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+Improved design and experimental
+demonstration of ultrahigh-Q C6-symmetric H1
+hexapole photonic crystal nanocavities
+KENTA TAKATA1,2,4, EIICHI KURAMOCHI1,2, AKIHIKO SHINYA1,2 AND
+MASAYA NOTOMI1,2,3,5
+1Nanophotonics Center, NTT Corporation, 3-1 Morinosato-Wakamiya, Atsugi, Kanagawa 243-0198, Japan
+2NTT Basic Research Laboratories, NTT Corporation, 3-1 Morinosato-Wakamiya, Atsugi, Kanagawa
+243-0198, Japan
+3Department of Physics, Tokyo Institute of Technology, 2-12-1 Ookayama, Meguro-ku, Tokyo 152-8551,
+Japan
+4kenta.takata.ke@hco.ntt.co.jp
+5masaya.notomi.mn@hco.ntt.co.jp
+Abstract:
+An H1 photonic crystal nanocavity is based on a single point defect and has
+eigenmodes with a variety of symmetric features. Thus, it is a promising building block for
+photonic tight-binding lattice systems that can be used in studies on condensed matter, non-
+Hermitian and topological physics. However, improving its radiative quality (𝑄) factor has been
+considered challenging. Here, we report the design of a hexapole mode of an H1 nanocavity with
+a 𝑄 factor exceeding 108. We achieved such extremely high-𝑄 conditions by designing only four
+structural modulation parameters thanks to the C6 symmetry of the mode, despite the need of
+more complicated optimizations for many other nanocavities. The fabricated silicon photonic
+crystal nanocavities exhibited a systematic change in their resonant wavelengths depending on the
+spatial shift of the air holes in units of 1 nm. Out of 26 such samples, we found eight cavities with
+loaded 𝑄 factors over one million (1.2 × 106 maximum). We examined the difference between
+the theoretical and experimental performances by conducting a simulation of systems with input
+and output waveguides and with randomly distributed radii of air holes. Automated optimization
+using the same design parameters further increased the theoretical 𝑄 factor by up to 4.5 × 108,
+which is two orders of magnitude higher than in the previous studies. Our work elevates the
+performance of the H1 nanocavity to the ultrahigh-𝑄 level and paves the way for its large-scale
+arrays with unconventional functionalities.
+© 2023 Optica Publishing Group
+1.
+Introduction
+Photonic crystal nanocavities (PCNs) in dielectric slabs are a particular series of optical
+resonators that exhibit both strong light confinement and small modal volumes [1–12]. These
+features enable intense light-matter interactions, which make PCNs very useful for extremely
+low-power photonics [13–15], on-chip nonlinear optics [16–18] and quantum optics [19–21].
+Integration of PCNs also opens a route to functional nanophotonic devices, such as slow light
+waveguides [22–24], and all-optical switches [25–27], memories [28–30], and transistors [31],
+which are potential for information processing.
+An H1 PCN comprises a vacancy of a single lattice element [32–35]. Such a point defect
+structure takes over the spatial symmetry of its host system. Thus, the eigenmodes of the
+Maxwell equations for the H1 nanocavity are also those for the symmetry operations in the
+entire point group of the photonic crystal (PhC) [36]. As a result, they are analogous to atomic
+orbitals in terms of their symmetric properties, and hence, coupled H1 PCNs work as good
+photonic emulators of molecules and tight-binding lattices including basis functions [22,37].
+arXiv:2301.02376v1  [physics.optics]  6 Jan 2023
+
+Because their evanescent couplings, resonant frequencies and radiation losses can be controlled by
+structural modulation, PCNs can also be combined with unconventional functionalities emerging
+in non-Hermitian and topological physics [38–47]. In particular, arrays of H1 PCNs may pave
+the way for large-scale two-dimensional crystalline systems [48–53]. This potential is in stark
+contrast to most other PCNs based on linear defects, which are less symmetric and thus limited
+in their coupling profiles.
+However, it is generally more difficult for a smaller PCN to have an ultrahigh 𝑄 factor. Narrower
+field distributions in real space result in broader ones in reciprocal space. Parts of such modes
+tend to reside in the light cone (LC) and hence turns into radiation fields, namely losses [3].
+We showed two decades ago that a hexapole mode of the H1 nanocavity in a triangular-lattice
+PhC slab could have a theoretical 𝑄 factor up to 3 × 106, unlike the other eigenmodes [32,33].
+However, this record was not broken even with an algorithmic optimization [54]. Moreover, the
+experimental counterpart was an order of magnitude smaller, namely 3 × 105 [34]. Unfortunately,
+there values compared disadvantageously to those of PCNs with larger defect regions [55–59].
+The lack of tightest light confinement seems to be a significant obstacle to using large-scale H1
+nanocavity arrays, for example, to enhance light-matter interactions with bulky coupled modes,
+and to make robust optical circuits with topological edge states.
+In this article, we design, analyze and experimentally examine the hexapole mode of an H1
+PCN with a theoretical 𝑄 factor (𝑄th) over 108, on the basis of our latest prototype for studying
+non-Hermitian physics [44]. Structural modulation in the design maintains the C6v symmetry of
+the PCN, which the hexapole mode also respects. As a result, we find that we can dramatically
+increase the 𝑄 factor just with four optimization parameters. By elaborating the dependence of
+𝑄th on major three parameters in a simulation, we clarify that such extremely high-𝑄 conditions
+form a region with some width in the parameter space. Here, we obtained a hexapole mode with
+𝑄th = 1.4 × 108 and a modal volume (𝑉) of 0.72(𝜆/𝑛)3. We also compare its field profiles with
+those of another H1 PCN based on a previous study in real and reciprocal spaces.
+We experimentally investigated a series of silicon (Si) H1 PCNs with different spatial shifts of
+air holes. These samples exhibited a systematic variation in their resonant wavelengths, indicating
+that undesired variations in the positions of air holes were restricted. We found that eight such
+PCNs out of 26 had loaded 𝑄 factors (𝑄exp), which include the effects of the input and output
+waveguides, of over one million. The best sample had 𝑄exp = 1.2 × 106, and the cavity’s intrinsic
+𝑄 factor (𝑄i) was estimated to be 𝑄i = 1.5 × 106. We also performed a simulation of the system
+with randomly varying radii of the air holes to close the gap between 𝑄th and 𝑄exp.
+Finally, we performed an automated optimization to further improve 𝑄th. Here, we added the
+hole radius of the background PhC as a parameter and found 𝑄th = 4.5 × 108, which is more
+than a hundred times those in the previous design. Our results show that the highly symmetric
+hexapole mode can achieve both an extremely high 𝑄th and a very small 𝑉 with an inexpensive
+optimization. It enables ultrahigh 𝑄exp (> 106) of H1 PCNs and will open up their various
+applications.
+The remainder of this paper is organized as follows. Section 2 shows the design and modal
+properties of our H1 PCN. Section 3 presents experimental results, and numerically analyzes
+and discusses them. The automated optimization and resultant impact on the hexapole mode
+are summarized in Sec. 4. Section 5 discusses fundamental limitations on the 𝑄 factors of
+nanocavities, including ours. Section 6 concludes this study.
+2.
+Cavity design
+2.1.
+Structure and scheme
+Figure 1(a) depicts the design of our PCN. The system is based on a Si slab with a refractive
+index of 𝑛Si = 3.47 and thickness 𝑡. The PhC here is a triangular lattice of circular air holes of
+radius 𝑅0 and lattice constant 𝑎. Triangular-lattice PhC slabs are widely used in experiments
+
+(a)
+(b)
+R0
+R0
+R1
+s1
+s2
+x
+y
+z
+Fig. 1. (a) Design of H1 PCN based on structural modulation of the innermost and
+second innermost layers of air holes with reference to the single point defect (colored
+red and orange, respectively). 𝑅0 is the radius of the holes for the background PhC
+and the second layer, and 𝑅1 that for the innermost holes. 𝑠1 is a radial shift of the
+innermost layer directed outward from the lattice points, and 𝑠2 is that for the second
+innermost layer with its regular hexagonal alignment kept. (b) 𝐻𝑧 field distribution of
+hexapole mode.
+because they have large photonic band gaps for TE-like modes. The lack of a single hole acts as a
+point defect and hence forms an H1 nanocavity, which is the simplest structure of PCNs that take
+over the C6v symmetry of the PhC. The six holes closest to the defect, which are colored red in
+the figure, have a smaller radius 𝑅1 than that of the background PhC (𝑅1 < 𝑅0). This innermost
+layer of holes is also shifted radially away from the lattice points by a distance 𝑠1. The second
+innermost hole layer comprises the twelve holes located one layer outward from the innermost
+ones and is drawn in orange. It is also translated in the radial direction so that it keeps the regular
+hexagonal alignment and its half diagonal is increased by a distance 𝑠2. In addition, it’s holes are
+of the same radius 𝑅0 as those of the PhC.
+We computed the complex eigenfrequencies 𝑓 of the hexapole eigenmode for various cases by
+using the finite element method on a commercial solver (COMSOL Multiphysics [60]). With the
+defect center defined as the coordinate origin, the system had 11 and 14 layers of holes in the ±𝑥
+and ±𝑦 directions, respectively. A rectangular air region with a height of 3 µm was placed on
+each side of the slab. A scattering boundary condition for plane waves is applied to every border
+of the computational domain. The 𝑥-𝑦 and 𝑦-𝑧 planes were set as perfect magnetic and electrical
+conductors, respectively, for reducing the computational cost. Any changes to these simulation
+conditions are noted in what follows. The theoretical 𝑄 factor is given by 𝑄th = Re 𝑓 /(2Im 𝑓 ).
+Figure 1(b) shows the 𝑧 component of the magnetic fields (𝐻𝑧) of the hexapole mode along the
+𝑥-𝑦 plane. This mode is TE-like and thus characterized by 𝐻𝑧. It is also an eigenmode for the C6
+rotation operator with an eigenvalue of −1. Such an odd parity of a symmetric two-dimensional
+multipole contributes to destructive interference in 𝐻𝑧 along the 𝑧 direction corresponding to
+Γ point [5, 61]. This feature suppresses radiation loss based on the transverse electric field
+components (𝐸𝑥, 𝐸𝑦), as they are linked to 𝐻𝑧 through the Maxwell equations. Thus, structural
+modulation maintaining the lattice-matched rotational symmetry is essential to achieving an
+ultrahigh 𝑄 factor of the hexapole mode. The other C6-symmetric eigenmode of this cavity is
+the monopole mode (not shown). It has an eigenvalue of +1 for the C6 operator and a far lower
+𝑄th < 3000 in our simulations.
+As illustrated in Fig. 1(a), our design uses only four parameters (𝑅0, 𝑅1, 𝑠1, 𝑠2) to improve
+
+85
+88
+91
+94
+97
+99
+101
+103
+105
+107
+R1 (nm)
+s1 (nm)
+105
+106
+107
+108
+Qth
+85
+88
+91
+94
+97
+99
+101
+103
+105
+107
+R1 (nm)
+s1 (nm)
+1.536
+1.547
+1.557
+1.568
+1.578
+λ (µm)
+85
+88
+91
+94
+16
+19
+22
+25
+s2 (nm)
+s1 (nm)
+105
+106
+107
+108
+Qth
+85
+88
+91
+94
+16
+19
+22
+25
+s2 (nm)
+s1 (nm)
+1.548
+1.553
+1.558
+1.563
+1.569
+λ (µm)
+(a)
+(b)
+(d)
+(c)
+Wavelength
+Q factor
+Fig. 2. Dependence of (a) resonant wavelength (𝜆) and (b) theoretical 𝑄 factor (𝑄th) of
+the hexapole mode on 𝑠1 and 𝑅1 for 𝑠2 = 20.5 nm. (c) 𝜆 and (d) 𝑄th dependent on 𝑠1
+and 𝑠2 for 𝑅1 = 102 nm. Black dots represent sample points in the simulation. The
+data among the points are linearly interpolated. A band of parameter conditions for
+𝑄th > 108 appears. 𝑅0 = 131 nm, 𝑎 = 426 nm, and 𝑡 = 250 nm.
+the 𝑄 factor, unlike recent designs based on costly optimizations of many variables [62–65].
+𝑅0 determines the filling factor of the PhC, which is related to its photonic band gap and thus
+the in-plane modal confinement. 𝑅1, 𝑠1 and 𝑠2 affect the local modal properties. The lattice
+constant 𝑎 can be varied to adjust the resonant wavelengths of the simulated modes to telecom
+ones around 1.55 µm.
+2.2.
+Resonance properties versus hole shifts
+First, let us study the resonance characteristics of the mode for constant 𝑅0 = 131 nm, 𝑎 = 426 nm,
+and 𝑡 = 250 nm. Figure 2(a) and (b) are two-dimensional color plots of the resonant wavelength
+𝜆 = 𝑐/Re 𝑓 and 𝑄th for isolated (unloaded) H1 PCNs depending on 𝑠1 and 𝑅1. Here, 𝑐 is the
+speed of light in vacuum and 𝑠2 = 20.5 nm. The plot of 𝜆 indicates that a small 𝑠1 and large 𝑅1
+squeeze the magnetic poles in Fig. 1(b) and thus yield a short 𝜆, whereas a large 𝑠1 and small 𝑅1
+broaden the magnetic poles and thus increase 𝜆. Remarkably, the 𝑄th plot exhibits a sequence of
+optimum points with 𝑄th > 108 forming a linear band. Such a peak distribution indicates that
+there is an optimal polar width for every 𝜆 that suppresses local scattering-induced radiation loss.
+There is a margin of about ±1.5 nm in 𝑅1 and a wider one in 𝑠1 from each optimum point to have
+a 𝑄th > 107. The largest 𝑄 factor here is 𝑄th = 1.43 × 108 for (𝑠1, 𝑅1) = (88.75 nm, 101.75 nm).
+In units of 0.5 nm for the parameters, 𝑄th = 1.41 × 108 for (𝑠1, 𝑅1) = (89.5 nm, 102 nm) was
+obtained.
+Figure 2(c) and (d) depict the dependence of 𝜆 and 𝑄th on 𝑠1 and 𝑠2 for 𝑅1 = 102 nm. There
+
+is a notable difference between Fig. 2(a) and (c) in the directions of the iso-wavelength contours.
+This difference is due to negative correlation between the effect of 𝑅1 and that of 𝑠2; a larger 𝑠2
+results in a longer 𝜆 because of the higher effective index of the cavity region. In contrast, Fig.
+2(b) and (d) appear to have more or less similar properties. As the mode wavelength increases
+with 𝑠1, the optimal 𝑠2 also becomes larger. 𝑠2 can be used to dramatically improve 𝑄th because
+it introduces a gradual variation in the effective potential barrier of the PhC [7,66]. However, the
+trace of the extremely high 𝑄 values in Fig. 2(d) is nearly perpendicular to the contour lines in
+Fig. 2(c), meaning that the conditions for a much improved 𝑄th are limited for each 𝜆. The peak
+value of 𝑄th decreases for large and small 𝑠1 because 𝑅1 is fixed. Overall, a global optimization
+for (𝑅1, 𝑠1, 𝑠2) enables us to find the continuous conditions for 𝑄th > 108 in the parameter space.
+The best 𝑄th here is 1.46 × 108 for (𝑠1, 𝑠2) = (90.25 nm, 20.75 nm).
+2.3.
+Modal properties
+Next, let us compare the modal shapes in real and reciprocal spaces of the design with
+𝑄th > 108 and that in the previous study. Figure 3(a) and (b) show the spatial magnetic intensity
+distributions on a common logarithmic scale (log10(|H(r)|2)) along 𝑧 = 0 for hexapole modes
+with 𝑄th = 2.0 × 106 and 1.4 × 108, respectively. The PCN shown in (a) is based on Ref. [33]
+and does not include 𝑠2 in its design with 𝑅0 = 109 nm, 𝑅1 = 100 nm, 𝑠1 = 78 nm, 𝑎 = 435 nm,
+and 𝑡 = 220 nm. The other PCN in (b) corresponds to (𝑠1, 𝑅1) = (89.5 nm, 102 nm) in Fig. 2(a)
+and (b). A sizable portion of (a) has evanescent fields with relative intensities of about 10−4, and
+visible components with intensities over 10−8 reach the boundaries of the entire geometry. In
+comparison, the optimal mode shown in (b) obviously decays faster from the center. This means
+that the current design provides stronger in-plane light confinement.
+Figure 3(c) and (d) depict the Fourier transforms of the 𝑥 component of the electric fields on a
+logarithmic scale (log10(|F (𝐸𝑥(r))|)) along 𝑧 = 0 for the hexapole modes in Fig. 3(a) and (b).
+Transverse electric field components lying within the LC measure the magnitude of radiation loss,
+because they can directly couple with radiative plane waves [3,67]. As shown in Fig. 3(c), the
+previously designed mode has relative Fourier amplitudes of about 10−2.5 distributed in the LC
+defined by the black dashed circle. In stark contrast, the radiative field amplitudes are suppressed
+over the entire LC for the optimized mode shown in Fig. 3(d). Their maximum value is about
+one order of magnitude smaller than that of Fig. 3(c), confirming an improvement in the 𝑄 factor
+due to the reduction of the radiation flux. A similar trend is seen in the case of 𝐸𝑦. These modal
+properties also support the discussion on Fig. 2(b) and (d).
+The standard Purcell mode volume 𝑉 for PCNs is given by [2]
+𝑉 =
+∫
+𝜖(r)|E(r)|2𝑑3r
+max{𝜖(r)|E(r)|2} .
+(1)
+This definition is accurate in estimating the Purcell effect for high-𝑄 cavities and has been used
+for comparison purposes in the literature. Interestingly, the effective volume 𝑉opt = 0.72(𝜆/𝑛)3
+for the mode with 𝑄th = 1.4 × 108 is larger by 9% than that of the previously studied one,
+𝑉p = 0.66(𝜆/𝑛)3. The electric energy densities of hexapole modes tend to concentrate mostly on
+the sides of the innermost air holes. However, the optimized mode distributes more electric energy
+around the point defect than the mode based on Ref. [33] because of the potential modulation by
+𝑠2. Thus, it has a reduced maximum energy density or denominator in Eq. (1).
+This result shows that we can dramatically improve 𝑄th of the hexapole mode without sacrificing
+its small 𝑉. 𝑉opt here is comparable with those of optimized L3 PCNs without hole radius
+modulation [64,67], while the hexapole mode has a larger 𝑄th. Thus, our H1 PCNs can be expected
+to have 𝑄exp values as high as those ones. In addition, our optimal 𝑄th/𝑉opt = 1.9 × 108(𝑛/𝜆)3 is
+slightly better than another L3 nanocavity with 𝑄th/𝑉 = 1.7 × 108(𝑛/𝜆)3 (𝑄th = 1.9 × 108 and
+𝑉 = 1.1(𝜆/𝑛)3) designed by the particle-swarm algorithm [65].
+
+(a)
+(b)
+(d)
+(c)
+a = 435 nm
+a = 426 nm
+0
+0
+4
+-4
+kx (units of π/a)
+-4
+4
+0
+kx (units of π/a)
+0
+-4
+4
+4
+-4
+ky (units of π/a)
+log10(|H|2)
+log10(|(Ex)|)
+Fig. 3. (a) Magnetic field intensity distribution in the logarithmic scale (log10(|H(r)|2))
+for the hexpole nanocavity based on the previous work [33] with 𝑎 = 435 nm and
+𝑄th = 2.0×106. (b) Same but for the hexpole mode designed in this study with 𝑎 = 435
+nm, 𝑠1 = 89.5 nm, 𝑅1 = 102 nm, and 𝑄th = 1.4 × 108, exhibiting more tightly confined
+in-plane evanescent fields than in (a). (c), (d) Absolute Fourier-space distributions of
+the 𝑥 components of the electric fields on a logarithmic scale (log10(|F (𝐸𝑥(r))|)) for
+the eigenmodes corresponding to (a) and (b), respectively. (d) has significantly reduced
+radiative components inside the light line that is marked by the black dashed curve.
+In summary, we showed designs of H1 PCNs based on a manual or brute-force search for
+extremely high-𝑄 hexapole modes. By focusing on the case for a constant 𝑅0, we found a series
+of conditions for 𝑄th > 108 with just three major optimization parameters (𝑅1, 𝑠1, 𝑠2), thanks to
+the C6 symmetry of the mode. Introduction of an optical potential modulation with 𝑠2 resulted
+in improved light confinement of the optimized mode in both the in-plane and out-of-plane
+directions. This point will be examined quantitatively in Sec. 4.
+3.
+Experimental result and numerical analysis
+3.1.
+Sample fabrication and measurement
+We fabricated Si H1 PCNs of our design for an experimental demonstration. The sample
+structures were patterned by electron beam (EB) lithography on a positive EB resist coated
+on a silicon-on-insulator (SOI) wafer. The mask pattern was projected to the Si film with a
+
+0
+-0.5
+-1
+-1.5
+-2
+-2.5
+-3
+-3.5
+-4
+-4.5
+-50
+-1
+-2
+-3
+-4
+-5
+-6
+-7
+-8190.4305
+190.4315
+190.4325
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Normalized power
+Frequency (THz)
+(b)
+(c)
+1 μm
+Qexp = 1.1×106
+(a)
+2 μm
+Fig. 4. (a) Laser scope image of a sample with 𝑑 = 5
+√
+3𝑎. The input and output Si
+waveguides are broadened and extended to both sides of the sample chip and coupled
+with lensed fibers. (b) Close-up SEM image of H1 PCN with 𝑎 = 434 nm. Typical radii
+of the small and large air holes are estimated as 𝑅1,s ≈ 106.8 nm and 𝑅0,s ≈ 133.1
+nm. (c) Transmission spectrum of sample with 𝑎 = 434 nm and 𝑠1 = 99.5 nm. The
+Lorentzian curve colored red matches the experimental data shown as blue points and
+indicates that the cavity has a loaded 𝑄exp of 1.1 × 106.
+nominal thickness of 250 nm by inductively coupled plasma etching. The buried oxide (BOX)
+layer beneath the PhCs was removed by wet etching with buffered hydrogen fluoride to obtain
+air-bridged samples. After the above device processes were completed, the wafer was cleaved so
+that the size of each sample chip was 5 mm × 15 mm.
+Figure 4(a) is a laser scope image of a PCN sample. The H1 cavity was butt-coupled (loaded)
+with two W1 PhC waveguides, each of which had a width of 𝑊0 =
+√
+3𝑎 based on the removal
+of a single row of air holes. The spatial interval 𝑑 between the cavity and them varied with the
+samples, and ones with 𝑑 = 5
+√
+3𝑎 exhibited ultrahigh-𝑄 resonances. Each W1 waveguide was
+broadened by 100 nm at either end of the PhC by shifting five pairs of air holes on the sides
+outward with a stepwise increment of 20 nm. Consequently, they were efficiently coupled with
+air-suspended wire waveguides with a width of 𝑊0. These optical channels were extended farther
+and connected to 8 µm-wide slab waveguides that were supported by the BOX layer and led to
+the edges of the chip.
+A close-up scanning electron microscope (SEM) image of an H1 nanocavity is shown as
+Fig. 4(b). Typical radii for the innermost and second innermost hole layers of the resist mask
+were estimated as 𝑅1,m ≈ 102.8 nm and 𝑅0,m ≈ 130.4 nm, respectively, which were close to
+the condition for 𝑄th > 108 found in Fig. 2. However, the radii of the fabricated samples
+became somewhat bigger in the etching process: 𝑅1,s ≈ 106.8 nm and 𝑅0,s ≈ 133.1 nm. We
+prepared PCN chips with five distinct lattice constants, 𝑎 = 418, 422, 426, 430, 434 nm. For
+the evaluations, we focused on the one with 𝑎 = 434 nm, because it best compensated for the
+discrepancies in hole radii between the design and fabrication.
+We performed transmission measurements on each sample chip by placing it on a metallic
+stage whose temperature was maintained at 25◦C by a Peltier element and a PID controller.
+Tapered optical fibers were carefully aligned by using three-axis nano-positioners equipped with
+fiber holder stages, so that they were coupled with the slab waveguides at both ends of the chip
+and hence formed a measurement channel. The typical coupling loss per such interface was about
+10 dB. A coherent transverse electric (TE) polarized light from a tunable laser was injected into
+each sample. The output was detected by a power meter synchronized with the wavelength sweep
+of the laser. The transverse magnetic (TM) field components of the input and output signals were
+filtered out by fiber polarizers. The entire system was based on polarization-maintaining fibers.
+We prepared and measured a pair of H1 nanocavity samples with nominally the same structure
+for each of 𝑠1; namely the shifts of the innermost holes varied from 89.5 to 101.5 nm in units
+
+of 1 nm. All these 26 samples had 𝑠2 = 20.5 nm and 𝑑 = 5
+√
+3𝑎. A transmission spectrum of
+an H1 nanocavity with 𝑠1 = 99.5 nm is plotted in Fig. 4(c). The experimental data shown as
+blue points match the Lorentzian curve (colored red) obtained by a least squares fitting. The
+peak frequency (wavelength) was 190.4315 THz (1575.370 nm), and the linewidth of the best-fit
+curve was 173.8 MHz. These values give an experimental loaded 𝑄 factor of 𝑄exp = 1.1 × 106.
+Here, we have excluded any arbitrariness in determining 𝑄exp of the measured resonance with
+discrete data points. The input power was attenuated so that thermal linewidth broadening and
+nonlinearity would be avoided. In this case, however, the detection power around resonance tails
+tended to be slightly reduced, as indicated by its visible drop near 190.4319 THz. This is because
+the power meter had a limited dynamic range with a minimum detectable power of -80 dBm.
+We can certainly identify this resonance to be the hexapole mode, because the other cavity
+modes typically have 𝑄th < 20000 in our simulations and their wavelength spacing with respect
+to the ultrahigh-𝑄 peak is 30 nm or larger.
+3.2.
+Measured wavelengths and quality factors of H1 PCNs
+Figure 5(a) presents the dependence of the measured resonance wavelengths 𝜆 of the hexapole
+modes on 𝑠1. To show the correspondence between the data of 𝜆 and 𝑄exp, we divided the
+samples into two sets according to their positions, so that each sample in set 1 is closer to the
+front edge of the chip than its counterpart in set 2 with the same 𝑠1. It can be clearly seen that
+𝜆 is positively correlated with ��1, as predicted in Fig. 2(a) and (c). The variation in 𝜆 within
+pairwise samples for each 𝑠1 is so weak that a linear regression of the entire data, shown by the
+red line, reproduces their average trend. The slope of the regression line is 1.55 ± 0.032 nm (𝜆) /
+nm (𝑠1), and its coefficient of determination is 𝑅2 = 0.990.
+Here we define the difference in resonant wavelength between set 1 and 2 as Δ𝜆(𝑠1) =
+𝜆1(𝑠1) − 𝜆2(𝑠1), where 𝜆1(𝑠1) and 𝜆2(𝑠1) are the wavelengths of the samples with 𝑠1 in set 1
+and 2, respectively. Δ𝜆 for all 𝑠1 in Fig. 5(a) are calculated, and then their standard deviation is
+found to be 𝜎Δ𝜆 = 0.848 nm. Because 𝜆1(𝑠1) and 𝜆2(𝑠1) ideally have the same value and their
+variations should stem from numerous independent and random processes during fabrication,
+we assume that they have no covariance. Thus, we can estimate the deviation in 𝜆 to be
+𝜎𝜆 = [𝜎2
+Δ𝜆/2]1/2 = 0.600 nm.
+This result implies that our nanocavities have highly accurate hole positions. Although the
+obtained value of 𝜎𝜆 corresponds to a change solely in 𝑠1 of 0.39 nm, in reality, there are other
+major factors that affect 𝜎𝜆, such as the hole radii, local Si slab thicknesses and surface roughness.
+In addition, the positioning accuracy of the electron beam used in patterning the resist mask is as
+small as 0.05 nm. Thus, undesired variations in hole positions, including those in 𝑠1 and 𝑠2, will
+be less significant in the actual samples.
+The measured loaded 𝑄 factors for the two sample sets are plotted in Fig. 5(b) as a function of
+𝑠1. They exhibit a gentle peak centered around 𝑠1 = 94.5 or 95.5 nm; 𝑄exp for these values of 𝑠1
+is significantly larger than that for 𝑠1 = 89.5 and 101.5 nm. The best sample here belongs to set
+2 and has 𝑠1 = 96.5 nm and 𝑄exp = 1.2 × 106 with an estimated linewidth of 160.4 MHz. Its
+transmission spectrum is shown in the inset of Fig. 5(b). Although the shape of the resonance is
+slightly asymmetric, it is still fitted by a Lorentzian function.
+Eight samples out of 26 had 𝑄exp > 106. Remarkably, they included ones with 𝑠1 = 90.5 and
+99.5 nm, namely off from the peak center. This trend implies that the 𝑄 factors for these PCNs
+are much larger in theory but were reduced because of fabrication imperfections. The effect of
+disorder is also reflected in the outlier sample with a low 𝑄exp = 3.0 × 105 and 𝑠1 = 96.5 nm in
+set 1.
+
+89.5
+91.5
+93.5
+95.5
+97.5
+99.5 101.5
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+ Set 1
+ Set 2
+Loaded Q factor (×106)
+s1 (nm)
+89.5
+91.5
+93.5
+95.5
+97.5
+99.5 101.5
+1.558
+1.562
+1.566
+1.570
+1.574
+1.578
+1.582
+ Set 1
+ Set 2
+Resonant wavelength (µm)
+s1 (nm)
+89.5
+91.5
+93.5
+95.5
+97.5
+99.5 101.5
+1.558
+1.562
+1.566
+1.570
+1.574
+1.578
+1.582
+Resonant wavelength (µm)
+s1 (nm)
+89.5
+91.5
+93.5
+95.5
+97.5
+99.5 101.5
+106
+107
+108
+ Unloaded
+ Loaded
+ WG coupling
+Q factor
+s1 (nm)
+(a)
+(b)
+(c)
+(d)
+191.111 191.112 191.113
+0.0
+0.5
+1.0
+Normalized power
+Frequency (THz)
+Qexp = 1.2×106
+Fig. 5. (a) Dependence of measured 𝜆 on 𝑠1 for two nominally duplicate sets of H1 PCN
+samples with 𝑎 = 434 nm, 𝑠2 = 20.5 nm, and 𝑑 = 5
+√
+3𝑎. The grouping of the samples
+into sets is based on their positions relative to the front edge of chip (the samples in set
+1 are closer to the edge). The red line is a linear regression of the experimental data.
+(b) Loaded 𝑄 factor (𝑄exp) as a function of 𝑠1 for the two sample sets. The inset is the
+transmission spectrum for the best sample that had 𝑄exp = 1.2 × 106 and 𝑠1 = 96.5
+nm. (c) Simulated 𝜆(𝑠1) for 𝑎 = 434 nm, 𝑡 = 241 nm, 𝑅1 = 106 nm, 𝑅0 = 134 nm,
+and 𝑠2 = 20.5 nm, which agrees well with the experimental data. (d) Simulated 𝑄
+factors for the same parameters on a semi-logarithmic scale. Squares show results for
+unloaded samples (𝑄th), while dots are for loaded ones (𝑄th,L) including two W1 PhC
+waveguides with 𝑑 = 5
+√
+3𝑎 that radiate out the light. Triangles show the 𝑄 factors
+𝑄WG due to the losses by the waveguides.
+3.3.
+Simulation of measured samples
+We performed simulations by varying the structural parameters around those estimated from the
+SEM image. Figure 5(c) shows the theoretical 𝜆 as a function of 𝑠1 for 𝑎 = 434 nm, 𝑡 = 241
+nm, 𝑅1 = 106 nm, 𝑅0 = 134 nm, and 𝑠2 = 20.5 nm. The theoretical values agree well with the
+experimental data. Although the simulation result is slightly convex upward, its average slope
+(1.55 nm (𝜆) / nm (𝑠1)) coincides with that of the experimental result. We emphasize that 𝑅1
+and 𝑅0 here are consistent with the measured 𝑅1,s and 𝑅0,s within an error of a few nanometers,
+as expected for the current measurement. The value of 𝑡 is smaller than the nominal thickness
+250 nm of the Si film, indicating that the PhC slabs were thinned down by the etching processes
+and/or that 𝑛Si in the simulation is slightly smaller than that of the actual material.
+Moreover, as shown in Fig. 5(d), the corresponding theoretical 𝑄 factors follow the trend seen
+
+in the experiment. The figure compares 𝑄th for the H1 PCNs with and without two W1 PhC
+waveguides with 𝑑 = 5
+√
+3𝑎 extending to the right and left sides of the simulation domain where
+the fields are scattered out. The plots are on a semi-logarithmic scale, with the horizontal axis
+depicting steps of 1 nm. The loaded 𝑄 factors, 𝑄th,L, are the black dots, and the unloaded ones, 𝑄th,
+are the purple squares. Both plots peak at 𝑠1 = 96.5, where 𝑄th,L = 5.9×106 and 𝑄th = 5.9×107.
+The loaded hexapole mode for this condition has a theoretical modal volume of 𝑉 = 0.74(𝜆/𝑛)3.
+Thus, our best experimental sample is expected to have had 𝑄exp/𝑉 = 1.6 × 106(𝑛/𝜆)3.
+The difference between 𝑄th,L and 𝑄th comes from the coupling with the environment via
+the waveguides. The impact of this coupling, 𝑄WG, can be derived from the relation 1/𝑄th,L =
+1/𝑄th + 1/𝑄WG. The resultant values are plotted as the triangles in Fig. 5(d). They exhibit a
+moderate variation with 𝑠1 probably due to the group velocity dispersion of the waveguides
+and are about 𝑄WG = 6.6 × 106 around the peak of 𝑄th. As a result, the intrinsic (unloaded) 𝑄
+factor of the optimum sample is estimated to be 𝑄i = [1/𝑄exp − 1/𝑄WG]−1 = 1.5 × 106. The
+correspondent 𝑄/𝑉 amounts to 𝑄i/𝑉 = 2.0 × 106(𝑛/𝜆)3, which is comparable with those of
+PCNs without having their surface Si passivated with hydrogen [28,56,57,68].
+3.4.
+Impact of varying hole radii
+We can see that 𝑄exp is still lower than 𝑄th,L and hence it is expected to be affected by reductive
+factors other than 𝑄WG. A simple but realistic cause of extra loss is radiative scattering induced
+by random variations in the radii and positions of the air holes [55, 69]. The hole radii can
+change on the atomic scale order because of stochastic processes in fabrication, such as in the
+EB exposure, resist development, and dry and wet etching. On the other hand, the EB shots are
+precisely aligned in our lithography process. Thus, the positions of the hole centers are mainly
+affected by the small and probabilistic anisotropy of etching or distortion in the shapes of the
+holes, part of which is also considered to impact the radii.
+Here, we simulated samples with air holes just of varying radii to statistically evaluate the
+effect of fabrication imperfections on the 𝑄 factor. The result estimates a dominant portion of the
+disorder-induced scatting loss denoted as 1/𝑄scat. We used the parameters that reproduce 𝜆 of
+the measured samples and set 𝑠1 = 96.5 nm for 𝑄th = 5.9 × 107 without structural imperfections
+or PhC waveguides. The PEC boundary condition of the 𝑦-𝑧 plane was removed so that the
+simulation explicitly included all the holes. The small and large holes were assumed to have
+random radii sampled from Gaussian distributions with means 𝑅1 and 𝑅0, respectively, and a
+common standard deviation (SD) of 𝜎𝑟. The 𝑄 factor obtained in each run is denoted as 𝑄th,F
+and satisfies 1/𝑄th,F = 1/𝑄th + 1/𝑄scat.
+Figure 6(a) and (b) show 𝜆 and 𝑄th,F for 100 random patterns with 𝜎𝑟 = 1.0 nm. The data
+points of both plots look randomly scattered. The mean and SD of the resonant wavelengths are
+(𝜇𝜆, 𝜎𝜆) = (1.57084 µm, 1.052 nm) and those of the 𝑄 factors are (𝜇𝑄, 𝜎𝑄) = (2.3 × 106, 1.07 ×
+106). The wavelengths tend to be distributed symmetrically around 𝜇𝜆, while the 𝑄 factors are
+specifically high for some sample points, indicating distinct statistical properties.
+We repeated the random simulations for different 𝜎𝑟. The dependence of (𝜇𝜆, 𝜎𝜆) on 𝜎𝑟 and
+that of (𝜇𝑄, 𝜎𝑄) are plotted in Fig. 6(c) and (d), respectively. The mean wavelength for each
+𝜎𝑟 is mostly convergent at 𝜆 = 1.5710 µm, which is obtained for the case of no disorder. The
+deviation in 𝜆 grows proportionally with 𝜎𝑟. The variance of the radii 𝜎2
+𝑟 is directly related to
+that of the effective dielectric constant of the PhC slab via the filling fraction of the air holes.
+Thus, 𝜎𝑟 affects the deviation of the effective index and has an approximately linear dependence
+on 𝜎𝜆. Its slope is estimated as 𝜎𝜆/𝜎𝑟 = 1.11.
