−ε [y − ˜r⊤x] ≤ 0
+⇐⇒ P(µ,Σ,g)
+�˜r⊤x ≤ y
+�
+≤ ε
+⇐⇒ P(µ,Σ,g)
+�˜r⊤x ≥ y
+�
+≥ 1 − ε,
+
+Ruan, Chen, Ho: Risk-Averse MDPs under Reward Ambiguity
+25
+where the last equivalence follows from P(µ,Σ,g) being a continuous distribution.
+Q.E.D.
+Proof of Proposition 2.
+By Lemma 1, the first constraint in (5) is the same as
+P(µ,Σ,g)
+�˜r⊤x ≥ y
+�
+≥ 1 − ε,
+where ε = 1 − Φ(¯η⋆) ≤ ε and ¯η⋆ is the smallest η ≥ Φ−1(1 − ε) that satisfies
+η(Φ(η) − (1 − ε)) −
+� η2/2
+(Φ−1(1−ε))
+2/2
+kg(z)dz ≥ θ.
+The constraint can then be further written as
+P(µ,Σ,g)[˜r⊤x ≥ y] ≥ 1 − ε ⇐⇒ Φ((µ⊤x − y)/
+√
+x⊤Σx) ≥ 1 − ε
+⇐⇒ µ⊤x − y ≥ Φ−1(1 − ε)
+√
+x⊤Σx
+⇐⇒ µ⊤x − y ≥ ∥Φ−1(1 − ε)Σ1/2x∥2,
+where the first equivalence holds by the linearity of elliptical distributions, the second one is because
+that Φ(·) is non-decreasing, and the last one is due to the fact that 1−ε ≥ 0.5 (which follows from
+ε ≤ ε < 0.5). Observe that the optimum is achieved at y⋆ = µ⊤x − ∥Φ−1(1 − ε)Σ1/2x∥2, plugging
+this in the objective of problem (5) then concludes our proof.
+Q.E.D.
+A.3.
+Proofs of Results in Section 5
+Proof of Proposition 3.
+By Proposition 1 and Proposition 2, we have
+inf
+P∈F(θ)EP[˜r⊤x] = −θ∥x∥2 + EˆP[˜r⊤x]
+and
+inf
+P∈F′(θ)P-VaR1−ε[˜r⊤x] = µ⊤x − ∥Φ−1(1 − ε)Σ1/2x∥2
+with ε as claimed. Substituting the above two equations into (7) and rearranging the terms then
+concludes our proof.
+Q.E.D.
+Proof of Proposition 4.
+By the definition of ˆT, problem (9) can be rewritten as:
+max
+π∈(∆A)S ψ
+�
+i∈[N]
+wi · g(π, ˆP i) + (1 − ψ)max
+η∈R
+�
+�
+�η −
+1
+1 − ι
+�
+i∈[N]
+wi(η − g(π, ˆP i))+
+�
+�
+�.
+By introducing auxiliary decision variables y ∈ RN, it can be further reformulated as:
+max ψ
+�
+i∈[N]
+wi · g(π, ˆP i) + (1 − ψ)
+�
+�η −
+1
+1 − ι
+�
+i∈[N]
+yi
+�
+�
+s.t. yi ≥ wi(η − g(π, ˆP i))
+∀i ∈ [N]
+π ∈ (∆A)S,y ∈ RN
++,η ∈ R.
+(13)
+
+Ruan, Chen, Ho: Risk-Averse MDPs under Reward Ambiguity
+26
+Here we can express
+wi · g(π,P ) = max µ⊤x − αθ · ∥x∥2 − (1 − α) · ∥Φ−1(1 − ε)Σ1/2x∥2
+s.t. xs,a = πs,a ·
+�
+a′∈A
+xs,a′
+∀(s,a) ∈ S × A
+(E − γ · ¯P )x = wi · p0
+x ∈ RSA
++
+(14)
+as in Lobo et al. (2020). We can then, by combining (13) and (14), reformulate problem (9) as:
+max ψ
+�
+i∈[N]
+(µ⊤xi − αθ · ∥xi∥2 − (1 − α) · ∥Φ−1(1 − ε)Σ1/2xi∥2) + (1 − ψ)(η −
+1
+1 − ι
+�
+i∈[N]
+yi)
+s.t. yi − wiη ≥ αθ · ∥xi∥2 + (1 − α) · ∥Φ−1(1 − ε)Σ1/2xi∥2 − µ⊤xi
+∀i ∈ [N]
+xi
+s,a = πs,a ·
+�
+a′∈A
+xi
+s,a′
+∀i ∈ [N],(s,a) ∈ S × A
+(E − γ · ¯P i)xi = wi · p0
+∀i ∈ [N]
+π ∈ (∆A)S,η ∈ R,xi ∈ RSA
++ ,y ∈ RN
++
+∀i ∈ [N].
+Now it is sufficient to focus on the second set of constraints
+xi
+s,a = πs,a ·
+�
+a′∈A
+xi
+s,a′ ∀i ∈ [N],(s,a) ∈ S × A.