+In contrast, both 𝜇𝑄 and 𝜎𝑄 tend to be inversely proportional to 𝜎2
+𝑟 . As discussed in Ref. [70],
+local variations in the dielectric constant affect the extra scattering rate and hence the loss. By
+
+20
+40
+60
+80
+100
+1
+0
+2
+4
+6
+8
+10
+Q factor (×106) 
+Fluctuation pattern index
+20
+40
+60
+80
+100
+1
+1.568
+1.569
+1.570
+1.571
+1.572
+1.573
+1.574
+Wavelength (µm)
+Fluctuation pattern index
+0.0
+0.5
+1.0
+1.5
+2.0
+0
+2
+4
+6
+Mean Q factor µQ (×106)
+Deviation of radii σr (nm)
+0
+1
+2
+3
+Deviation of Q factor σQ (×106)
+0.0
+0.5
+1.0
+1.5
+2.0
+1.568
+1.569
+1.570
+1.571
+1.572
+1.573
+1.574
+Mean wavelength µλ(µm)
+Deviation of radii σr (nm)
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+Deviation of wavelength σλ (nm)
+(a)
+(b)
+(c)
+(d)
+Fig. 6. (a) Simulated resonant wavelengths and (b) unloaded 𝑄 factors of H1 PCNs with
+100 different random patterns of hole radii for 𝜎𝑟 = 1.0 nm. (c) Mean and standard
+deviation of the resonant wavelength (𝜇𝜆, 𝜎𝜆) and (d) those of the 𝑄 factor (𝜇𝑄, 𝜎𝑄)
+of the random simulation for different 𝜎𝑟. 𝜇𝜆(𝜎𝑟) converges at the result without
+any disorder shown as the black line, while 𝜎𝜆(𝜎𝑟) grows linearly, as indicated by
+the regression line in red. Both 𝜇𝑄 and 𝜎𝑄 are inversely proportional to 𝜎2𝑟 . The
+approximate statistical properties of the scattering loss are given by Eqs. (2) and (3).
+The mean 𝑅0 and 𝑅1 are 134 nm and 106 nm, respectively. The other parameters are
+the same as those used for Fig. 5.
+subtracting 1/𝑄th from 1/𝑄th,F of the data, the approximate mean and SD of 1/𝑄scat are given by
+𝜇[1/𝑄scat] = 6.3 × 10−7𝜎2
+𝑟 ,
+(2)
+𝜎[1/𝑄scat] = 3.3 × 10−7𝜎2
+𝑟 ,
+(3)
+where 𝜎𝑟 is measured in nanometers.
+Similar properties have been reported in multi-
+heterostructure nanocavities with variations in the positions and radii of the air holes [55,69].
+As mentioned in the discussion of Fig. 5(a), the experimental data suggest 𝜎𝜆 = 0.600 nm.
+This value corresponds to 𝜎𝑟 = 0.54 nm via the proportional relation between 𝜎𝜆 and 𝜎𝑟. By
+substituting the value of 𝜎𝑟 into Eqs. (2) and (3), we obtain 𝜇[1/𝑄scat] = 1.8 × 10−7 and
+𝜎[1/𝑄scat] = 9.6 × 10−8, as the estimated statistical properties of the scattering loss for the
+measured samples. The resultant mean 𝑄scat is 5.4 × 106. We should emphasize that we did not
+underestimate 𝑄scat by neglecting inaccuracies in the hole positions. The variation in wavelength
+in the experiment is attributed solely to 𝜎𝑟, and its entire impact is hence taken into consideration
+in obtaining 𝑄scat.
+Because the mean 𝑄exp is 𝜇[𝑄exp] ≈ 106 around the optimal condition, this result indicates the
+existence of further loss in the experiment with an average 𝑄 factor of (𝜇[1/𝑄exp] − 𝜇[1/𝑄scat] −
+
+12
+14
+16
+18
+20
+22
+24
+26
+28
+0
+1
+2
+3
+4
+5
+Qth (×108)
+s2 (nm)
+12
+14
+16
+18
+20
+22
+24
+26
+28
+1.50
+1.52
+1.54
+1.56
+1.58
+1.60
+1.62
+Resonant wavelength (µm)
+s2 (nm)
+log10(|(Ex)|)
+0
+kx (units of π/a)
+0
+-4
+4
+4
+-4
+ky (units of π/a)
+88
+90
+92
+94
+96
+98
+94.6
+96.8
+99.0
+101.2
+103.4
+105.6
+125.0
+127.5
+130.0
+132.5
+135.0
+R 1 (nm)
+R0 (nm)
+s1 (nm)
+Start
+Qth = 9.2×10
+6
+Optimum
+Qth = 3.1×10
+8
+Qth = 1.3×10
+8
+Qth = 2.1×10
+8
+(a)
+(b)
+(c)
+(d)
+Fig. 7. (a) Evolution of (𝑅0, 𝑅1, 𝑠1) in the Nelder-Mead optimization of 𝑄th for
+𝑠2 = 23 nm. Blue arrows indicate the direction of the parameter variation. (b)
+log10(|F (𝐸𝑥(r))|) for the optimized hexapole mode for 𝑠2 = 23 nm. The radiative
+component lying inside the LC is reduced, compared with Fig. 3. The black dashed
+circle denotes the light line. (c) 𝜆 and (d) 𝑄th of the optimized H1 PCNs for different
+𝑠2. Both of them tend to be positively correlated with 𝑠2. We obtained 𝑄th = 4.5 × 108
+for the optimized variables (𝑅0, 𝑅1, 𝑠1) ≈ (115.92 nm, 90.258 nm, 85.773 nm) for
+𝑠2 = 26 nm. Other fixed parameters are 𝑎 = 426 nm and 𝑡 = 250 nm.
+1/𝑄WG)−1 ≈ 1.5 × 106. We attribute part of this loss to a slight amount of EB resist remaining
+on the sample. Considering that the laser scope comes into focus twice in scanning the surface,
+it is expected to form a very thin layer over the chip. This results in structural asymmetry in
+the out-of-plane direction and hence induces extra radiation loss, as is the case with samples
+fabricated on sacrificial layers. Its unevenness, which can be seen at the top right of Fig. 4(b) for
+example, could also be a source of scattering. We did not try to remove the resist layer from the
+chip, because such a process unavoidably thins down the Si layer and thus alters the dependence
+of the resonance properties on 𝑠1. The sample quality will be improved in future studies.
+4.
+Automated optimization
+Recent studies have used various automated optimization algorithms to achieve high theoretical
+𝑄 factors in PCNs [54, 56, 62–65]. We used the built-in optimization module of COMSOL
+Multiphysics and found that the performance of the H1 PCN can further be improved. Here, we
+chose the Nelder-Mead method [71], which prepares a symplex in a parameter space and repeats
+
+0
+-0.5
+-1
+-1.5
+-2
+-2.5
+-3
+-3.5
+-4
+-4.5
+-5its update based on the reflection, expansion, contraction, or shrink process, depending on the
+value of the function 𝐹 to be optimized. This scheme does not use any gradient or assume any
+approximate form of the function. Thus, it is expected to work regardless of the actual landscape
+of 𝐹. We fixed 𝑠2 and obtain a maximal 𝑄th by varying 𝑅0, 𝑅1 and 𝑠1 in each optimization run,
+namely 𝐹 = 𝑄th(𝑅0, 𝑅1, 𝑠1).
+Figure 7(a) shows the evolution of the parameters in the optimization for 𝑠2 = 23 nm, 𝑎 = 426
+nm, and 𝑡 = 250 nm. Here, the initial point was set as (𝑅0, 𝑅1, 𝑠1) = (128.3 nm, 99.5 nm, 89.4 nm)
+with 𝑄th = 9.2 × 106. The variables undergo substantial changes at steps in the early stage
+of the operation. The state passes through a condition for 𝑄th > 108 and is then bound in a
+region of suboptimal points with 𝑄th < 2 × 108. After a while, however, the algorithm finds
+a direction in which 𝑄th is improved beyond 2 × 108. It eventually settles at (𝑅0, 𝑅1, 𝑠1) ≈
+(125.18 nm, 97.421 nm, 89.024 nm) exhibiting the optimum objective, 𝑄th = 3.1 × 108. The
+normalized absolute Fourier amplitudes of 𝐸𝑥 for this optimal mode are depicted on a logarithmic
+scale in Fig. 7(b). Compared with Fig. 3(d), the domain with the relative amplitudes below 10−5
+in the LC is doubly extended in the 𝑘𝑥 direction. This feature confirms that the light confinement
+of this H1 PCN is stronger than that of the manually designed ones shown in Sec. 2.
+We repeated the optimization routine with different values of 𝑠2, which is the additional
+factor not in the former design examined in Fig. 3(a) and (c). To understand quantitatively
+the impact of 𝑠2, we plot the dependences of 𝜆 and 𝑄th of the optimized PCN in Fig. 7(c)
+and (d). The resonant wavelength is monotonically red-shifted as 𝑠2 increases. Accordingly,
+a larger 𝑠2 results in a higher optimal 𝑄 factor. We find that 𝑄th = 4.5 × 108 for 𝑠2 = 26 nm,
+which is more than a hundred-times the values in the previous reports [33, 54]. Remarkably,
+the optimized mode also has a small volume of 𝑉opt = 0.71(𝜆/𝑛)3, and thus its 𝑄/𝑉 is as large
+as 𝑄th/𝑉opt = 6.3 × 108(𝑛/𝜆)3. This result confirms the striking contribution of the gradual
+variation in the optical potential introduced by 𝑠2 to 𝑄th, as mentioned in Sec. 2.
+The optimal structural parameters vary greatly with 𝑠2.
+We obtained (𝑅0, 𝑅1, 𝑠1) ≈
+(144.23 nm, 111.61 nm, 86.020 nm) and (115.92 nm, 90.258 nm, 85.773 nm) for 𝑠2 = 13
+nm and 26 nm, respectively. 𝑅0 and 𝑅1 tend to be negatively correlated with 𝑠2 and 𝜆, while 𝑠1
+oscillates gently between 82 nm and 92 nm with respect to 𝑠2. Optimization with more parameters
+such as (𝑅0, 𝑅1, 𝑠1, 𝑠2, 𝑎) might result in a better 𝑄th. In that case, however, the parameter space
+would become larger and contain more local minima of 𝑄th. Thus, the computation would be
+much harder in terms of both its convergence and the probability of finding a good solution. We
+leave that consideration out of the scope of this study.
+5.
+Discussion
+Experimental 𝑄 factors of PCNs are generally limited by many kinds of defects. Discussing their
+impact will allow us to predict how high 𝑄exp could be made in a real PCN device.
+A major cause of the reduction of 𝑄 factors is structural imperfections. In our result, the
+variations in 𝜆 and 1/𝑄th,F were attributed to those in the hole radii, and 𝜎𝑟 = 0.54 nm and
+𝜇[1/𝑄scat] = 1.8 × 10−7 were obtained. A groundbreaking report by Asano et al. on multi-
+heterostructure PCNs [72], including one with 𝑄exp = 1.1 × 107, considered the same deviation
+𝜎hole in both the positions and radii of the air holes. They estimated 𝜎hole to be 0.25 nm and the
+corresponding 𝜇[1/𝑄scat] to be 4.7 × 10−8 for their PCN samples. A monolayer of Si is about
+0.135-nm-thick and an air hole has two side walls in the radial direction. Thus, 𝜎hole = 0.25 nm
+seems to indicate that the etching process just leaves the uncertainty at the level where a single
+atomic layer is removed or not at every Si surface, including the resultant hole displacement.
+Both Eq. (2) and the dependence of 𝜇[1/𝑄scat] on 𝜎hole in Ref. [72] are quadratic equations
+and have similar coefficients. Even though 𝜎𝑟 and 𝜎hole of the two PCNs can be reduced to the
+monolayer level (= 0.135 nm), a dimensionless loss of about 𝜇[1/𝑄scat] ≈ 10−8 remains. This
+implies that it is hard to achieve 𝜇[𝑄scat] > 108 for PCNs.
+
+Another limiting factor is the formation of surface oxidation layers on Si. Every Si/SiO𝑥
+interface has a few kinds of surface states whose spectral densities of states are within the band
+gap of Si [73]. They exhibit optical absorption at telecommunication wavelengths (≈ 0.8 eV) and
+are known to significantly increase loss in Si photonic devices [74]. This detrimental effect can
+be circumvented by passivating Si surfaces with hydrogen via HF etching [75,76]. However, the
+Si-H termination is not stable and the surfaces hence suffer from natural oxidation in ambient
+conditions. Thus, a combination of this process and subsequent measurement of the samples
+in an inert gas-purged chamber seems to be needed in order to achieve 𝑄exp > 107 [72]. For
+heterostructure PCNs with oxide layers [77], the inverse of the 𝑄 factor based on absorption
+(1/𝑄abs) was estimated to be about 1/(7 × 106) = 1.43 × 10−7, and a large part of it seemed
+to stem from the surface states. Although water molecules that adhere to sample surfaces also
+induce absorption loss, their impact appears to be an order of magnitude smaller. Repeating the
+formation and removal of SiO𝑥 layers can also reduce the surface roughness and hence suppress
+extra scattering loss [78,79]. Performing such a process on the bottom surface of Si may also be
+helpful in removing dopant contamination that could concentrate around the interface between
+the Si and BOX layers [72,80].
+Overall, the 𝑄exp achievable for practical PCNs in air seems to be limited to below 107; with
+the hydrogen passivation 𝑄exp may reach on the order of 107. Because PCNs can have such a
+high 𝑄/𝑉 coefficient, we should mention that they would also be subject to fluctuations in the
+refractive index caused by thermal noise, which induce their linewidth broadening [81]. Although
+PCNs are not so affected by ambient temperature, thermal noise may become a problem when
+they absorb the injected light. Our experiment showed a symptom of the linewidth broadening,
+when the measured transmission power exceeded 1 nW. This feature is attributed to heat, since it
+appears as a precursor of bistable transmission based on thermo-optic nonlinearity. A similar
+result was seen in a previous report [34]. PCNs with larger 𝑄exp than ours might need a smaller
+probe power to avoid it. In that case, a time-resolved ("ring-down") measurement with a pulsed
+excitation might be useful [82].
+6.
+Conclusion
+The theoretical and experimental 𝑄 factors of our hexapole H1 PCNs were 𝑄th > 108 and
+𝑄exp > 106. Thanks to the 𝐶6 symmetry of the hexapole mode, our design required optimization
+of only four structural modulation parameters. Bands of valid conditions for 𝑄th ⪆ 108 were
+found in both the (𝑠1, 𝑅1) and (𝑠1, 𝑠2) parameter spaces. The field distributions of such modes
+indicated stronger light confinement in both the in-plane and out-of-plane directions compared
+with the previous design that did not use 𝑠2. In the experimental demonstration, the Si H1 PCN
+samples exhibited a systematic change in their resonant wavelengths when varying the radial shift
+of the innermost holes 𝑠1 in steps of 1 nm. Their maximum loaded 𝑄 factor was 𝑄exp = 1.2 × 106,
+and the corresponding cavity’s intrinsic 𝑄 factor was 𝑄i = 1.5 × 106. Repeating an automated
+optimization with (𝑅0, 𝑅1, 𝑠1) for different values of the radial shift of the second innermost
+holes 𝑠2 resulted in 𝑄th = 4.5 × 108, a more than a hundred-fold improvement compared with
+the previous studies. We also discussed some of the major elements that degrade 𝑄exp in reality
+and estimated the order of practically obtainable 𝑄exp. Our work spotlights the power of modal
+symmetry for improving the performance of nanocavities. It also shows the potential of the H1
+PCN in various applications such as functional photonic devices, quantum information processing,
+and large-scale one- and two-dimensional resonator lattices for studying non-Hermitian and
+topological photonics and other emergent topics.
+Funding.
+JSPS KAKENHI Grant Number JP20H05641.
+Acknowledgements.
+We thank Toshiaki Tamamura, Junichi Asaoka, Osamu Moriwaki, Toshifumi
+Watanabe and Mizuki Ikeya for support with the sample fabrication. We are also grateful to Hideaki
+Taniyama for support with the complemental FDTD simulation and Shota Kita for fruitful discussion.
+
+Disclosures.
+The authors declare no conflicts of interest.
+Data availability.
+Data underlying the results presented in this paper are not publicly available at this
+time but may be obtained from the authors upon reasonable request.
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+

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+ON THE COMPLEXITY OF SUB-TREE SCHEDULING FOR
+WIRELESS SENSOR NETWORKS WITH PARTIAL COVERAGE
+Michele Barbato∗
+Dipartimento di Informatica
+Universit`a degli Studi di Milano
+via Celoria 18, 20133 Milano
+michele.barbato@unimi.it
+Nicola Bianchessi
+Dipartimento di Informatica
+Universit`a degli Studi di Milano
+via Celoria 18, 20133 Milano
+nicola.bianchessi@unimi.it
+ABSTRACT
+Given an undirected graph G whose edge weights change over s time slots, the sub-tree scheduling
+for wireless sensor networks with partial coverage asks to partition the vertices of G in s non-empty
+trees such that the total weight of the trees is minimized. In this note we show that the problem is NP-
+hard in both the cases where s (i) is part of the input and (ii) is a fixed instance parameter. In both
+our proofs we reduce from the cardinality Steiner tree problem. We additionally give polynomial-
+time algorithms for structured inputs of the problem.
+Keywords Wireless sensor network, Sub-tree scheduling, Partial coverage, Complexity
+1
+Introduction
+A central problem in the management of wireless sensor networks is to extend the lifetime of wireless sensors through
+operating policies ensuring energy efficiency and/or balancing. Its importance stems from the fact that even a single
+failure of a wireless sensor can in principle compromise the effectiveness of the whole network. From the viewpoint
+of energy balancing, a general approach to minimize energy consumption is to split the set of sensors into several
+non-empty subsets and to subdivide the planning horizon into as many slots, so that the subsets of sensors are operated
+sequentially, one at each time slot.
+The sub-tree scheduling for wireless sensor networks with partial coverage (STSWSN-PC), introduced by Adasme
+(2019), is a particular implementation of such an approach, with the additional requirements that the sensors operated
+simultaneously are mutually connected under a tree topology, and each sensor must be active in a unique time slot.
+Namely, the STSWSN-PC is defined on an undirected graph G = (V, E) representing the network of sensors, a
+number s, 1 ≤ s ≤ |V |, of time slots, and vectors w1, w2, . . . , ws ∈ RE
++ of edge-weights (one for each time slot). The
+aim is to find a set T1, T2, . . . , Ts of non-empty vertex-disjoint trees of G covering V and minimizing �s
+i=1 wi(Ti).
+In the above description, the vertices of G represent the sensors of the network, the edges represent direct links among
+sensors, and the weights represent the time slot-dependent power for transmitting information over the corresponding
+edges.
+The input of the STSWSN-PC is simultaneously defined by the graph G, the number of time slots s, and the values
+of the edge weight vectors. The STSWSN-PC may admit efficient optimization algorithms for structured inputs.
+For example, when the weights are constant throughout the time slots (i.e., wi ≡ wj for all i, j = 1, 2, . . . , s), the
+STSWSN-PC is solvable in polynomial time, e.g., by using Kruskal’s algorithm (Kruskal, 1956) and terminating it at
+the first iteration yielding a spanning forest with s trees; when s = 1, the STSWSN-PC boils down to the minimum
+spanning tree (MST) problem on general graphs and, as such, is solvable in polynomial time; when s = |V |, the
+optimal solution consists of arbitrarily assigning one vertex to each time slot.
+However, unstructured instances of the STSWSN-PC have been tackled in Adasme (2019) and Bianchessi (2022) by
+means of branch-and-bound and branch-and-cut algorithms, respectively. These approaches implicitly suggest that the
+∗Corresponding author
+arXiv:2301.00739v1  [cs.CC]  2 Jan 2023
+
+On the complexity of STSWSN-PC
+1
+3
+6
+7
+2
+4
+5
+8
+(a)
+v1
+1
+v1
+2
+v1
+3
+v3
+1
+v3
+2
+v3
+3
+v6
+1
+v6
+2
+v6
+3
+v7
+1
+v7
+2
+v7
+3
+1
+3
+6
+7
+2
+4
+5
+8
+(b)
+Figure 1: Example of a CST instance (a), in which terminal vertices are squared-shaped, and of the corresponding
+STSWSN-PC instance for k = 5 (b), in which fictitious vertices are diamond-shaped.
+problem is theoretically intractable, although its computational complexity is unknown to the best of our knowledge.
+The purpose of this note is to fill in this gap.
+In Sect. 2 we study the complexity of the STSWSN-PC when s is part of the input, that is, s is not fixed in
+{2, 3, . . . , |V | − 1}; in Sect. 3 we study the complexity under the assumption that s is an instance parameter with
+a prescribed value ¯s ≥ 2. Through reductions from the (minimum weight) Steiner tree problem (Garey and Johnson,
+1990, p. 208), we show that the STSWSN-PC is NP-hard in both cases, thus justifying the usage of implicit enumera-
+tion schemes to solve it. Finally, in Sect. 4 we discuss additional structured inputs, other than those mentioned above,
+for which the STSWSN-PC is solvable in polynomial time.
+2
+NP-hardness when the number of time slots is not fixed
+Given an undirected connected graph G = (V, E) with |V | = n vertices and a subset R ⊂ V of terminal vertices, a
+Steiner tree is a subtree T of G such that R ⊆ V (T). Given also a weight w(e) ∈ Z+ for each e ∈ E, computing the
+Steiner tree of minimum total edge-weight is in general NP-hard, and the problem remains NP-hard if all weights are
+equal (Garey and Johnson, 1990, p. 209). In particular, given w(e) = 1 for each e ∈ E, the pair (G, R), and k ∈ Z+
+with |R| − 1 ≤ k ≤ n − 2, the cardinality Steiner tree (CST) problem consisting of determining the existence of a
+Steiner tree of G with at most k edges is NP-complete.
+We now show that an oracle solving the STSWSN-PC in polynomial time allows to solve the CST in polynomial time,
+thus obtaining that the STSWSN-PC is NP-hard. We point out the CST with at most three terminals can be solved
+in polynomial time (Arrighi and de Oliveira Oliveira, 2021), therefore we restrict ourselves to CST instances with
+|R| ≥ 4.
+Given k and (G, R) defining a CST instance as above, we construct a new graph ¯G = ( ¯V , ¯E) obtained from G by
+introducing n−k fictitious vertices vr
+1, vr
+2, . . . , vr
+n−k for each terminal vertex r ∈ R and defining ¯E = E ∪ER, where
+ER = {(vr
+j, r): j = 1, 2, . . . , n − k, r ∈ R}; that is, each fictitious vertex is connected precisely to the corresponding
+terminal vertex. An example of such a construction is given in Figure 1.
+Next, we define a STSWSN-PC instance I on graph ¯G and s = n−k+1 time slots. Note that, since |R|−1 ≤ k ≤ n−2
+and |R| ≥ 4, then 3 ≤ s ≤ n − 2 < | ¯V |, hence our definition of the number of time slots excludes the polynomially
+solvable cases of the STSWSN-PC.
+The weights of the time slots are defined as follows:
+w1
+e =
+�0
+if e ∈ ER
+1
+otherwise
+(1)
+wj
+e =
+�n
+if e ∈ ER
+1
+otherwise
+∀j = 2, 3, . . . , n − k + 1
+(2)
+Lemma 1. Let T ⋆
+1 , T ⋆
+2 , . . . , T ⋆
+n−k+1 be an optimal solution to I. The restriction of T ⋆
+1 to the vertices in V is a Steiner
+tree of G.
+2
+
+On the complexity of STSWSN-PC
+Proof. Assume that the restriction of T ⋆
+1 to the vertices of G is not a Steiner tree. Then there is at least a terminal
+vertex r⋆ contained in a tree of a time slot after the first one. Since all trees T ⋆
+1 , T ⋆
+2 , . . . , T ⋆
+n−k+1 are connected, at
+least one edge of ER belongs to that time slot. By (2) the optimal solution to I has value at least n. Now we show the
+existence of a solution with better value. Namely, in the first time slot we consider the tree T1 spanning all vertices
+of ¯G except the n − k fictitious vertices linked to r⋆ and we set Tj = {vr⋆
+j−1} for j = 2, 3, . . . , n − k + 1. Then
+T1, T2, . . . , Tn−k+1 is a feasible solution whose value is n − 1 by (1).
+Now we can prove the main result. In the proof, given S ⊆ ¯V , we denote by δ(S) its cut, namely, the set of edges
+having one endpoint in S and the other endpoint outside S.
+Proposition 1. There exists a solution to the CST instance given by k and (G, R) if and only if the optimal solution to
+I has value at most k. Therefore the STSWSN-PC is NP-hard.
+Proof. For the “if” part assume that there exists an optimal solution having value at most k; denoting by T ⋆ the
+restriction of its tree of the first time slot to the vertices in V , the nonnegativity of the weights in (1) yields |T ⋆| ≤ k.
+Then the result follows from Lemma 1.
+Now, let us prove the “only if” part. Assume that there exists a Steiner tree T of G such that |T| ≤ k. We assume,
+without loss of generality, that |T| = k: otherwise we repeatedly update T by adding one edge of G belonging to
+δ(T), until reaching the required cardinality (this is always possible as G is connected and since the update always
+returns a Steiner tree). Then, let ¯v ∈ ¯V \ V be an arbitrary fictitious vertex, define ˆV = ¯V \ {V ∪ {¯v}} as the set of
+remaining fictitious vertices, and let V C = V \ V (T) = {v1, v2, . . . , vn−k−1} be the vertices in the complement of
+T in G (as |T| = k, T comprises k + 1 vertices). We consider the feasible solution for I given by T1 = T ∪ δ( ˆV ),
+T2 = {¯v} and Tj = {vj−2 ∈ V C} for every j = 3, 4, . . . , n − k + 1. By (1)–(2) such a solution has value k. Then the
+optimal solution to I has value at most k.
+In the above construction, ¯G is obtained from G by appending leaves to its terminal vertices. This is a minor modifi-
+cation of the initial graph, hence the STSWSN-PC remains difficult on those classes of graphs which are closed under
+such modification and on which the CST is NP-complete. It is the case of chordal bipartite graphs, that is, bipartite
+graphs whose cycles C of length at least 6 induce a subgraph with at least |C| + 1 edges. More precisely we have:
+Corollary 1. The STSWSN-PC is NP-hard on bipartite chordal graphs.
+Proof. Appending leaves to a subset of vertices of a bipartite chordal graph maintains the chordal bipartiteness. Then
+the result follows from the NP-completeness of the CST on bipartite chordal graphs proved by M¨uller and Brandst¨adt
+(1987).
+3
+NP-hardness when the number of time slots is fixed
+In this section we consider the complexity of the STSWSN-PC by assuming that we have s = ¯s time slots, with ¯s ≥ 2
+fixed, and we show that the problem remains NP-hard.
+We modify the approach of previous section as follows. Let us consider a graph G = (V, E) with |V | = n vertices
+and a set R ⊂ V , |R| ≥ 4, of terminal vertices defining an instance of the CST problem. We define a graph
+G⋆ = (V ⋆, E⋆) where V ⋆ = V ∪ V R, with V R = {vr
+1, vr
+2, . . . , vr
+¯s−1 : r ∈ R} being a set of fictitious vertices
+associated with those in R, and where E⋆ = E ∪ EC ∪ ER, with EC = {(v, w): v, w ∈ V s.t. (v, w) ̸∈ E} and
+ER = {(r, vr
+j): r ∈ R, j = 1, 2, . . . , ¯s − 1}. That is, G⋆ is obtained by extending G to a complete graph and by
+linking each terminal vertex in G to the corresponding ¯s − 1 fictitious vertices (see Figure 2 for an example).
+For every e ∈ E⋆ we define the following edge weights:
+w1
+e =
+�
+�
+�
+0
+if e ∈ ER
+1
+if e ∈ E
+n
+otherwise,
+(3)
+wj
+e =
+�n
+if e ∈ ER
+0
+otherwise.
+∀j = 2, 3, . . . , ¯s
+(4)
+Let I⋆ be the resulting STSWSN-PC instance.
+3
+
+On the complexity of STSWSN-PC
+1
+3
+6
+7
+2
+4
+5
+8
+(a)
+v1
+1
+v1
+2
+v3
+1
+v3
+2
+v6
+1
+v6
+2
+v7
+1
+v7
+2
+1
+3
+6
+7
+2
+4
+5
+8
+(b)
+Figure 2: Example of a CST instance (a), in which terminal vertices are squared-shaped, and of the corresponding
+STSWSN-PC instance for ¯s = 3 (b), in which fictitious vertices are diamond-shaped.
+A Steiner tree T of G with k edges corresponds to a solution T1, T2, . . . , T¯s of I⋆ having value k. We distinguish two
+cases:
+1. if T is not spanning, let r ∈ R be an arbitrary terminal vertex of G and let vr
+1, vr
+2, . . . , vr
+¯s−2 be ¯s − 2 arbitrary
+fictitious vertices linked to r. One obtains T1 by extending T with all vertices in V R \ {vr
+1, vr
+2, . . . , vr
+¯s−2}
+(whose linking edges in ER have weight 0 in the first time slot, by (3)), by defining T2 as the spanning tree
+of the complete graph G⋆ \ V (T1) involving only edges in E ∪ EC (which have weight 0 in the second time
+slot, by (4)) and by defining Tj = {vr
+j−2} for every j = 3, 4, . . . ¯s;
+2. if T is spanning, let r ∈ R be an arbitrary terminal vertex of G and let vr
+1, vr
+2, . . . , vr
+¯s−1 be the ¯s − 1 fictitious
+vertices linked to r. One obtains T1 by extending T with all vertices in V R \ {vr
+1, vr
+2, . . . , vr
+¯s−1}, and by
+defining Tj = {vr
+j−1} for every j = 2, 3, . . . , ¯s.
+Note that, since a spanning tree of G is also a Steiner tree, the construction in the above case 2 shows that an optimal
+solution to I⋆ has value at most n − 1. Then, as in Lemma 1 and Prop. 1, it is possible to state that if T ⋆
+1 , T ⋆
+2 , . . . , T ⋆
+¯s
+is an optimal solution to I⋆, the restriction of T ⋆
+1 to the vertices in V is a Steiner tree of G having the same value.
+Indeed, we first observe that T ⋆
+1 has its edges in E ∪ ER, as otherwise (3) would imply that the considered solution
+has weight at least n, contradicting its optimality; moreover, if T ⋆
+1 is not a Steiner tree of G, there should be a vertex
+r ∈ R belonging to T ⋆
+j with 2 ≤ j ≤ ¯s and, since T ⋆
+1 , T ⋆
+2 , . . . , T ⋆
+¯s are connected, we have that at least one edge of
+ER is taken outside the first time slot; then by (4), the considered solution has value at least n, again contradicting its
+optimality.
+The above arguments prove that the considered CST instance admits a solution if and only if the corresponding
+STSWSN-PC instance has value at most k, hence we have:
+Proposition 2. The STSWSN-PC with a fixed number ¯s ≥ 2 of time slots is NP-hard.
+We remark that the transformation from G to G⋆ used in the above reduction does not allow to state a result similar
+to Cor. 1.
+4
+Structured polynomially-solvable cases
+The results of Prop. 1 and Prop. 2 hold without making any assumption on the structure of the STSWSN-PC instances.
+Here we present two polynomially-solvable cases when the input is structured. The first one generalizes the approach
+described in the Introduction for the case s = |V |.
+Observation 1. When |V | − s is constant the STSWSN-PC is solvable in polynomial time.
+Proof. When s = |V | − 1, a feasible solution contains one edge in a time slot and single vertices in all remaining time
+slots; then an optimal solution can be determined in O(|V ||E|) time by exhaustively listing all values wj
+e for e ∈ E
+and 1 ≤ j ≤ |V | − 1 and considering the minimum one. A similar algorithm (of higher time complexity) can be
+exhibited for any constant value of |V | − s.
+4
+
+On the complexity of STSWSN-PC
+The second polynomially-solvable case relates to the graph topology:
+Observation 2. If G = (V, E) is a tree, the STSWSN-PC with a fixed number ¯s ≥ 2 of time slots is solvable in
+polynomial time.
+Proof. We can list in O(n¯s−1) all subsets of ¯s − 1 edges whose removal decomposes G into a forest with ¯s trees. For
+each such a subset we assign in polynomial time the corresponding trees to the ¯s time slots solving a perfect matching
+on the weighted complete bipartite graph B = (T ; S, W) where each vertex in T represents a tree, each vertex of S
+represents a time slot and edge eτσ ∈ W linking τ ∈ T to σ ∈ S has weight wσ(τ).
+Obs. 2 motivates the following questions that we leave open: (i) When the number of time slots is not fixed, which
+is the complexity of the STSWSN-PC defined on trees? (ii) Are there any other graph families (other than trees) for
+which the STSWSN-PC is solvable in polynomial time, at least when the number of time slots is fixed?
+Acknowledgments
+The authors are grateful to Alberto Ceselli and to Emiliano Lancini for their comments on the manuscript.
+References
+Adasme, P. (2019). Optimal sub-tree scheduling for wireless sensor networks with partial coverage. Computer Stan-
+dards & Interfaces, 61, 20–35.
+Arrighi, E. and de Oliveira Oliveira, M. (2021). Three Is Enough for Steiner Trees. In D. Coudert and E. Natale, ed-
+itors, 19th International Symposium on Experimental Algorithms (SEA 2021), volume 190 of Leibniz International
+Proceedings in Informatics (LIPIcs), pages 5:1–5:15, Dagstuhl, Germany. Schloss Dagstuhl – Leibniz-Zentrum f¨ur
+Informatik.
+Bianchessi, N. (2022). On optimally solving sub-tree scheduling for wireless sensor networks with partial coverage.
+Universit`a degli Studi di Milano, http://hdl.handle.net/2434/934107.
+Garey, M. R. and Johnson, D. S. (1990). Computers and Intractability; A Guide to the Theory of NP-Completeness.
+W. H. Freeman & Co., USA.
+Kruskal, J. B. (1956). On the shortest spanning subtree of a graph and the traveling salesman problem. Proceedings
+of the American Mathematical Society, 7(1), 48–50.
+M¨uller, H. and Brandst¨adt, A. (1987). The NP-completeness of Steiner tree and dominating set for chordal bipartite
+graphs. Theoretical Computer Science, 53(2-3), 257–265.