+(15)
+Since we only consider deterministic policy π ∈ {0,1}SA and �
+a∈A xi
+s,a ∈ [0,wi/(1 − γ)] (see, e.g.,
+lemma C.10 in Petrik (2010)), we have the McCormick relaxation (see, e.g., Petrik and Luss (2016))
+of (15) as:
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+xi
+s,a ≤
+�
+a′∈A
+xi
+s,a′
+xi
+s,a ≤
+wi
+1 − γ πs,a
+xi
+s,a ≥ 0
+xi
+s,a ≥
+wi
+1 − γ (πs,a − 1) +
+�
+a′∈A
+xi
+s,a′
+for all i ∈ [N],(s,a) ∈ S × A. Our conclusion then follows from the fact that the McCormick
+relaxation is precise when π ∈ {0,1} (i.e., the extreme values of the interval [0,1]).
+Q.E.D.
+A.4.
+Proofs of Results in Section 6
+Proof of Proposition 5.
+By (11), it is sufficient to focus on solving ProjBℓΣ(·)(x). By eigenvalue
+decomposition, we have Σ = G⊤DG4 with D = diag(d1,··· ,dSA), thus we have:
+ProjBℓΣ(·)(x) = arg min 1
+2 · ∥v − x∥2
+2
+s.t.
+v⊤G⊤DGv ≤ 1
+v ∈ RSA.
+4 The eigenvalue decomposition here is not counted in the time complexity of the bisection method (or the AD-LPMM
+algorithm), since this process is carried out for computing Σ1/2 in (8) (before we solve (8)).
+
+Ruan, Chen, Ho: Risk-Averse MDPs under Reward Ambiguity
+27
+By change of variable u = Gv and let b = Gx, it is sufficient to focus on the equivalent problem:
+arg min 1
+2 · ∥u − b∥2
+2
+s.t.
+u⊤Du ≤ 1
+u ∈ RSA,
+(16)
+where we can retrieve v⋆ = G⊤u⋆. The Lagrangian function of(16) (with the introduced dual
+variable ζ ∈ R+) is
+L(u;ζ) = 1
+2 · ∥u − b∥2
+2 + ζ(u⊤Du − 1).
+Since (16) is a convex optimization problem, the KKT condition is the sufficient condition for the
+optimality of the primal and dual solutions:
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+u⊤Du ≤ 1
+ζ ≥ 0
+ζ(u⊤Du − 1) = 0
+∇uL(u;ζ) = u − b + 2ζ · Du = 0,
+where for ζ = 0, we have
+�
+�
+�
+u⊤Du ≤ 1
+u − b = 0;
+while when ζ > 0, we have
+�
+�
+�
+u⊤Du = 1
+(I + 2ζ · D)u − b = 0.
+Therefore, if b⊤Db ≤ 1, we have u⋆ = b; if b⊤Db > 1, it is sufficient to solve the equation g(ζ) = 1
+where
+g(ζ) =
+�
+i∈[SA]
+dib2
+i
+(1 + 2ζdi)2 .
+The function g is monotonically decreasing function on [0,+∞) and limζ→+∞ g(ζ) = 0, thus we can
+apply the bisection method to search on the interval [0, ¯ζ] (where ¯ζ : g(¯ζ) ≤ 1 is the upper bound
+for the search which we provide in Lemma 2) to locate ζ⋆ and retrieve u⋆
+i = bi/(1+2ζ⋆di) ∀i ∈ [SA].
+The pseudocode is provided in Algorithm 2.
+The time complexity of solving Py(x,ξ;c) is dominated by the bisection method, which has
+time complexity O(log(1/δ′)). Our conclusion follows from the fact that the computation in each
+iteraion of the bisection takes time O(SA).
+Q.E.D.
+Lemma 2. The inequality g(ζ) ≤ 1 holds for all ζ ≥ (1/(2di′′))(bi′√SAdi′ − 1), where i′ ∈
+arg maxi∈[SA] dib2
+i and i′′ ∈ arg mini∈[SA] di
+
+Ruan, Chen, Ho: Risk-Averse MDPs under Reward Ambiguity
+28
+Algorithm 2: Bisection for Problem (16)
+Input: Desired precision δ′, initial lower bound ζ ← 0 and upper bound ζ > 0
+if g(0) ≤ 1 then
+u ← b;
+end
+else
+while |ζ − ζ| ≥ δ′ do
+ζ ← 0.5(ζ + ζ);
+if
+g(ζ) >= 1 then
+ζ ← ζ;
+end
+else
+ζ ← ζ;
+end
+end
+for i = 1,··· ,SA do
+ui = bi/(1 + 2ζdi);
+end
+end
+Output: Solution u
+Proof. Observe that,
+g(ζ) ≤
+�
+i∈[SA]
+di′b2
+i′
+(1 + 2ζdi)2
+≤
+SAdi′b2
+i′
+(1+2ζdi′′)2 ,
+from which we have
+SAdi′b2
+i′
+(1 + 2ζdi′′)2 ≤ 1 ⇒ g(ζ) ≤ 1.
+Our conclusion thus follows by rearranging the terms of the inequality on the left-hand side.
+Q.E.D.
+By Lemma 2, one can choose ζ = (1/(2di′′))(bi′√SAdi′ − 1), where i′ ∈ arg maxi∈[SA] dib2
+i and
+i′′ ∈ arg mini∈[SA] di for Algorithm 2.
+Proof of Proposition 6.
+Notice that, it is sufficient to solve the ith subproblem:
+arg min
+z≥0
+c
+2z2 − (cxi + µi + ηi)z = max
+�
+0, 1
+c(cxi + µi + ηi)
+�
+for all i ∈ [SA], where our conclusion follows.
+Q.E.D.
+
+Ruan, Chen, Ho: Risk-Averse MDPs under Reward Ambiguity
+29
+Proof of Proposition 7.
+By the definition of Q(·,·), we have
+Px(y,z,λ,ξ,η;c,ν, ˆx)
+= arg min
+x
+αθ · ∥x∥2 + x⊤((E − γ · ¯P )⊤λ + ξ + η) + c
+2 ·
+��������
+(E − γ · ¯P )(x − ˆx) + (E − γ · ¯P )ˆx − p0
+x − ˆx + ˆx − y
+x − ˆx + ˆx − z
+��������
+2
+2
++ 1
+2 · ℓ2
+Q(c,ν)(x − ˆx)
+= arg min
+x
+αθ · ∥x∥2 + x⊤((E − γ · ¯P )⊤λ + ξ + η) + c
+2 ·
+��������
+(E − γ · ¯P )(x − ˆx)
+x − ˆx
+x − ˆx
+��������
+2
+2
++c · x⊤ �
+(E − γ · ¯P )⊤ �
+(E − γ · ¯P )ˆx − p0
+�
++ 2 · ˆx − y − z
+�
++ 1
+2 · ℓ2
+Q(c,ν)(x − ˆx)
+= arg min
+x
+αθ
+cν · ∥x∥2 + x⊤w + 1
+2 · ∥x − ˆx∥2
+2
+= arg min
+x
+αθ
+cν · ∥x∥2 + 1
+2 · ∥x − (ˆx − w)∥2
+2
+=
+�
+1 −
+αθ
+cν
+max{∥w∥2, αθ
+cν }
+�
+· (ˆx − w)
+where
+we
+denote
+w
+=
+1
+cν
+·
+��
+E − γ · ¯P
+�⊤ λ + ξ + η
+�
++
+1
+ν
+·
+��
+E − γ · ¯P
+�⊤ ��
+E − γ · ¯P
+� ˆx − p0
+�
++ 2 · ˆx − y − z
+�
+, and the last equality holds by, e.g., exam-
+ple 6.1.9 in Beck (2017).
+The computation time is dominated by computing ∥w∥2, which is O(SA).
+Q.E.D.
+B.
+Evaluation of VaR and CVaR of Student’s t-Distribution
+The VaR of a Student’s t-distribution with threshold ε is in fact the lower-ε percentile of its
+probability density function (PDF), which can be looked up in table in, e.g., Hogg and Craig (1995)
+(under some common values of ε < 0.5). We provide the calculation of CVaR as follows (with degree
+of freedom δ > 1 and v := Pt-dist-VaRε(˜r) assumed known):
+Pt-dist-CVaRε(˜r) = 1
+ε ·
+Γ( δ+1
+2
+)
+(πδ)
+1
+2 Γ( δ
+2 )
+� v
+−∞
+r
+(1+ r2
+δ )
+δ+1
+2 dr
+= 1
+ε ·
+δ
+1
+2 ·Γ( δ+1
+2
+)
+2π
+1
+2 Γ( δ
+2 )
+� 1+ v2
+δ
+−∞
+u− k+1
+2 du
+= −
+δ
+1
+2 ·Γ( δ+1
+2
+)
+επ
+1
+2 (δ−1)Γ( δ
+2 ) ·
+�
+1 + v2
+δ
+�− k−1
+2 ,
+where the first equality follows from the definition of the CVaR and the PDF of the t-distribution
+herein, the second equality holds by the technique of integration by substitution.
+C.
+Preliminaries on Elliptical Distributions
+The probability density distribution of an elliptical reference distribution P(µ,Σ,g) is given by
+f(r) = k · g
+�1
+2(r − µ)⊤Σ−1(r − µ)
+�
+,
+
+Ruan, Chen, Ho: Risk-Averse MDPs under Reward Ambiguity
+30
+where k is a positive normalization scalar, µ is a mean vector, Σ is a positive definite matrix and g
+is a generating function. Elliptical distribution is a broad family of distributions that includes for
+example, the multivariate normal distribution, multivariate t-distribution and multivariate logistic
+distribution, as special cases. One notable property of the elliptical distribution is the linearity: any
+linear combination of elliptically distributed random variables still follows an elliptical distribution.
+That is, for any random vector ˜r ∼ P(µ,Σ,g), it holds that ˜r⊤x ∼ P(µx,σ2x,g) with µx = µ⊤x and
+σx =
+√
+x⊤Σx. Indeed, we can express the combination as ˜r⊤x = µx + σx˜z, where ˜z ∼ P(0,1,g) is a
+standard elliptically distributed random variable whose probability density function and cumulative
+distribution function are φ(z) = k·g (z2/2) and Φ(x) =
+� x
+−∞ k·g(z2/2)dz, respectively. For a concrete
+example we take a closer look at a standard normal distribution, for which the normalization scalar
+and generating function are k = 1/
+√
+2π and g(x) = exp(−x), respectively.
+D.
+Distributionally Optimistic MDPs
+In contrast to the robust model, sometimes the decision maker prefers exploration over exploitation
+if she would like to learn more information about the MDP. As such, we could instead adopt an
+optimistic counterpart where we focus on the best case, motivating the following distributionally
+optimistic MDP:
+ℓO(θ) = max
+x∈X
+sup
+P∈F(θ)
+EP[˜r⊤x].
+(17)
+In contrast to the robust case, here our decision depends instead on the best possible (expected)
+outcome, which exactly embodies optimism. We summarize the reformulation of (17) as follows.