+5
+

2tAyT4oBgHgl3EQf1vk5/content/tmp_files/load_file.txt

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+page_content='ON THE COMPLEXITY OF SUB-TREE SCHEDULING FOR  WIRELESS SENSOR NETWORKS WITH PARTIAL COVERAGE  Michele Barbato∗  Dipartimento di Informatica  Universit`a degli Studi di Milano  via Celoria 18, 20133 Milano  michele.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content='barbato@unimi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content='it  Nicola Bianchessi  Dipartimento di Informatica  Universit`a degli Studi di Milano  via Celoria 18, 20133 Milano  nicola.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content='bianchessi@unimi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content='it  ABSTRACT  Given an undirected graph G whose edge weights change over s time slots, the sub-tree scheduling  for wireless sensor networks with partial coverage asks to partition the vertices of G in s non-empty  trees such that the total weight of the trees is minimized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content=' In this note we show that the problem is NP-  hard in both the cases where s (i) is part of the input and (ii) is a fixed instance parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content=' In both  our proofs we reduce from the cardinality Steiner tree problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content=' We additionally give polynomial-  time algorithms for structured inputs of the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content='  Keywords Wireless sensor network, Sub-tree scheduling, Partial coverage, Complexity  1  Introduction  A central problem in the management of wireless sensor networks is to extend the lifetime of wireless sensors through  operating policies ensuring energy efficiency and/or balancing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content=' Its importance stems from the fact that even a single  failure of a wireless sensor can in principle compromise the effectiveness of the whole network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content=' From the viewpoint  of energy balancing, a general approach to minimize energy consumption is to split the set of sensors into several  non-empty subsets and to subdivide the planning horizon into as many slots, so that the subsets of sensors are operated  sequentially, one at each time slot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content='  The sub-tree scheduling for wireless sensor networks with partial coverage (STSWSN-PC), introduced by Adasme  (2019), is a particular implementation of such an approach, with the additional requirements that the sensors operated  simultaneously are mutually connected under a tree topology, and each sensor must be active in a unique time slot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content='  Namely, the STSWSN-PC is defined on an undirected graph G = (V, E) representing the network of sensors, a  number s, 1 ≤ s ≤ |V |, of time slots, and vectors w1, w2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content=' , ws ∈ RE  + of edge-weights (one for each time slot).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content=' The  aim is to find a set T1, T2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content=' , Ts of non-empty vertex-disjoint trees of G covering V and minimizing �s  i=1 wi(Ti).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content='  In the above description, the vertices of G represent the sensors of the network, the edges represent direct links among  sensors, and the weights represent the time slot-dependent power for transmitting information over the corresponding  edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content='  The input of the STSWSN-PC is simultaneously defined by the graph G, the number of time slots s, and the values  of the edge weight vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content=' The STSWSN-PC may admit efficient optimization algorithms for structured inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content='  For example, when the weights are constant throughout the time slots (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content=', wi ≡ wj for all i, j = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content=' , s), the  STSWSN-PC is solvable in polynomial time, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content=', by using Kruskal’s algorithm (Kruskal, 1956) and terminating it at  the first iteration yielding a spanning forest with s trees;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content=' when s = 1, the STSWSN-PC boils down to the minimum  spanning tree (MST) problem on general graphs and, as such, is solvable in polynomial time;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content=' when s = |V |, the  optimal solution consists of arbitrarily assigning one vertex to each time slot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content='  However, unstructured instances of the STSWSN-PC have been tackled in Adasme (2019) and Bianchessi (2022) by  means of branch-and-bound and branch-and-cut algorithms, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content=' These approaches implicitly suggest that the  ∗Corresponding author  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content='00739v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content='CC]  2 Jan 2023  On the complexity of STSWSN-PC  1  3  6  7  2  4  5  8  (a)  v1  1  v1  2  v1  3  v3  1  v3  2  v3  3  v6  1  v6  2  v6  3  v7  1  v7  2  v7  3  1  3  6  7  2  4  5  8  (b)  Figure 1: Example of a CST instance (a), in which terminal vertices are squared-shaped, and of the corresponding  STSWSN-PC instance for k = 5 (b), in which fictitious vertices are diamond-shaped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content='  problem is theoretically intractable, although its computational complexity is unknown to the best of our knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content='  The purpose of this note is to fill in this gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content='  In Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content=' 2 we study the complexity of the STSWSN-PC when s is part of the input, that is, s is not fixed in  {2, 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content=' , |V | − 1};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content=' in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content=' 3 we study the complexity under the assumption that s is an instance parameter with  a prescribed value ¯s ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content=' Through reductions from the (minimum weight) Steiner tree problem (Garey and Johnson,  1990, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content=' 208), we show that the STSWSN-PC is NP-hard in both cases, thus justifying the usage of implicit enumera-  tion schemes to solve it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content=' Finally, in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content=' 4 we discuss additional structured inputs, other than those mentioned above,  for which the STSWSN-PC is solvable in polynomial time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content='  2  NP-hardness when the number of time slots is not fixed  Given an undirected connected graph G = (V, E) with |V | = n vertices and a subset R ⊂ V of terminal vertices, a  Steiner tree is a subtree T of G such that R ⊆ V (T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content=' Given also a weight w(e) ∈ Z+ for each e ∈ E, computing the  Steiner tree of minimum total edge-weight is in general NP-hard, and the problem remains NP-hard if all weights are  equal (Garey and Johnson, 1990, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content=' 209).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content=' In particular, given w(e) = 1 for each e ∈ E, the pair (G, R), and k ∈ Z+  with |R| − 1 ≤ k ≤ n − 2, the cardinality Steiner tree (CST) problem consisting of determining the existence of a  Steiner tree of G with at most k edges is NP-complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content='  We now show that an oracle solving the STSWSN-PC in polynomial time allows to solve the CST in polynomial time,  thus obtaining that the STSWSN-PC is NP-hard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content=' We point out the CST with at most three terminals can be solved  in polynomial time (Arrighi and de Oliveira Oliveira, 2021), therefore we restrict ourselves to CST instances with  |R| ≥ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content='  Given k and (G, R) defining a CST instance as above, we construct a new graph ¯G = ( ¯V , ¯E) obtained from G by  introducing n−k fictitious vertices vr  1, vr  2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content=' , vr  n−k for each terminal vertex r ∈ R and defining ¯E = E ∪ER, where  ER = {(vr  j, r): j = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content=' , n − k, r ∈ R};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content=' that is, each fictitious vertex is connected precisely to the corresponding  terminal vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content=' An example of such a construction is given in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content='  Next, we define a STSWSN-PC instance I on graph ¯G and s = n−k+1 time slots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content=' Note that, since |R|−1 ≤ k ≤ n−2  and |R| ≥ 4, then 3 ≤ s ≤ n − 2 < | ¯V |, hence our definition of the number of time slots excludes the polynomially  solvable cases of the STSWSN-PC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content='  The weights of the time slots are defined as follows:  w1  e =  �0  if e ∈ ER  1  otherwise  (1)  wj  e =  �n  if e ∈ ER  1  otherwise  ∀j = 2, 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content=' , n − k + 1  (2)  Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content=' Let T ⋆  1 , T ⋆  2 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content=' , T ⋆  n−k+1 be an optimal solution to I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content=' The restriction of T ⋆  1 to the vertices in V is a Steiner  tree of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content='  2  On the complexity of STSWSN-PC  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content=' Assume that the restriction of T ⋆  1 to the vertices of G is not a Steiner tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content=' Then there is at least a terminal  vertex r⋆ contained in a tree of a time slot after the first one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content=' Since all trees T ⋆  1 , T ⋆  2 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content=' , T ⋆  n−k+1 are connected, at  least one edge of ER belongs to that time slot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content=' By (2) the optimal solution to I has value at least n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content=' Now we show the  existence of a solution with better value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content=' Namely, in the first time slot we consider the tree T1 spanning all vertices  of ¯G except the n − k fictitious vertices linked to r⋆ and we set Tj = {vr⋆  j−1} for j = 2, 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content=' , n − k + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content=' Then  T1, T2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content=' , Tn−k+1 is a feasible solution whose value is n − 1 by (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content='  Now we can prove the main result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content=' In the proof, given S ⊆ ¯V , we denote by δ(S) its cut, namely, the set of edges  having one endpoint in S and the other endpoint outside S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content='  Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content=' There exists a solution to the CST instance given by k and (G, R) if and only if the optimal solution to  I has value at most k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content=' Therefore the STSWSN-PC is NP-hard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content=' For the “if” part assume that there exists an optimal solution having value at most k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content=' denoting by T ⋆ the  restriction of its tree of the first time slot to the vertices in V , the nonnegativity of the weights in (1) yields |T ⋆| ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content='  Then the result follows from Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content='  Now, let us prove the “only if” part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content=' Assume that there exists a Steiner tree T of G such that |T| ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content=' We assume,  without loss of generality, that |T| = k: otherwise we repeatedly update T by adding one edge of G belonging to  δ(T), until reaching the required cardinality (this is always possible as G is connected and since the update always  returns a Steiner tree).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content=' Then, let ¯v ∈ ¯V \\ V be an arbitrary fictitious vertex, define ˆV = ¯V \\ {V ∪ {¯v}} as the set of  remaining fictitious vertices, and let V C = V \\ V (T) = {v1, v2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content=' , vn−k−1} be the vertices in the complement of  T in G (as |T| = k, T comprises k + 1 vertices).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content=' We consider the feasible solution for I given by T1 = T ∪ δ( ˆV ),  T2 = {¯v} and Tj = {vj−2 ∈ V C} for every j = 3, 4, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content=' , n − k + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content=' By (1)–(2) such a solution has value k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content=' Then the  optimal solution to I has value at most k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content='  In the above construction, ¯G is obtained from G by appending leaves to its terminal vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content=' This is a minor modifi-  cation of the initial graph, hence the STSWSN-PC remains difficult on those classes of graphs which are closed under  such modification and on which the CST is NP-complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content=' It is the case of chordal bipartite graphs, that is, bipartite  graphs whose cycles C of length at least 6 induce a subgraph with at least |C| + 1 edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content=' More precisely we have:  Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content=' The STSWSN-PC is NP-hard on bipartite chordal graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content=' Appending leaves to a subset of vertices of a bipartite chordal graph maintains the chordal bipartiteness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content=' Then  the result follows from the NP-completeness of the CST on bipartite chordal graphs proved by M¨uller and Brandst¨adt  (1987).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content='  3  NP-hardness when the number of time slots is fixed  In this section we consider the complexity of the STSWSN-PC by assuming that we have s = ¯s time slots, with ¯s ≥ 2  fixed, and we show that the problem remains NP-hard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content='  We modify the approach of previous section as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content=' Let us consider a graph G = (V, E) with |V | = n vertices  and a set R ⊂ V , |R| ≥ 4, of terminal vertices defining an instance of the CST problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content=' We define a graph  G⋆ = (V ⋆, E⋆) where V ⋆ = V ∪ V R, with V R = {vr  1, vr  2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content=' , vr  ¯s−1 : r ∈ R} being a set of fictitious vertices  associated with those in R, and where E⋆ = E ∪ EC ∪ ER, with EC = {(v, w): v, w ∈ V s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content=' (v, w) ̸∈ E} and  ER = {(r, vr  j): r ∈ R, j = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content=' , ¯s − 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content=' That is, G⋆ is obtained by extending G to a complete graph and by  linking each terminal vertex in G to the corresponding ¯s − 1 fictitious vertices (see Figure 2 for an example).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content='  For every e ∈ E⋆ we define the following edge weights:  w1  e =  �  �  �  0  if e ∈ ER  1  if e ∈ E  n  otherwise,  (3)  wj  e =  �n  if e ∈ ER  0  otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content='  ∀j = 2, 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content=' , ¯s  (4)  Let I⋆ be the resulting STSWSN-PC instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content='  3  On the complexity of STSWSN-PC  1  3  6  7  2  4  5  8  (a)  v1  1  v1  2  v3  1  v3  2  v6  1  v6  2  v7  1  v7  2  1  3  6  7  2  4  5  8  (b)  Figure 2: Example of a CST instance (a), in which terminal vertices are squared-shaped, and of the corresponding  STSWSN-PC instance for ¯s = 3 (b), in which fictitious vertices are diamond-shaped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content='  A Steiner tree T of G with k edges corresponds to a solution T1, T2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content=' , T¯s of I⋆ having value k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content=' We distinguish two  cases:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content=' if T is not spanning, let r ∈ R be an arbitrary terminal vertex of G and let vr  1, vr  2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content=' , vr  ¯s−2 be ¯s − 2 arbitrary  fictitious vertices linked to r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content=' One obtains T1 by extending T with all vertices in V R \\ {vr  1, vr  2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content=' , vr  ¯s−2}  (whose linking edges in ER have weight 0 in the first time slot, by (3)), by defining T2 as the spanning tree  of the complete graph G⋆ \\ V (T1) involving only edges in E ∪ EC (which have weight 0 in the second time  slot, by (4)) and by defining Tj = {vr  j−2} for every j = 3, 4, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content=' ¯s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content=' if T is spanning, let r ∈ R be an arbitrary terminal vertex of G and let vr  1, vr  2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content=' , vr  ¯s−1 be the ¯s − 1 fictitious  vertices linked to r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content=' One obtains T1 by extending T with all vertices in V R \\ {vr  1, vr  2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content=' , vr  ¯s−1}, and by  defining Tj = {vr  j−1} for every j = 2, 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content=' , ¯s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content='  Note that, since a spanning tree of G is also a Steiner tree, the construction in the above case 2 shows that an optimal  solution to I⋆ has value at most n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content=' Then, as in Lemma 1 and Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content=' 1, it is possible to state that if T ⋆  1 , T ⋆  2 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content=' , T ⋆  ¯s  is an optimal solution to I⋆, the restriction of T ⋆  1 to the vertices in V is a Steiner tree of G having the same value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content='  Indeed, we first observe that T ⋆  1 has its edges in E ∪ ER, as otherwise (3) would imply that the considered solution  has weight at least n, contradicting its optimality;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content=' moreover, if T ⋆  1 is not a Steiner tree of G, there should be a vertex  r ∈ R belonging to T ⋆  j with 2 ≤ j ≤ ¯s and, since T ⋆  1 , T ⋆  2 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content=' , T ⋆  ¯s are connected, we have that at least one edge of  ER is taken outside the first time slot;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content=' then by (4), the considered solution has value at least n, again contradicting its  optimality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content='  The above arguments prove that the considered CST instance admits a solution if and only if the corresponding  STSWSN-PC instance has value at most k, hence we have:  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content=' The STSWSN-PC with a fixed number ¯s ≥ 2 of time slots is NP-hard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content='  We remark that the transformation from G to G⋆ used in the above reduction does not allow to state a result similar  to Cor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content='  4  Structured polynomially-solvable cases  The results of Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content=' 1 and Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content=' 2 hold without making any assumption on the structure of the STSWSN-PC instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content='  Here we present two polynomially-solvable cases when the input is structured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content=' The first one generalizes the approach  described in the Introduction for the case s = |V |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content='  Observation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content=' When |V | − s is constant the STSWSN-PC is solvable in polynomial time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content=' When s = |V | − 1, a feasible solution contains one edge in a time slot and single vertices in all remaining time  slots;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content=' then an optimal solution can be determined in O(|V ||E|) time by exhaustively listing all values wj  e for e ∈ E  and 1 ≤ j ≤ |V | − 1 and considering the minimum one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content=' A similar algorithm (of higher time complexity) can be  exhibited for any constant value of |V | − s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content='  4  On the complexity of STSWSN-PC  The second polynomially-solvable case relates to the graph topology:  Observation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content=' If G = (V, E) is a tree, the STSWSN-PC with a fixed number ¯s ≥ 2 of time slots is solvable in  polynomial time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content=' We can list in O(n¯s−1) all subsets of ¯s − 1 edges whose removal decomposes G into a forest with ¯s trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content=' For  each such a subset we assign in polynomial time the corresponding trees to the ¯s time slots solving a perfect matching  on the weighted complete bipartite graph B = (T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content=' S, W) where each vertex in T represents a tree, each vertex of S  represents a time slot and edge eτσ ∈ W linking τ ∈ T to σ ∈ S has weight wσ(τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content='  Obs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content=' 2 motivates the following questions that we leave open: (i) When the number of time slots is not fixed, which  is the complexity of the STSWSN-PC defined on trees?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content=' (ii) Are there any other graph families (other than trees) for  which the STSWSN-PC is solvable in polynomial time, at least when the number of time slots is fixed?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content='  Acknowledgments  The authors are grateful to Alberto Ceselli and to Emiliano Lancini for their comments on the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content='  References  Adasme, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
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+page_content=' and de Oliveira Oliveira, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
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+page_content=' In D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content=' Coudert and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
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+page_content=' Schloss Dagstuhl – Leibniz-Zentrum f¨ur  Informatik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content='  Bianchessi, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content=' On optimally solving sub-tree scheduling for wireless sensor networks with partial coverage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content='  Universit`a degli Studi di Milano, http://hdl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content='handle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content='net/2434/934107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content='  Garey, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content=' and Johnson, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content=' (1990).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content=' Computers and Intractability;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content=' A Guide to the Theory of NP-Completeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content='  W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content=' Freeman & Co.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content=', USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content='  Kruskal, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content=' (1956).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content=' On the shortest spanning subtree of a graph and the traveling salesman problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content=' Proceedings  of the American Mathematical Society, 7(1), 48–50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content='  M¨uller, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content=' and Brandst¨adt, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content=' (1987).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content=' The NP-completeness of Steiner tree and dominating set for chordal bipartite  graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content=' Theoretical Computer Science, 53(2-3), 257–265.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}
+page_content='  5' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'}

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+arXiv:2301.02624v1  [math.QA]  6 Jan 2023
+Shapovalov elements of classical and quantum groups
+Andrey Mudrov
+In memorium of Vladimir Lyachovsky
+Moscow Institute of Physics and Technology,
+9 Institutskiy per., Dolgoprudny, Moscow Region, 141701, Russia,
+University of Leicester,
+University Road, LE1 7RH Leicester, UK,
+e-mail: am405@le.ac.uk
+Abstract
+Shapovalov elements θβ,m of the classical or quantized universal enveloping algebra of a
+simple Lie algebra g are parameterized by a positive root β and a positive integer m. They
+relate the highest vector of a reducible Verma module with highest vectors of its submodules.
+We obtain a factorization of θβ,m to a product of θβ,1 and calculate θβ,1 as a residue of a
+matrix element of the inverse Shapovalov form via a generalized Nigel-Moshinsky algorithm.
+This way we explicitly express θβ,m of a classical simple Lie algebra through the Cartan-Weyl
+basis in g. In the case of quantum groups, we give an analogous formulation through the
+entries of the R-matrix (quantum L-operator) in fundamental representations.
+Key words: Shapovalov elements, Shapovalov form, Verma modules, singular vectors, Hasse diagrams,
+R-matrix
+AMS classification codes: 17B10, 17B37
+1
+
+1
+Introduction
+Category O introduced in [1] for semi-simple Lie algebras and later defined for many other classes
+of algebras including quantum groups plays a fundamental role in various fields of mathematics and
+mathematical physics. In particular, it accommodates finite-dimensional and numerous important
+infinite dimensional representations like parabolic Verma modules and their generalizations [2].
+There are distinguished objects in O called Verma modules that feature a universality property:
+all simple modules in O are their quotients. The maximal proper submodule in a Verma module
+is generated by extremal vectors [3], which are invariants of the positive triangular subalgebra.
+This makes extremal vectors critically important in representation theory.
+Extremal vectors in a Verma module Vλ are related with the vacuum vector of highest weight
+λ via special elements θβ,m of the (classical or quantum) universal enveloping of the negative Borel
+subalgebra that are called Shapovalov elements [4, 5]. They are parameterized with a positive
+root β and an integer m ∈ N validating a De Concini-Kac-Kazhdan condition on λ.
+In the
+classical version, it is 2(λ + ρ, β) − m(β, β) = 0 with ρ being the half-sum of positive roots. This
+condition guarantees that the Verma module is reducible. In the special case when the root β is
+simple, the element θβ,m is a power f m
+β of the simple root vector fβ of weight β. For non-simple
+β, the Shapovalov elements are complicated polynomials in negative Chevalley generators with
+coefficients in the Cartan subalgebra. It can be viewed as a function θβ,m(λ) of the highest weight
+of a generic Verma module Vλ with values in the subalgebra generated by negative root vectors.
+A description of extremal vectors in Verma modules over classical Kac-Moody algebras was
+obtained in [6] via an interpolation procedure resulting in a calculus of polynomials with complex
+exponents. Another approach based on extremal projectors [7] was employed by Zhelobenko in
+[8]. He obtained θβ,m for simple Lie algebras as a product of m copies of θβ,1 with shifted weights.
+The idea of factorization was also used in a construction of Shapovalov elements for contragredient
+Lie superalgebras in [9].
+Factorization of θβ,m into a product of polynomials of lower degree is a great simplification
+that is convenient for their analysis. For example it is good for the study of the classical limit in
+the case of quantum groups, which is crucial for quantization of conjugacy classes [10].
+With regard to quantum groups, an inductive construction of extremal vectors in Verma mod-
+ules was suggested in [11]. Explicit expressions for Shapovalov elements for the A-type appeared
+in [12] and recently were obtained in [13] by other methods. It is worthy to note that ordered
+PBW-like monomials in θβ,1 deliver an orthogonal basis in generic irreducible Verma modules [12].
+While Zhelobenko’s factorization via extremal projectors simplifies construction of Shapovalov
+2
+
+elements in the case of classical simple Lie algebras, there remains a problem of explicit descrip-
+tion of the factors.
+In this paper, we pursue an alternative approach based on the canonical
+contravariant bilinear form on Verma modules. It gives expressions for all Shapovalov elements in
+a factorized form through root vectors in the classical case and through elements of the R-matrix
+in the adjoint representation in the case of quantum groups.
+Extremal vectors generate the kernel of the canonical contravariant form on Vλ, which is a
+specialization at λ of the ”universal” Shapovalov form on the Borel subalgebra [4] with values in
+the Cartan subalgebra. This contravariant form itself is extremely important and has numerous
+applications, see e.g. [14, 15, 16, 17]. For generic λ, the module Vλ is irreducible and the form
+is non-degenerate. The inverse form gives rise to an element S of extended tensor product of
+positive and negative subalgebras of the (quantized) universal enveloping algebra [18]. Sending
+the positive leg of S to an auxiliary representation yields a matrix with entries in the negative
+subalgebra which we call Shapovalov matrix.
+Its explicit description was obtained in [18] by
+generalization of Nagel-Moshinski formulas for the lowering operators of sl(n) [19]. They can also
+be derived (in the quantum setting) from the ABRR equation [20] on dynamical twist [14].
+Our method relates θβ,m with certain entries of the Shapovalov matrix. This point of view is
+quite natural because the kernel of the contravariant form on Vλ results in poles of S. Our approach
+not only provides a factorization of θβ,m to a product of θβ,1 but also an efficient description of
+θβ,1 in a very elementary way, by a generalized Nagel-Moshinsky rule (3.5) using a technique of
+Hasse diagrams. We do it by evaluating residues of matrix elements of S that go singular at a De
+Concini-Kac-Kazhdan ”hyperplane”.
+Our approach is absolutely parallel for a classical semi-simple Lie algebra g and its quantum
+group Uq(g). The classical case can be processed directly or obtained as the limit case q → 1 of
+the deformation parameter. Let us describe the method in more detail.
+With a module V from the category O and a pair of non-zero vectors vb, va ∈ V we associate
+a Shapovalov matrix element, ⟨vb|va⟩, which belongs to the negative Borel (universal enveloping)
+subalgebra ˆUq(b−) rationally extended over the Cartan subalgebra. Under certain assumptions
+on vb and va, such matrix elements deliver factors in θβ,m. These factors normalize positive root
+vectors of a reductive Lie subalgebra l ⊂ g whose negative counterparts annihilate vb. This way
+they become lowering operators in the Mickelsson algebras of the pair (g, l), [21]. When λ satisfies
+the De Concini-Kac-Kazhdan condition, the factors become θβ,1 shifted by certain weights.
+The vector vb should be highest for the minimal simple Lie subalgebra in g that accommodates
+the root β and and its weight should satisfy the condition (νb, β) = (β,β)
+2 . For finite dimensional
+V the latter is equivalent to saying that vb generates a 2-dimensional submodule of the sl(2)-
+3
+
+subalgebra generated by the root spaces g±β.
+The vector va determines a homomorphism Vλ2 → V ⊗ Vλ1, where Vλi are irreducible Verma
+modules of highest weights λi and λ2 − λ1 equals the weight of va. Iteration of this construction
+yields a chain of homomorphisms
+Vλm → V ⊗ Vλm−1 → . . . → V ⊗m ⊗ Vλ0,
+where each mapping V ⊗i ⊗ Vλm−i → V ⊗i ⊗ (V ⊗ Vλm−i−1) is identical on the factor V ⊗i. We prove
+factorization of ⟨v⊗m
+b
+|v⊗m
+a
+⟩ to a product of ⟨vb|va⟩. Then we demonstrate that, under the specified
+conditions, that matrix element is proportional to θβ,m(λ) with λ0 = λ.
+As a result, we obtain θβ,m(λ) as a product �m−1
+i=0 θβ,1(λi). The factors θβ,1 are calculated by
+a general rule (3.5) specialized to the case in Section 5. Viewed as an element of ˆUq(b−), θβ,m
+becomes a product of θβ,1 shifted by the integer multiple weights of vb. This shift can be made
+trivial by a choice of V if β contains a simple root α with multiplicity 1. Then θβ,m becomes the
+m-th power of θβ,1.
+It is worthwhile mentioning that θβ,1 can be obtained via an arbitrary auxiliary module V with
+a pair of vectors (va, vb = eβva). They all coincide up to a scalar factor on the De Concini-Kac-
+Kazhdan ”hyperplane” and generally differ away from it. The problem is to use θβ,1 as a factor
+block for constructing θβ,m of higher m. That is why we choose (V, vb, va) in a special way as
+described above. On the other hand, since the left tensor leg of S is in the positive subalgebra
+Uq(g+) ⊂ Uq(g), it is the structure of Uq(g+)-submodule on V that determines θβ,1. A remarkable
+fact is that the cyclic submodule Uq(g+)va in an admissible V turns out to be isomorphic to a
+subquotient of the Uq(g+)-module corresponding to g/g+ in the classical limit. This means that
+θβ,1 in each case can be calculated via Shapovalov matrix elements from End(g/g+) ⊗ Uq(b−), by
+Theorem 5.3.
+Except for Section 5, we present only the q-version of the theory. The classical case can be
+obtained by sending q to 1. However the final expression for θβ,1 is greatly simplified when q = 1,
+so we give a special consideration to it in the Section 5.
+2
+Preliminaries
+Let g be a simple complex Lie algebra and h ⊂ g its Cartan subalgebra. Fix a triangular de-
+composition g = g− ⊕ h ⊕ g+ with maximal nilpotent Lie subalgebras g±. Denote by R ⊂ h∗ the
+root system of g, and by R+ the subset of positive roots with basis Π of simple roots. This basis
+generates a root lattice Γ ⊂ h∗ with the positive semigroup Γ+ = Z+Π ⊂ Γ.
+4
+
+For a positive root β ∈ R+ and a simple root α ∈ Π denote by ℓα,β ∈ Z+ the multiplicity with
+which α enters β, that is the α-coefficient in the expansion of β over the basis Π.
+Choose an ad-invariant form ( . , . ) on g, restrict it to h, and transfer to h∗ by duality. For
+every λ ∈ h∗ there is a unique element hλ ∈ h such that µ(hλ) = (µ, λ), for all µ ∈ h∗. For a
+non-isotropic µ ∈ h∗ set µ∨ =
+2
+(µ,µ)µ and h∨
+µ =
+2
+(µ,µ)hµ.
+Fundamental weights are denoted by ωα, α ∈ Π.
+They are determined by the system of
+equations (ωα, β∨) = δα,β, for all α, β ∈ Π.
+We assume that q ∈ C is not a root of unity and we understand that when saying ”all q”. By
+almost all q we mean all q excepting maybe a finite set of values distinct from q = 1.
+The standard Drinfeld-Jimbo quantum group Uq(g) is a complex Hopf algebra with the set of
+generators eα, fα, and q±hα labeled by simple roots α and satisfying relations [22, 23]
+qhαeβ = q(α,β)eβqhα,
+[eα, fβ] = δα,β[hα]q,
+qhαfβ = q−(α,β)fβqhα,
+α, β ∈ Π.
+The symbol [z]q, where z ∈ h + C, stands for
+qz−q−z
+q−q−1 .
+The elements qhα are invertible, with
+qhαq−hα = 1, while {eα}α∈Π and {fα}α∈Π also satisfy quantized Serre relations. Their exact form
+is not important for this presentation, see [24] for details.
+A Hopf algebra structure on Uq(g) is introduced by the comultiplication
+∆(fα) = fα ⊗ 1 + q−hα ⊗ fα,
+∆(q±hα) = q±hα ⊗ q±hα,
+∆(eα) = eα ⊗ qhα + 1 ⊗ eα
+set up on the generators and extended as a homomorphism Uq(g) → Uq(g)⊗Uq(g). The antipode is
+an algebra and coalgebra anti-automorphism of Uq(g) that acts on the generators by the assignment
+γ(fα) = −qhαfα,
+γ(q±hα) = q∓hα,
+γ(eα) = −eαq−hα.
+The counit homomorphism ǫ: Uq(g) → C returns
+ǫ(eα) = 0,
+ǫ(fα) = 0,
+ǫ(qhα) = 1.
+We extend the notation fα, eα to all α ∈ R+ meaning the Lusztig root vectors with respect to
+some normal ordering of R+, [24]. They are known to generate a Poincare-Birkhoff-Witt (PBW)
+basis in Uq(g±).
+Denote by Uq(h), Uq(g+), and Uq(g−) subalgebras in Uq(g) generated by {q±hα}α∈Π, {eα}α∈Π,
+and {fα}α∈Π, respectively. The quantum Borel subgroups are defined as Uq(b±) = Uq(g±)Uq(h);
+they are Hopf subalgebras in Uq(g). We will also need their extended version ˆUq(b±) = Uq(g±) ˆUq(h),
+where ˆUq(h) is the ring of fractions of Uq(h) over the multiplicative system generated by [hα − c]q
+with α ∈ Γ+ and c ∈ Q.
+5
+
+Given a Uq(g)-module V , a non-zero vector v is said to be of weight µ if qhαv = q(µ,α)v for all
+α ∈ Π. The linear span of such vectors is denoted by V [µ]. A module V is said to be of highest
+weight λ if it is generated by a weight vector v ∈ V [λ] that is killed by all eα. Such vector v is
+called highest; it is defined up to a non-zero scalar multiplier.
+We define an involutive coalgebra anti-automorphism and algebra automorphism σ of Uq(g)
+setting it on the generators by the assignment
+σ: eα �→ fα,
+σ: fα �→ eα,
+σ: qhα �→ q−hα.
+The involution ω = γ−1 ◦ σ = σ ◦ γ is an algebra anti-automorphism of Uq(g) and preserves the
+comultiplication.
+A symmetric bilinear form (., .) on a g-module V is called contravariant if
+�
+xv, w
+�
+=
+�
+v, ω(x)w
+�
+for all x ∈ Uq(g), v, w ∈ V . A module of highest weight has a unique C-valued contravariant form
+such that squared norm of the highest vector is 1. We call this form canonical and extend this
+term to a form on tensor products that is the product of canonical forms on tensor factors. Such
+a form is contravariant because ω is a coalgebra map.
+Let us recall the definition of Uq(h)-valued Shapovalov form on the Borel subalgebra Uq(b−)
+that was introduced for U(g) and studied in [4]. Regard Uq(b−) as a free right Uq(h)-module gen-
+erated by Uq(g−). The triangular decomposition Uq(g) = Uq(g−)Uq(h)Uq(g+) facilitates projection
+℘: Uq(g) → Uq(h) along the sum g−Uq(g) + Uq(g)g+, where g−Uq(g) and Uq(g)g+ are right and
+left ideals generated by positive and negative root vectors, respectively. Set
+(x, y) = ℘
+�
+ω(x)y
+�
+,
+x, y ∈ Uq(g).
+Thus defined the form is Uq(h)-linear and contravariant. It follows that the left ideal Uq(g)g+ is in
+the kernel, so the form descends to a Uq(h)-linear form on the quotient Uq(g)/Uq(g)g+ ≃ Uq(b−).
+A Verma module Vλ = Uq(g) ⊗Uq(b+) Cλ of highest weight λ ∈ h∗ is induced from the 1-
+dimensional Uq(b+)-module Cλ that is trivial on Uq(g+) and returns q(λ,α) on qhα ∈ Uq(h), α ∈ Π.
+Its highest vector is denoted by vλ, which is also called vacuum vector. It freely generates Vλ over
+Uq(g−).
+Specialization of the Shapovalov form at λ ∈ h∗ yields the canonical contravariant C-valued
+form (x, y)λ = λ
+�
+℘
+�
+ω(x)y
+��
+on Vλ, upon a natural Uq(g−)-module isomorphism Uq(g−) ≃ Vλ
+extending the assignment 1 �→ vλ. Conversely, the canonical contravariant form on Vλ regarded as
+a function of λ descends to the Shapovalov form if one views Uq(h) as the algebra of polynomial
+functions on h∗. By an abuse of terminology, we also mean by Shapovalov form the canonical
+contravariant form on Vλ.
+6
+
+It is known from [25] that the contravariant form on Vλ module goes degenerate if and only if
+its highest weight is in the union of
+Hβ,m = {λ ∈ h∗ | q2(λ+ρ,β)−m(β,β) = 1}
+(2.1)
+over β ∈ R+ and m ∈ N, where ρ = 1
+2
+�
+α∈R+ α. In the classical case q = 1, Hβ,m becomes a
+Kac-Kazhdan hyperplane of weights satisfying 2(λ + ρ, β) = m(β, β).
+Recall that a vector v ∈ Vλ of weight λ−µ with µ ∈ Γ+, µ ̸= 0, is called extremal if eαv = 0 for
+all α ∈ Π. Extremal vectors are in the kernel of the contravariant form and generate submodules
+of the corresponding highest weights. We will be interested in the special case when µ = mβ with
+β ∈ R+ and m ∈ N. Then the highest weight λ has to be in Hβ,m. The image θβ,m of v under the
+isomorphism Vλ → Uq(g−) is called Shapovalov element of a positive root β and degree m.
+For simple β the element θβ,m is just the m-th power of the root vector, θβ,m = f m
+β . For
+non-simple β, it is a rational trigonometric function Hβ,m → Uq(g−). The goal of this work is to
+find explicit expressions for θβ,m with non-simple β.
+3
+Shapovalov inverse form and its matrix elements
+Define an opposite Verma Uq(g)-module V ′
+λ of lowest weight −λ as follows. The underlying vector
+space of V ′
+λ is taken to be Vλ, while the representation homomorphism π′
+λ is twisted by σ, that is
+π′
+λ = πλ ◦ σ. The module V ′
+λ is freely generated over Uq(g+) by its lowest vector v′
+λ.
+Let σλ : Vλ → V ′
+λ denote the isomorphism of vector spaces, xvλ �→ σ(x)v′
+λ,
+x ∈ Uq(g−). It
+intertwines the representations homomorphisms π′
+λ◦σ = σλ◦πλ. This map relates the contravariant
+form on Vλ with a Uq(g)-invariant pairing Vλ ⊗ V ′
+λ → Vλ ⊗ Vλ → C.
+Suppose that the module Vλ is irreducible. Then its invariant pairing is non-degenerate (as well
+as the contravariant form on Vλ). The inverse form belongs to a completed tensor product V ′
+λ ˆ⊗Vλ.
+Under the isomorphisms Vλ → Uq(g−), V ′
+λ → Uq(g+), it goes to an element that we denote by
+S ∈ Uq(g+)ˆ⊗Uq(g−) and call universal Shapovalov matrix. Given a Uq(g+)-locally nilpotent Uq(g)-
+module V with representation homomorphism π: Uq(g) → End(V ) the image S = (π ⊗ id)(S)
+is a matrix with entries in Uq(g−). It features a rational trigonometric (rational in the classical
+case) dependance on λ ∈ h∗. We will assume that V is diagonalizable with finite dimensional
+weight spaces. We will also assume that V is endowed with a non-degenerate contravariant form,
+for instance, if V is a tensor power of an irreducible module of highest weight. Using terminology
+adopted in the quantum inverse scattering theory, we call the module V auxiliary.
+7
+
+Varying the highest weight λ we get a rational trigonometric dependance of S. As a function of
+λ, S is regarded as an element of Uq(g+)ˆ⊗ ˆUq(b−), where ˆUq(b−) is viewed as a right ˆUq(h)-module
+freely generated by Uq(g−). This way the weight dependance is accommodated by the right tensor
+leg of S.
+An explicit expression of S in a weight basis {vi}i∈I ⊂ V , vi ∈ V [νi], can be formulated in
+terms of Hasse diagram, H(V ). Such a diagram is associated with any partially ordered sets. In
+our case the partial ordering is induced by the Uq(g+)-action on V . Nodes are elements of the
+basis {vi}i∈I. Arrows are simple root vectors eα connecting the nodes vi
+eα
+←− vj whose weight
+difference is νi − νj = α. Then a node vi is succeeding a node vj if νi − νj ∈ Γ+\{0}. The matrix
+S is triangular: sii = 1 and sij = 0 if νi ̸≻ νj. The entry sij is a rational trigonometric function
+h∗ → Uq(g−) taking values in the subspace of weight νj − νi ∈ −Γ+. It is also convenient to
+introduce a stronger partial ordering as we will explain below.
+Clearly the matrix S depends only on the Uq(b+)-module structure on V . In order to calculate
+a particular element sij, we can choose a weight basis that extends a basis in the cyclic submodule
+Uq(g+)vj. Then, in particular, sij = 0 if vi ̸∈ Uq(g+)vj.
+We define a Hasse sub-diagram H(vi, vj) ⊂ H(V ) that comprises all possible routes from vj to
+vi. A node vk ∈ H(V ) is in H(vi, vj) if and only if vi ⪰ vk ⪰ vj. The sub-diagram H(vi, vj) is
+associated with a Uq(g+)-module V (vi, vj) that is the quotient of Uq(g+)vj by the sum of cyclic
+submodules Uq(g+)vk ⊂ Uq(g+)vj where vk ̸∈ H(vi, vj). It is the module V (vi, vj) that is needed
+to calculate a matrix element sij.
+We recall a construction of S following [18]. Let {hi}rkg
+i=1 ∈ h be an orthonormal basis. The
+element q
+�
+i hi⊗hi belongs to a completion of Uq(h)⊗Uq(h) in the ℏ = ln q-adic topology. Choose an
+R-matrix R of Uq(g) such that ˇR = q− �
+i hi⊗hiR ∈ Uq(g+)ˆ⊗Uq(g−) and set C =
+1
+q−q−1( ˇR − 1 ⊗ 1).
+The key identity on C that facilitates the q-version of the theory is [18]
+[1 ⊗ eα, C] + (eα ⊗ q−hα)C − C(eα ⊗ qhα) = eα ⊗ [hα]q,
+∀α ∈ Π.
+(3.2)
+In the classical limit, C = �
+α∈R+ eα ⊗ fα becomes the polarized split Casimir of g without its
+Cartan part. One then recovers an identity
+[1 ⊗ eα, C] + [eα ⊗ 1, C] = eα ⊗ hα
+(3.3)
+for each simple root α.
+Let cij ∈ Uq(g−) denote the entries of the matrix (π ⊗ id)(C) ∈ End(V ) ⊗ Uq(g−). We rectify
+the partial ordering and the Hasse diagram H(V ) by removing arrows vi ← vj if cij = 0. This will
+not affect the formula (3.5) for matrix elements of S.
+8
+
+For each weight µ ∈ Γ+ put
+ηµ = hµ + (µ, ρ) − 1
+2(µ, µ) ∈ h ⊕ C.
+(3.4)
+Regard ηµ as an affine function on h∗ by the assignment ηµ : ζ �→ (µ, ζ + ρ) − 1
+2(µ, µ), ζ ∈ h∗.
+Observe that ηmβ = m
+�
+hβ + (β, ρ) − m
+2 (β, β)
+�
+. That is, [ηmβ(λ)]q vanishes on Hβ,m (and only on
+Hβ,m in the classical case).
+For a pair of non-zero vectors v, w ∈ V define a matrix element ⟨w|v⟩ = (w, S1v)S2 ∈ ˆUq(b−),
+where S1 ⊗ S2 stands for a Sweedler-like notation for S and the pairing is with respect to a non-
+degenerate contravariant form on V . Its specialization at a weight λ is denoted by ⟨w|v⟩λ, which
+can be determined from the equality ⟨w|v⟩λvλ = ⟨w|v⟩vλ ∈ Vλ. For each w from V , the map
+V → Vλ, v �→ ⟨v|w⟩vλ satisfies: eα⟨v|w⟩vλ = ⟨σ(eα)v|w⟩vλ for all α ∈ Π. This is a consequence of
+Uq(g+)-invariance of the tensor S(1 ⊗ vλ) ∈ Uq(g+)ˆ⊗Vλ.
+The matrix element ⟨vi|vj⟩ equals sij if (vi, vk) = δik for all k ∈ I. It will be always the case in
+what follows.
+Fix a ”start” node va and an ”end” node vb such that vb ≻ va. Then a re-scaled matrix element
+ˇsab = −sba[ηνb−νa]qq−ηνb−νa can be calculated by the formula
+ˇsba = cba +
+�
+k⩾1
+�
+vb≻vk≻...≻v1≻va
+cbk . . . c1a
+(−1)kqηµk . . . qηµ1
+[ηµk]q . . . [ηµ1]q
+∈ ˆUq(b−),
+(3.5)
+where µl = νl − νa ∈ Γ+, l = 1, . . . , k. Here the summation is performed over all possible routes
+(sequences of ordered nodes) from va to vb, see [18] for details.
+It is straightforward that Uq(g+)-invariance of the tensor S(va ⊗ vλ) implies
+eαˇsba(λ)vλ ∝ [ηνb−νa(λ)]q
+�
+k
+π(eα)bkska(λ)vλ.
+(3.6)
+The matrix entries ska(λ) carry weight −(νb − νa − α). It follows that ˇsba(λ)vλ is an extremal
+vector in Vλ for λ satisfying [ηνb−νa(λ)]q = 0 provided
+1. ˇsba(λ) ̸= 0,
+2. λ is a regular point for all ska(λ) and all α.
+We aim to find an appropriate matrix element for θβ,m that satisfies these conditions.
+Let V be a Uq(g)-module with a pair of vectors va, vb ∈ V such that eβva = vb for β ∈ R+. We
+call the triple (V, vb, va) a β-representation.
+Proposition 3.1. Let (V, vb, va) be a β-representation for β ∈ R+. Then for generic λ ∈ Hβ,1 the
+vector ˇsba(λ)vλ ∈ Vλ is extremal.
+9
+
+Proof. The factors
+qηµk
+[ηµk ]q in (3.5) go singular on the union of a finite number of the null-sets
+{λ ∈ h∗ | [ηµk(λ)]q = 0}. None of µk is collinear to β, hence ˇsba(λ) is regular at generic λ ∈ Hβ,1.
+By the same reasoning, all ska(λ) in (3.6) are regular at such λ. Finally, the first term cba (and
+only this one) involves the Lusztig root vector fβ, a generator of a PBW basis in Uq(g−). It is
+therefore independent of the other terms, and ˇsba(λ) ̸= 0.
+Upon identification of ˆUq(b−) with rational Uq(g−)-valued functions on h∗ we conclude that ˇsba
+is a Shapovalov element θβ,1 and denote it by θβ. Uniqueness of extremal vector of given weight
+implies that all matrix elements ˇsba with vb = eβva deliver the same θβ, up to a scalar factor.
+However, they are generally different at λ ̸∈ Hβ,m. When we aim at θβ,m with m > 1, we have to
+choose matrix elements for θβ more carefully in order to use them as building blocks.
+Note that it was relatively easy to secure the above two conditions in the case of m = 1. For
+higher m we will opt a different strategy: we will satisfy the first condition by the very construction
+and bypass a proof of the second with different arguments.
+4
+Factorization of Shapovalov elements
+For a positive root β ∈ Π denote by Πβ ⊂ Π the set of simple roots entering the expansion of
+β over the basis Π with positive coefficients. A simple Lie subalgebra, g(β) ⊂ g, generated by
+eα, fα with α ∈ Πβ is called support of β. Its universal enveloping algebra is quantized as a Hopf
+subalgebra in Uq(g).
+Definition 4.1. Let β ∈ R+ be a positive root and (V, vb, va) a β-representation such that eαvb = 0
+for all α ∈ Π, and (νb, β∨) = 1. We call such β-representation admissible.
+If a triple is (V, vb, va) is admissible then vb is the highest vector of a Uq(g)-submodule in V .
+For finite dimensional dim V < ∞, vb generates a 2-dimensional submodule of the sl(2)-subalgebra
+generated by fβ, eβ. The vector vb can be included in an orthonormal basis in V , as required.
+Lemma 4.2. Let (V, vb, va) be an admissible β-representation. Set vm
+b = v⊗m
+b
+∈ V ⊗m for m ∈ N.
+Pick up λ ∈ h∗ such that all Verma modules Vλk with λk = λ + kνa, k = 0, . . . , m − 1, are
+irreducible. Then there is vm
+a ∈ V ⊗m of weight mνa such that
+⟨vm
+b |vm
+a ⟩λ0 = ⟨vb|va⟩λm−1 . . . ⟨vb|va⟩λ0.
+(4.7)
+Proof. Let λ satisfy the required conditions. There is an equivariant map ϕk : Vλk → V ⊗ Vλk−1
+sending the highest vector vλk to an extremal vector S(va ⊗ vλk−1) ∈ V ⊗ Vλk−1. Here S is the
+10
+
+universal Shapovalov matrix of Vλk−1. Consider a chain of module homomorphisms
+Vλm
+ϕm
+−→ V ⊗ Vλm−1
+id1⊗ϕm−1
+−→
+V ⊗ (V ⊗ Vλm−2) → . . .
+idm−1⊗ϕ1
+−→
+V ⊗(m−1) ⊗ (V ⊗ Vλ0),
+where idk are the identity operators on V ⊗k. The vector vλm eventually goes over to S(˜vm
+a ⊗ vλ0),
+where ˜vm
+a ∈ V ⊗m is of weight mνa. It is related with v⊗m
+a
+by an invertible operator from End(V ⊗m),
+which is m − 1-fold dynamical twist [14].
+Let us calculate ⟨vm
+b |˜vm
+a ⟩λ0 by pairing the tensor leg of S(˜vm
+a ⊗vλ0) with vm
+b = vb ⊗vm−1
+b
+. Using
+equality S(˜vm
+a ⊗ vλ0) = S
+�
+va ⊗ S(˜vm−1
+a
+⊗ vλ0)
+�
+we reduce ⟨vm
+b |˜vm
+a ⟩λ0 to
+�
+vm−1
+b
+, ⟨vb|va⟩(1)
+λm−1S1˜vm−1
+a
+�
+⟨vb|va⟩(2)
+λm−1 S2(λ0) = ⟨vb|va⟩(2)
+λm−1
+�
+ω
+�
+⟨vb|va⟩(1)
+λm−1
+�
+vm−1
+b
+|˜vm−1
+a
+�
+λ0,
+where we use the Sweedler notation ∆(x) = x(1) ⊗ x(2) ∈ Uq(b−) ⊗ Uq(g−) for the coproduct of
+x ∈ Uq(g−). Since yqhαvb = ǫ(y)q(α,β)vb for all y ∈ Uq(g+) and α ∈ Γ+, we arrive at
+⟨vm
+b |˜vm
+a ⟩λ0 = q−(β,νb)⟨vb|va⟩λm−1⟨vm−1
+b
+|˜vm−1
+a
+⟩λ0.
+Proceeding by induction on m we conclude that ⟨vm
+b |˜vm
+a ⟩λ0 equals the right-hand side of (4.7), up
+to the factor q−m(β,νb). Finally, set vm
+a = qm(β,νb)˜vm
+a . This proves the lemma for generic and hence
+for all λ where the right-hand side of (4.7) makes sense.
+It follows from the above factorization that the least common denominator of the extremal
+vector u = S(vm
+a ⊗ vλ) ∈ V ⊗m ⊗ Vλ contains
+d(λ) = [ηβ(λ + (m − 1)νa)]q = [(λ + ρ, β) − m
+2 (β, β)]q.
+It comes from the leftmost factor ⟨vb|va⟩λm−1 in the right-hand side of (4.7). Denote by svm
+b ,vm
+a (λ)
+the matrix element ⟨vm
+b |vm
+a ⟩λ. Since d divides [ηmβ]q, the re-scaled matrix element
+ˇsvm
+b ,vm
+a (λ) = c(λ)d(λ)svm
+b ,vm
+a (λ) ∝
+m−1
+�
+k=0
+θ(λk),
+where c(λ) = −q−ηmβ(λm−1) [ηmβ(λ)]q
+d(λ)
+, is regular and does not vanish at generic λ ∈ Hβ,m because
+d(λ) cancels the pole in ⟨vb|va⟩λm−1. Put ˇu = dk(λ)u, where k ⩾ 1 is the maximal degree of this
+pole in u. It is an extremal vector in V ⊗m ⊗ Vλ that is regular at generic λ ∈ Hβ,m.
+Indeed, let Hµ denote the null set {λ ∈ h∗|[ηµ(λ)]q = 0} for µ ∈ Γ+. Then the Vλ-components
+of ˇu may have poles only at λ ∈ ∪µ<βHµ. But each µ is either not collinear to β or µ = lβ with
+l < m. In both cases the complement to Hβ,m ∩ Hµ is dense in Hβ,m because q is not a root of
+unity.
+11
+
+Proposition 4.3. For generic λ ∈ Hβ,m, θβ,m(λ) ∝ ˇsvm
+b ,vm
+a (λ).
+Proof. The singular vector ˇu is presentable as
+ˇu = vm
+a ⊗ dk(λ)vλ + . . . + vm
+b ⊗ dk−1(λ)c(λ)ˇsvm
+b ,vm
+a (λ)vλ.
+We argue that ˇu = vm
+b ⊗ c(λ)ˇsvm
+b ,vm
+a (λ)vλ for generic λ in Hβ,m, where d(λ) = 0. Indeed, the
+Vλ-components of ˇu span a Uq(g+)-submodule in Vλ. A vector of maximal weight in this span is
+extremal and distinct from vλ. But θβ,m(λ)vλ is the only, up to a factor, extremal vector in Vλ,
+for generic λ. Therefore k = 1 and θβ,m ∝ ˇsvm
+b ,vm
+a .
+An admissible β-representation can be associated with every simple root α ∈ Πβ if one sets V
+to be the irreducible module of highest weight
+(β,β)
+ℓ(α,α)ωα, where ℓ = ℓα,β is the multiplicity of α with
+which it enters β. We denote this module by Vα,β. It is finite dimensional if
+(β,β)
+ℓ(α,α) ∈ N. Otherwise
+it is a parabolic Verma module relative to a Levi subalgebra with the root basis Π\{α}, cf. the
+next section.
+One can pass to the ”universal form” of θβ regarding it as an element of ˆUq(b−). Then
+θβ,m = (τ m−1
+νb
+θβ) . . . (τνbθβ) θβ,
+(4.8)
+where τν is an automorphism of ˆUq(h) generated by the affine shift of h∗ by the weight ν, that is,
+(τνϕ)(µ) = ϕ(µ + ν), ϕ ∈ ˆUq(h), µ ∈ h∗. One may ask when the shift is trivial, τνbθβ = θβ, and
+θβ,m is just the m-th power of θβ.
+Proposition 4.4. Let β be a positive root. Suppose that there is α ∈ Πβ with ℓα,β = 1. Then
+θβ,m = θm
+β ∈ Uq(b−).
+Proof. Let s ⊂ g be a semi-simple subalgebra generated by simple root vectors fµ, eµ with µ ̸= α.
+Take for V the module Vα,β with highest weight φ = (β,β)
+(α,α)ωα. Put vb to be the highest vector and
+va ∝ fβvb.
+Both va and vb can be included in an orthonormal basis because they span their weight sub-
+spaces in V . Therefore ⟨vb|va⟩ = sba can be calculated by formula (3.5). We write it as
+θβ(λ) = cba +
+�
+vb≻vi≻va
+cbisia(λ).
+The highest vector vb is killed by s−, therefore the Hasse diagram between va and vb is
+vb
+eα
+←−
+fαvb
+. . .
+va,
+12
+
+where arrows in the suppressed part are simple root vectors from Uq(s+). But then the only copy
+of fα is in cbi while all sia belong to Uq(s−) ˆUq(hs), the extended Borel subalgebra of Uq(s).
+Finally, since Πs is orthogonal to νb, we have (µ, νa) = −(µ, β) for all µ ∈ R+
+s . Therefore
+θβ(λk) = θβ(λ − kβ),
+θβ,m(λ) =
+m−1
+�
+k=0
+θβ(λ − kβ),
+where the product is taken in the descending order from left to right. This proves the plain power
+factorization because each θβ carries weight −β.
+Conditions of the above proposition are fulfilled for all pairs α, β in the case of sl(n).
+5
+Shapovalov elements of degree 1
+In this section we describe the factor θβ entering (4.8), for a particular admissible β-representation
+(V, vb, va). We give a complete solution to the problem in the classical case. In the case of q ̸= 1,
+we do it up to calculation of the entries of the matrix C in a simple finite dimensional module ˜g
+that is a q-deformation of the adjoint module g. Its highest weight is the maximal root, ξ ∈ R+.
+To achieve our goals, we need to figure out the Hasse sub-diagram H(vb, va) ⊂ H(V ) that
+comprises all possible routes from va to vb. We argue that H(vb, va) can be extracted from a
+diagram H(b−) which we introduce below, and the underlying Uq(g+)-modules are isomorphic.
+The Uq(g+)-module associated with H(b−) is constructed from ˜g by factoring out the span
+of positive weight spaces. In order to distinguish the case of q ̸= 1 from classical and to avoid
+confusion with root vectors, we will mark the nodes with tilde. Vectors ˜fη of weights −η ∈ −R+
+are defined uniquely up to a sign if we normalize them by ( ˜fη, ˜fη) = 1. We may assume that they
+are deformations of classical root vectors. We take ˜hα = eα ˜fα, α ∈ Π, for basis elements of zero
+weight.
+For example, the diagram H(b−) in the case of g = g2 is
+b−
+eα2
+eα1
+eα2
+eα2
+eα1
+˜hα2
+˜fα2
+˜fα1+α2
+˜fα1+2α2
+˜fα1+3α2
+˜f2α1+3α2
+❜
+❜
+❜
+❜
+❜
+❜
+✛
+✟
+✟
+✟
+✟
+✙
+✛
+✛
+✛
+eα1
+eα2
+˜hα1
+˜fα1
+❜
+❜
+✛
+❍
+❍
+❍
+❍
+❨
+From now on we fix V = Vα,β with highest weight φ =
+(β,β)
+ℓα,β(α,α)ωα and highest vector vb. We
+denote by l ⊂ g a reductive Lie subalgebra of maximal rank whose root system is Πl = R\{α}
+and by p = l + g+ its parabolic extension.
+13
+
+In order to construct the start node va ∈ V , we will use the following observation. Recall that
+a singular vector �
+i wi ⊗ vi in a tensor product W ⊗ V of two irreducible modules of highest
+weight defines a Uq(g+)-homomorphism W ∗ → V (and respectively V ∗ → W).
+Here W ∗ is
+an irreducible Uq(g)-module of lowest weight, which is negative the highest weight of W. The
+dual action is defined with the help of antipode γ in the standard way: (xϕ)(w) = ϕ
+�
+γ(x)w
+�
+,
+for x ∈ Uq(g+), w ∈ W, and ϕ ∈ W ∗. The homomorphism W ∗ → V is implemented via the
+assignment ϕ �→ �
+i ϕ(wi)vi. We will apply this construction to W = ˜g.
+Lemma 5.1. There exists a unique, up to a scalar factor, singular vector u ∈ ˜g ⊗ V of weight φ.
+Proof. Let J ⊂ Uq(g−) be the annihilator of the highest vector vb ∈ V . Singular vectors in ˜g ⊗ V
+of weight φ are in bijection with vectors ˜h ∈ ˜g of zero weight killed by the left ideal σ(J) ⊂ Uq(g+).
+Pick up ˜h ̸= 0 orthogonal to all µ ∈ Πl; it is unique up to a scalar factor.
+The ideal J is generated by elements θ ∈ Uq(g+) such that θvb are singular vectors in the
+Verma module Vφ covering V . By construction, ˜u is killed by eα ∈ J with α ∈ Πl. If θvφ ∈ Vφ
+is a singular vector of weight φ − mη with η ∈ R+\R+
+l , then m > 1. Indeed, since φ = lωα with
+positive rational l =
+(β,β)
+ℓα,β(α,α), we have an inequality l(ωα, η∨) + (ρ, η∨) > 1. Then the condition
+(2.1), where λ is replaced with φ and β with η, is fulfilled only if m > 1, since q is not a root of
+unity. Then the element σ(θ) kills ˜h because mη with m > 1 is not a weight of ˜g.
+Remark that V is finite dimensional if
+(β,β)
+ℓα,β(α,α) ∈ Z and a parabolic Verma module otherwise
+because its highest weight is away from De Concini-Kac-Kazhdan hyperplanes Hη,m with η ∈
+R+\R+
+l .
+Now let va ∈ V be the vector of minimal weight in the expansion u = ˜eξ ⊗ va + . . . over the
+chosen basis in ˜g (we have omitted the terms of lower weights in the ˜g-factor). Notice that in
+the classical case the vector fηvb does not vanish if η ∈ R+\R+
+l because (η, φ) > 0. In particular,
+va ∝ fξvb ̸= 0 for the maximal root ξ. For general q, va is killed by the left ideal in Uq(g+)
+annihilating the lowest vector ˜fξ ∈ ˜g ≃ ˜g∗, Such va is unique in V up to a scalar factor, because
+of Lemma 5.1.
+Introduce a partial order on positive roots by writing µ ≺ ν iff fµ ≻ fν in H(b−). This is in
+agreement with the partial order on H(g+) ⊂ H(g), which is exactly the Hasse diagram of the root
+system R+, [26]. Note that α ≺ β for simple α if and only if α ∈ Πβ.
+Proposition 5.2. Let u = ˜eξ ⊗ va + . . . be the singular vector from Lemma 5.1 with va ∈ V of
+minimal weight in the expansion over a weight basis in ˜g. Then the Uq(g+)-module generated by
+va ∈ V is isomorphic to ˜g(˜hα, ˜fξ), for almost all q.
+14
+
+Proof. The Uq(g+)-module homomorphism ˜g → V determined by the assignment ˜fξ �→ va factors
+through the quotient g(˜hα, ˜fξ) because the kernel includes all ˜fη with η ∈ R+
+l , all ˜hη = eµ ˜fη with
+η ∈ Πl, and all negative weight spaces. We are left to prove that it is an isomorphism on g(˜hα, ˜fξ)
+for almost all q. It is sufficient to check that it is injective for q = 1 because V rationally depends
+on q. But then for each positive root η subject to α ⪯ η ⪯ ξ the vector fηvb is in U(g+)fξvb and
+is not zero, because (η, φ) > 0.
+It follows that eβva ̸= 0 because eβ ˜fξ ̸= 0. Therefore (V, vb, va) is an admissible β-representation
+for almost all q.
+Let us consider the classical case in more detail. We choose h∨
+α =
+2
+(α,α)hα, α ∈ Π, as a basis
+in h ⊂ b−, so that α(h∨
+α) = 2. The root vectors fµ with µ ∈ R+ form a basis in g−. Arrows
+labeled by α ∈ Π are h∨
+α
+eα
+←− fα and fµ
+eα
+←− fν if µ = ν − α is a positive root. The U(g+)-module
+underlying H(b−) is g/g+.
+Specialization of the formula (3.5) for θβ requires the knowledge of matrix C = (π ⊗ id)(C) ∈
+End(V ) ⊗ Uq(g−), which is readily available for q = 1. For ν, γ ∈ R+, denote by Cν,γ ∈ C the
+scalars such that [eν, fγ] = Cν,γfγ−ν, if γ − ν ∈ R+, Cγ,γ = (β,β)
+2
+ℓα,γ
+ℓα,β , and Cν,γ = 0 otherwise. Then
+(π ⊗ id)(C)(fγvb ⊗ 1) = vb ⊗ Cγ,γfγ +
+�
+ν≺γ
+fγ−νvb ⊗ Cν,γfν,
+for all γ satisfying α ⪯ γ ⪯ β. This equality yields all entries of the matrix C needed. The
+formula (3.5) becomes
+θβ = Cβ,βfβ +
+�
+k⩾1
+�
+ν1+...+νk+1=β
+(Cνk+1,γk . . . Cν1,γ0)(fνk+1 . . . fν1)
+(−1)k
+ηµk . . . ηµ1
+.
+(5.9)
+The internal summation is performed over all partitions of β to a sum of νi ∈ R+ such that all
+γi = γi−1 − νi for i = 1, . . . , k with γ0 = β are in R+ and subject to α ⪯ γi. In particular,
+γk = νk+1. The weights µi are defined to be µi = γ0 − γi = ν1 + . . . + νi. Note that in the q ̸= 1
+case the corresponding sum may involve terms with entries of C whose weights are not roots.
+Now we summarise the results of this paper.
+Theorem 5.3. For each α ≺ β, the rescaled matrix element ⟨˜hα| ˜fβ⟩[ηβ]q with ˜hα, ˜fβ ∈ ˜g, is a
+Shapovalov element θβ,1. For general degree m > 1, θβ,m is given by the factorization formula
+(4.8) with θβ = θβ,1 and the shift weight νb =
+(β,β)
+ℓα,β(α,α)ωα.
+Proof. Observe that summation formula (3.5) involves only the structure of Uq(g+)-module deter-
+mined by the initial and final nodes. That is straightforward with regard to the matrix elements
+15
+
+of C and also true for the Cartan factors, which depend only on weight differences (mind that
+weights in a cyclic Uq(g+)-module generated by a weight vector are fixed up to a constant weight
+summand). Furthermore, the nodes of the sub-diagram H(va, vb) can be included in an orthonor-
+mal basis whence sba ∝ ⟨vb|va⟩. Now, for almost all q, the theorem follows from Proposition 5.2
+and Proposition 4.3 with Lemma 4.2. Therefore it is true for all q where the factors (4.8) are
+defined.
+We remark in conclusion that for fixed β ∈ R+ one can pick up α ∈ Πβ delivering the simplest
+Hasse diagram H(˜hα, ˜fβ), e.g. with the smallest fundamental group. Such diagrams can be found
+amongst subdiagrams in fundamental auxiliary modules of minimal dimension. That also applies
+to their associated Uq(g+)-modules. For all non-exceptional types of g, the entries of the matrix C
+participating in the route summation formula are calculated in [27], Proposition 2.2. That is also
+done for g2 in [28]. This makes the above description of Shapovalov elements for such quantum
+groups absolutely explicit. For exceptional g of rank > 2, the problem reduces to calculation of
+relevant entries of C.
+In the context of quantization of semi-simple conjugacy classes [10], it is crucial to make
+sure that θβ,m(λ) tends to f m
+β as q → 1. Factorization (4.8) together with the route summation
+formula for θβ,1 gives important information about possible singularities of θβ,m(λ) and facilitate
+the analysis even without knowing the matrix elements of C.
+Acknowledgement
+This work is partially supported by the Moscow Institute of Physics and Technology under the
+Priority 2030 Strategic Academic Leadership Program and by Russian Science Foundation grant
+23-21-00282. The author thanks Vadim Ostapenko and Vladimir Stukopin for stimulating discus-
+sions.
+References
+[1] Bernstein, J. H., Gelfand, I. M., Gelfand, S. I.: On some category of g-modules, Funct. Anal.
+Appl. 10 no. 2 (1976), 87–92.
+[2] Humphreys, J. Representations of Semisimple Lie Algebras in the BGG Category O, Graduate
+Studies in Mathematics 94, AMS, 2008.
+16
+
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+

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+DMOps: Data Management Operation and Recipes
+Eujeong Choi1, Chanjun Park 1 †
+1 Upstage
+{eujeong, chanjun.park}@upstage.ai
+Abstract
+Data-centric AI has shed light on the signif-
+icance of data within the machine learning
+(ML) pipeline. Acknowledging its importance,
+various research and policies are suggested
+by academia, industry, and government depart-
+ments. Although the capability of utilizing ex-
+isting data is essential, the capability to build a
+dataset has become more important than ever.
+In consideration of this trend, we propose a
+"Data Management Operation and Recipes"
+that will guide the industry regardless of the
+task or domain. In other words, this paper
+presents the concept of DMOps derived from
+real-world experience. By offering a baseline
+for building data, we want to help the industry
+streamline its data operation optimally.
+1
+Introduction
+With the emergence of Data-centric AI (Polyzotis
+and Zaharia, 2021; Mazumder et al., 2022), various
+in-depth research has been introduced in academia
+alongside the wide range of policies from indus-
+try and government departments (Pencheva et al.,
+2020).
+In the case of academia, there are studies
+boosting model performance through large-scale
+datasets (Liu et al., 2021; Costa-jussà et al., 2022)
+along with the production of benchmark datasets
+for objective performance comparison between
+models (Wang et al., 2018; Ruder, 2021). Further-
+more, there are also benchmark datasets that spe-
+cialize in specific tasks (Rajpurkar et al., 2016; Alt
+et al., 2020). The government contributes to the
+field by implementing public data open policies
+and releasing datasets from the National Statistics
+department (Panagos et al., 2012).
+However, the industry frequently dives into an
+untapped and specialized domain, where there is
+rarely a ready-to-go dataset. Especially for B2B
+companies, there is usually an urgent demand
+†Corresponding author.
+for data that meets the requirements of their cus-
+tomers or their business items (Pustejovsky and
+Stubbs, 2012). Since the open source and bench-
+mark datasets are normally insufficient to meet
+these specific demands, additional data production
+is always a necessary step to specialize in a par-
+ticular task. As a result, the majority of the AI
+businesses started to build their own task-specific
+datasets, alongside the emergence of companies
+that specialize in operating crowd workers to meet
+these demands, and research on efficient data pro-
+duction on human-in-the-loop started to make ap-
+pearance (Doan, 2018; Wu et al., 2022).
+Despite its necessity, there has been a paucity of
+studies in the field of data production. To the best of
+our knowledge, there has not yet been research that
+proposes the entire process starting from analyz-
+ing the business standpoint to data annotation and
+evaluation. Therefore, we propose a "Data Man-
+agement Operation and Recipes" that will assist
+in building a dataset efficiently and economically
+regardless of task and domain. Specially, we pro-
+pose a DMOps that can produce high-quality data
+needed in manufacturing deep learning models.
+2
+Proposed Data Management Operation
+and Recipes (DMOps)
+Data management operations involve the integra-
+tion of human input and decision-making into a
+data management pipeline or system. This involves
+tasks such as data annotation, data quality assur-
+ance, and other activities that require a human
+touch. One way to implement a data management
+operation is through the use of recipes. Recipes are
+step-by-step instructions for performing a specific
+task or set of tasks, and can be used to guide human
+workers through the data management process.
+Our Data Recipes consists of 12 steps. Through
+these steps, we go over the entire process of data op-
+eration : from establishing the goal of the project to
+delivering the final data to the modeling team. The
+arXiv:2301.01228v1  [cs.DB]  2 Jan 2023
+
+name and explanation of each step is as follows.
+1. Establish the Project Goal: Analyzing the
+purpose and requirements of data production
+is the first step of the recipes. This step re-
+quires collaboration with ML engineer teams
+and business operation teams. Through com-
+munication, we can decide the input and out-
+put format of data that is suitable to the model
+of choice, and also set data milestones that fit
+the timeline of the business operation team.
+2. Secure Raw Data: Researching and collect-
+ing raw data is the second step of the recipes.
+Three possible cases of collecting raw data are
+1) the client providing the raw data, 2) using
+open-sourced public data, and 3) purchasing
+the raw data from its source platform. The
+key issue here is the copyright of each data
+source. License information must be checked
+thoroughly, and getting a legal review is rec-
+ommended before its usage.
+3. Data Pre-processing: The third step is im-
+proving the quality of the raw data through
+pre-processing. Basically the pre-processing
+consists of two main tasks: first, adjusting the
+format of data regarding its requirements, sec-
+ond, filtering non-ethical, privacy invading,
+and noisy data (Wiegand et al., 2018; Park
+et al., 2020). This step is all about practicing
+quality over quantity.
+4. Design a Data Schema: Fourth step is de-
+signing an annotation system that is efficient
+while containing all the information required.
+We need to come up with a label system that
+can represent human perception by digging
+through the collected data with the aid of ML
+methods such as unsupervised learning. Also,
+figuring out parts that can be somewhat auto-
+mated (pseudo-labeling) and parts that need
+human intervention (annotating) is essential
+in making the process efficient and moreover,
+accurate. With few pilot annotation iterations,
+the data scheme is expected to reach its opti-
+mal design.
+5. Prepare a Guideline: Fifth step is the doc-
+umentation of the data scheme. Its purpose
+is to deliver the designed labeling system to
+the expected annotators. The difficulty of the
+guideline should be monitored with caution
+since the clarity and detailed explanation may
+be in a trade-off relationship.
+6. Recruit Annotators: Sixth step is recruiting
+the annotators. The key is to select workers
+that are fit for the task for an efficient and
+accurate outcome. The best case would be se-
+lecting those who scored high on a test similar
+to the actual labeling task.
+7. Instruct Annotators: Seventh step is instruct-
+ing the annotators with the guideline made
+above. In this stage, two-way communication
+that draws out questions and debates is the key
+whereas one-sided communication is discour-
+aged.
+8. Data Annotation: This is the step where
+data annotators annotate the actual data. It
+is the process of transferring the linguis-
+tic/cognitive/visual intuition of the construc-
+tor into data. Therefore, the data construction
+manager must devise a way to unify the differ-
+ent intuitions of different builders in a more
+general line. When constructing data, it is also
+key to continuously respond to the QA of data
+builders.
+9. Data Inspection: This ninth step is inspecting
+the annotated data. During this step, inspec-
+tors must identify commonly occurring human
+errors and sort out the edge cases through
+discussions. Considering the nature of the
+Human-in-the-loop process, this step is essen-
+tial to ensure the fidelity of the dataset.
+10. Data Verification: The tenth step is verifying
+the data. When inspecting data, it is necessary
+to first determine whether the work has been
+completed by observing the given guideline.
+Also, 1) data sufficiency, 2) data diversity, 3)
+data trustworthiness, 4) data privacy and se-
+curity 5) data ethics suitability should be re-
+viewed (Roh et al., 2019; Koo et al., 2022). Fi-
+nally, data consistency can be identified based
+on the inter-annotator agreement (IAA) score.
+11. Data Evaluation: Eleventh step is verifying
+the quality of data through actual modeling. In
+order to quantitatively verify whether the data
+is made as planned, various experiments are
+conducted such as checking data efficiency by
+increasing the amount of data or sectioning
+
+the data to check the consistency of its qual-
+ity (Moon et al., 2021; Park et al., 2021). It is
+natural to find artifacts within one’s data; after
+identifying the repeated errors, revisiting the
+recipes from step 5 is frequently required to
+enhance the quality of data. If there are parts
+that do not match our purpose while proceed-
+ing the steps, we should return to stage 5 and
+revise the guideline for another iteration.
+12. Data Deliverables: Final step of the recipes
+is delivering the final data outcome. In other
+words, it is the process of delivering annotated
+data to the modeler or customer. When deliv-
+ering, the versioning must be adapted to the
+protocol, and it is important to reveal the fea-
+tures of the data with its sample. Furthermore,
+after going through the EDA process, it is rec-
+ommended to deliver the data analysis and the
+quality evaluation document together.
+Figure 1: Process of the Data Management Operation
+and Recipes (DMOps)
+Why DMOps?
+Due to the absence of a standard
+data-building process, there are many cases where
+the order of steps is mixed up or cannot be applied
+task-agnostically. The "DMOps" we propose offers
+a fixed process of data production, and at the same
+time can be used universally regardless of the task
+or domain. Therefore, our recipes can serve as a
+baseline for data production.
+Data is built through several stages. However the
+industry does not have a unified standard of the
+order to construct data, so there are many cases
+where the stages are scattered or mixed up. How-
+ever, when the proposed process is applied, it not
+only corrects the scattered order but is also task
+agnostic and can be universally applied to any do-
+main. In other words, our methodology can serve
+as a baseline for data construction.
+3
+Conclusion and Future Works
+In this paper, we proposed a DMOps that can effi-
+ciently produce high-quality data with human an-
+notation. The methodology is task agnostic which
+allows it to serve as a baseline for any data produc-
+tion. In the future, we plan to increase the reliability
+of the proposed process through quantitative verifi-
+cation at each stage of the process. In addition, we
+intend to conduct a study to verify the difference in
+data quality depending on whether the data recipes
+is applied or not.