+Proposition 8. The distributionally optimistic MDP (17) is equivalent to an optimization prob-
+lem
+ℓO(θ) = max
+x∈X EˆP[˜r⊤x] + θ∥x∥∗.
+Proof. It is sufficient to rewrite the objective of (17) as follows:
+sup
+P∈F(θ)
+EP[˜r⊤x] = − inf
+P∈F(θ)EP[−˜r⊤x] = −(EˆP[−˜r⊤x] − θ∥x∥∗) = EˆP[˜r⊤x] + θ∥x∥∗,
+where the second identity follows similar lines as in the proof of Proposition 1.
+Q.E.D.
+The reformulation in Proposition 8 is a reverse conic program that is, in general, non-convex.
+However, it can be recast as a mixed-integer linear program, provided that ∥ · ∥∗ is the commonly
+used L1-norm or L∞-norm. Such a mixed-integer linear program can be solved by the state-of-the-
+art approaches.
+
+Ruan, Chen, Ho: Risk-Averse MDPs under Reward Ambiguity
+31
+E.
+Distributionally Optimistic Chance-Constrained Model
+In a distributionally optimistic chance-constrained MDP model, where we focus on the best case
+that with high probability, the reward is no smaller than some lower bound that we maximize.
+Formally, the distributionally optimistic chance-constrained MDP model is formulated as follows:
+ℓDOCC(θ,ε) =
+�
+�
+�
+�
+�
+�
+�
+�
+�
+max y
+s.t.
+sup
+P∈F(θ)
+P[˜r⊤x ≥ y] ≥ 1 − ε
+x ∈ X, y ∈ R.
+(18)
+The optimistic chance-constrained model (18) is also equivalent to a nominal chance-constrained
+model, however, at a less risky level. Before formally establishing this argument, two lemmas are
+introduced as follows.
+Lemma 3. The worst (largest) probability of the random vector ˜r attaining a value in the set R,
+sup
+P∈F(θ)
+P[˜r ∈ R],
+(19)
+is equivalent to
+min
+λ≥0
+�
+λθ +
+�
+r∈RSA(λ · dist(r,R) − 1)−dˆPr
+�
+.
+Here, we use dist(r,R) = inf{∥r − ˆr∥ | ˆr ∈ R} to denote the distance from the vector r ∈ RSA to
+the set R ⊆ RSA.
+Proof. Using theorem 1 in Gao and Kleywegt (2016) or theorem 1 in Blanchet and Murthy (2019),
+the uncertainty quantification problem (19) is equal to
+min
+λ≥0
+�
+λθ −
+�
+r∈RSA
+inf
+w∈RSA{λ∥w − r∥ − I[w ∈ R]}dˆPr
+�
+,
+(20)
+where I is the 0-1 indicator function. Consider the second term in the objective of the above
+minimization problem, we have
+inf
+w∈RSA{λ∥w − r∥ − I[w ∈ R]} = −(λ · dist(r,R) − 1)−.
+(21)
+Indeed, if r ∈ R (for which, dist(r,R) = 0), then by choosing w = v, it holds that
+inf
+w∈RSA{λ∥w − r∥ − I[w ∈ R]} = −1 = −(λ · dist(r,R) − 1);
+whereas if r /∈ R, then it holds that
+inf
+w∈RSA{λ∥w − r∥ − I[w ∈ R]} = min
+�
+inf
+w∈R{λ∥w − r∥ − 1}, inf
+w /∈Rλ∥w − r∥
+�
+= min
+�
+inf
+w∈R{λ∥w − r∥ − 1},0
+�
+= −(λ · dist(r,R) − 1)−.
+Plugging expression (21) into problem (20) gives the desired result, which, by proposition 3 in Gao
+and Kleywegt (2016), holds regardless of whether R is open or closed.
+Q.E.D.
+
+Ruan, Chen, Ho: Risk-Averse MDPs under Reward Ambiguity
+32
+Lemma 4. The distributionally optimistic chance constraint
+inf
+P∈F(θ)P[˜r ∈ R] ≤ ε
+(22)
+with a risk threshold ε ∈ (0,1) is satisfiable if and only if
+P-CVaRε[−dist(˜r, ¯R)] ≥ −
+θ
+1 − ε,
+where ¯R = RSA \ R is the complement of the set of undesired events R.
+Proof. We first re-express (22) as
+sup
+P∈F(θ)
+P[˜r ∈ ¯R] ≥ 1 − ε.
+Using Lemma 3, the above constraint is equivalent to
+min
+λ≥0
+�
+λθ +
+�
+r∈RSA(λ · dist(r, ¯R) − 1)−dˆPr
+�
+≥ 1 − ε.
+(23)
+The left-hand side problem can be presented by
+min
+�
+min
+λ>0
+�
+λθ +
+�
+r∈RSA(λ · dist(r, ¯R) − 1)−dˆPr
+�
+,1
+�
+.
+Since 1 ≥ 1 − ε, the above re-expression implies that constraint (23) is equivalent to
+min
+λ>0
+�
+λθ +
+�
+r∈RSA(λ · dist(r, ¯R) − 1)−dˆPr
+�
+≥ 1 − ε.
+Multiplying both sides by (λ(1 − ε))−1 > 0, we arrive at
+min
+τ<0
+�
+1
+1 − ε
+�
+r∈RSA(−dist(r, ¯R) − τ)+dˆPr + τ
+�
+≥ −
+θ
+1 − ε,
+which, together with the fact
+min
+τ≥0
+�
+1
+1 − ε
+�
+r∈RSA(−dist(r, ¯R) − τ)+dˆPr + τ
+�
+≥ 0 ≥ −
+θ
+1 − ε,
+is equivalent to
+min
+τ∈R
+�
+1
+1 − ε
+�
+r∈RSA(−dist(r, ¯R) − τ)+dˆPr + τ
+�
+≥ −
+θ
+1 − ε,
+where the left-hand side is essentially ˆP-CVaRε[−dist(˜r, ¯R)].
+Q.E.D.
+Now we are ready to establish the equivalence between the chance-constrained model and its
+optimistic counterpart (with an adjusted risk threshold).
+Lemma 5. Suppose in the Wasserstein ambiguity set (3), the reference distribution is an ellip-
+tical distribution ˆP = P(µ,Σ,g) and the Wasserstein distance is equipped with a Mahalanobis norm
+associated with the positive definite matrix Σ. The distributionally optimistic robust chance con-
+straint
+∃ P ∈ F(θ) : P[˜r⊤x ≥ y] ≥ 1 − ε
+is satisfiable if and only if P(µ,Σ,g)[˜r⊤x ≥ y] ≥ 1 − ¯ε, where ¯ε = 1 − Φ(η⋆) ≥ ε with η⋆ being the
+smallest η ≤ Φ−1(1 − ε) that satisfies η(Φ(η) − (1 − ε)) +
+� (Φ−1(1−ε))
+2/2
+η2/2
+kg(z)dz ≤ θ.
+
+Ruan, Chen, Ho: Risk-Averse MDPs under Reward Ambiguity
+33
+Proof. We first look at the individual distributionally optimistic robust chance constraint
+∃ P ∈ F(θ) : P[˜r⊤x ≥ y] ≥ 1 − ε
+for some generic coefficient vector x ∈ RSA. The above chance constraint is equivalent to
+sup
+P∈F(θ)
+P[˜r⊤x ≥ y] ≥ 1 − ε ⇐⇒
+sup
+P∈F(θ)
+P[˜r⊤x > y] ≥ 1 − ε ⇐⇒
+inf
+P∈F(θ)P[˜r⊤x ≤ y] ≤ ε,
+where for the first equivalence, by using proposition 3 in Gao and Kleywegt (2016) , it is indifferent
+to replace the strict inequality with a weak one. Exploring the definition of VaR, we note that
+inf
+P∈F(θ)P[˜r⊤x ≤ y] ≤ ε ⇐⇒ inf
+P∈F(θ)P-VaR1−ε[y − ˜r⊤x] ≤ 0.
+Hence, with the translation invariance of VaR, it is sufficient to show that
+inf
+P∈F(θ)P-VaR1−ε[−˜r⊤x] ≜ inf
+v∈R
+�
+v |
+inf
+P∈F(θ)P[−˜r⊤x > v] ≤ ε
+�
+.
+(24)
+By Lemma 4 and the assumption of Mahalanobis norm, we have
+inf
+P∈F(θ)P
+�
+−˜r⊤x > v
+�
+≤ ε ⇐⇒ P(µ,Σ,g)-CVaRε[−dist(˜r, ¯R)] ≥ −
+θ
+1 − ε
+⇐⇒ −P(µ,Σ,g)-CVaRε[−(−˜r⊤x − v)+] ≤ θ∥x∥Σ−1
+1 − ε
+,
+where ¯R =
+�
+r | − r⊤x ≤ v
+�
+and we leverage the closed form solution
+dist(˜r, ¯R) =
+�
+−˜r⊤x − v
+�+ /∥x∥Σ−1;
+see, e.g., lemma 2 in Chen et al. (2018).
+Let PS = P(µ,Σ,g) for simplicity. By the property of elliptical distribution, for ˜r ∼ PS and any real
+vector x, we have −˜r⊤x ∼ P(µS,σ2
+S,g) = P(−µ⊤x,x⊤Σx,g). We denote its probability density function
+as
+h(z) = k
+σS
+· g
+�
+(z − µS)
+2
+2σ2
+S
+�
+.
+The left-hand side of the constraint can be further transformed as
+−PS-CVaRε[−(−˜r⊤x − v)+]
+= −EPS[−(−˜r⊤x − v)+ | − (−˜r⊤x − v)+ ≥ PS-VaRε[−(−˜r⊤x − v)+]]
+= −
+1
+1 − ε
+� sup{z|−(z−v)+≥PS-VaRε[−(−˜r⊤x−v)+]}
+−∞
+−(z − v)+h(z)dz
+=
+1
+1 − ε
+� sup{z|−(z−v)+≥PS-VaRε[−(−˜r⊤x−v)+]}
+v
+(z − v)h(z)dz
+=
+1
+1 − ε
+� PS-VaR1−ε[−˜r⊤x]
+v
+(z − v)h(z)dz,
+
+Ruan, Chen, Ho: Risk-Averse MDPs under Reward Ambiguity
+34
+in which the last equality holds from
+sup{z | − (z − v)+ ≥ PS-VaRε[−(−˜r⊤x − v)+]}
+= sup{z | min{v − z,0} ≥ PS-VaRε[min{v + ˜r⊤x,0}]}
+= sup{z | min{−z,−v} ≥ PS-VaRε[min{˜r⊤x,−v}]}
+= sup{z | − z ≥ PS-VaRε[min{˜r⊤x,−v}]}
+= sup{z | z ≤ PS-VaR1−ε[max{−˜r⊤x,v}]}
+= sup{z | z ≤ PS-VaR1−ε[−˜r⊤x]}
+= PS-VaR1−ε[−˜r⊤x].