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+Guideline
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+

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+page_content='  1  Introduction  With the emergence of Data-centric AI (Polyzotis  and Zaharia, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'}
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+page_content=', 2022), various  in-depth research has been introduced in academia  alongside the wide range of policies from indus-  try and government departments (Pencheva et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'}
+page_content=',  2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'}
+page_content='  In the case of academia, there are studies  boosting model performance through large-scale  datasets (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'}
+page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'}
+page_content=' Costa-jussà et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'}
+page_content=', 2022)  along with the production of benchmark datasets  for objective performance comparison between  models (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'}
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+page_content=' Further-  more, there are also benchmark datasets that spe-  cialize in specific tasks (Rajpurkar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'}
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+page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'}
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+page_content='  However, the industry frequently dives into an  untapped and specialized domain, where there is  rarely a ready-to-go dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'}
+page_content=' Especially for B2B  companies, there is usually an urgent demand  †Corresponding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'}
+page_content='  for data that meets the requirements of their cus-  tomers or their business items (Pustejovsky and  Stubbs, 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'}
+page_content=' Since the open source and bench-  mark datasets are normally insufficient to meet  these specific demands, additional data production  is always a necessary step to specialize in a par-  ticular task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'}
+page_content=' As a result, the majority of the AI  businesses started to build their own task-specific  datasets, alongside the emergence of companies  that specialize in operating crowd workers to meet  these demands, and research on efficient data pro-  duction on human-in-the-loop started to make ap-  pearance (Doan, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'}
+page_content=' Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'}
+page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'}
+page_content='  Despite its necessity, there has been a paucity of  studies in the field of data production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'}
+page_content=' To the best of  our knowledge, there has not yet been research that  proposes the entire process starting from analyz-  ing the business standpoint to data annotation and  evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'}
+page_content=' Therefore, we propose a "Data Man-  agement Operation and Recipes" that will assist  in building a dataset efficiently and economically  regardless of task and domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'}
+page_content=' Specially, we pro-  pose a DMOps that can produce high-quality data  needed in manufacturing deep learning models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'}
+page_content='  2  Proposed Data Management Operation  and Recipes (DMOps)  Data management operations involve the integra-  tion of human input and decision-making into a  data management pipeline or system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'}
+page_content=' This involves  tasks such as data annotation, data quality assur-  ance, and other activities that require a human  touch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'}
+page_content=' One way to implement a data management  operation is through the use of recipes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'}
+page_content=' Recipes are  step-by-step instructions for performing a specific  task or set of tasks, and can be used to guide human  workers through the data management process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'}
+page_content='  Our Data Recipes consists of 12 steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'}
+page_content=' Through  these steps, we go over the entire process of data op-  eration : from establishing the goal of the project to  delivering the final data to the modeling team.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'}
+page_content=' The  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'}
+page_content='01228v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'}
+page_content='DB]  2 Jan 2023  name and explanation of each step is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'}
+page_content=' Establish the Project Goal: Analyzing the  purpose and requirements of data production  is the first step of the recipes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'}
+page_content=' This step re-  quires collaboration with ML engineer teams  and business operation teams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'}
+page_content=' Through com-  munication, we can decide the input and out-  put format of data that is suitable to the model  of choice, and also set data milestones that fit  the timeline of the business operation team.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'}
+page_content=' Secure Raw Data: Researching and collect-  ing raw data is the second step of the recipes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'}
+page_content='  Three possible cases of collecting raw data are  1) the client providing the raw data, 2) using  open-sourced public data, and 3) purchasing  the raw data from its source platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'}
+page_content=' The  key issue here is the copyright of each data  source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'}
+page_content=' License information must be checked  thoroughly, and getting a legal review is rec-  ommended before its usage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'}
+page_content=' Data Pre-processing: The third step is im-  proving the quality of the raw data through  pre-processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'}
+page_content=' Basically the pre-processing  consists of two main tasks: first, adjusting the  format of data regarding its requirements, sec-  ond, filtering non-ethical, privacy invading,  and noisy data (Wiegand et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'}
+page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'}
+page_content=' Park  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'}
+page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'}
+page_content=' This step is all about practicing  quality over quantity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'}
+page_content=' Design a Data Schema: Fourth step is de-  signing an annotation system that is efficient  while containing all the information required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'}
+page_content='  We need to come up with a label system that  can represent human perception by digging  through the collected data with the aid of ML  methods such as unsupervised learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'}
+page_content=' Also,  figuring out parts that can be somewhat auto-  mated (pseudo-labeling) and parts that need  human intervention (annotating) is essential  in making the process efficient and moreover,  accurate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'}
+page_content=' With few pilot annotation iterations,  the data scheme is expected to reach its opti-  mal design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'}
+page_content=' Prepare a Guideline: Fifth step is the doc-  umentation of the data scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'}
+page_content=' Its purpose  is to deliver the designed labeling system to  the expected annotators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'}
+page_content=' The difficulty of the  guideline should be monitored with caution  since the clarity and detailed explanation may  be in a trade-off relationship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'}
+page_content=' Recruit Annotators: Sixth step is recruiting  the annotators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'}
+page_content=' The key is to select workers  that are fit for the task for an efficient and  accurate outcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'}
+page_content=' The best case would be se-  lecting those who scored high on a test similar  to the actual labeling task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'}
+page_content=' Instruct Annotators: Seventh step is instruct-  ing the annotators with the guideline made  above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'}
+page_content=' In this stage, two-way communication  that draws out questions and debates is the key  whereas one-sided communication is discour-  aged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'}
+page_content='  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'}
+page_content=' Data Annotation: This is the step where  data annotators annotate the actual data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'}
+page_content=' It  is the process of transferring the linguis-  tic/cognitive/visual intuition of the construc-  tor into data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'}
+page_content=' Therefore, the data construction  manager must devise a way to unify the differ-  ent intuitions of different builders in a more  general line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'}
+page_content=' When constructing data, it is also  key to continuously respond to the QA of data  builders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'}
+page_content='  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'}
+page_content=' Data Inspection: This ninth step is inspecting  the annotated data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'}
+page_content=' During this step, inspec-  tors must identify commonly occurring human  errors and sort out the edge cases through  discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'}
+page_content=' Considering the nature of the  Human-in-the-loop process, this step is essen-  tial to ensure the fidelity of the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'}
+page_content='  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'}
+page_content=' Data Verification: The tenth step is verifying  the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'}
+page_content=' When inspecting data, it is necessary  to first determine whether the work has been  completed by observing the given guideline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'}
+page_content='  Also, 1) data sufficiency, 2) data diversity, 3)  data trustworthiness, 4) data privacy and se-  curity 5) data ethics suitability should be re-  viewed (Roh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'}
+page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'}
+page_content=' Koo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'}
+page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'}
+page_content=' Fi-  nally, data consistency can be identified based  on the inter-annotator agreement (IAA) score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'}
+page_content='  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'}
+page_content=' Data Evaluation: Eleventh step is verifying  the quality of data through actual modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'}
+page_content=' In  order to quantitatively verify whether the data  is made as planned, various experiments are  conducted such as checking data efficiency by  increasing the amount of data or sectioning  the data to check the consistency of its qual-  ity (Moon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'}
+page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'}
+page_content=' Park et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'}
+page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'}
+page_content=' It is  natural to find artifacts within one’s data;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'}
+page_content=' after  identifying the repeated errors, revisiting the  recipes from step 5 is frequently required to  enhance the quality of data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'}
+page_content=' If there are parts  that do not match our purpose while proceed-  ing the steps, we should return to stage 5 and  revise the guideline for another iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'}
+page_content='  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'}
+page_content=' Data Deliverables: Final step of the recipes  is delivering the final data outcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'}
+page_content=' In other  words, it is the process of delivering annotated  data to the modeler or customer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'}
+page_content=' When deliv-  ering, the versioning must be adapted to the  protocol, and it is important to reveal the fea-  tures of the data with its sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'}
+page_content=' Furthermore,  after going through the EDA process, it is rec-  ommended to deliver the data analysis and the  quality evaluation document together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'}
+page_content='  Figure 1: Process of the Data Management Operation  and Recipes (DMOps)  Why DMOps?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'}
+page_content='  Due to the absence of a standard  data-building process, there are many cases where  the order of steps is mixed up or cannot be applied  task-agnostically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'}
+page_content=' The "DMOps" we propose offers  a fixed process of data production, and at the same  time can be used universally regardless of the task  or domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'}
+page_content=' Therefore, our recipes can serve as a  baseline for data production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'}
+page_content='  Data is built through several stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'}
+page_content=' However the  industry does not have a unified standard of the  order to construct data, so there are many cases  where the stages are scattered or mixed up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'}
+page_content=' How-  ever, when the proposed process is applied, it not  only corrects the scattered order but is also task  agnostic and can be universally applied to any do-  main.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'}
+page_content=' In other words, our methodology can serve  as a baseline for data construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'}
+page_content='  3  Conclusion and Future Works  In this paper, we proposed a DMOps that can effi-  ciently produce high-quality data with human an-  notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'}
+page_content=' The methodology is task agnostic which  allows it to serve as a baseline for any data produc-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'}
+page_content=' In the future, we plan to increase the reliability  of the proposed process through quantitative verifi-  cation at each stage of the process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'}
+page_content=' In addition, we  intend to conduct a study to verify the difference in  data quality depending on whether the data recipes  is applied or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'}
+page_content='  References  Christoph Alt, Aleksandra Gabryszak, and Leonhard  Hennig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'}
+page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'}
+page_content=' Tacred revisited: A thorough evalu-  ation of the tacred relation extraction task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'}
+page_content='  arXiv  preprint arXiv:2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'}
+page_content='14855.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'}
+page_content='  Marta R Costa-jussà, James Cross, Onur Çelebi, Maha  Elbayad, Kenneth Heafield, Kevin Heffernan, Elahe  Kalbassi, Janice Lam, Daniel Licht, Jean Maillard,  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'}
+page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'}
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+page_content=' IEEE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'}
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+page_content=' arXiv preprint arXiv:2207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'}
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+page_content='  Start  文  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'}
+page_content=' Establish the  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'}
+page_content=' Secure Raw Data  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'}
+page_content=' Data  Project Goal  Pre-processing  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'}
+page_content=' Design a Data  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'}
+page_content=' Prepare a  Schema  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'}
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+page_content=' Data Verification  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'}
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9dAyT4oBgHgl3EQfdPeW/content/tmp_files/2301.00299v1.pdf.txt

@@ -0,0 +1,1090 @@
+arXiv:2301.00299v1  [stat.AP]  31 Dec 2022
+Definition and clinical validation of Pain Patient
+States from high-dimensional mobile data:
+application to a chronic pain cohort
+1st Jenna M. Reinen
+Digital Health
+IBM Research
+Yorktown Heights, NY
+jenna.reinen@ibm.com
+2nd Carla Agurto
+Digital Health
+IBM Research
+Yorktown Heights, NY
+carla.agurto@ibm.com
+3rd Guillermo Cecchi
+Digital Health
+IBM Research
+Yorktown Heights, NY
+gcecchi@us.ibm.com
+4th Jeffrey L. Rogers
+Digital Health
+IBM Research
+Yorktown Heights, NY
+jeffrogers@us.ibm.com
+5th NAVITAS and ENVISION Studies Physician Author Group
+Clinical Research
+Boston Scientific
+Valencia, CA
+6th Boston Scientific Research Scientists Consortium
+Data Research and Engineering
+Boston Scientific
+Valencia, CA
+Abstract—The technical capacity to monitor patients with a
+mobile device has drastically expanded, but data produced from
+this approach are often difficult to interpret. We present a
+solution to produce a meaningful representation of patient status
+from large, complex data streams, leveraging both a data-driven
+approach, and use clinical knowledge to validate results. Data
+were collected from a clinical trial enrolling chronic pain patients,
+and included questionnaires, voice recordings, actigraphy, and
+standard health assessments. The data were reduced using a
+clustering analysis. In an initial exploratory analysis with only
+questionnaire data, we found up to 3 stable cluster solutions
+that grouped symptoms on a positive to negative spectrum.
+Objective features (actigraphy, speech) expanded the cluster
+solution granularity. Using a 5 state solution with questionnaire
+and actigraphy data, we found significant correlations between
+cluster properties and assessments of disability and quality-
+of-life. The correlation coefficient values showed an ordinal
+distinction, confirming the cluster ranking on a negative to
+positive spectrum. This suggests we captured novel, distinct Pain
+Patient States with this approach, even when multiple clusters
+were equated on pain magnitude. Relative to using complex time
+courses of many variables, Pain Patient States holds promise as
+an interpretable, useful, and actionable metric for a clinician or
+caregiver to simplify and provide timely delivery of care.
+Index Terms—chronic pain, digital health, clustering, medical
+decision making
+I. INTRODUCTION
+Recent advances in digital medicine have provided the
+opportunity to collect large sets of clinical data to evaluate and
+predict critical medical outcomes. For instance, mobile-based
+applications, accelerometers, and biosensors are now ubiqui-
+tous in phones and watches, enabling one to longitudinally
+track variables like mobility and speech, and facilitate patient
+symptom self-report. Importantly, these features may associate
+with clinical meaning. Large-scale studies have shown that
+data from mobile applications tracking daily activity may
+predict outcomes relevant to health and illness, such as in
+geriatric care and diabetes [1], [2]. Further, language can
+assess affective, psycholinguistic, physiological, and cognitive
+features can predict physiological and pharmacological [3],
+psychiatric [4], and cognitive disease states [5]. These types
+of findings have demonstrated the promise of digital health
+profiles in understanding patient experience and predicting
+important clinical outcomes.
+Despite these advances, the size and complexity of the
+clinical data generated by mobile applications is nontrivial to
+interpret and apply for several reasons. First, digital healthcare
+data can exist in multiple formats, creating the need to fuse
+vast amount of diverse information [6]. Second, there is a
+need for methods that can obtain clear data representations.
+These methods should provide interpretation that are manage-
+able in size, yet can maintain the characteristics of the raw
+information, allowing for patients and healthcare professionals
+to interpret and use the output [7]. Attempts to reduce and
+understand such data in a biological context have commonly
+used data-driven methods, especially those using machine
+learning algorithms. This approach offers the advantage of
+being able to handle large, multidimensional data sets through
+the ability to recognize patterns or joint representations that
+are otherwise difficult to identify using standard statistical
+approaches, providing knowledge discovery about a particular
+Copyright © 2022 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media,
+including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or
+lists, or reuse of any copyrighted component of this work in other works.
+
+topic that can span time, location, and scales [8]. In particular,
+clustering analysis offers the ability to collapse across oth-
+erwise incomprehensible multidimensional data and observe
+how features co-occur. In the case of spectrum illnesses that
+incorporate a range and variety of symptoms, decomposition
+can be helpful, with some outputs having an advantage in
+outcomes prediction [9]. But not all results allow for interpre-
+tation, and it is particularly susceptible to problems in small,
+unvalidated datasets which may result in overfitting and thus
+results that are not replicable or generalizable. Further, while
+results from unsupervised approaches may reveal meaningful
+clinical patterns, few methods exist to formally assign labels,
+rank, or identify qualitative aspects from the results of data-
+driven approaches through independent validation.
+A prime illustration of this problem is in chronic pain, a dis-
+ease affecting a substantial percentage of the population [10]
+that significantly impacts general function including employ-
+ment, mental health, and social interaction. This heterogeneous
+condition interacts with well-characterized facets of health,
+including mood, sleep, psychosocial function, medication use,
+and mobility. However, the current practice for most pain
+studies is to evaluate outcomes based on pain magnitude
+alone, which does not consider all of the variance shown to
+predict treatment success, quality-of-life (QoL), or other mea-
+surements of physical, psychological, and social well-being
+[11]. But, using all of these features as outcome variables
+is nontrivial to compute, conceptualize, and interpret. Few
+standard approaches have been developed that incorporate both
+the computational methodology required to complete such
+a task, and the ability to provide a clinically interpretable
+summary of the output. To date, machine learning has been
+used to predict pain outcomes, identify clinical subgroups
+[12], extract knowledge, and detect structure in biological
+and clinical features [13]. In chronic pain, while artificial
+intelligence (AI) has been applied to improve diagnoses, fewer
+studies apply it to the treatment and management of pain
+patients [14], and analyses that use longitudinal data or clinical
+validation are extremely limited.
+Given the quickly expanding capacity of digital health and
+learning algorithms to inform treatment outcomes in complex
+illnesses, there is a benefit to developing an approach to
+validate health states from multidimensional data. While it
+is known that various chronic pain symptoms can co-occur,
+it remains currently unknown whether symptom profiles may
+be successfully organized into distinct health states. Here, we
+propose a method by which we aim to identify clusters from
+high-dimensional, longitudinal data in chronic pain patients,
+and label them as Pain Patient States that may be operational-
+ized for clinical application and decision making [15] [16].
+To this end, we examined data from chronic pain patients in
+three subsets of data: 1) with questionnaires only; 2) with
+questionnaires plus voice data; and 3) with questionnaires
+plus actigraphy data. The dimensionality of each dataset
+was reduced into stable clusters using standard unsupervised
+clustering algorithms. Next, we quantitatively evaluated the
+clusters based on relationships to established health metrics,
+using standard assessments as clinical benchmarks in chronic
+pain to compare the data-driven results. A clear ordinal rank of
+states emerged, allowing us to assign unique qualitative labels
+even in clusters that were nearly identical in pain magnitude,
+so that they may be used as clinically-informed states. This
+system serves as an example of organizing diverse types of
+large datasets and anchoring them to known metrics as to
+evaluate treatment or assess function. Here, these formerly
+convoluted data patterns may now act to contextualize signal,
+rank results, track longitudinal health changes, and monitor
+meaningful medical outcomes.
+II. METHODS
+A. Participants and Data Collection
+Participants were recruited from pain clinics in on-going,
+longitudinal, multi-center, clinical studies (Clinicaltrials.gov
+ID: NCT01719055) aimed to understand chronic lower back
+and leg pain patients who are candidates for spinal cord
+stimulator (SCS) treatment (Boston Scientific, Valencia, CA).
+Participants were recruited and enrolled in the NAVITAS
+and/or ENVISION studies at multiple United States clinical
+sites if they intended to receive or had already received an
+SCS trial or implant, were at least 18 years old, and had
+been diagnosed with intractable chronic neuropathic pain.
+Additionally, subjects may have been previously enrolled in
+the RELIEF study (Clinicaltrials.gov ID: NCT01719055). Data
+were included in this analysis from each study a subject was
+enrolled in. Health-related questionnaires were administered
+via an at-home, custom-designed clinical study version of a
+digital health ecosystem (Boston Scientific, Valencia, CA) for
+up to 36 months. The questions chosen included pain-related
+subjective ratings, symptoms hypothesized to contribute to
+variability in pain ratings, as well as symptoms hypothesized to
+be impacted by pain, specifically pain magnitude, mood, sleep,
+alertness, medication use, and activity. Following enrollment,
+data were collected in separate in-clinic and at-home data
+streams. Mobile data analyzed here included voice recordings,
+as well as daily, self-reported symptom monitoring, with the
+option to respond more frequently if participants wished. In
+addition, subjects were asked to wear a smartwatch to assess
+mobility using accelerometer data (Galaxy Watch S2, Samsung
+USA, Menlo Park, CA with custom watch application, Boston
+Scientific, Valencia, CA). In-clinic assessments were collected
+at the baseline (enrollment) visit, and at 1-month, 3-month,
+12-month, and optionally 24-month and 36-month visits fol-
+lowing enrollment. In the present analysis, we used in-clinic
+assessments to evaluate QoL [17] and disability measured by
+the Oswestry Disability Index, or ODI [18] questionnaires.
+B. Voice data processing
+Voice recordings were collected from weekly recordings
+based on prompts aimed to understand the participant’s experi-
+ence with pain. Speech features for psycholinguistic, sentiment
+[19], and acoustic characteristics [20], [21] were extracted
+from the audio files using in-house and standard code. Age
+
+and sex were regressed from all features. Next, to reduce di-
+mensionality of these features, a principal components analysis
+(PCA) was used (var ≥ 2%) to identify the decomposed
+components. These components were later included in a
+clustering analysis alongside the 6 features derived from the
+questionnaires.
+C. Actigraphy data processing
+Effective mobility was derived from the watch-based actig-
+raphy data. It is a novel metric of physical function and activity
+meant to reflect the duration and type of activity a person
+experiences beyond steps or activities of daily life. Rates of
+activity were calculated into categories for each participant
+throughout the day. These categories ranged from Zone 0
+(e.g., resting, using a mobile device while seated) to Zone
+4 (e.g., intense or repetitive motion or vigorous exercise) and
+were used along with the questionnaire data in the clustering
+analysis.
+D. Data and Clustering Analysis
+For each participant, all available data was downloaded
+and selected based on days for which all subjective features
+from questionnaires (e.g., overall/leg/back pain, mood, sleep
+hours, sleep quality alertness, medication use for opioid/over-
+the- counter/non-opioid pain medication, activity interference
+due to pain, and activities of daily life), as well as actigraphy
+and voice data (where applicable) were present. Patients were
+included in the analysis regardless of time point in the study
+(e.g., baseline/enrollment, SCS trial period, follow-up), in the
+interest of observing a spectrum of pain-related variability and
+experience. However, the criteria for removing samples from
+the analysis consisted of: 1) any day missing a single data
+point, 2) any individual having fewer than 10 total complete
+data points, and when applicable 3) individuals who wore the
+smartwatch for less than 10 days. All question value responses
+were normalized prior to cluster analysis to equate the different
+subjective feature values across the individual question, and
+data distributions were inspected for abnormalities. Next, each
+question categorized to assess pain, sleep, and medication use
+were averaged to produce single composite scores for each
+modality; for activity, a difference score was taken between
+the two questions, in which we include a penalty that account
+with pain interferes with any overall activity. If any participant
+had answered more than one question on a certain day, the
+average of those responses was used to represent the daily
+value for that category. Participants were assessed for their
+average responses over time in order to determine the extent
+to which some participants responded more frequently than
+others, and the analysis was rerun without outliers to further
+ensure cluster stability.
+Cluster definitions were calculated using a k-means cluster-
+ing algorithm with Euclidean distance exploring up to cluster
+solutions for k = 10. Optimal k was determined using multiple
+methods including sum of squares distances and silhouette
+values, agglomerative analysis, and consensus clustering. To
+ensure clusters were similar across subsamples of participants
+exhibiting variability in number of responses included in the
+analysis, we repeated the analyses in varying samples of
+participants in which highly contributing participants (those
+with higher daily average responses) were excluded. Next, we
+employed an analogous approach to examine cluster solutions
+over the course of time. Generally, we expected the clusters
+to remain similar over time with some slight changes (e.g.,
+higher pain prior to therapy) that would be evident in the
+cluster. With this in mind, cluster solution results were then
+visually inspected in order to ensure similarities in qualitative
+characteristics and are discussed in the results section.
+Fig. 1. Conceptual data and methods overview. (A) Data were collected from
+a multi-center clinical trial recruiting participant with chronic low back and
+leg pain seeking spinal cord stimulator (SCS) treatment. Both in-clinic and at-
+home data collection were used to record 1) questionnaire-based daily reports
+of pain, mood, activity, medication, alertness, sleep; 2) standard assessments
+of QoL (EQ5D) and disability (ODI); 3) voice responses to open-ended
+questions about their pain; and 4) actigraphy from a smartwatch. (B) Data from
+questionnaires, voice, and actigraphy were subjected to a k-means clustering
+analysis and the (C) resulting cluster representation was examined across
+features. To validate these clusters, (D) centroid distance to each cluster was
+compared to the clinical scores for disability and QoL allowing for (E) an
+interpretation and label to be assigned to each cluster.
+III. RESULTS
+A. Sample demographics and data chronology
+In the primary analysis including questionnaires only, 121
+individuals with 11,763 samples of data were used (40.5%
+male, mean age 59.4 years old, 17.6 years since pain onset).
+In the analysis examining the addition of actigraphy data to
+the questionnaire data, 116 individuals with 11,286 samples of
+data were used (39.7% male, mean age 59.3 years old, 17.8
+years since pain onset). For the analysis including voice, 2,080
+samples were included.
+B. Clustering results and characteristics for questionnaire
+data only
+Cluster definition was examined for the questionnaire-only
+data for k = 2 to 10. Sum of squares distances and silhouette
+analyses indicated that a cluster solution of k = 2 or 3 was
+stable. Agglomerative hierarchical clustering was repeated to
+validate k with cross-methodological clustering, which also
+converged on a solution of k = 2 or 3. Given the relative
+stability of smaller cluster solutions, we first examined a
+
+(A)
+Clinic Data
+Day 5
+ay 55
+Mobile Data
+Day 1
+Day 60
+(B)
+Decomposed representation
+(C)
+ Examine clusters
+(D)
+Compare to clinical scores
+(E)  Interpretation
+Mooc
+Disability Score = 27
+Medic
+Disability Score = 11
+Disability Score = 52
+Mobilitysimple and stable solution of k = 2. Feature characteristics
+of the cluster solution for k = 2 were examined by inspecting
+mean values for each feature in each cluster (Figure 2A). Re-
+sults indicated a clear negative-to-positive grouping of health
+features, such that the questionnaire responses of one cluster
+appeared to represent a superior health state represented by
+better mood, sleep, alertness and activity, and lower ratings
+of pain and medication use. The other cluster appeared to
+represent an inferior health state, characterized by higher pain
+and medication use, with lower ratings for alertness, mood,
+sleep, and activity. This analysis was repeated to exclude the
+high-responder group in order to ensure that the clusters were
+not being driven by the high-responders. Results indicated that
+the clusters were very similar both in all participants, and
+without the high-responders. Finally, the cluster solutions were
+re-examined over the course of time, such that the analysis
+was repeated in the baseline period prior to SCS activation,
+during the first 6 months of treatment, and the subsequent 6
+months of treatment. Results indicated that the cluster solution
+was very similar over time, with some indication of higher
+pain prior to treatment. An examination of a 3-cluster solution
+revealed a third, intermediate cluster that represented a health
+state similar to or in between the two states represented in
+the two-cluster solution (Figure 2B). This cluster showed
+relatively high ratings of alertness, mood, and sleep, but with
+intermediate values for pain, activity and medication use.
+A repeated analysis excluding high-responders also showed
+an intermediate cluster, with values for each feature with a
+magnitude between the previous two clusters.
+Fig. 2. Cluster analysis for questionnaire data reveals negative and positive
+symptom groups. (A) A two cluster (k = 2) solution resulting from the k-means
+analysis of the questionnaire data revealed two clusters of symptoms that
+stratified on a negative-to-positive spectrum of pain-related health, in which
+one cluster revealed a better health state of better mood, sleep, more reported
+activity and alertness, less medication usage, and lower pain. Conversely, the
+other cluster depicted a worse health based on the feature means. (B) A
+three cluster (k=3) solution also revealed a spectrum of positive to negative
+symptom groupings, including superior and inferior states similar to k=2, with
+an additional intermediate state showing moderate pain and medication use
+but with high mood, sleep, activity, and alertness scores.
+C. Decomposing and clustering questionnaire and voice data
+Prior to clustering, the results of the principal components
+analysis (PCA) of the voice features were inspected. The
+results showed that 7 components were present, and character-
+ized features such as voiced and unvoiced energy in a speech
+signal, negative sentiment, emotional content, and acoustic
+voice properties (Table 1). We repeated the clustering analysis
+with each of these 7 components included along with the 6
+questionnaire components. With the addition of the 7 compo-
+nents, solutions for k of 2 or 5 were possible. For the k = 2
+solution, results showed that in particular, component 4, which
+was characterized by high loadings of negative sentiment and
+acoustic features associated with emotion, tracked well with
+the inferior health cluster (Figure 3A). Further, while not all
+components showed the same discrimination between states as
+did component 4, there was evidence that the addition of the
+voice data expanded the granularity of the state solutions. This
+was illustrated by the comparison of a cluster solution with
+only questionnaires in which pain was stratified across 3 levels
+in all states (Figure 3B). When the cluster solution included
+both voice and questionnaires, pain across states expanded to
+5 levels (Figure 3C).
+TABLE I
+DECOMPOSITION OF VOICE FEATURES INTO 7 COMPONENTS
+Fig. 3. Adding voice features to cluster analysis improves pain granularity
+in state solutions. (A) Clustering analysis was run including 6 questionnaire
+components and 7 voice components for a 2-state solution, indicating that
+voice features denoting negative sentiment were associated with the poorer
+health cluster. (B) A 5-state cluster solution without voice features reveals
+three levels of pain magnitude across clusters, while the (C) addition of voice
+to a 5-state cluster solution adds further granularity to pain magnitude across
+clusters.
+D. Clustering results and characteristics for questionnaire
+and actigraphy data
+Actigraphy data downloaded from the watch were parsed
+into mobility Zones 0 - 4 of effective mobility. Inspection of
+results indicated that these zones indeed provided granularity
+that added description beyond number of steps or self-reported
+ADLs (Figure 4). The clustering analysis included the 6
+categories derived from the questionnaires along with the
+
+(A) 
+(B)
+(C) 
+MOOD
+MOOD
+MOOD
+ACOUSTIC 3
+ALERTNESS
+ACOUSTIC 3
+ALERTNESS
+ACOUSTIC 2
+SLEEP
+ACOUSTIC 2
+SLEEP
+ ALERTNESS
+8337S
+VOICE QUAL
+ACTIVITY
+ VOIGE QUAL
+- ACTIVITY
+ NEGATIVE
+ PAIN
+NEGATIVE
+PAIN
+ EMOT
+MEDS USE
+ACTIVITY
+ EMOT
+ACOUSTIC
+MEDS USE
+ACOUSTIC
+MEDS USE
+ENERGY
+ENERGY
+MFCC
+(ACOUSTIC 1
+ PAIN
+MECC
+ACOUSTIC 1PCA
+Largest Loadings
+Smallest/Negative Loadings
+Component Name
+Content: typetoken/speech richness (psycholinguistic), Fisher SWB
+Acoustic 1
+Acoustic shape and characteristics of voice spectrum, RASTA #10
+Acoustic: energy, spectral roll off, voicing probability,
+Formant 1 (bandwidth)
+MFCC
+Acoustic: MFcC #2, voicing probability - may be related to how much
+Acoustic: MFCC #2, RASTA #2, spectral roll off/voicing probability (how
+speech is present (vowel voicing)
+ much a person is talking)
+Acoustic: Formants 1 and 2 (modulation/harmonics of voice), energy, and
+Acoustic energy
+Acoustic: energy, spectral flux (voice timbre)
+ RASTA #5
+Content: negative sentiment (VADER), negative emotion (LIWC),
+Content: positive sentiment (VADER), positive emotion, tone, reward
+Negative emot
+"feel" words 
+(LIWC), compound (positive valence/intensity)
+Acoustic: MFCC #3 (frequency band ~250 Hz)
+Acoustic voice quality/properties: MFCC #2, spectral entropy, HNR.
+Content: tone, compound (positive valence/intensity
+Voice quality
+unvoiced (% voice not in recording)
+Acoustic: jitter features (characteristic of voice time)
+Acoustic: formant 1 (modulation in larynx), unvoiced frames (% voice
+Acoustic 2
+Acoustic: MFCC 12, 13, 14 (higher frequencies)
+ not in recording), formant 2 (bandwidth), frequency at max energy
+Acoustic: Formant 2 (indicative of emotional content),
+Content: tone, compound (positive valence/intensity)
+Acoustic 3
+HNR (voice quality)
+Acoustic: MFCC #4, slope of LTAS (avg spectrum)(A)
+(B)
+MOOD
+MOOD
+ALERTNESS
+SLEEP
+ALERTNESS
+SLEEP
+8 0102 0384 0506 07
+0 0.11 0.2: 0.3
+0.6 :0.
+MED.USE
+-ACTIVITY
+MED. USE
+ ACTIVITY
+PAIN
+PAINeffective mobility. An analysis for optimal k showed that state
+solutions of up to 5 clusters was possible. These clusters
+appeared to range from a ”best” state that included low pain
+and medication use, and high reports of mood, sleep, alertness,
+and effective mobility, to an inferior state that is associated
+with high levels of pain and medication use, and low reports of
+activity, mood, sleep, alertness, and effective mobility (Figure
+5).
+Fig. 4. Description of effective mobility zones. Mobility data was parsed into
+zones of “effective mobility” based on rates of activity calculated at regular
+time window intervals throughout the day. When compared to step counts
+and self-reported activities of daily life (ADLs), effective mobility showed
+additional computational granularity of participant mobility.
+Fig. 5.
+Adding mobility features contributes to cluster dimensionality. (A)
+A cluster solution including effective mobility identified 5 stable clusters for
+which the addition of effective mobility may contribute to additional clusters
+relative to the questionnaire-only solutions, still ranging from a negative-to-
+positive spectrum and including a best and a worst state. (B) States from the
+5-cluster solution show further granularity as it pertains to patient experience
+beyond the 2- and 3-state model.
+E. Cluster validation and state classification
+For the validation analysis, we obtained pairs of metrics
+comprised of 1) distances from the cluster centroids on a given
+day; and 2) responses to standard assessments (disability, or
+ODI, and QoL, or EQ5D measurements focusing on Pain,
+Activities, and VAS Health). These two metrics were collected
+within one week of each other; any pairs with collection dates
+outside of the week window were dropped from analysis. We
+first calculated correlations between centroid distances of each
+cluster in the two-state solution, and found that the correlations
+were statistically significant and consistent in terms of direc-
+tion and magnitude for the two states, indicating a clear best
+and inferior state (in cluster 1 values were: disability/ODI, r =
+0.42, EQ5D Pain, r = 0.47, EQ5D Activities r = 0.37, EQ5D
+VAS Health r = −0.32; all p-values <0.001, for cluster 2
+values were: disability/ODI, r = −0.41, EQ5D Pain, r = −0.43,
+EQ5D Activities r = −0.38, EQ5D VAS Health r = 0.28; all
+p-values < 0.001). This indicated that larger centroid distances
+were associated with higher values for the outcomes. Critically,
+while most of the validation metric outcomes represented neg-
+ative health values with increasing severity including disability,
+EQ5D-Pain, EQ5D-Activities, etc., the EQ5D measure of VAS
+Health represents health on a positive scale, and as expected
+showed an inverse relationship to the findings above. Given
+that each cluster was associated with consistent directionality
+across all of the standard assessments, we were able to infer
+that each of the clusters represented distinct health states,
+aligned with what we would have expected to find in patients
+across time.
+F. Cluster validation with voice data
+A similar analysis was repeated using the k = 2 cluster solu-
+tion that included voice data. Results indicated that generally
+the directionality of the correlations was consistent relative
+to prior analyses. However, for several validation metrics, the
+magnitude of the r values increased with the addition of voice
+features (for disability/ODI, r = 0.47, EQ5D VAS Health r =
+−0.49). In particular, assessments that may take into account
+negative affect showed an increase in the correlation across
+these metrics. Notably, because voice data is collected less
+frequently, there was a decrease in sample size relative to
+the prior analysis. That said, permutation tests were used to
+compare across the two approaches and to ensure that there
+were no meaningful differences due to sample size. In all
+instances, permutation tests confirmed the significance of prior
+findings at p < 0.05.
+G. Cluster validation with actigraphy data
+Next, we aimed to determine whether correlations between
+centroids from a more highly dimensional state solution com-
+pared to the standard assessments could provide further ordinal
+information about the states. To do this, we ran a similar
+analysis using the 5-state solution that was obtained with the
+cluster solution including effective mobility. Here, we found
+that the correlations across the 5 states also provided evidence
+for a consistent ranking of those states from best to worst
+(Table 2).
+TABLE II
+CLUSTER CHARACTERISTICS INCLUDING EFFECTIVE MOBILITY
+
+e D
+State E
+31**
+r = -0.46**
+25**
+r = -0.32**
+24**
+r = -0.35**
+19**
+r = 0.23**
+.2**
+r = -0.37**Metric
+State A
+State B
+State C
+State
+ODI Total
+r = 0.46**
+r = 0.41**
+r = -0.06*
+r = -0.
+EQ5DActivities
+r = 0.28**
+r = 0.26**
+r = -0.09**
+r = -0.
+EQ5D Pain
+r = 0.42**
+r = 0.41**
+r = -0.09**
+r= -0.
+EQ5D HealthVAS
+r = -0.18**
+r = -0.13**
+r = 0.04 ns
+r= 0.
+EQ5D - Normed Score
+r = 0.4**
+r = 0.32**
+r = -0.12**
+r=-0Mood
+(A)
+(B)
+State A
+ StateB State C State D
+StateE
+ Average
+ Better
+Effective
+ State A
+Pain 
+mobility.
+ Sleep
+State B
+State C
+State D
+Medication
+ State E
+Activities of
+daily living
+Mood
+70) 0.1 0.2 0.3 0. 4 0.5 0.6 0.7
+Alertness
+-Activity.
+Sleep
+Alertness
+Effective
+mobility
+Worse
+Medication
+Average
+PainZone 0
+Resting, using a mobile phone, remote control
+Zone 1
+Dressing, moving around, slowing walking, stretching
+Zone 2
+Walking briskly, light exercise
+Zone 3
+Running, swimming or exercising
+Zone 4
+Intense or repetitive motion or vigorous exercise
+Number of Steps
+Self reported number of ADLs16.0
+14.3
+3000
+13.8
+13.1
+14.0
+12.9
+2425
+12.0
+2500
+11.4 Worn Hours
+11.15
+12.0
+1925
+9.3 1787
+1.2
+2000
+10.0
+8.26
+7.15 Active Hours
+7.01
+7.22
+8.0
+6.75
+4.0
+6.77
+1.3
+1500
+803
+1160
+1.2
+1.0
+1117
+1.3
+6.0
+12
+1.2
+Steps
+3.3
+1000
+2.5
+2.6
+4.0
+3.3
+574
+2.6
+3.0
+3.3
+1.3
+1.4
+500
+2.0
+1.1
+12
+3.4
+1.3
+1.2
+1.5
+2.2
+2.2
+1.8
+1.3
+1.5
+0.0
+0
+Day 1
+Day 2
+Day 3
+Day 4
+Day 5
+Day 6
+Day7H. Comparison of state timecourse to health events
+In an exploratory analysis, we examined the relationship
+between state expression change over time relative to known
+health events. Here (see Figure 6), we first show that states
+represent a more interpretable visualization of health changes
+across time relative to examining the timecourse of all vari-
+ables at once. Second, several exemplar patients show expected
+changes in states before and after implantation of the SCS de-
+vice, a procedure that involves surgery and probable eventual
+pain relief.
+Fig. 6. Examples of patient experiences show that states track with meaningful
+clinical events. Top time course for each patient denotes state assignment,
+whereas lower time course shows changes in multiple variables. Bar graphs
+show the dwell time change before and after a notable event, which here
+involves the implantation of a SCS device hypothesized to bring about eventual
+pain relief and improvement in QoL. (Here, data in the time courses included
+overall, leg, and back pain, sleep hours and quality, number of activities, pain
+interference, medication usage for opioid, over-the-counter, and non-opioid
+pain medications, alertness, mood, and effective mobility. States are ranked
+as A > B > C > D > E, as shown in Table 2.)
+IV. DISCUSSION
+A. High-dimensional health data can be decomposed mean-
+ingfully
+Using a unique set of longitudinal questionnaire, mobility,
+and speech data, we have developed a novel method to decom-
+pose, group, and validate large amounts of chronic pain digital
+health data. This study marks one of the only approaches to
+create clinically usable pain-related categories from complex
+questionnaire, mobility, and speech data across time. This
+approach demonstrates that high dimensional, longitudinal
+health data from chronic pain patients may be decomposed into
+clusters and used to classify patients according to a holistic
+status named Pain Patient States. These states have an ordinal
+ranking based on clinically-validated standard health assess-
+ments. Specifically, we demonstrated that in chronic pain,
+we can take multiple streams of information including sleep
+hours and quality, mood, pain magnitude at multiple sites,
+alertness, multiple types of medication use, ADLs, actigraphy,
+and speech in order to represent 3-5 Pain Patient States over
+the course of time. The stable solutions that emerged from this
+method suggest the discovery of distinct clinical states with
+non-obvious properties that may serve as new knowledge that
+informs biological mechanisms and clinical care. In addition
+to the identification of these Patient Pain States, this improves
+upon prior assessments and clinical trials that only use pain
+magnitude as an outcome evaluation by considering a much
+more comprehensive picture of patient experience in a way
+that is clinically interpretable. This approach leverages both
+data- and clinically-driven analyses by first using powerful
+learning algorithms, and then comparing the output to standard
+clinical metrics. Consequently, we are able to transform what
+was previously multiple, complex time courses for hundreds
+of patients into 3-5 states that are clinically contextualized,
+straightforward, and meaningful.