+Here, the second equality is due to the translation invariance of VaR, the third one follows from
+−v ≥ PS-VaRε[min{˜r⊤x,−v}], the fifth one is because that for any ε ∈ (0,1), the distributionally
+optimistic robust VaR satisfies
+v = inf
+P∈F(θ)P-VaR1−ε[−˜r⊤x] ≤ PS-VaR1−ε[−˜r⊤x],
+(25)
+thus the 1 − ε quantiles of −˜r⊤x and max{−˜r⊤x,v} coincide.
+Let us denote q1−ε = PS-VaR1−ε[−˜r⊤x], which, by its definition, satisfies
+q1−ε − µS
+σS
+= PS-VaR1−ε
+�−˜r⊤x − µS
+σS
+�
+= P0
+(0,1,g)-VaR1−ε[˜z] = Φ−1(1 − ε),
+Here, the first equality holds for the translation invariance and the positive homogeneity of VaR,
+while the last one follows from the definition of VaR under the standard elliptical distribution
+P(0,1,g).
+Following the last reformulation of the constraint, we further have
+1
+1 − ε
+� q1−ε
+v
+(z−v)h(z)dz =
+1
+1 − ε
+� q1−ε
+v
+z· k
+σS
+·g
+�
+(z − µS)
+2
+2σ2
+S
+�
+dz−
+v
+1 − ε
+� q1−ε
+v
+k
+σS
+·g
+�
+(z − µS)
+2
+2σ2
+S
+�
+dz.
+
+Ruan, Chen, Ho: Risk-Averse MDPs under Reward Ambiguity
+35
+For its first component, we have
+1
+1 − ε
+� q1−ε
+v
+z · k
+σS
+· g
+�
+(z − µS)
+2
+2σ2
+S
+�
+dz
+=
+1
+1 − ε
+� q1−ε
+v
+z − µS
+σS
+· k · g
+�
+(z − µS)
+2
+2σ2
+S
+�
+dz +
+1
+1 − ε
+� q1−ε
+v
+µS
+σS
+· k · g
+�
+(z − µS)
+2
+2σ2
+S
+�
+dz
+=
+σS
+1 − ε
+� q1−ε
+v
+z − µS
+σS
+· k · g
+�
+(z − µS)
+2
+2σ2
+S
+�
+d
+�z − µS
+σS
+�
++
+µS
+1 − ε
+�
+Φ
+�q1−ε − µS
+σS
+�
+− Φ
+�v − µS
+σS
+��
+=
+σS
+1 − ε
+�
+q1−ε−µS
+σS
+v−µS
+σS
+t · k · g
+�t2
+2
+�
+d
+�z − µS
+σS
+�
++ µS
+1 − ε
+�
+Φ
+�q1−ε − µS
+σS
+�
+− Φ
+�v − µS
+σS
+��
+=
+σS
+1 − ε
+�
+(q1−ε−µS)2
+2σ2
+S
+(v−µS)2
+2σ2
+S
+k · g(z)dz + µS
+1 − ε
+�
+Φ
+�q1−ε − µS
+σS
+�
+− Φ
+�v − µS
+σS
+��
+,
+while for the second component, it holds that
+v
+1 − ε
+� q1−ε
+v
+k
+σS
+· g
+�
+(z − µS)
+2
+2σ2
+S
+�
+dz =
+v
+1 − ε
+�
+q1−ε−µS
+σS
+v−µS
+σS
+k · g
+�z2
+2
+�
+dz
+=
+v
+1 − ε
+�
+Φ
+�q1−ε − µS
+σS
+�
+− Φ
+�v − µS
+σS
+��
+.
+Hence, combine the constraint with (25), we have the following equivalent expression for prob-
+lem (24):
+inf v
+s.t.
+�
+(q1−ε−µS)2
+2σ2
+S
+(v−µS)2
+2σ2
+S
+k · g(z)dz + µS − v
+σS
+�
+Φ
+�q1−ε − µS
+σS
+�
+− Φ
+�v − µS
+σS
+��
+≤ θ∥x∥Σ−1
+σS
+= θ
+v ≤ PS-VaR1−ε[−˜r⊤x]
+v ∈ R,
+where the equality follows from the definition of the Mahalanobis norm. Let η = (v − µS)/σS, the
+best-case VaR now becomes
+inf µS + σSη
+s.t.
+� (Φ−1(1−ε))2/2
+η2/2
+k · g(z)dz − η · (1 − ε − Φ(η)) ≤ θ
+η ≤ Φ−1(1 − ε)
+η ∈ R.
+(26)
+The function
+V (η) ≜
+� (Φ−1(1−ε))2/2
+η2/2
+k · g(z)dz − η · (1 − ε − Φ(η))
+
+Ruan, Chen, Ho: Risk-Averse MDPs under Reward Ambiguity
+36
+is monotonically decreasing on (−∞,Φ−1(1 − ε)) since for any η < Φ−1(1 − ε), it holds that
+V ′(η) = −η · k · g
+�η2
+2
+�
+− (1 − ε) + Φ(η) + ηφ(η) = Φ(η) − (1 − ε) < 0.
+Thus problem (26) can be efficiently solved be a bisection algorithm and the optimal η⋆ as claimed
+can be obtained. Finally the result can be obtained as follows:
+∃ P ∈ F(θ) : P[˜r⊤x ≥ y] ≥ 1 − ε ⇐⇒ −y ≥ σSη⋆ + µS
+⇐⇒ −y − µS
+σS
+≥ η⋆
+⇐⇒ Φ
+�−y − µS
+σS
+�
+≥ Φ(η⋆)
+⇐⇒ P(µ,Σ,g)
+� ˜r⊤x − µS
+σS
+≥ y − µS
+σS
+�
+≥ 1 − ¯ε
+⇐⇒ P(µ,Σ,g)[˜r⊤x ≥ y] ≥ 1 − ¯ε.
+Q.E.D.
+With ¯ε in Lemma 5, we are now ready to derive a second-order cone reformulation of the
+distributionally optimistic chance-constrained model (18).
+Proposition 9. Suppose in the Wasserstein ambiguity set (3), the reference distribution is an
+elliptical distribution ˆP = P(µ,Σ,g) and the Wasserstein distance is equipped with a Mahalanobis
+norm associated with the positive definite matrix Σ. If the risk threshold satisfies ε ≤ ¯ε < 0.5, then
+the distributionally optimistic chance-constrained MDP (18) is equivalent to the second-order cone
+program
+ℓDOCC(θ,ε) = max
+x∈X µ⊤x − ∥Φ−1(1 − ¯ε)Σ1/2x∥2,
+where ¯ε = 1 − Φ(η⋆) ≥ ε with η⋆ being the smallest η ≤ Φ−1(1 − ε) that satisfies
+η(Φ(η) − (1 − ε)) +
+� (Φ−1(1−ε))
+2/2
+η2/2
+kg(z)dz ≤ θ.
+Proof. By Lemma 5, the first constraint in (18) is equivalent to
+P(µ,Σ,g)[˜r⊤x ≥ y] ≥ 1 − ¯ε,
+where ¯ε = 1 − Φ(η⋆) ≥ ε with η⋆ being the smallest η ≤ Φ−1(1 − ε) that satisfies
+η(Φ(η) − (1 − ε)) +
+� (Φ−1(1−ε))
+2/2
+η2/2
+kg(z)dz ≤ θ,
+which can be further transformed as follows:
+P(µ,Σ,g)[˜r⊤x ≥ y] ≥ 1 − ¯ε ⇐⇒ Φ((µ⊤x − y)/
+√
+x⊤Σx) ≥ 1 − ¯ε
+⇐⇒ µ⊤x − y ≥ Φ−1(1 − ¯ε)
+√
+x⊤Σx
+⇐⇒ µ⊤x − y ≥ ∥Φ−1(1 − ¯ε)Σ1/2x∥2,
+
+Ruan, Chen, Ho: Risk-Averse MDPs under Reward Ambiguity
+37
+where the first equivalence holds by the linearity of elliptical distributions, the second one holds
+because of the non-decreasing cumulative distribution function Φ(·), and the third one holds as
+¯ε < 0.5. Since the optimal value is achieved with y = µ⊤x − ∥Φ−1(1 − ¯ε)Σ1/2x∥2, plugging this
+equation in the objective of (18) then concludes our proof.
+Q.E.D.
+F.
+Additional Details on Robust MDPs
+As introduced in Delage and Mannor (2010), robust MDPs maximizes the total expected return
+considering the worst-case realization of the uncertain parameter within a predefined ambiguity
+set:
+max
+π∈Π
+min
+r0∈R,r1∈R,···E
+� ∞
+�
+t=0
+γtrt(st) | s0 ∝ p0,π
+�
+,
+(27)
+where Π is the set of all the stationary randomized policies, rt and st are the reward and state at
+time stage t, respectively. As in Delage and Mannor (2010), we set R to be the 99% confidence
+ellipsoid of the random reward vector as the uncertainty set.
+G.
+Additional Details on BROIL
+Similar to our return-risk model, BROIL (Brown et al. 2020) also seeks a policy that maximizes
+the weighted average of the mean and percentile performances:
+max
+π∈Π λ · E
+� ∞
+�
+t=0
+γtrt(st) | s0 ∝ p0,π
+�
++ (1 − λ) · CVaRε
+� ∞
+�
+t=0
+γtrt(st) | s0 ∝ p0.π
+�
+,
+(28)
+where λ ∈ [0,1] is the weight. Given R ∈ RSA×n as the matrix of (n) reward samples, BROIL can
+be expressed as a linear program as follows:
+max
+x∈X,y∈Rλ · 1
+ne⊤R⊤x + (1 − λ) ·
+�
+y − 1
+ε · 1
+ne⊤(y · e − R⊤x)
+�
+.
+Observe that, there are two major differences between BROIL and our return-risk model: first,
+BROIL use CVaR as its risk measure, while VaR is applied in our return-risk model; second, while
+distributionally robustness is considered in (both the mean and VaR of return in) our objective
+function, BROIL only computes the nominal mean and CVaR of the return.
+H.
+Additional Details and Results on the Experiments
+H.1.
+Additional Details of Parameter Selection
+We use cross validation for parameter selection in both the simulation and empirical studies.