+B. The decomposition can be externally validated and ranked
+We found that the resulting clusters from our analysis strat-
+ified on a negative-to-positive spectrum of health in chronic
+pain, and that these clusters were reliable across subsets of
+individuals and over time. Importantly, these states provide
+valuable, novel information per se, representing new findings
+that may define patient experience. Nevertheless, because they
+were derived from a purely data-driven analysis, we chose
+to compare cluster characteristics to independent standard as-
+sessments of disability and QoL. We found not only that good
+and bad clusters associate with better and worse disability and
+QoL, but that more granular state solutions had a clear ordinal
+rank which contextualized the data-driven output (Table 2).
+Further, in a 5-state solution (see figure 5A), only 2 levels
+of discriminable pain were present for 4 states. This adds
+clear dimensionality beyond what pain alone may indicate
+about a patient’s well-being. Thus, we were able to assess
+5 ordinal steps of health based on multidimensional aspects,
+providing evidence that we can offer a more full picture of
+patient experience yet preserve interpretability, making these
+states meaningful and actionable clinical information. This can
+improve precision in outcomes assessments, especially as it
+pertains to pain research and clinical trials.
+C. Objective data adds granularity to state solutions
+In particular, raw objective metrics such as actigraphy
+and speech features are too complex to use without some
+dimension reduction. However, actigraphy and speech offer
+insight into patient experience both because they reflect a novel
+behavioral measure and because they involve limited self-
+assessment, which is known to be susceptible to psychological
+biases. Here we showed that we were able to quantify and se-
+lect features from these objective measures in a preprocessing
+step, and then incorporate them into a clustering analysis. We
+found that one benefit of this approach is that these types of
+features indeed add dimensionality to a state solution, and the
+preprocessing in this case allowed for the derived features to
+add some biological interpretation. Additionally, we identified
+speech features that capture negative sentiment, possibly aug-
+menting the ability for the states to detect disability versus
+wellness as indicated by higher correlation values between
+those states and the independent assessments.
+
+dynamic cnanges in states
+State B
+tateD
+following implant moving
+73.3%
+0.0%
+between multiple states.
+State C
+State D
+13.3% 
+toteE
+6.7%
+6.7%
+to Implant 
+ Post-Implantt30 Days
+Time (Major Ticks Marked every 14 days)
+PATIENT 3: DE NOVO SCS PA
+A) Longitudinal State Plot for Patient 3
+C) D
+State A
+State B
+State :C
+State D
+State E
+mplar
+B) Health Outcomes Plot for Patient 3
+Normalized Values
+0.8
+0.6
+0.2
+Time (Major Ticks Markedevery 14 days)
+Trial EndTIENT
+ellTimeChange
+Patient 1 achieves the State A
+State A
+with SCS therapy during trial and
+ateC
+10.0%
+1.6%
+following implant eventually
+remaining stable in State B.
+StateB
+86.7%
+A marked reduction of dwell
+ateD
+time in State C and D is
+3.4%
+State D
+3.3%
+observed post-implant.
++ Post-implant defined as days 14 to 44 days after implant to
+oImplant
+Post-Implantt
+account for postsurgical healing.
+TIENT
+vellTimeChange
+State C
+Patient 2 achieves cycles
+10.0%
+State A
+State B
+13.3%
+between State A and State B
+36.7%
+tateD
+following ScS therapy.
+80.0%
+StateC
+20.0%
+State E
+10.0%
+StateD
+30.0%
+Pre-Implant
+Post-Implant+
+TIENT
+well TimeChange
+tateC
+State B
+State A
+Patient 3 has more
+13.3%
+10.0%
+6.7%PATIENT 1: DE NOVO SCS PA
+A) Longitudinal State Plot for Patient 1
+C) Dwe
+State A
+State B
+State C
+State D
+State E
+Trial
+Trial
+5.
+Start
+Normalized Values 
+B) Health Outcomes Plot for Patient 1
+Sta
+0.2
+Time (Major Ticks Marked every 14 days)
+Trial End to
+PATIENT 2: DE NOVO SCS PA
+Aj Longitudinal State Plot f
+State A.
+C) Dw
+State B
+State C
+State D
+State E
+B) Health Outcomes Plot for Patient 2
+0.8
+0.6
+0.4D. Conclusions
+Ultimately, this analysis combined AI and clinical knowl-
+edge to successfully reduce complex mobile data into useful
+health states that reflect important clinical time points and
+changes in patient experience (Figure 6). While all approaches
+should be tested and verified broadly across additional popu-
+lations and data sets, this approach lays a solid foundation
+by which complex datastreams may be reduced into and
+authenticated as useful wellness information. We were able to
+show that we could successfully use this method in patients
+undergoing treatment for chronic pain, with results yielding
+new, distinct representations of patient experience. These find-
+ings imply it is possible to expand this approach to other
+illnesses associated with heterogeneous sets of symptoms.
+Finally, while we were able to compare our findings to known
+metrics, the health states provide deep insights in and of
+themselves that could aid a clinician in medical decision
+making and patient care. Given the growing use of digital
+health solutions, this approach to define Pain Patient States
+holds great promise in harnessing AI-driven solutions to aid
+in the care of large groups of chronic pain patients.
+ACKNOWLEDGMENT
+The NAVITAS and ENVISION Studies Physician Author
+Group includes Richard Rauck (The Center for Clinical Re-
+search), Eric Loudermilk (PCPMG Clinical Research Unit),
+Julio Paez (South Lake Pain Institute), Louis Bojrab (Forest
+Health Medical Center), John Noles (River Cities Interven-
+tional Pain), Todd Turley (Hope Research Institute), Mohab
+Ibrahim (Banner University Medical Center), Amol Patward-
+han (Banner University Medical Center), James Scowcroft
+(KC Pain Centers), Rene Przkora (University of Florida),
+Nathan Miller (Coastal Pain and Spinal Diagnostics), and
+Gassan Chaiban (Ochsner Clinic Foundation).
+The Boston Scientific Research Scientists Consortium in-
+cludes Dat Huynh (Boston Scientific, Data Research and
+Engineering), Kristen Lechleiter (Clinical Research, Boston
+Scientific), Brad Hershey (Data Research and Engineering,
+Boston Scientific), Rex Woon (Data Research and Engineer-
+ing, Boston Scientific), and Matt McDonald (Boston Scientific,
+Data Research and Engineering).
+We wish to acknowledge work by Erhan Bilal (IBM, Digital
+Health) for his work on consensus clustering.
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+

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+A Comparison of Fundamental Methods for Iso-surface 
+Extraction 
+ 
+JAN PATERA1, VÁCLAV SKALA2 
+Department of Computer Science and Engineering 
+Faculty of Applied Sciences, University of West Bohemia 
+Univerzitní 22, Plzeň 
+CZECH REPUBLIC 
+hopatera@kiv.zcu.cz, skala@kiv.zcu.cz 
+http://herakles.zcu.cz 
+ 
+Abstract: In this paper four fundamental methods for an iso-surface extraction are compared, 
+based on cell decomposition to tetrahedra. The methods are compared both on mathematically 
+generated data sets as well as on real data sets. The comparison using mathematical data is 
+made from different points of view such as area approximation, volume approximation. On 
+the other hand, the Hausdorff distance and root mean square are used to compare methods on 
+real data sets. The presented comparison can be helpful when deciding among tested methods 
+which one to choose, as well as when we need to compare a newly developed method with 
+other existing approaches. 
+ 
+Key-Words: Comparison, Iso-surface extraction, Error, Hausdorff distance, Volume data, 
+Computer graphics. 
+ 
+1   Introduction 
+ 
+In the recent period of time volume data have started to play a significant role in many 
+scientific areas and are spread across many professions. In medical field, various devices, 
+such as Computed Tomography (CT) scanners, Magnetic Resonance Imaging (MRI) scanners 
+produce volume data. The volume data are also produced as a result of mathematical or 
+physical simulations and experiments and researchers need to visualize such data. 
+                                                
+1   Supported by the Ministry of Education of Czech Republic; project number MSM 
+235200005 (Information Systems and Technologies) 
+2    Supported by project NoE – 3DTV PLT 511568 
+Machine Graphics and Vision, Polish Academy of Sciences, Vol.13, No.4., pp.329-344, ISSN 1230-0535, 2004
+
+There are two main techniques for the volume data visualization. The first approach is 
+based on volume rendering (ray-tracing-like methods), the second one on surface rendering 
+(iso-surface-extraction-like methods). The volume rendering methods are complex and work 
+with the whole volume data. This paper is concentrated on surface rendering methods that 
+visualizes surfaces stored in the volume data (so called iso-surfaces). The extracted 
+iso-surface is determined by a threshold value. All the points on the iso-surface have their 
+value equal to the threshold.  
+The field of the iso-surface extraction is quite large. There are many approaches used 
+for the iso-surface extraction such as view-dependent techniques, parallel or distributed 
+approaches, external memory (or sometimes called I/O) techniques, multiresolution (LOD) 
+based extractions and others. In general, we can describe the iso-surface generation and 
+visualization process with the following steps: 
+1. Search for all active cells (cells that are intersected by the iso-surface) 
+2. The iso-surface and normal vectors approximation within these cells (e.g. by a triangle 
+set) 
+3. Iso-surfaces visualization (visualization of a set of triangles; different iso-surfaces can 
+be visualized with different colors depending on a selected threshold value, alpha 
+blending, etc.) 
+The first phase of the iso-surface extraction can be accelerated using a wide set of speed up 
+algorithms [7], [9], [10], [11], [17] or [18]. However, we are interested not that much in speed 
+of the extraction process but in properties of the output set of triangles. 
+As there are many various methods for the iso-surface generation and each such a 
+method generates generally different approximation of a searched iso-surface for a given 
+threshold, there is no way how to compare the resulting iso-surfaces to each other unless we 
+know how the iso-surface should look like. We try to compare generated iso-surfaces 
+produced by different methods. 
+ 
+Such a comparison can be made with respect to the volume data. When we generate 
+the volume data using some mathematical or physical model, we are able to gain some 
+additional information concerning the object that is utilized to make a comparison more 
+informative and objective. As additional information, we assume e.g. possibility to compute 
+area or volume of such an object. For real data sets, when we do not have any additional 
+information concerning the scanned object, we can just use general approaches for 
+comparison, such as Hausdorff distance or root mean square (RMS) distance. 
+Machine Graphics and Vision, Polish Academy of Sciences, Vol.13, No.4., pp.329-344, ISSN 1230-0535, 2004
+
+ 
+This paper is organized in the following way. At first, compared methods are 
+described. Afterwards, we will explain used approaches for the comparison and how the data 
+are generated. The last two sections are devoted to the error analysis, methods comparisons 
+and conclusion. 
+ 
+2   Method Description 
+ 
+2.1   Marching Cubes 
+ 
+There are many kinds of volume data. From simulations, we often get unstructured volume 
+data. In the other hand from medical imaging the output data is structured one. We aimed at 
+comparison of iso-surface generation methods that are used for structured data, especially for 
+regular grids. Compared methods do not differ in the kind of used interpolation but only in the 
+way they divide a cell into tetrahedra. The well-known method is Marching Cubes (MC) 
+method that was firstly published by Lorensen and Cline [12].  
+ 
+The input volume data consist of samples organized into a regular 3D Cartesian grid. 
+From such a grid, we can easily obtain a set of cells. The cell has in this case a cube shape and 
+consists of eight corresponding samples from two adjacent sample planes. Four samples are 
+from the first plane and four samples are from the second plane. MC method processes 
+sequentially all the cells that can be found in volume data. The iso-surface, which we are 
+looking for, is specified by a threshold value. 
+ 
+Each cell is processed separately. Firstly, the cell index is computed. The cell has eight 
+vertices, let us name them from A to H, and each vertex has its data value. Depending on a 
+selected threshold the vertex is assigned a binary value index = ABCDEFGHB. Each bit of the 
+index is 0 when the data value in the corresponding vertex is less than the threshold and 1 
+otherwise. 
+ 
+Based on the index, we are able to distinguish 256 cases how the iso-surface can 
+intersect the cell, because each vertex can be inside or outside of the iso-surface. When the 
+index is 0 or 255 the cell is not intersected by the iso-surface, otherwise such a cell is called 
+an active cell.  The purpose of the index will be described later. For an active cell, normal 
+vectors are computed in all its vertices using symmetric or asymmetric difference of data 
+samples.  
+Machine Graphics and Vision, Polish Academy of Sciences, Vol.13, No.4., pp.329-344, ISSN 1230-0535, 2004
+
+ 
+Each index represents a different case how the iso-surface can intersect the cell. All 
+these cases can be tabularized and easily triangulated using linear interpolation. The triangles 
+vertices lay on the cell edges. Note, that triangles vertices are interpolated only on the cell 
+edges, this will not be true for other methods. Maximum of four triangles per the cell is 
+needed to approximate the iso-surface. For each triangle vertex a normal vector is computed 
+from normal vectors in the cell vertices, using linear interpolation as well. 
+ 
+Each cell face is shared by another cell. Due to such sharing, the iso-surface is 
+continuous among adjacent cells. Note that there can be ambiguous faces at which the 
+triangulation proposed by [12] will produce holes. There are few approaches how to avoid the 
+holes. Ambiguous cases can be detected and a special triangulation can be applied [16]. The 
+cells can be divided into tetrahedra and resulting simplices triangulated in a little bit different 
+way as we will describe in the next section. Other approaches are out of the scope of this 
+paper, see [2], [3], [6], [13], [14], [15].  
+ 
+The algorithm complexity of MC method is O(N), where N is the number of all cells. 
+ 
+2.1   Marching Tetrahedra 
+ 
+Marching Tetrahedra (MT) method is based on the same principle as MC method. The 
+significant difference is that the cube cell is furthermore split into tetrahedra. There are two 
+main splitting schemes. The cell is divided into five tetrahedra (MT5) [8], [15] or the cell is 
+divided into six tetrahedra (MT6) [15]. There are several ways how the cube cell can be 
+divided into five (e.g. Fig. 1) or six tetrahedra (e.g. Fig. 2).  
+ 
+For the five tetrahedra scheme, it is necessary to alternate two different splitting 
+schemes. Otherwise, the continuity of the extracted iso-surface will not be maintained 
+properly. 
+ 
+Fig. 1 - MT5 tetrahedra division of the cell 
+ 
+Machine Graphics and Vision, Polish Academy of Sciences, Vol.13, No.4., pp.329-344, ISSN 1230-0535, 2004
+
+ 
+Fig. 2 - Three tetrahedra from a half of the cube, the second half is divided in similar way 
+ 
+After the cell is split into tetrahedra (four vertices), the index=ABCDB for each tetrahedron is 
+computed separately and tetrahedron is processed separately in the similar way as the cube 
+cell in the MC method. There are only 16 possibilities how the tetrahedron can be intersected 
+with iso-surface. These methods generate at most two triangles per tetrahedron. 
+ 
+Five or six tetrahedra decomposition introduces new edges at which the triangles 
+vertices are to be interpolated. For five tetrahedra the interpolation will be held on face 
+diagonals of the cube cell, for six tetrahedra both face and internal diagonals are used. 
+ 
+If we look at five tetrahedra division, there is one tetrahedron with different shape and 
+size. For six tetrahedra splitting, all the tetrahedra are the same. 
+ 
+2.3   Centered Cubic Lattice 
+ 
+The last method that will be compared is Centered Cubic Lattice (CCL) method, see [5]. This 
+method is little bit different, because it splits the cube cell into 24 tetrahedra.  
+ 
+The difference is that the resulting tetrahedra are partially shared between adjacent 
+cells and a new data value is introduced to the center of gravity of the processed cell, Fig. 3. 
+There are several ways how to compute the value of the central sample, e.g. the arithmetic 
+mean or weighted mean. 
+ 
+Each tetrahedron is then processed separately in the same way as in MT5 or MT6 
+methods. 
+ 
+As well as in previous methods this kind of splitting introduces new edges at which 
+the interpolation will be made. These are edges among adjacent central points. 
+ 
+In this division scheme, all the 24 tetrahedra are the same as to the dimensions 
+(similarly to MT6 method). 
+Machine Graphics and Vision, Polish Academy of Sciences, Vol.13, No.4., pp.329-344, ISSN 1230-0535, 2004
+
+ 
+There are also other possible decompositions of the cube cell, e.g. [19] that 
+decomposes parallelepiped cell into two tetrahedra and one octahedron. These techniques 
+were not included into our study. 
+ 
+ 
+Fig. 3 - Centered Cubic Lattice division for one cell face 
+ 
+3   Comparison Approaches 
+ 
+3.1   Hausdorff Distance 
+ 
+As mentioned before, we use Hausdorff distance [20] for comparisons mainly for iso-surfaces 
+that are extracted from real data sets. At first, we define a distance between a point p (from 
+surface S) and a surface S’ (with points p’) as 
+d(p, S’)=min||p-p’||, 
+for all p’ from S’. Now we can define Hausdorff distance between two surfaces S and S’ as  
+dH(S,S’)=max d(p,S’), 
+for all p from S. Note important thing that Hausdorff distance is not symmetrical 
+d(S,S’)≠d(S’,S). When we call d(S,S’) a forward and d(S’,S) a backward distance, we can 
+define a symmetrical Hausdorff distance [1] as  
+dSH(S,S’)=max(d(S,S’), d(S’,S)). 
+The symmetrical difference provides better error measurement for two surfaces. We utilized a 
+METRO software tool (described in [4]) for accurate computation of Hausdorff distance of 
+two discrete surfaces (triangle meshes). The METRO tool was mainly used to compare 
+original mesh with its simplified (e.g. decimated) version. We use it for comparison of two 
+iso-surfaces, each generated with different method. 
+ 
+Machine Graphics and Vision, Polish Academy of Sciences, Vol.13, No.4., pp.329-344, ISSN 1230-0535, 2004
+
+3.2   Root Mean Square Distance 
+ 
+We use also the Root Mean Square (RMS) of computed distances. RMS distance in discrete 
+case is defined as [20] 
+n
+x
+x
+S
+S
+RMS
+n
+2
+2
+1
+...
+)'
+,
+(
++
++
+=
+, 
+where n is a number of points of a mesh S’, xi (where i=1.. n) represents the distance of 
+corresponding point pi’ from S xi=d(pi’, S). We compare S’ to S. 
+ 
+Note that RMS is not symmetrical as well as Hausdorff distance. We do not use 
+symmetrical RMS distance in our tests, thus it is not defined here. This measurement is 
+computed with METRO tool as well. 
+ 
+Both the Hausdorff distance and the RMS distance are calculated according to some 
+source mesh using METRO tool. As such a mesh, we use a mesh generated with MC method. 
+ 
+3.3   Mathematical Data 
+ 
+At first, we should mention how the testing data are generated from basic mathematical 
+objects. For such objects we need to know an equation. Let us consider for example a sphere. 
+Each vertex of a regular grid has its coordinates and we have to assign it a value. The vertex 
+value is computed as a distance of the grid vertex (known coordinates) from the object surface 
+(known equation). The zero threshold then represents the object surface in volume data. 
+ 
+As we know the object equation and its dimensions, we are able to compute some 
+additional information concerning the object, such as surface area, object volume, triangles 
+position difference from the object surface, etc. We believe that these properties are worth to 
+compute, because they can help us to differentiate among the quality of methods. 
+ 
+Surface area – the iso-surface is generated by an extraction method in a form of a set 
+of triangles. We compute the total area as a sum of all triangles area. Than we can compute 
+the area of mathematical object and compare it with iso-surface area obtained. For special 
+objects such as sphere, we are able to track the error behaviour dependency on the sphere 
+radius. 
+ 
+Volume enclosed with the iso-surface – for basic objects the volume is computed 
+using appropriate formula. The volume enclosed with the iso-surface is computed in the 
+following manner (for tetrahedra only). There are three cases for a tetrahedron: 
+Machine Graphics and Vision, Polish Academy of Sciences, Vol.13, No.4., pp.329-344, ISSN 1230-0535, 2004
+
+1. The whole tetrahedron is outside of the iso-surface – does not affect the total volume 
+computation. 
+2. The whole tetrahedron is inside – the whole tetrahedron contributes to the total 
+volume. The tetrahedron volume is computed easily. 
+3. The tetrahedron is intersected with the iso-surface – we have to compute the part of the 
+tetrahedron which is inside of the iso-surface. As there are at most two triangles 
+generated per tetrahedron, these triangles form two small tetrahedra with appropriate 
+tetrahedron vertex and we are able to compute the volume of the tetrahedron part 
+which contributes to the total volume. 
+Triangles position difference – we measure the difference between triangle center of gravity 
+and object surface. This gives us information about triangles position difference compared to 
+the object surface. 
+ 
+The three mentioned geometric properties are the main aspects that we used for 
+extraction methods output comparison. The obtained results are showed in the next section. 
+ 
+4   Results 
+ 
+At first, we should describe the data sets used for our comparisons and give the reasons why 
+we chose them. The main part of the used data set is a set of mathematically generated 
+objects, Fig. 4. A real data set was used to show how the Hausdorff distance is dependent on 
+applied iso-surface extraction method. The brief description of used data sets follows in 
+upcoming paragraphs. 
+ 
+ 
+Fig. 4 - Objects (csph, torus, sombrero, cube, sphere and noisedsph) 
+ 
+Machine Graphics and Vision, Polish Academy of Sciences, Vol.13, No.4., pp.329-344, ISSN 1230-0535, 2004
+
+4.1   Used Objects 
+ 
+Sphere – sphere is an example of an object that we use to follow the error behaviour 
+dependency on sphere radius. The sphere equation used for data generation is a modified 
+implicit equation  
+r
+s
+z
+s
+y
+s
+x
+z
+y
+x
+F
+Z
+Y
+X
+−
+−
++
+−
++
+−
+=
+2
+2
+2
+)
+(
+)
+(
+)
+(
+)
+,
+,
+(
+ 
+where x, y and z are samples coordinates, sx, sy and sz are the sphere centre coordinates, r is 
+sphere radius and F(x,y,z) is a corresponding sample value. This equation assigns data value 
+to all the volume data samples. The sphere is then represented with a zero threshold 
+iso-surface. The samples that are inside of the sphere have negative value, on the sphere zero 
+value and samples placed out of the sphere have positive value. The sample value represents 
+the distance of the sample from the sphere surface. The radius was 25 in our experiments.  
+ 
+Cell edge has a length 1 for our purposes. The object dimensions (e.g. radius, edge 
+length) are then related to a cell edge length.  
+ 
+Noised sphere – (noisedsph) to study the influence of the noise to the shape of the 
+output set of triangles we generate a noised sphere. The random noise is introduced (added) to 
+all samples of the volume data. The size of the noise is given in percentage from the sphere 
+radius size. We used radius 25 and 10% noise.  
+ 
+Cube – this kind of an object we use to follow the behaviour and properties of the 
+iso-surface on edges. We will show the iso-surface difference mainly visually. Data are 
+generated similarly as in the previous case using the distance of sample from the closest face, 
+edge or vertex. The inner, on surface and outer samples have the negative, zero and positive 
+value respectively. Cube was generated using a=b=c=42.  
+ 
+Cube minus sphere – (csph) such an object was constructed to combine both 
+properties of the sphere (r=25) and cube (a=b=c=42). The generation of it is a little bit 
+complicated. At first, the cube is generated in the volume data. Afterwards, the values of all 
+samples that are closer to the sphere than to the cube are modified to the new distance.  
+ 
+Torus – is the typical mathematically generated object. Torus is defined with the 
+following equation [20] 
+a
+z
+y
+x
+c
+z
+y
+x
+F
+−
++
++
+−
+=
+2
+2
+2
+2
+)
+(
+)
+,
+,
+(
+ 
+where x, y and z are samples coordinates, c is a torus main radius, a is a torus secondary 
+radius and F(x,y,z) is a corresponding sample value. The samples value are negative, zero or 
+positive as well. Torus dimensions are c=20 and a=42 in our case.  
+Machine Graphics and Vision, Polish Academy of Sciences, Vol.13, No.4., pp.329-344, ISSN 1230-0535, 2004
+
+ 
+Sombrero – is the last mathematically generated object we use. It is a surface defined 
+with the mathematical equation (taken from Derive mathematical program) 
+ 
+2
+2
+2
+2
+))
+(
+cos(
+)
+,
+,
+(
+z
+x
+c
+z
+x
+b
+a
+y
+z
+y
+x
+F
++
++
++
+⋅
+⋅
+−
+=
+ 
+where x, y and z are sample coordinates and F(x,y,z) is a corresponding sample value and a, b 
+and c are constants modifying the shape of the function. Sombrero parameters we used are 
+a=12, b=0.25 and c=3.  
+ 
+Real data sets – Samples of real data set have only positive values that represent a 
+density of the space in the sample position (we used engine.vol, ctmayo.vol and hplogo.vol 
+sets). 
+ 
+4.2   Tests and Results 
+ 
+For all our mathematically generated objects, we are able to compute the triangles position 
+difference compared to the mathematical object. Firstly, a triangle center of gravity is 
+computed. As we have the routines for point to object distance computation, we can compute 
+the distance of the center of gravity of the triangle from the appropriate object. The overall 
+position difference PERR is computed as 
+ 
+n
+objDist
+P
+n
+i
+ERR
+∑
+=
+=
+1
+|)
+,
+(
+|
+iT
+O
+ 
+where Ti (i goes from 1 to n) is the center of gravity of the i-th triangle, n is the number of 
+triangles and objDist(O, X) is the distance of point X from an object O surface. 
+ 
+Triangles Position Difference
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+cube
+csph
+sphere
+torus
+TPD
+MC
+MT5
+MT6
+CCL
+ 
+Fig. 5 - Triangles position difference comparison (edge vs. smooth object) 
+ 
+Machine Graphics and Vision, Polish Academy of Sciences, Vol.13, No.4., pp.329-344, ISSN 1230-0535, 2004
+
+The position difference for a sombrero object was slightly smaller and similar to the results 
+obtained for a sphere. For a cube the CCL method gives the worst results, see Fig. 5. This is 
+probably due to different interpolation of the cube edges (Fig. 6). A csph object has more 
+edges than a cube itself. The more tetrahedra we create the worse results we get. Surprisingly 
+for a torus the MT6 method gives the greatest position difference. We think this is because of 
+the interpolation at a cell interior edge (the longest one). 
+ 
+ 
+Fig. 6 - Iso-surface on edges (MT5, MC, MT6, CCL) 
+ 
+Note that RMS distance is related to the MC method. For a sphere and a torus the obtained 
+results were slightly less than results for a sombrero. Again, when the object has edges the 
+CCL method is the worst from the view of RMS distance, see Fig. 7. For noisedsph object the 
+CCL method gives the best results. We suppose that the central cell sample value computation 
+(using arithmetic mean) filters data a little bit as well. 
+ 
+RMS Comparison
+0.00
+0.05
+0.10
+0.15
+0.20
+0.25
+cube
+csph
+noisedsph
+sombrero
+RMS
+MT5
+MT6
+CCL
+ 
+Fig. 7 - RMS distance histogram 
+ 
+Again, a sphere and sombrero give approximately similar results compared to torus. From the 
+view of Hausdorff distance the MT6 method gives the worst results for all tested objects, see 
+Fig. 8. As you can see for noisedsph the CCL method is the best choice. The best choice in 
+Machine Graphics and Vision, Polish Academy of Sciences, Vol.13, No.4., pp.329-344, ISSN 1230-0535, 2004
+
+this case is probably MT5 method because it does not generate as much triangles as CCL 
+method. 
+ 
+Hausdorff Distance Comparison
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+1.4
+cube
+csph
+noisedsph
+torus
+Hausdorff dist.
+MT5
+MT6
+CCL
+ 
+Fig. 8 - Hausdorff distance histogram 
+ 
+The more tetrahedra is used the larger area is extracted for all tested objects that have edges, 
+see Fig. 9. The results in Fig. 9 and Fig. 10 are relative due to mathematical results. For 
+objects like torus (does not have edges) the results were approximately the same as for a 
+sphere. We think that for the area approximation purposes the best choice is MC method. 
+ 
+Area Comparison
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+1.4
+1.6
+1.8
+cube
+csph
+sphere
+Area
+Math
+MC
+MT5
+MT6
+CCL
+ 
+Fig. 9 - Area comparison (relative to mathematical volume) 
+ 
+The MT5 method is in most cases slightly better than MT6 method and both methods are 
+approaching to the original volume from below, see Fig. 10. The CCL method in the other 
+hand is in most cases approaching mathematically computed volume from above. MC method 
+is not included because it is hard to compute the volume enclosed with the iso-surface (due to 
+256 cases). 
+ 
+Machine Graphics and Vision, Polish Academy of Sciences, Vol.13, No.4., pp.329-344, ISSN 1230-0535, 2004
+
+Volume Comparison
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+cube
+csph
+sphere
+torus
+Volume
+Math
+MT5
+MT6
+CCL
+ 
+Fig. 10 - Volume comparison (relative to mathematical volume) 
+ 
+4.3   Sphere Additional Test 
+ 
+A relative volume error is defined in a following way 
+V
+V
+V
+Error
+TR −
+=
+ 
+where VTR is a volume enclosed with iso-surface triangles, V is mathematically computed 
+volume of the sphere. 
+ 
+The CCL method is the best choice for the volume approximation, see Fig. 11. We 
+assume that it is due to high number of tetrahedra. The CCL method error oscillates about 
+zero value. MT5 gives slightly better results than MT6 method. The progress of error is 
+similar. Both methods are approaching the zero error from below. Another thing we compare 
+is a number of extracted triangles. 
+ 
+Error of Volume Approximation (Sphere, r=10 to 100)
+-0.06
+-0.05
+-0.04
+-0.03
+-0.02
+-0.01
+0.00
+0.01
+0.02
+0.03
+0
+20
+40
+60
+80
+100
+120
+Radius
+Error[%]
+MT5
+MT6
+CCL  
+Fig. 11 - Sphere volume error graph 
+ 
+Machine Graphics and Vision, Polish Academy of Sciences, Vol.13, No.4., pp.329-344, ISSN 1230-0535, 2004
+
+It is a known fact that a number of generated triangles is mainly dependent on the type of the 
+cell division, see Fig. 12. MC works with a cube cell (at most four triangles per cell) and it 
+does not divide it into tetrahedral (at most two triangles per tetrahedron). MT5 divides the 
+cube cell into 5 tetrahedra, MT6 into 6 tetrahedra. In fact, CCL divides the cube cell into 24 
+tetrahedra, but these tetrahedra also contain parts of adjacent cube cells. When we sum the 
+volume of all 24 tetrahedra, we obtain two cube cells volume, so on average 12 tetrahedra per 
+cube cell. 
+ 
+Number of Extracted Triangles
+0
+200000
+400000
+600000
+800000
+1000000
+1200000
+1400000
+1600000
+1800000
+0
+20
+40
+60
+80
+100
+Radius
+Triangles
+MC
+MT5
+MT6
+CCL
+ 
+Fig. 12 - Number of extracted triangles 
+ 
+4   Conclusions 
+ 
+We compared fundamental methods for the iso-surface extraction evaluating Hausdorff 
+distance, RMS distance, triangles position difference and iso-surface area and volume. 
+ 
+Hausdorff distance is in fact the biggest distance between two compared surfaces 
+(extreme distance). In general, we are more interested in average distance between two 
+surfaces (the RMS distance). In this case, the CCL method generally gives worse results 
+compared to other methods. If we look at a position difference, the MC method seems to be 
+generally the best one.  The quality of the extracted set of triangles for noised sphere was in 
+general bad. Interesting is that a volume of objects is approximated with the similar difference 
+no matter of method used except for csph object. 
+ 
+It is important to realize that for real data we do not know the exact area or volume of 
+the object. Hence, the speculations such that the Hausdorff distance is bigger or lower are not 
+completely correct.  
+ 
+Machine Graphics and Vision, Polish Academy of Sciences, Vol.13, No.4., pp.329-344, ISSN 1230-0535, 2004
+
+Acknowledgements 
+ 
+We want to thank to Dr. Ivana Kolingerová for her help and support during preparation of this 
+paper. 
+ 
+References 
+ 
+[1] 
+Aspert,N., Santa-Cruz,D., Ebrahimi,T.: Mesh Measuring Errors Between Surfaces 
+Using The Hausdorff Distance, In Proceedings of the IEEE International Conference in 
+Multimedia and Expo (ICME) 2002, Vol. 1, pages 705-708, Lausanne, Switzerland, 
+August 26-29, 2002 
+[2] 
+Bonnel,K.S., Duchaineau,M.A., Schikore,D.R., Hamann,B., Joy,K.I.: Material Interface 
+Reconstruction, IEEE Transactions on Visualization and Computer Graphics, Vol. 9, 
+No. 4, pages 500-511, 2003 
+[3] 
+Cignoni,P., Ganovelli,F., Montani,C., Scopigno,R.: Reconstruction of Topologically 
+Correct and Adaptive Trilinear Isosurfaces, Computers & Graphics, Vol. 24, No. 3, 
+pages 399-418, 2000 
+[4] 
+Cignoni,P., Rocchini,C., Scopigno,R.: Metro: Measuring Error on Simplified Surfaces, 
+Computer Graphics Forum, Blackwell Publishers, Vol. 17, No. 2, pages 167-174, June 
+1998 
+[5] 
+Chan,S.L., Purisima,E.O.: A New Tetrahedral Scheme for Iso-surface Generation, 
+Computers & Graphics, Vol. 22, No. 1, pages 82-90, Elsevier Science Limited, 1998 
+[6] 
+Chernyaev,E.V.: Marching Cubes 33: Construction of Topologically Correct 
+Isosurfaces, Institute for High Energy Physics, Moscow, Russia, Report CN/95-17, 
+1995 
+[7] 
+Giles,M., Haimes,R.: Advanced Interactive Visualization for CFD, Computing Systems 
+in Engineering, Vol. 1, No.1, pages 51-62, 1990 
+[8] 
+Hall,M. Warren,J.: Adaptive Polygonalization of Implicitly Defined Surfaces, IEEE 
+Computer Graphics and Applications, Vol. 10, No. 6, pages 33-42, November 1990 
+[9] 
+Itoh,T., Yamaguchi,Y., Koyamada,K.: Fast Isosurface Generation Using the Volume 
+Thinning Algorithm, IEEE Transactions on Visualization and Computer Graphics, Vol.  
+7, No. 1, pages 32-46, 2001 
+Machine Graphics and Vision, Polish Academy of Sciences, Vol.13, No.4., pp.329-344, ISSN 1230-0535, 2004
+
+[10] Van Kreveld,M., van Oostrum,R., Bajaj,C., Pascucci,V., Schikore,D: Contour Trees 
+and Small Seed Sets for Iso-surface Traversal, In Proceedings 13th Annual Symposium 
+Computational Geometry, pages 212-220, 1997 
+[11] Livnat,Y., Parker,S.G., Johnson,C.R.: Fast Iso-surface Extraction Methods for Large 
+Imaging Data Sets, Center for Scientific Computing and Imaging, Department of 
+Computer Science, University of Utah, Salt Lake City, USA, 1999 
+[12] Lorensen,W.E., Cline,H.E.: Marching Cubes: A High Resolution 3D Surface 
+Construction Algorithm, Computer Graphics, Vol. 21, No. 4, July 1987 
+[13] Lopez,A., Brodlie,K.: Improving the Robustness and Accuracy of the Marching Cubes 
+Algorithm for Isosurfacing, IEEE Transactions on Visualization and Computer 
+Graphics, Vol. 9, No. 1, January-March 2003 
+[14] Natarajan,B.K.: On Generating Topologically Consistent Isosurfaces from Uniform 
+Samples, The Visual Computer, Vol. 11, pages 52-62, 1994 
+[15] Ning, P. and Bloomenthal, J.: An Evaluation of Implicit Surface Tilers, Computer 
+Graphics and Applications 13(6), pages 33-41, November 1993 
+[16] Schroeder,W., Martin,K., Lorensen,B.: The Visualization Toolkit, 2nd Edition, Prentice 
+Hall PTR, ISBN 0-13-954694-4, 1998 
+[17] Shen,H.-W., Hansen,C.D.,  Livnat,Y.,  Johnson,C.R: Isosurfacing in Span Space with 
+Utmost Efficiency (ISSUE), IEEE Visualization 96, pages 287-294, 1996 
+[18] Shen,H., Johnson,C.R.: Sweeping Simplicies: A Fast Iso-surface Extraction Algorithm 
+for Unstructured Grids, Proceedings of Visualisation '95, IEEE Computer Society 
+Press, Los Alamos, CA, 1995 
+[19] Takahashi,T., Yonekura,T.: Isosurface Construction From a Data Set Sampled On a 
+Face-Centered-Cubic Lattice, Proceedings of  ICCVG 2002, No. 2, pages 754-763, 
+September 2002 
+[20] Weisstein,E.W.: MathWorld, A Wolfram Web Resource, 
+ 
+http://mathworld.wolfram.com 
+ 
+Machine Graphics and Vision, Polish Academy of Sciences, Vol.13, No.4., pp.329-344, ISSN 1230-0535, 2004
+
+ 
+Ing. Jan Patera (http://zcu.cz/~hopatera) is a PhD student and a part-time 
+tutor at the Department of Computer Sciences at the University of West 
+Bohemia in Plzeň in Czech Republic. He graduated in the field of computer 
+graphics at the University of West Bohemia in 2002. He is a member of the 
+Center of Computer Graphics and Data Visualization (CGDV). His research 
+activities concern volume data, iso-surface extraction, algorithms and data 
+visualization.  
+ 
+ 
+Vaclav Skala is a full professor of Computer Science at the Faculty of 
+Applied Sciences at the University of West Bohemia in Plzen, Czech 
+Republic. He is responsible for courses on Computer Graphics, Algorithms 
+for Computer Graphics, Visualization, Multimedia Systems, Programming 
+in Windows, .NET Technologies at the Department of Computer Science. 
+He is a member of The Visual Computer and Computers&Graphics 
+editorial boards, Eurographics Executive Committee and member of 
+program committees of established international conferences. He has been a 
+research fellow or lecturing at the Brunel University (London, U.K.), 
+Moscow Technical University (Russia), Gavle University (Sweden) and 
+others institutions in Europe. He organizes the WSCG International 
+Conferences in Central Europe on Computer Graphics, Visualization and 
+Computer Vision (http://wscg.zcu.cz) held annually since 1992 and .NET 
+Technologies conferences (http://dotnet.zcu.cz).  He is interested in 
+algorithms, data structures, mathematics, computer graphics, computer 
+vision and visualization.  He has been responsible for several research 
+projects as well. Currently he is a director of the Center of Computer 
+Graphics and Visualization (http://herakles.zcu.cz).  
+ 
+ 
+Machine Graphics and Vision, Polish Academy of Sciences, Vol.13, No.4., pp.329-344, ISSN 1230-0535, 2004
+

AtE1T4oBgHgl3EQfDQOp/content/tmp_files/2301.02875v1.pdf.txt

@@ -0,0 +1,1449 @@
+arXiv:2301.02875v1  [math.NA]  7 Jan 2023
+SCIENCE CHINA Mathematics
+1
+XXXX Vol. XX No. XX XX–XX
+www.SciChina.com
+www.springerlink.com
+An iterative two-grid method for strongly non-
+linear elliptic boundary value problems
+Jiajun Zhan1, Lei Yang1, Xiaoqing Xing2,†, Liuqiang Zhong2
+1 School of Computer Science and Engineering, Faculty of Innovation Engineering, Macau University of Science
+and Technology, Macao SAR 999078, China;
+2 School of Mathematical Sciences, South China Normal University, Guangzhou 510631, China
+Email: 2109853gii30011@student.must.edu.mo, leiyang@must.edu.mo, xingxq@scnu.edu.cn, zhong@scnu.edu.cn
+Abstract
+We design and analyze an iterative two-grid algorithm for the finite element discretizations
+of strongly nonlinear elliptic boundary value problems in this paper. We propose an iterative two-grid
+algorithm, in which a nonlinear problem is first solved on the coarse space, and then a symmetric positive
+definite problem is solved on the fine space. The innovation of this paper lies in the establishment
+of a first convergence analysis, which requires simultaneous estimation of four interconnected error
+estimates.
+We also present some numerical experiments to confirm the efficiency of the proposed
+algorithm.
+Keywords:
+iterative two-grid method, convergence, strongly nonlinear elliptic problems.
+MSC(2020):
+65N30, 65M12, 35J60
+1
+Introduction
+The two-grid methods are first proposed for nonselfadjoint problems and indefinite elliptic
+problems [6, 10]. Then, the two-grid methods are extended to solve semiliinear elliptic problems
+[7], quasi-linear and nonlinear elliptic problems [8, 9], respectively. Especially, for nonlinear
+elliptic problems, the basic idea of two-grid methods is to first obtain a rough solution by
+solving the original problem in a “coarse mesh” with mesh size H, and then correct the rough
+solution by solving a symmetric positive definite (SPD) system in a “fine mesh” with mesh size
+h. Noticing the mesh size of “coarse mesh” is much smaller than that of “fine mesh”, it is not
+difficult to solve an original problem in “coarse mesh”. Therefore, two-grid methods reduce
+the computational complexity of solving the original problem to solving a SPD problem and
+dramatically improve the computational speed. Recently, Bi, Wang and Lin [1] presented a
+two-grid algorithm to solve the strongly nonlinear elliptic problems and provided a posteriori
+error estimator for the two-grid methods. It’s noted that the literature mentioned above is all
+about non-iterative two-grid methods.
+As is well-known, the mesh size H of “coarse mesh” and h of “fine mesh” should satisfy a
+certain relationship for the optimal convergence order in non-iterative two-grid methods. The
+iterative two-grid methods have the advantage over the non-iterative two-grid methods in that,
+the distance between the mesh sizes H and h can be enlarged by increasing the iteration counts
+† Corresponding author
+
+2
+Jiajun Zhan & et al.
+with the same accuracy. However, there is only a small amount of literature on iterative two-grid
+methods of conforming finite element discretization for elliptic problems. Xu [9] first proposed
+and analyzed an iterative two-grid method for non-symmetric positive definite elliptic problems.
+Zhang, Fan and Zhong [11] designed some iterative two-grid algorithms for semilinear elliptic
+problems and provided the corresponding convergence analysis. To our knowledge, there is
+not any published literature on the iterative two-grid algorithm of conforming finite element
+discretization for strongly nonlinear elliptic boundary value problems.
+In this paper, an iterative two-grid algorithm for solving strongly nonlinear elliptic problems
+is studied. The discrete system of strongly nonlinear elliptic problems is presented at first. And
+then, an iterative two-grid algorithm is proposed for the discrete system, which is obtained by
+applying a non-iterative two-grid algorithm of [8] in a successive fashion. Finally, a challenging
+convergence analysis of the proposed algorithm is provided. Despite the fact that our algorithm
+is simply obtained by [8], the convergence analysis of the non-iterative two-grid algorithm could
+not be directly applied to the iterative two-grid algorithm. Here we complete this challenging
+convergence analysis by mathematical induction which can also be used in solving semilinear
+elliptic problems by iterative two-grid algorithms in [11]. However, we must emphasize that the
+convergence analysis of our algorithm is significantly different from the one of [11]. Compared
+with the current work [11], our convergence analysis is far more difficult and complex, and
+specific challenges could be reflected in: (1) the higher order derivative component of our model
+problem is still nonlinear; (2) the interconnectedness of the error estimates causes formidable
+obstacle for the convergence analysis (See the proof of Lemma 4.7).
+To avoid the repeated use of generic but unspecified constants, x ≲ y is used to denote x ⩽
+Cy, where C are some positive constants which do not depend on the mesh size. Furthermore
+the constants C may denote different values under different circumstances. For some specific
+constants, we use the constant C with some subscript to denote.
+2
+Model problems and discrete systems
+In this section, we present the continuous and discrete variational problems of strongly nonlinear
+elliptic problems, and provide the corresponding well-posedness and priori error estimates.
+Given a bounded convex polygonal domain Ω ⊂ R2 with the boundary ∂Ω. We denote
+W m,p(Ω) as the standard Sobolev space with norm ∥ · ∥m,p,Ω and seminorm | · |m,p,Ω, where the
+integers m ⩾ 0 and p ⩾ 1. For convenience, we also denote Hm(Ω) = W m,2(Ω), ∥·∥m = ∥·∥m,2,Ω
+and H1
+0(Ω) := {u ∈ H1(Ω) : u|∂Ω = 0}.
+We consider the following strongly nonlinear elliptic problems:
+�
+−∇ · a(x, u, ∇u) + f(x, u, ∇u) = 0, in Ω,
+u = 0, on ∂Ω,
+(2.1)
+where a(x, y, z) : ¯Ω × R × R2 → R2 and f(x, y, z) : ¯Ω × R × R2 → R. When a(x, u, ∇u) and
+f(x, u, ∇u) take different functions, different problems are available, such as mean curvature
+flow, Bratu’s problem and so on(See [3]).
+We assume that a(x, y, z) and f(x, y, z) are second order continuous differentiable functions.
+For simplicity, we denote that ay(w) = Dya(x, w, ∇w), az(w) = Dza(x, w, ∇w), fy(w) =
+Dyf(x, w, ∇w) and fz(w) = Dzf(x, w, ∇w), and similar notations are applied to the second
+order derivatives of a(x, y, z) and f(x, y, z).
+
+An iterative two-grid method for strongly nonlinear elliptic boundary value problems
+3
+Remark 2.1
+Since a(x, y, z) and f(x, y, z) are second order continuous differentiable func-
+tions, there exists a positive constant ˜C as upper bound with respect to all the first and second
+order derivatives of a(·, ·, ·) and f(·, ·, ·).
+We denote
+A(v, ϕ) = (a(x, v, ∇v), ∇ϕ) + (f(x, v, ∇v), ϕ),
+∀v, ϕ ∈ H1
+0(Ω).
+(2.2)
+By Green formula, it’s easy to see that the solution u ∈ H1
+0(Ω) of (2.1) satisfies
+A(u, v) = 0,
+∀v ∈ H1
+0(Ω).
+(2.3)
+The Fr´echet derivative L′ of (2.1) at w is given by
+L′(w)v = −∇ · (ay(w)v + az(w)∇v) + fy(w)v + fz(w)∇v.
+In the following, we gives some of our basic assumptions (Similar assumptions also could be
+found in [9] or [3]). Firstly, the problem (2.3) has a solution u ∈ H1
+0(Ω)∩Hr+1(Ω)∩W 2,2+ε(Ω)
+(ε > 0 and integer r ⩾ 1). Secondly, for the solution u of (2.3), there exists a positive constant
+α0 such that
+ξT az(u)ξ ⩾ α0|ξ|2,
+∀ξ ∈ R2, x ∈ ¯Ω.
+(2.4)
+Finally, L′(u) : H1
+0(Ω) → H−1(Ω) is an isomorphism. These assumptions guarantee that u is
+an isolated solution of (2.3).
+Let Th be a conforming quasi-uniform triangulation on Ω, where the mesh size h denotes
+the maximum of the circumscribed circle diameters of element K ∈ Th. By this, any element
+K ∈ Th is contained in (contains) a circle of radius ˆC1h (respectively, ˆC2h), where the constant
+ˆC1 and ˆC2 do not depend on mesh size h, and there is no hanging node on Th.
+The finite element space Vh on Th is defined as
+Vh = {vh ∈ H1
+0(Ω) : vh|K ∈ Pr(K), ∀ K ∈ Th},
+where Pr(K) is the set of polynomials of degree at most integer r on K.
+Here is the discrete system of (2.3): Find uh ∈ Vh such that
+A (uh, vh) = 0,
+∀vh ∈ Vh.
+(2.5)
+The following lemma presents the well-posedness of the variational problem (2.5) and its
+priori error estimates, which can be found in Lemma 3.2 and Theorem 3.4 of [9], respectively.
+Lemma 2.2
+Assume u is the solution of problem (2.3), then when h is small enough, the
+discrete variational problem (2.5) exists a unique solution uh ∈ Vh, and the following priori
+error estimate holds
+∥u − uh∥1,p ≲ hr,
+if u ∈ W r+1,p(Ω), 2 ⩽ p ⩽ ∞.
+(2.6)
+3
+Iterative two-grid algorithms
+
+4
+Jiajun Zhan & et al.
+In this section, we present an iterative two-grid algorithm for the variational problems (2.3).
+Let Th and TH be two quasi-uniform, conforming and nested mesh in Ω. Furthermore the
+mesh size h of Th and H of TH satisfy, for some 0 < λ < 1,
+H = O(hλ)
+and
+h < H < 1.
+For present the iterative two-grid algorithm, we introduce the form B(w; v, χ) (induced by
+L′) by , for a fixed w and any v, χ ∈ H1
+0(Ω),
+B(w; v, χ) = (ay(w)v, ∇χ) + (az(w)∇v, ∇χ) + (fy(w)v, χ) + (fz(w)∇v, χ).
+(3.1)
+Remark 3.1
+The form B(w; ·, ·) is a bilinear form for fixed w.
+To our knowledge, the two-grid algorithms of strongly nonlinear problems are firstly pro-
+posed in [8]. Here one of two-grid algorithms from Algorithm 3.3 of [8] is given.
+Algorithm 3.1
+1. Find uH ∈ VH, such that
+A(uH, vH) = 0,
+∀vH ∈ VH.
+2. Find uh ∈ Vh, such that
+B(uH; uh, vh) = B(uH; uH, vh) − A(uH, vh),
+∀vh ∈ Vh.
+Remark 3.2
+In the Algorithm 3.1, we first solve a nonlinear problem in a coarse space
+VH. However, because dim(VH) is relatively small, the calculated amount of solving a nonlinear
+problem in VH is not excessive. As for the second step of Algorithm 3.1, noticing that B(uH; ·, ·)
+is a bilinear form with given uH, we simply need to solve a linear problem in Vh, for which there
+are numerous concerning fast algorithms.
+In [8], Xu had showed that the solution uh of Algorithm 3.1 could be a good approximation
+with respect to finite element solution uh at a low cost, namely,
+∥uh − uh∥1 ≲ H2.
+(3.2)
+Using triangle inequality, (2.6) with r = 1 and (3.2), we obtain the error estimate of Algorithm
+3.1,
+∥u − uh∥1 ⩽ ∥u − uh∥1 + ∥uh − uh∥1 ≲ h + H2.
+(3.3)
+Next, putting the Algorithm 3.1 into a successive fashion, we obtain our iterative two-grid
+algorithm.
+Algorithm 3.2
+Let u0
+h = uH be the solution of (2.5) in VH. Assume that uk
+h ∈ Vh has been
+obtained, then uk+1
+h
+∈ Vh can be obtained by the following two steps.
+Step 1. Find ek
+H ∈ VH such that, for any vH ∈ VH,
+A(uk
+h + ek
+H, vH) = 0.
+(3.4)
+Step 2. Find uk+1
+h
+∈ Vh such that, for any vh ∈ Vh,
+B(uk
+h + ek
+H; uk+1
+h
+, vh) = B(uk
+h + ek
+H; uk
+h + ek
+H, vh) − A(uk
+h + ek
+H, vh).
+(3.5)
+
+An iterative two-grid method for strongly nonlinear elliptic boundary value problems
+5
+Remark 3.3
+Noticing the uniqueness of finite element solution (See Lemma 2.2), (2.5),
+u0
+h = uH and (3.4) with k = 0, we can see that e0
+H = 0, which means u0
+h + e0
+H = uH. By
+observing the the Step 2 of Algorithm 3.2 and the second step of Algorithm 3.1, the conclusion
+is that Algorithm 3.2 is same with Algorithm 3.1 when k = 0.
+In comparison to [8], our method is still valid for high order conforming finite elements,
+whereas [8] only considered piecewise linear finite element space. Here gives the error estimate
+of our algorithm (See Theorem 4.9),
+∥u − uk
+h∥1 ≲ hr + Hr+k.
+(3.6)
+Specially, if we choose finite element space Vh as piecewise linear finite element space, i.e. r = 1,
+the error estimate (3.6) of Algorithm 3.2 could be written as
+∥u − uk
+h∥1 ≲ h + H1+k.
+To achieve the optimal convergence order, the relationship h = H2 should be satisfied in
+Algorithm 3.1 (See (3.3)). But in Algorithm 3.2, we could expand the distance between the
+mesh size H and h by increasing the iteration counts k.
+4
+Convergence analysis
+In this section, we provide the corresponding convergence analysis of Algorithm 3.2. To this
+end, we need to introduce some preliminaries based on form B(w; v, χ) at first.
+4.1
+Some preliminaries based on form B(w; v, χ)
+In this subsection, we present some properties of form B(w; v, χ) and introduce two discrete
+Green function.
+Firstly, with fixed w, by Remark 2.1 and Cauchy-Schwarz inequality, it’s easy to obtain
+that the form B(w; ·, ·) is continuous, i.e.,
+B(w; v, χ) ≲ ∥v∥1∥χ∥1,
+∀ v, χ ∈ H1
+0(Ω).
+(4.1)
+Secondly, we present a lemma which provides the Babuˇska-Brezzi(BB) conditions of form
+B(·; ·, ·) in Vh. And this lemma can be proved using similar arguments in Lemma 2.2 of [9].
+Lemma 4.1
+Assume u is the solution of problem (2.3), then when h is small enough, we
+have, for any wh ∈ Vh,
+∥wh∥1 ≲ sup
+vh∈Vh
+B (u; wh, vh)
+∥vh∥1
+and
+∥wh∥1 ≲ sup
+vh∈Vh
+B (u; vh, wh)
+∥vh∥1
+.
+(4.2)
+Proof.
+For the solution u of (2.3), a projection operator Ph : H1
+0(Ω) → Vh is defined by
+(az(u)∇Phv, ∇χh) = (az(u)∇v, ∇χh),
+∀v ∈ H1
+0(Ω), χh ∈ Vh.
+(4.3)
+By (2.4), we can know that the projection operator Ph is well-defined. Taking v = vh ∈ Vh ⊂
+H1
+0(Ω) and χh = Phvh − vh, and using (2.4), we could prove that the projection operator Ph
+is identity operator for space Vh. Substituting χh = Phv into (4.3), and using (2.4), Poincar´e
+inequality, Remark 2.1 and Cauchy–Schwarz inequality, it holds that
+∥Phv∥1 ≲ ∥v∥1,
+∀v ∈ H1
+0(Ω).
+(4.4)
+
+6
+Jiajun Zhan & et al.
+By (2.4), duality argument and (4.4), we can obtain (See Theorem 3.2.5 in [2])
+∥v − Phv∥0 ≲ h∥v∥1,
+∀v ∈ H1
+0(Ω).
+(4.5)
+For any wh ∈ Vh, v ∈ H1
+0(Ω), by (3.1), Green formula, (4.3), Remark 2.1, Cauchy-Schwarz
+inequality and (4.5), we have
+B(u; wh, v − Phv)
+=
+(ay(u)wh, ∇(v − Phv)) + (az(u)∇wh, ∇(v − Phv))
++(fy(u)wh, v − Phv) + (fz(u)∇wh, v − Phv)
+=
+((∇ · ay(u))wh, v − Phv) + (ay(u) · ∇wh, v − Phv)
++(fy(u)wh, v − Phv) + (fz(u)∇wh, v − Phv)
+≲
+∥wh∥1∥v − Phv∥0
+≲
+h∥wh∥1∥v∥1.
+(4.6)
+Noticing that L′(u) : H1
+0(Ω) → H−1(Ω) is an isomorphism, using (4.6) and (4.4), we obtain
+that
+∥wh∥1
+≲
+sup
+v∈H1
+0 (Ω)
+B(u; wh, v)
+∥v∥1
+≲
+sup
+v∈H1
+0 (Ω)
+B(u; wh, v − Phv)
+∥v∥1
++ sup
+v∈H1
+0 Ω
+B(u; wh, Phv)
+∥v∥1
+≲
+h∥wh∥1 +
+sup
+v∈H1
+0 (Ω)
+B(u; wh, Phv)
+∥Phv∥1
+.
+Taking h sufficiently small in the above inequality with projection operator Ph being identity
+operator for Vh, we could obtain the first estimate of (4.2). The proof of the second estimate
+of (4.2) is similar.
+Next, we provide another BB condition of the form B(·; ·, ·).
+Lemma 4.2
+Assume u is the solution of (2.3) and Ψ satisfying ∥u − Ψ∥1,∞ ≲ H, then when
+H is small enough, for any wh ∈ Vh, it holds that
+∥wh∥1 ≲ sup
+vh∈Vh
+B (Ψ; wh, vh)
+∥vh∥1
+and
+∥wh∥1 ≲ sup
+vh∈Vh
+B (Ψ; vh, wh)
+∥vh∥1
+.
+(4.7)
+Proof.
+Using the definition (3.1) of form B, Taylor expansion h(y, z) = h(y0, z0)+∂yh(˜θ1, ˜θ2)(y−
+y0) + ∂zh(˜θ1, ˜θ2)(z − z0), where ˜θ1 is between y and y0 and ˜θ2 is between z and z0, Remark 2.1,
+and H¨older inequality, we obtain
+B(u; wh, vh) − B(Ψ; wh, vh)
+=
+((ay(u) − ay(Ψ))wh, ∇vh) + ((az(u) − az(Ψ))∇wh, ∇vh)
++((fy(u) − fy(Ψ))wh, vh) + ((fz(u) − fz(Ψ))∇wh, vh)
+=
+(ayy(θ1)(u − Ψ)wh, ∇vh) + (ayz(θ1)∇(u − Ψ)wh, ∇vh)
++(azy(θ2)(u − Ψ)∇wh, ∇vh) + (∇(u − Ψ)T azz(θ2)∇wh, ∇vh)
++(fyy(θ3)(u − Ψ)wh, vh) + (fyz(θ3) · ∇(u − Ψ)wh, vh)
++(fzy(θ4) · ∇wh(u − Ψ), vh) + (∇(u − Ψ)T fzz(θ4)∇wh, vh)
+≲
+∥u − Ψ∥1,∞∥wh∥1∥vh∥1,
+(4.8)
+
+An iterative two-grid method for strongly nonlinear elliptic boundary value problems
+7
+where θi (i = 1, 2, 3, 4) are between u and Ψ.
+By Lemma 4.1, (4.8) and ∥u − Ψ∥1,∞ ≲ H, it is obtained that
+∥wh∥1
+≲
+sup
+vh∈Vh
+B(u; wh, vh) − B (Ψ; wh, vh)
+∥vh∥1
++ sup
+vh∈Vh
+B (Ψ; wh, vh)
+∥vh∥1
+≲
+∥u − Ψ∥1,∞∥wh∥1 + sup
+vh∈Vh
+B (Ψ; wh, vh)
+∥vh∥1
+≲
+H∥wh∥1 + sup
+vh∈Vh
+B (Ψ; wh, vh)
+∥vh∥1
+.
+Taking H sufficiently small into the above inequality, we can derive the first estimate of (4.7).
+The proof of the second estimate of (4.7) is similar.
+Remark 4.3
+According to (2.6), Lemma 4.2 still holds with replacing Ψ by the finite element
+solution uh of (2.5).
+For more concise notations and the subsequent analysis, we denote
+Ek = uh − uk
+h,
+(4.9)
+uk,1
+h
+= uk
+h + ek
+H,
+(4.10)
+where uh is the solution of problem (2.5) and, uk
+h and ek
+H are given by Algorithm 3.2. It’s noted
+that these notation will be used frequently in the rest of this paper.
+Remark 4.4
+For k ⩾ 0, assume that Ek, uk,1
+h
+and ek
+H are given by (4.9), (4.10) and Algorithm
+3.2, respectively. If both ∥Ek∥1,∞ ≲ H and ∥ek
+H∥1,∞ ≲ H are provided, the Lemma 4.2 still
+holds with replacing Ψ by uk,1
+h . In fact, using (4.10), (4.9), triangle inequality, (2.6) with r ⩾ 1
+and h < H, ∥Ek∥1,∞ ≲ H and ∥ek
+H∥1,∞ ≲ H, we derive that
+∥u − uk,1
+h ∥1,∞ ⩽ ∥u − uh∥1,∞ + ∥Ek∥1,∞ + ∥ek
+H∥1,∞ ≲ H.
+Therefore, the Lemma 4.2 still holds with Ψ = uk,1
+h .
+And then, for the finite element solution uh of (2.5) and any fixed x ∈ Ω, we introduce the
+Green functions gx
+H ∈ VH, which be defined by
+B(uh; vH, gx
+H) = ∂vH(x),
+∀vH ∈ VH,
+(4.11)
+where ∂ denotes either
+∂
+∂x1 or
+∂
+∂x2 . It’s easy to see that the Green function gx
+H is well-defined
+by Remark 4.3.
+Assume uk,1
+h
+is given by (4.10), similarly, for any fixed x ∈ Ω, we introduce the Green
+functions gk,x
+h
+∈ Vh by
+B(uk,1
+h ; vh, gk,x
+h
+) = ∂vh(x),
+∀vh ∈ Vh.
+(4.12)
+By Remark 4.4, we also can see that Green function gk,x
+h
+is well-defined.
+Here give some estimates of the above two Green functions gx
+H and gk,x
+h
+(See Lemma 3.3 of
+[4], or (2.10) and (2.11) of [9])
+∥gx
+H∥1,1 ≲ | log H|
+and
+∥gk,x
+h
+∥1,1 ≲ | log h|.
+(4.13)
+
+8
+Jiajun Zhan & et al.
+At last, for any v ∈ H1
+0(Ω) ∩ W 1,∞(Ω), using (3.1.11) of [2], it could be obtained that
+∥v∥1,∞ ≲ |v|1,∞.
+(4.14)
+4.2
+Error estimate
+In this subsection, we present the convergence analysis of Algorithm 3.2 by a series of lemmas.
+Lemma 4.5
+Assume uk,1
+h , Ek and ek
+H are given by (4.10), (4.9) and Algorithm 3.2, respec-
+tively, then we have, for any vh ∈ Vh,
+B(uk,1
+h ; Ek+1, vh) ≲ (∥Ek∥1,∞ + ∥ek
+H∥1,∞)(∥Ek∥1 + ∥ek
+H∥1)∥vh∥1,
+(4.15)
+B(uk,1
+h ; Ek+1, vh) ≲ (∥Ek∥2
+1,∞ + ∥ek
+H∥2
+1,∞)∥vh∥1,1.
+(4.16)
+Proof.
+Using (4.9), Remark 3.1, (3.5), (2.5) and (2.2), it is obtained that
+B(uk,1
+h ; Ek+1, vh)
+=
+B(uk,1
+h ; uh, vh) − B(uk,1
+h ; uk+1
+h
+, vh)
+=
+B(uk,1
+h ; uh, vh) − B(uk,1
+h ; uk
+h + ek
+H, vh)
++A(uk,1
+h , vh) − A(uh, vh).
+=
+B(uk,1
+h ; Ek − ek
+H, vh) + (a(uk,1
+h , ∇uk,1
+h ), vh) + (f(uk,1
+h , ∇uk,1
+h ), vh)
+−(a(uh, ∇uh), vh) − (f(uh, ∇uh), vh)
+:=
+A1 − A2 − A3,
+(4.17)
+where
+A1
+=
+B(uk,1
+h ; Ek − ek
+H, vh),
+A2
+=
+(a(uh, ∇uh), vh) − (a(uk,1
+h , ∇uk,1
+h ), vh),
+A3
+=
+(f(uh, ∇uh), vh) − (f(uk,1
+h , ∇uk,1
+h ), vh).
+For A1, using the definition (3.1) of B, we have
+A1
+=
+(ay(uk,1
+h )(Ek − ek
+H), ∇vh) + (az(uk,1
+h )∇(Ek − ek
+H), ∇vh)
++(fy(uk,1
+h )(Ek − ek
+H), vh) + (fz(uk,1
+h )∇(Ek − ek
+H), vh).
+(4.18)
+For A2, using second order Taylor expansion, (4.10) and (4.9), we obtain
+A2
+=
+(ay(uk,1
+h )(Ek − ek
+H), ∇vh) + (az(uk,1
+h )∇(Ek − ek
+H), ∇vh)
++(ayy(θ5)(Ek − ek
+H)2, ∇vh) + 2(ayz(θ5)∇(Ek − ek
+H)(Ek − ek
+H), ∇vh)
++(∇(Ek − ek
+H)T azz(θ5)∇(Ek − ek
+H), ∇vh),
+(4.19)
+where θ5 is between uh and uk,1
+h .
+Similarly for A3, using second order Taylor expansion, (4.10) and (4.9), it is obtained that
+A3
+=
+(fy(uk,1
+h )(Ek − ek
+H), vh) + (fz(uk,1
+h )∇(Ek − ek
+H), vh)
++(fyy(θ6)(Ek − ek
+H)2, vh) + 2(fyz(θ6) · ∇(Ek − ek
+H)(Ek − ek
+H), vh)
++(∇(Ek − ek
+H)T fzz(θ6)∇(Ek − ek
+H), vh),
+(4.20)
+
+An iterative two-grid method for strongly nonlinear elliptic boundary value problems
+9
+where θ6 is between uh and uk,1
+h .
+Noticing the sum of the first order derivative items about a(·, ·, ·) and f(·, ·, ·) in (4.19) and
+(4.20) exactly equal A1. Substituting (4.18), (4.19) and (4.20) into (4.17), it’s obtained that
+B(uk,1
+h ; Ek+1, vh)
+=
+−(ayy(θ5)(Ek − ek
+H)2, ∇vh) − 2(ayz(θ5)∇(Ek − ek
+H)(Ek − ek
+H), ∇vh)
+−(∇(Ek − ek
+H)T azz(θ5)∇(Ek − ek
+H), ∇vh) − (fyy(θ6)(Ek − ek
+H)2, vh)
+−2(fyz(θ6) · ∇(Ek − ek
+H)(Ek − ek
+H), vh)
+−(∇(Ek − ek
+H)T fzz(θ6)∇(Ek − ek
+H), vh).
+(4.21)
+Applying Remark 2.1, H¨older inequality and triangle inequality into (4.21), we could obtain
+B(uk,1
+h ; Ek+1, vh)
+≲
+∥Ek − ek
+H∥1,∞∥Ek − ek
+H∥1∥vh∥1
+⩽
+(∥Ek∥1,∞ + ∥ek
+H∥1,∞)(∥Ek∥1 + ∥ek
+H∥1)∥vh∥1,
+which completes the proof of (4.15). Similarly, we could obtain (4.16) by (4.21).
+Lemma 4.6
+Assume that uk,1
+h , ek
+H and Ek are defined by (4.10), Algorithm 3.2 and (4.9),
+respectively, then we have
+B(uk,1
+h ; ek
+H, vH) ≲ (∥Ek∥1 + ∥Ek+1∥1)∥vH∥1,
+∀vH ∈ VH.
+(4.22)
+Proof.
+Taking vh = vH into (3.5) and using (3.4), we obtain
+B(uk,1
+h ; uk+1
+h
+, vH) = B(uk,1
+h ; uk
+h + ek
+H, vH).
+Rewriting the the above equation with Remark 3.1, and then using (4.9), (4.1) and triangle
+inequality, we have
+B(uk,1
+h ; ek
+H, vH)
+=
+B(uk,1
+h ; uk+1
+h
+− uk
+h, vH)
+=
+B(uk,1
+h ; uk+1
+h
+− uh + uh − uk
+h, vH)
+=
+B(uk,1
+h ; Ek − Ek+1, vH)
+≲
+∥Ek − Ek+1∥1∥vH∥1
+⩽
+(∥Ek∥1 + ∥Ek+1∥1) ∥vH∥1,
+which completes the proof.
+Lemma 4.7
+Assume that Ek and ek
+H are given by (4.9) and Algorithm 3.2, respectively, and
+r ⩾ 1, when h is small enough, then for any integer k ⩾ 1,
+∥Ek∥1 ≲ Hr+k,
+∥Ek∥1,∞ ≲ | log h|H2,
+∥ek
+H∥1,∞ ≲ H,
+∥ek
+H∥1 ≲ Hr+k.
+(4.23)
+Proof.
+Here we use mathematical induction to prove that (4.23) is true.
+By (3.4), u0
+h = uH, (2.5) and the uniqueness of finite element solution (See Lemma 2.2), it
+could be seen that e0
+H = 0.
+
+10
+Jiajun Zhan & et al.
+Making use of triangle inequality, (2.6) and h ⩽ H, we have
+∥E0∥1 ⩽ ∥u − uh∥1 + ∥u − uH∥1 ≲ hr + Hr ⩽ Hr,
+(4.24)
+∥E0∥1,∞ ⩽ ∥u − uh∥1,∞ + ∥u − uH∥1,∞ ≲ hr + Hr ⩽ Hr.
+(4.25)
+Next, we will prove (4.23) is true when k = 1.
+(i) For ∥E1∥1 ≲ Hr+1.
+Noticing that r ⩾ 1 and e0
+H = 0, and using (4.25), we have
+∥E0∥1,∞ ≲ H and ∥e0
+H∥1,∞ ≲ H , which could derive the BB condition of form B(u0,1
+h ; ·, ·) (See
+Remark 4.4). Using the BB condition of form B(u0,1
+h ; ·, ·), (4.15), (4.25), ek
+H = 0, (4.24), r ⩾ 1
+and H < 1, it’s obtained that
+∥E1∥1
+≲
+sup
+vh∈Vh
+B(u0,1
+h ; E1, vh)
+∥vh∥1
+≲
+(∥E0∥1,∞ + ∥e0
+H∥1,∞)(∥E0∥1 + ∥e0
+H∥1)
+≲
+(Hr + 0)(Hr + 0)
+≲
+Hr+1.
+(4.26)
+(ii) For ∥E1∥1,∞ ≲ | log h|H2. For k = 0 and any fixed x ∈ Ω, taking vh = E1 into (4.12),
+using (4.16), (4.25), e0
+H = 0, (4.13), r ⩾ 1 and H < 1, we obtain
+∂E1(x)
+=
+B(u0,1
+h ; E1, g0,x
+h )
+≲
+(∥E0∥2
+1,∞ + ∥e0
+H∥2
+1,∞)∥g0,x
+h ∥1,1
+≲
+(H2r + 0)| log h|
+≲
+| log h|H2.
+Further using the arbitrariness of x and (4.14), we derive that
+∥E1∥1,∞ ≲ | log h|H2.
+(iii) For ∥e1
+H∥1,∞ ≲ H. Using ∥E1∥1,∞ ≲ | log h|H2 and Lemma A.1 (The specific content
+of lemma and proof are referred to Appendix), we obtain
+∥e1
+H∥1,∞ ≲ H.
+(iv) For ∥e1
+H∥1 ≲ Hr+1. Noticing that ∥E1∥1,∞ ≲ | log h|H2 and ∥e1
+H∥1,∞ ≲ H are satisfied,
+therefore the BB condition of form B(u1,1
+h ; ·, ·) holds (See Remark 4.4). Using the BB condition
+of B(u1,1
+h ; ·, ·) and (4.15), it’s obtained that
+∥E2∥1
+≲
+sup
+vh
+B(u1,1
+h ; E2, vh)
+∥vh∥1
+≲
+(∥E1∥1,∞ + ∥e1
+H∥1,∞)(∥E1∥1 + ∥e1
+H∥1).
+(4.27)
+Using the BB condition of form B(u1,1
+h ; ·, ·), (4.22) with k = 1, (4.27), ∥E1∥1,∞ ≲ | log h|H2
+and ∥e1
+H∥1,∞ ≲ H, we have
+∥e1
+H∥1
+≲
+sup
+vH∈VH
+B(u1,1
+h ; e1
+H, vH)
+∥vH∥1
+≲
+∥E1∥1 + ∥E2∥1
+≲
+∥E1∥1 + (∥E1∥1,∞ + ∥e1
+H∥1,∞)(∥E1∥1 + ∥e1
+H∥1)
+≲
+∥E1∥1 + (| log h|H2 + H)(∥E1∥1 + ∥e1
+H∥1).
+
+An iterative two-grid method for strongly nonlinear elliptic boundary value problems
+11
+Taking H be small enough in the above inequality, and using (4.26), it’s obtained that
+∥e1
+H∥1 ≲ ∥E1∥1 ≲ Hr+1.
+We assume (4.23) is true when k = l, i.e.,
+∥El∥1 ≲ Hr+l,
+∥El∥1,∞ ≲ | log h|H2,
+∥el
+H∥1,∞ ≲ H,
+∥el
+H∥1 ≲ Hr+l.
+(4.28)
+Next, we will prove (4.23) also holding when k = l + 1.
+(i) For ∥El+1∥1 ≲ Hr+l+1.
+Noticing that ∥El∥1,∞ ≲ | log h|H2 and ∥el
+H∥1,∞ ≲ H are
+satisfied, therefore the BB condition of form B(ul,1
+h ; ·, ·) holds (See Remark 4.4). Using the BB
+condition of form B(ul,1
+h ; ·, ·), (4.15), (4.28) and H < 1, we obtain
+∥El+1∥1
+≲
+sup
+vh∈Vh
+B(ul,1
+h ; El+1, vh)
+∥vh∥1
+≲
+(∥El∥1,∞ + ∥el
+H∥1,∞)(∥El∥1 + ∥el
+H∥1)
+≲
+(| log h|H2 + H)(Hr+l + Hr+l)
+≲
+Hr+l+1.
+(4.29)
+(ii) For ∥El+1∥1,∞ ≲ | log h|H2. Taking vh = El+1 into (4.12) with k = l, using (4.16),
+(4.28) and (4.13), we obtain
+∂El+1(x)
+=
+B(ul,1
+h ; El+1, gl,x
+h )
+≲
+(∥El∥2
+1,∞ + ∥el
+H∥2
+1,∞)∥gl,x
+h ∥1,1
+≲
+(| log h|2H4 + H2)| log h|
+≲
+| log h|H2,
+which combining the arbitrariness of x and (4.14), it could be derived that
+∥El+1∥1,∞ ≲ | log h|H2.
+(iii) For ∥el+1
+H ∥1,∞ ≲ H. Using ∥El+1∥1,∞ ≲ | log h|H2 and Lemma A.1, we obtain
+∥el+1
+H ∥1,∞ ≲ H.