+For DRMDPs (4), the candidate set for θ is {0,2,··· ,18}; for CC (2), the candidate set for ε
+is {iε′/5}i∈[5]; for RR (7), we select θ such that ε varies among {iε′/5}i∈[5], and we select α ∈
+{0,0.25,0.5,0.75,1}; for BROIL (28), we select λ × ε ∈ {0,0.25,0.5,0.75,1} × {0.05,0.1,0.15}; for
+RMDPs (27), as in Delage and Mannor (2010), we set R to be the 99% confidence ellipsoid of the
+random reward vector as the uncertainty set.
+
+Ruan, Chen, Ho: Risk-Averse MDPs under Reward Ambiguity
+38
+Figure 6
+A machine replacement problem with fixed Gaussian rewards.
+H.2.
+Additional Details of the Simulation Study
+We consider S = 10 states, A = 10 actions, a uniform initial state distribution, and a discount
+factor γ = 0.95. For each state s ∈ [S], the number of reachable next-state is ⌈log S⌉. We sample
+the true reward from a multivariate normal distribution N(µ′,Σ′), where for each k ∈ [SA], µ′
+k
+is generated as follows: first we sample a number (0 or 1) from a discrete uniform distribution in
+{0,1}. If the result is 0, we generate µ′
+k from the normal distribution N(50,100); otherwise we
+generate it from N(90,100). Standard deviations of rewards are generated in the same manner
+with another two normal distributions N(3,9) and N(18,9). Both standard deviations and means
+are trimmed to be non-negative after the above procedure. The correlation matrix of rewards
+is generated as follows: we first sample a matrix R ∈ RSA×SA with all its entries independently
+sampled in [0.25,1] uniformly, and then obtain our correlation matrix diag(d)V diag(d), where
+V = R⊤R and d = {di}i∈[SA] = {1/√Vii}i∈[SA].
+H.3.
+Additional Details of the Empirical Study
+In this experiment, each machine is subject to the same underlying MDP with a state set S = [S]
+with S = 50 and an action set with only two actions: repair the machine or not. The transition is
+deterministic and the discount factor is 0.8. The reward depends on both the current state and
+action, and all the rewards are independently and normally distributed. Figure 6 illustrates the
+true underlying distribution that generates the random rewards.
+H.4.
+Additional Results of the Simulation Study
+H.5.
+Additional Results of the Empirical Study
+
+130,1)
+N(-130,1)
+-130,1)
+N(-130,20)
+2
+N(0,10)
+N(0,10
+N(0,10-4)
+V0.10
+- Repair
+N(-100,800)
+. Not RepairRuan, Chen, Ho: Risk-Averse MDPs under Reward Ambiguity
+39
+100
+200
+300
+400
+500
+Sample size
+1500
+1600
+1700
+VaR ( '=0.05)
+ 
+DRMDP
+CC
+RR
+BROIL
+RMDP
+100
+200
+300
+400
+500
+Sample size
+1550
+1600
+1650
+1700
+1750
+VaR ( '=0.1)
+ 
+DRMDP
+CC
+RR
+BROIL
+RMDP
+Figure 7
+Simulation. Models DRMDP (4), CC (2), RR (7), RMDP and BROIL evaluated by VaR (risk thresh-
+old ε′ ∈ {5%,10%}). The upper and lower edges of the shaded areas are respectively the 95% and 5%
+percentiles of the 100 performances, while the solid lines are the medians.
+100
+200
+300
+400
+500
+Sample size
+15.5
+15.0
+14.5
+14.0
+13.5
+VaR ( '=0.05)
+ 
+DRMDP
+CC
+RR
+BROIL
+RMDP
+100
+200
+300
+400
+500
+Sample size
+15.5
+15.0
+14.5
+14.0
+13.5
+VaR ( '=0.1)
+ 
+DRMDP
+CC
+RR
+BROIL
+RMDP
+Figure 8
+Empirical. Models DRMDP (4), CC (2), RR (7), RMDP and BROIL evaluated by VaR (risk threshold ε′ ∈
+{5%,10%}). The upper and lower edges of the shaded areas are respectively the 95% and 5% percentiles
+of the 100 performances, while the solid lines are the medians.
+I.
+Related Works
+Table 2 summarizes literature that is related to our work. We remark that, compared to its related
+works in Table 2, our return-risk model is the only one that considers risk ambiguity, and we have
+also designed a fast first-order algorithm to obtain its solution, which enhance the practicality of
+our model for large-scale problems.
+
+Ruan, Chen, Ho: Risk-Averse MDPs under Reward Ambiguity
+40
+Table 2
+Related works.
+Paper
+Uncertainty
+Robustness
+Ambiguity set
+Risk measure Soft-robustness
+Delage and Mannor (2010)
+Rewards
+and
+transition kernel
+-
+-
+VaR
+No
+Xu and Mannor (2010)
+Rewards
+and
+transition kernel
+DRO
+Nested
+-
+No
+Yu and Xu (2015)
+Rewards
+and
+transition kernel
+DRO
+(General) Nested
+-
+No
+Brown et al. (2020)
+Rewards
+-
+-
+CVaR
+Yes
+Gilbert et al. (2017)
+Rewards
+-
+-
+VaR
+No
+Lobo et al. (2020)
+Transition kernel
+-
+-
+CVaR
+Yes
+Yang (2020)
+Transition kernel
+DRO
+Wasserstein
+-
+No
+This paper
+Rewards
+DRO
+Wasserstein
+VaR
+Yes
+

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