+(4.30)
+(iv) For ∥el+1
+H ∥1 ≲ Hr+l+1. Noticing that ∥El+1∥1,∞ ≲ | log h|H2 and ∥el+1
+H ∥1,∞ ≲ H are
+satisfied, therefore the BB condition of form B(ul+1,1
+h
+; ·, ·) holds (See Remark 4.4). Using the
+BB condition of form B(ul+1,1
+h
+; ·, ·), (4.15), it’s obtained that
+∥El+2∥1
+≲
+sup
+vh
+B(ul+1,1
+h
+; El+2, vh)
+∥vh∥1
+≲
+(∥El+1∥1,∞ + ∥el+1
+H ∥1,∞)(∥El+1∥1 + ∥el+1
+H ∥1).
+(4.31)
+Using the BB condition of form B(ul+1,1
+h
+; ·, ·), (4.22) with k = l + 1, (4.31), ∥El+1∥1,∞ ≲
+| log h|H2 and ∥el+1
+H ∥1,∞ ≲ H, we have
+∥el+1
+H ∥1
+≲
+sup
+vH∈VH
+B(ul+1,1
+h
+; el+1
+H , vH)
+∥vH∥1
+≲
+∥El+1∥1 + ∥El+2∥1
+≲
+∥El+1∥1 + (∥El+1∥1,∞ + ∥el+1
+H ∥1,∞)(∥El+1∥1 + ∥el+1
+H ∥1)
+≲
+∥El+1∥1 + (| log h|H2 + H)(∥El+1∥1 + ∥el+1
+H ∥1).
+
+12
+Jiajun Zhan & et al.
+Taking H be small enough in the above inequality and using (4.29), it’s obtained that
+∥el+1
+H ∥1 ≲ ∥El+1∥1 ≲ Hr+l+1.
+By mathematical induction, the conclusion is obtained.
+Remark 4.8
+Although we just use the estimation ∥Ek+1∥1 ≲ Hr+k+1 in our main result
+(See Theorem 4.9), the availability of ∥Ek+1∥1 ≲ Hr+k+1 requires the support of ∥Ek∥1,∞ ≲
+| log h|H2, ∥ek
+H∥1,∞ ≲ H and ∥ek
+H∥1 ≲ Hr+k.
+Here gives the main result of this paper.
+Theorem 4.9
+Assume that u is the solution of (2.3) and uk
+h is given by Algorithm 3.2, then
+we have
+∥u − uk
+h∥1 ≲ hr + Hr+k.
+(4.32)
+Proof.
+Using triangle inequality, (4.9), (2.6) and Lemma 4.7, we could obtain that
+∥u − uk
+h∥1 ⩽ ∥u − uh∥1 + ∥Ek∥1 ≲ hr + Hr+k,
+which completes the proof.
+5
+Numerical experiments
+In this section, we present some numerical experiments to show the efficiency of the pro-
+posed iterative two-grid algorithm. We implemented these experiments by the software package
+FEALPy of programming language Python [5]. Specially in the Step 1 of Algorithm 3.2, we
+solve the nonlinear systems by Newton iteration methods with relative residual 10−8.
+We adopt the following mean curvature flow problem as our model problem:
+−∇ ·
+�
+∇u
+(1 + |∇u|2)1/2
+�
+= g in Ω,
+u = 0 on ∂Ω,
+where the computational domain Ω = (0, 1)× (0, 1), the exact solution u = x(1 − x)2y(1 − y)ex,
+and g is so chosen according to the exact solution.
+Firstly, we choose conforming piecewise linear finite element space as Vh, namely choose
+r = 1. According to Theorem 4.9, we should keep hr = Hr+k hold in order to achieve the
+optimal convergence order. Therefore in Table 1, we present some numerical results in different
+mesh size with h = H2 for k = 1. In this case, our algorithm is same with Algorithm 3.1 (See
+Remark 3.3). Furthermore, we could observe that ∥u−u1
+h∥1 ∗max{H2, h}−1 are stable in Table
+1, which agrees with (4.32) in Theorem 4.9.
+Table 1: k = 1, r = 1
+H
+h
+∥u − u1
+h∥1
+∥u − u1
+h∥1 ∗ max{H2, h}−1
+1/9
+1/81
+2.74E-03
+0.221953
+1/10
+1/100
+2.22E-03
+0.221967
+1/11
+1/121
+1.83E-03
+0.221977
+1/12
+1/144
+1.54E-03
+0.221983
+
+An iterative two-grid method for strongly nonlinear elliptic boundary value problems
+13
+And then, we increase the iterative counts k to expand the distance between H and h, which
+is shown in Tables 2 and 3. We also observe that ∥u − u1
+h∥1 ∗ max{H1+k, h}−1 are stable.
+Table 2: k = 2, r = 1
+H
+h
+∥u − u2
+h∥1
+∥u − u2
+h∥1 ∗ max{H3, h}−1
+1/3
+1/27
+8.20E-03
+0.221524
+1/4
+1/64
+3.47E-03
+0.221784
+1/5
+1/125
+1.77E-03
+0.221826
+1/6
+1/216
+1.03E-03
+0.221836
+Table 3: k = 3, r = 1
+H
+h
+∥u − u3
+h∥1
+∥u − u3
+h∥1 ∗ max{H4, h}−1
+1/2
+1/16
+1.38E-02
+0.220944
+1/3
+1/81
+2.74E-03
+0.221805
+1/4
+1/256
+8.67E-04
+0.221837
+At last, we implement similar numerical experiments for high order finite element space
+with r = 2 and r = 3 in Tables 4-9. By observation, all these numerical experiments are in
+support of (4.32) in Theorem 4.9.
+Table 4: k = 1, r = 2
+H
+h
+∥u − u1
+h∥1
+∥u − u1
+h∥1 ∗ max{H3, h2}−1
+1/4
+1/8
+2.57E-03
+0.164578
+1/9
+1/27
+2.30E-04
+0.167320
+1/16
+1/64
+4.09E-05
+0.167553
+1/25
+1/125
+1.07E-05
+0.167590
+1/36
+1/216
+3.59E-06
+0.167599
+Acknowledgements
+The work of the first and second authors were partially funded by the
+Science and Technology Development Fund, Macau SAR (Nos. 0070/2019/A2, 0031/2022/A1).
+The third author was supported by the National Natural Science Foundation of China (Grant
+No.
+11901212). The third and fourth authors are also supported by the National Natural
+Science Foundation of China (Grant No. 12071160).
+References
+[1]
+Bi C J, Wang C, Lin Y P. A posteriori error estimates of two-grid finite element methods for nonlinear
+elliptic problems. J Sci Comput, 2018, 74: 23–48
+[2]
+Ciarlet P G. The Finite Element Method for Elliptic Problems. Classics in Applied Mathematics, No. 40.
+SIAM, Philadelphia, 2002
+[3]
+Gudi T, Nataraj N, Pani A. hp-discontinuous Galerkin methods for strongly nonlinear elliptic boundary
+value problems. Numer Math, 2008, 109: 233–268
+[4]
+Thom´ee V, Xu J C, Zhang N Y. Superconvergence of the gradient in piecewise linear finite-element approx-
+imation to a parabolic problem. SIAM J Numer Anal, 1989, 26: 553–573
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+Wei H Y, Huang Y Q. Fealpy: Finite element analysis library in python. https://github.com/weihuayi/
+fealpy, Xiangtan University, 2017-2021
+[6]
+Xu J C. Iterative methods by SPD and small subspace solvers for nonsymmetric or indefinite problems. In:
+Proceedings of the 5th International Symposium on Domain Decomposition Methods for Partial Differential
+
+14
+Jiajun Zhan & et al.
+Table 5: k = 2, r = 2
+H
+h
+∥u − u2
+h∥1
+∥u − u2
+h∥1 ∗ max{H4, h2}−1
+1/8
+1/64
+4.09E-05
+0.167552
+1/9
+1/81
+2.55E-05
+0.167571
+1/10
+1/100
+1.68E-05
+0.167582
+1/11
+1/121
+1.14E-05
+0.167589
+1/12
+1/144
+8.08E-06
+0.167593
+Table 6: k = 3, r = 2
+H
+h
+∥u − u3
+h∥1
+∥u − u3
+h∥1 ∗ max{H5, h2}−1
+1/5
+1/55
+5.54E-05
+0.167534
+1/6
+1/90
+2.07E-05
+0.160874
+1/7
+1/126
+1.06E-05
+0.167590
+1/8
+1/184
+4.95E-06
+0.162211
+1/9
+1/243
+2.84E-06
+0.167600
+Table 7: k = 1, r = 3
+H
+h
+∥u − u1
+h∥1
+∥u − u1
+h∥1 ∗ max{H4, h3}−1
+1/8
+1/16
+1.83E-05
+0.075054
+1/27
+1/81
+1.40E-07
+0.074662
+1/64
+1/256
+4.44E-09
+0.074543
+Table 8: k = 2, r = 3
+H
+h
+∥u − u2
+h∥1
+∥u − u2
+h∥1 ∗ max{H5, h3}−1
+1/8
+1/32
+2.28E-06
+0.074870
+1/9
+1/36
+1.60E-06
+0.074838
+1/10
+1/40
+1.17E-06
+0.074810
+1/11
+1/55
+4.49E-07
+0.072343
+1/12
+1/60
+3.46E-07
+0.074716
+Table 9: k = 3, r = 3
+H
+h
+∥u − u3
+h∥1
+∥u − u3
+h∥1 ∗ max{H6, h3}−1
+1/8
+1/64
+2.85E-07
+0.074703
+1/9
+1/81
+1.40E-07
+0.074662
+1/10
+1/100
+7.46E-08
+0.074630
+1/11
+1/121
+4.21E-08
+0.074606
+1/12
+1/144
+2.50E-08
+0.074588
+
+An iterative two-grid method for strongly nonlinear elliptic boundary value problems
+15
+Equations. Siam, Philadelphia, 1992, 106–118
+[7]
+Xu J C. A novel two-grid method for semilinear elliptic equations. SIAM J Sci Comput, 1994, 15: 231–237
+[8]
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+neering (Quarteroni, Alfio and P´eriaux, Jacques and Kuznetsov, Yuri A and Widlund, Olof B eds). Contemp
+Math, vol. 157, Amer Math Soc, 1994, 79–87
+[9]
+Xu J C. Two-grid discretization techniques for linear and nonlinear PDEs. SIAM J Numer Anal, 1996, 33:
+1759–1777
+[10]
+Xu J C, Cai X C. A preconditioned GMRES method for nonsymmetric or indefinite problems. Math Comp,
+1992, 59: 311–319
+[11]
+Zhang W F, Fan R H, Zhong L Q. Iterative two-grid methods for semilinear elliptic equations. Comput
+Math Appl, 2020, 80: 522–530
+Appendix A
+The purpose of this appendix is to provide the proof of Lemma A.1.
+Lemma A.1
+Assume ek
+H is given in (3.4) and ∥Ek∥1,∞ ≲ | log h|H2, when H is small enough, it holds that
+∥ek
+H∥1,∞ ≲ H.
+(A.1)
+Before we present the proof of Lemma A.1, we need to introduce some preliminaries and lemmas.
+For the finite element solution uh of (2.5), we introduce a projection operator ˆPH : H1
+0(Ω) → VH, which be
+defined by,
+B(uh; ˆPHw, vH) = B(uh; w, vH),
+∀w ∈ H1
+0(Ω), vH ∈ VH.
+(A.2)
+It’s easy to derive that ˆPH is well-defined by the BB-conditions of form B(uh; ·, ·) which could be obtained by
+Remark 4.3. Furthermore, the projection operator ˆPH satisfies the following estimate
+∥ ˆPHw∥1,∞ ≲ | log H|∥w∥1,∞,
+∀w ∈ W 1,∞(Ω).
+(A.3)
+In fact, taking vH = ˆPHw in (4.11), and using (A.2), (3.1), Remark 2.1, H¨older inequality and (4.13), we obtain
+∂ ˆPHw(x)
+=
+B(uh; ˆPHw, gx
+H)
+=
+B(uh; w, gx
+H)
+=
+(ay(uh)w, ∇gx
+H) + (az(uh)∇w, ∇gx
+H) + (fy(uh)w, gx
+H) + (fz(uh)∇w, gx
+H)
+≲
+∥w∥1,∞∥gx
+H∥1,1
+≲
+| log H|∥w∥1,∞.
+Finally using of the arbitrariness of x and (4.14), we could obtain (A.3).
+By Taylor expansion, we have (the detailed proof can be found in Lemma 3.1 of [9])
+A(w, χ) = A(v, χ) + B(v; w − v, χ) + R(η; v, w, χ),
+∀w, v, χ ∈ H1
+0(Ω),
+(A.4)
+where the forms A(·, ·) and B(·; ·, ·) are given by (2.2) and (3.1), respectively, η = v + t(w − v) and
+R(η; v, w, χ)
+=
+� 1
+0
+�
+(ayy(η)(v − w)2, ∇χ) + 2(ayz(η)∇(v − w)(v − w), ∇χ)
++(∇(v − w)T azz(η)∇(v − w), ∇χ) + (fyy(η)(v − w)2, χ)
++2(fyz(η) · ∇(v − w)(v − w), χ) + (∇(v − w)T fzz(η)∇(v − w), χ)
+�
+(1 − t)dt.
+For the proof of Lemma A.1, we introduce a operator Φ as follow. Assume uh is the solution of (2.5), Ek,
+R, uk
+h are given in (4.9), (A.4) and Algorithm 3.2, respectively, we defined operator Φ : VH → VH by, for any
+wH ∈ VH,
+B(uh; Φ(wH), vH) = B(uh; Ek, vH) − R(uh + t(wH − Ek); uh, uk
+h + wH, vH),
+∀vH ∈ VH.
+(A.5)
+By the BB-conditions of form B(uh; ·, ·) (See Remark 4.3), it’s easy to prove that operator Φ is well-defined.
+We define a space
+QH = {vH ∈ VH : ∥vH − ˆPHEk∥1,∞ ⩽ H},
+(A.6)
+where ˆPH is a projection operator defined by (A.2). Since QH is a finite dimensional space, it’s easy to see that
+QH is a non-empty compact convex subset.
+Next, we will use Brouwer fixed point theorem to prove that (A.5) has a fixed point ¯wH in QH.
+Lemma A.2
+Assume ∥Ek∥1,∞ ≲ | log h|H2, then when H is small enough, we have Φ(QH) ⊂ QH.
+
+16
+Jiajun Zhan & et al.
+Proof.
+For any wH ∈ QH, vH ∈ VH, rewriting (A.5) with (A.2), we have
+B(uh; Φ(wH) − ˆPHEk, vH) = −R(uh + t(wH − Ek); uh, uk
+h + wH, vH).
+(A.7)
+Substituting vH = Φ(wH) − ˆPHEk into (4.11) and using (A.7), Remark 2.1, H¨older inequality, (4.9), triangle
+inequality, (4.13), (A.3), (A.6) and ∥Ek∥1,∞ ≲ | log h|H2, it is obtained that
+∂(Φ(wH) − ˆPHEk)(x)
+=
+B(uh; Φ(wH) − ˆPHEk, gx
+H)
+=
+−R(uh + t(wH − Ek); uh, uk
+h + wH, gx
+H)
+≲
+∥Ek − wH∥2
+1,∞∥gx
+H∥1,1
+≲
+(∥Ek − ˆPHEk∥2
+1,∞ + ∥ ˆPHEk − wH∥2
+1,∞)| log H|
+≲
+((1 + | log H|)2∥Ek∥2
+1,∞ + H2)| log H|
+≲
+((1 + | log H|)2| log h|2H4 + H2)| log H|.
+Further using the arbitrariness of x and (4.14), the proof is finished.
+Lemma A.3
+Assume ∥Ek∥1,∞ ≲ | log h|H2, then the operator Φ is continuous in VH.
+Proof.
+For any w1, w2 ∈ QH, by (A.5), we have
+B(uh; Φ(w1) − Φ(w2), vH)
+=
+R(uh + t(w2 − Ek); uh, uk
+h + w2, vH)
+−R(uh + t(w1 − Ek); uh, uk
+h + w1, vH).
+(A.8)
+Noticing that the definition of R in (A.4), for the terms concerning ayy on the right hand side of (A.8), we can
+use Remark 2.1 and H¨older inequality to obtain that
+(ayy(uh + t(w2 − Ek))(Ek − w2)2, ∇vH) − (ayy(uh + t(w1 − Ek))(Ek − w1)2, ∇vH)
+=
+(ayy(uh + t(w2 − Ek))(Ek − w2)2, ∇vH)
+−(ayy(uh + t(w1 − Ek))(Ek − w2)2, ∇vH)
++(ayy(uh + t(w1 − Ek))(Ek − w2)2, ∇vH)
+−(ayy(uh + t(w1 − Ek))(Ek − w1)2, ∇vH)
+=
+([ayy(uh + t(w2 − Ek)) − ayy(uh + t(w1 − Ek))] (Ek − w2)2, ∇vH)
++(ayy(uh + t(w1 − Ek))
+�
+−2Ekw2 + w2
+2 + 2Ekw1 − w2
+1
+�
+, ∇vH)
+=
+([ayy(uh + t(w2 − Ek)) − ayy(uh + t(w1 − Ek))] (Ek − w2)2, ∇vH)
++(ayy(uh + t(w1 − Ek)) (2Ek − w1 − w2) (w1 − w2), ∇vH)
+≲
+∥ayy(uh + t(w2 − Ek)) − ayy(uh + t(w1 − Ek))∥0,∞∥(Ek − w2)2∥0∥vH∥1
++∥2Ek − w1 − w2∥0∥w1 − w2∥0,∞∥vH∥1.
+(A.9)
+For ∥(Ek − w2)2∥0, we use triangle inequality, (A.3), (A.6) and ∥Ek∥1,∞ ≲ | log h|H2, it’s obtained that
+∥(Ek − w2)2∥0
+⩽
+∥Ek − w2∥2
+1,∞
+≲
+∥Ek∥2
+1,∞ + ∥ ˆPHEk∥2
+1,∞ + ∥ ˆPHEk − w2∥2
+1,∞
+≲
+∥Ek∥2
+1,∞ + | log H|2∥Ek∥2
+1,∞ + H2
+≲
+| log h|2H4 + | log H|2| log h|2H4 + H2
+:=
+C1(H),
+(A.10)
+where C1(H) is a constant depending on H.
+Similarly, for ∥2Ek − w1 − w2∥0, there also exists a constant C2(H) such that
+∥2Ek − w1 − w2∥0 ≲ C2(H).
+(A.11)
+Substituting (A.10) and (A.11) into (A.9), it’s could be obtained that
+(ayy(uh + t(w2 − Ek))(Ek − w2)2, ∇vH) − (ayy(uh + t(w1 − Ek))(Ek − w1)2, ∇vH)
+≲
+C(H)
+�
+∥ayy(uh + t(w2 − Ek)) − ayy(uh + t(w1 − Ek))∥0,∞
++∥w1 − w2∥0,∞
+�
+∥vH∥1,
+where C(H) = max{C1(H), C2(H)} .
+The rest of the items on the right hand side of (A.8) have similar
+results, and here is omitted.
+The conclusion follows from the above discussion, (A.8), the BB-conditions of
+
+An iterative two-grid method for strongly nonlinear elliptic boundary value problems
+17
+form B(uh; ·, ·) (See Remark 4.3) and the continuity of second order derivatives of a(·, ·, ·) and f(·, ·, ·) (See the
+assumptions about a(·, ·, ·) and f(·, ·, ·) in Section 2).
+At last, we present the proof of Lemma A.1 by Brouwer fixed point theorem.
+Proof of Lemma A.1.
+Making use of Lemmas A.2 and A.3 and Brouwer fixed point theorem, we know that
+(A.5) exists a fixed point ¯wH in QH.
+Taking w = uk
+h + ¯wH, v = uh and χ = vH into (A.4), and then using (2.5) with VH ⊂ Vh, Remark 3.1,
+(4.9), ¯wH = Φ( ¯wH) and (A.5), we obtain that
+A(uk
+h + ¯wH, vH)
+=
+A(uh, vH) + B(uh; uk
+h + ¯wH − uh, vH) + R(η; uh, uk
+h + ¯wH, vH)
+=
+B(uh; ¯wH, vH) − B(uh; Ek, vH) + R(η; uh, uk
+h + ¯wH, vH)
+=
+B(uh; Φ( ¯wH), vH) − B(uh; Ek, vH) + R(η; uh, uk
+h + ¯wH, vH)
+=
+0,
+(A.12)
+where η = uh + t( ¯wH − Ek). By the uniqueness of finite element solution (See Lemma 2.2), (3.4) and (A.12),
+we can see that ¯wH = ek
+H, which implies ek
+H ∈ QH.
+At last, using triangle inequality, (A.6), (A.3) and ∥Ek∥1,∞ ≲ | log h|H2, we obtain
+∥ek
+H∥1,∞
+⩽
+∥ek
+H − ˆPHEk∥1,∞ + ∥ ˆPHEk∥1,∞
+≲
+H + | log H|∥Ek∥1,∞
+≲
+H + | log H|| log h|H2,
+which completes the proof.
+

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+Magnetization dynamics with time-dependent spin-density functional theory:
+significance of exchange-correlation torques
+Daniel Hill, Justin Shotton, and Carsten A. Ullrich∗
+Department of Physics and Astronomy, University of Missouri, Columbia, Missouri 65211, USA
+(Dated: January 5, 2023)
+In spin-density-functional theory (SDFT) for noncollinear magnetic materials, the Kohn-Sham
+system features exchange-correlation (xc) scalar potentials and magnetic fields. The significance of
+the xc magnetic fields is not very well explored; in particular, they can give rise to local torques
+on the magnetization, which are absent in standard local and semilocal approximations.
+Exact
+benchmark solutions for a five-site extended Hubbard lattice at half filling and in the presence of
+spin-orbit coupling are compared with SDFT results obtained using orbital-dependent exchange-
+only approximations. The magnetization dynamics following short-pulse excitations is found to be
+reasonably well described in the exchange-only approximation for weak to moderate interactions.
+For stronger interactions and near transitions between magnetically ordered and frustrated phases,
+exchange and correlation torques tend to compensate each other and must both be accounted for.
+I.
+INTRODUCTION
+Spin dynamics in magnetic systems is a research area
+of much current activity. Spintronics [1], which is con-
+cerned with the manipulation of electronic spins, spin
+currents, spin textures, and spin excitations, has created
+a wealth of scientific knowledge and many avenues for
+new technologies.
+Prominent examples are spin waves
+for encoding and transmitting information (magnonics)
+[2, 3], skyrmions for magnetic information storage [4–
+8], and single-spin qubits for quantum computation [9].
+Another related area of much interest is ultrafast demag-
+netization induced by femtosecond laser pulses [10–14].
+Computational approaches to simulate magnetization
+dynamics in a wide variety of systems are typically based
+on the Landau-Lifshitz-Gilbert (LLG) equation of motion
+[15, 16]. The LLG equation provides a classical descrip-
+tion of the time evolution of the magnetization vector
+m(t) in response to a time-dependent perturbation (typ-
+ically, a short pulse or a periodic driving field) or evolving
+from a nonequilibrium initial state.
+Materials proper-
+ties such as anisotropy, deformations, strain, and various
+forms of damping can be built into the LLG approach
+via phenomenological or “second-principles” parameters.
+In this paper, we are less concerned with these spe-
+cific materials properties; instead of LLG we will use a
+fully quantum mechanical description of the electronic
+charge and spin degrees of freedom, and our focus will
+be specifically on the impact of electron-electron interac-
+tions on the magnetization dynamics. To be more clear,
+we consider a system of N interacting electrons under
+the influence of a time-dependent scalar potential V (r, t)
+and a time-dependent magnetic field B(r, t) which cou-
+ples only to the electron spin (and not to orbital motion).
+∗ ullrichc@missouri.edu
+The associated many-body Hamiltonian is given by
+ˆH =
+N
+�
+j
+�
+−∇2
+j
+2 + V (rj, t) + σj · B(rj, t)
+�
++ 1
+2
+N
+�
+j̸=k
+1
+|rj − rk| ,
+(1)
+where σj is the vector of Pauli matrices acting on the
+spin of the jth electron, and we define the magnetic field
+strength such that the Bohr magneton, µB = e¯h/2m,
+does not explicitly appear in the Hamiltonian ˆH.
+We
+use atomic units (e = m = ¯h = 4πϵ0 = 1) throughout.
+From the Heisenberg equation of motion for ˆH, Capelle
+et al. showed that the magnetization has the following
+time evolution [17]:
+dm(r, t)
+dt
++ ˆ∇ · J(r, t) = m(r, t) × B(r, t) ,
+(2)
+where J(r, t) is the spin-current tensor. Equation (2) is
+exact but not very helpful in practice since J(r, t) re-
+quires the many-body wave function associated with ˆH.
+A more practical (but still in principle exact) alternative
+is time-dependent spin-density functional theory (TD-
+SDFT). The idea of TD-SDFT is to consider an auxiliary
+system of noninteracting fermions, acted upon by an “ef-
+fective” scalar potential and magnetic field, Veff(r, t) and
+Beff(r, t), such that the same density n(r, t) and magne-
+tization m(r, t) are produced as in the physical system.
+The resulting equation of motion, the TD-SDFT coun-
+terpart to Eq. (2), is [17]
+dm(r, t)
+dt
++ ˆ∇ · JKS(r, t) = m(r, t) × Beff(r, t) .
+(3)
+Here, JKS(r, t) is the Kohn-Sham spin-current tensor,
+which is easily determined from the noninteracting wave
+function, and the effective magnetic field is defined as
+Beff(r, t) = B(r, t) + Bxc(r, t), where the exchange-
+correlation (xc) magnetic field Bxc is a functional of the
+arXiv:2301.01509v1  [cond-mat.str-el]  4 Jan 2023
+
+2
+density and magnetization. Formally, m(r, t) is the same
+in Eqs. (2) and (3), but J and JKS are in general dif-
+ferent (the difference lies in the transverse component).
+Thus, the so-called xc torque,
+τxc(r, t) = m(r, t) × Bxc(r, t) ,
+(4)
+ensures that TD-SDFT produces the correct magnetiza-
+tion dynamics [17].
+While all of this is clear at the formal level, the ex-
+act form of Bxc is unknown and must be approximated
+in practice. This immediately raises several questions:
+which approximations of Bxc are available, and do they
+produce xc torques?
+And, how important are the xc
+torques for the magnetization dynamics?
+A number of approximations for Bxc have been derived
+within ground-state SDFT for noncollinear magnetism
+[18–20]; via the adiabatic approximation, they immedi-
+ately carry over to TD-SDFT. The most widely used ap-
+proach, pioneered by K¨ubler et al. [21, 22] and imple-
+mented in many popular electronic structure codes, is
+to use standard local or semilocal xc functionals such as
+the local spin-density approximation (LSDA) or general-
+ized gradient approximations (GGAs), and assume a lo-
+cal spin quantization axis which is aligned with the local
+magnetization vector m(r, t); this produces a Bxc(r, t)
+that is parallel to m(r, t) everywhere. We see right away
+from Eq. (4) that this class of approximations does not
+produce any xc torques.
+Approximations for Bxc that do include xc torque ef-
+fects can be constructed in several ways.
+Existing lo-
+cal and semilocal functionals (LSDA and GGAs) have
+been modified [23–26] or used in a source-free construc-
+tion [27], and new gradient-corrected functionals were
+constructed based using the spin-spiral state of the elec-
+tron gas as reference system [28–30].
+More consistent
+derivations of xc meta-GGAs, starting from noncollinear
+generalizations of the exchange hole and the two-body
+density matrix, were recently presented [31, 32]. Vari-
+ous orbital-dependent functionals were generalized to the
+case of noncollinear magnetization [33–35].
+Existing applications of ground-state SDFT to non-
+collinear magnetic materials [25, 26, 33] and model sys-
+tems [36] seem to suggest that xc torques are of relatively
+minor importance for magnetic structure and energetics,
+although the torques themselves may not be insignificant
+[32]. On the other hand, there are good reasons to expect
+that xc torques will be more impactful for magnetization
+dynamics: they explicitly appear in the equation of mo-
+tion, Eq. (3), and even if τxc(r, t) is relatively small at
+a given r and t, its effect can accumulate over time. So
+far, however, there has been no systematic attempt to
+assess this hypothesis. We are only aware of one study
+in the literature, where Dewhurst et al. [37] used their
+source-free Bxc functional to simulate laser-induced spin
+dynamics in bulk Co and Ni and Co-Pt and Ni-Pt inter-
+faces. They found that xc torques were significant only
+if they are not overshadowed by magnetic anisotropy ef-
+fects (i.e., in bulk, and not at interfaces), and that they
+FIG. 1.
+Geometry of the 5-site Hubbard cluster used in
+this work. The arrows indicate the ordering of the nearest-
+neighbor sum in Eq. (6), accounting for the directional hop-
+ping due to SOC.
+give rise to rather slow spin rotation compared to other
+forms of spin dynamics, induced optically or via spin-
+orbit coupling (SOC).
+In this paper, our goal is to assess the importance of
+xc torques in frustrated magnetic systems.
+Exchange-
+frustrated solids such as spin glasses and kagome antifer-
+romagnetic lattices are characterized by many competing
+noncollinear spin configurations and quantum spin liquid
+phases [38–40], and may therefore exhibit an enhanced
+sensitivity to subtle xc torque effects. Needless to say, ex-
+tended spin frustrated solids are challenging to describe,
+and exact or quasi-exact benchmark results are hard to
+come by. We will therefore limit ourselves to small model
+systems which capture the spirit of spin frustration and
+yet are computationally manageable.
+Here, we will consider small Hubbard-type model
+systems along similar lines as in our earlier studies
+[35, 36, 41]; by including SOC we can generate intrin-
+sically noncollinear ground states. In particular, we will
+focus on a five-site half-filled Hubbard bowtie as a mini-
+mal model for studying xc torque effects in the presence
+of magnetic frustration. We will generate both exact and
+SDFT phase diagrams of spin configurations for this sys-
+tem and explore the spin dynamics for different config-
+urations in the phase diagram.
+The TD-SDFT treat-
+ment will be based on orbital-dependent exchange-only
+functionals, and we will compare with exact solutions
+of the many-body time-dependent Schr¨odinger equation.
+Focusing on a few representative case studies, we will
+gain insight into the significance of xc torques in differ-
+ent regimes.
+The paper is organized as follows. In Sec. II the ex-
+tended Hubbard model and the SDFT framework are in-
+troduced and the exact and SDFT magnetic phase dia-
+grams are discussed. In Sec. III we describe some techni-
+cal aspects of the TD-SDFT modeling such as the choice
+of initial state.
+In Sec. IV the results of exact diago-
+nalization and SDFT models are compared for the cases
+with moderate to strong correlations and non-local inter-
+actions. Conclusions are given in Sec. V.
+
+2
+4
+3
+53
+II.
+EXACT AND SDFT MAGNETIC
+STRUCTURE OF HUBBARD CLUSTERS
+A.
+Definition of the model
+In this paper we limit ourselves to (TD-)SDFT in the
+exchange-only approximation. As discussed earlier [36],
+the standard Hubbard model with on-site interactions
+does not give rise to any exchange torques. If one wishes
+to study exchange torque effects it is necessary to work
+with an extended Hubbard model instead. We will con-
+sider, in the following, a half-filled 5-site Hubbard cluster
+in a bowtie shape, as shown in Fig. 1. Here, we go be-
+yond Ref. [36] and include SOC through a modification
+of the kinetic-energy operator, where the hopping term
+becomes complex and the hopping acquires a directional-
+ity [42, 43]. Thus, our inhomogeneous extended Hubbard
+model with SOC is described by the Hamiltonian
+ˆHmodel = ˆHT + ˆHU + ˆHext .
+(5)
+The first term is a hopping term with SOC absorbed into
+a spin dependent phase factor,
+ˆHT = −th
+�
+⟨j,j′⟩
+�
+σ
+e−iσθc†
+jσcj′σ + h.c.,
+(6)
+where h.c. stands for Hermitian conjugate. Here, th =
+√
+T 2 + C2 is the generalized hopping strength parameter
+which depends on nearest neighbor hopping strength T
+and spin orbit coupling C, j is the site index for the
+geometry shown in Fig. 1, cjσ is the annihilation operator
+for an electron of spin σ at site j, the brackets ⟨. . .⟩ denote
+an ordered sum over nearest neighbors with the order
+indicated by the arrows in Fig. 1, and σ = ±1 labels
+spin-up and -down.
+Furthermore, θ is the SOC angle
+which parameterizes the strength of the SOC parameter
+C relative to the conventional hopping term T [44–46].
+The second term in the model Hamiltonian (5) com-
+prises the on-site and nearest-neighbor interaction terms,
+ˆHU = U0
+�
+j
+nj↑nj↓ + U1
+�
+⟨j,j′⟩
+�
+σ,σ′
+njσnj′σ′ ,
+(7)
+where njσ = c†
+jσcjσ is the spin σ particle number density
+at site j, and U0 and U1 are the on-site and nearest-
+neighbor repulsion strengths, respectively. For the pur-
+poses of this paper, we set U1 =
+1
+2U0, a fairly typical
+choice for modeling real materials [47], and we restrict
+the hopping parameter th and on-site interaction param-
+eter U0 to be of similar orders of magnitude. Finite non-
+local interactions are necessary for nontrivial exchange
+torques, but we avoid the much stronger interactions
+regime because the charge degrees of freedom tend to
+freeze out as U0 and U1 become large, resulting in the
+dynamics being dominated by a simpler pure-spin low-
+energy effective model.
+Lastly, ˆHext contains the couplings to the external po-
+tential and external magnetic field,
+ˆHext =
+�
+j
+(Vjnj + Bj · mj) ,
+(8)
+where Vj is the scalar potential and Bj is the magnetic
+field on site j, the total density is nj = nj↑ + nj↓, and
+the magnetization is given by mj = �
+σ,σ′ c†
+jσ⃗σσσ′cjσ′
+with ⃗σ = (σx, σy, σz) denoting a vector composed of
+the Pauli matrices. We keep the external field param-
+eters each less than the on-site interaction and hopping,
+Vj, |Bj| < U0, th.
+These external field parameters are
+not strictly set to zero because they can be used to break
+degeneracy in order to fix a symmetry breaking state,
+and because, as discussed in Section III, small variation
+of these parameters in the exact model is found to be
+useful in matching the SDFT initial state and the exact
+initial state more accurately.
+B.
+Magnetic phase diagram of the Hubbard bowtie
+We use exact diagonalization of ˆHmodel to construct
+benchmark solutions with which to compare our SDFT
+results. Figure 2a shows the exact phase diagram of the
+half-filled Hubbard bowtie in a plane whose x − y axes
+are defined by C = (th/U0) sin θ and T = (th/U0) cos θ;
+the SOC angle θ is here measured with respect to the
+kinetic energy axis. Similar phase diagrams for the half-
+filled Hubbard trimer were obtained by Tabrizi et al.
+[43]. Within the above specified regime the model has
+a phase transition at θc = nπ/3 for any integer n. For
+the case of zero external fields, the ground state of the
+5-site model at half filling is degenerate and magnetically
+ordered with a nontrivial noncollinear spin structure (ex-
+cept at isolated points in the phase diagram where the
+spins are ferromagnetically aligned) indicating magnetic
+frustration.
+On the phase boundary, θc, the ground state exhibits
+a symmetry breaking charge density wave (CDW) in the
+form of a spontaneous charge polarization along the x-
+axis of Fig. 1. In Fig. 2a the states shown outside the
+phase diagram image are the states at the critical angles
+θc. A specific choice of charge polarization is depicted
+in order to show the corresponding spin state. The sites
+with no spin indicated do not necessarily have zero mag-
+netic moment, but it tends to be orders of magnitude
+smaller. The states shown inside the shaded segments of
+the phase diagram are those of the midpoint angles be-
+tween the phase boundaries, e.g. θ = 30◦. As θ changes,
+the relative angles of the spins change as well, with the
+fastest changes occurring in the vicinity of the phase tran-
+sitions. Thus, the phase transitions at θc are not discon-
+tinuous, rather they appear to be a zero temperature,
+finite model analog of a second order phase transition,
+although the continuous transition occurs over a rather
+narrow range of θ.
+
+4
+T
+C
+60∘
+T
+60∘
+C
+a
+b
+CDW
+CDW
+CDW
+CDW
+CDW
+CDW
+FIG. 2.
+(a) Magnetic phase diagram of the half-filled 5-site Hubbard model, obtained using exact diagonalization.
+The
+red arrows indicate the relative in-plane spin direction of the state depicted (taken at the midpoint angle between the phase
+boundaries). The blue pluses and minuses indicate the direction of electric polarization for the CDW critical angle states
+for the specific spin arrangement shown. (b) Corresponding magnetic phase diagram using exchange-only SDFT, showing the
+broadening of the phase boundary states. The phase diagram has approximately the same states as for the exact diagonalization
+phase diagram, but the critical angles, where a CDW occurs, acquire a width of a few degrees.
+The complete phase diagram of the ground state of
+our 5-site Hubbard bowtie and other finite and extended
+triangular lattice systems is of interest in and by itself,
+especially with respect to their symmetries.
+A more
+complete formal analysis of the phase boundaries and
+other symmetry-related properties will be the subject of
+a forthcoming study.
+C.
+Exchange-only SDFT
+Exact exchange in noncollinear SDFT has been defined
+in Ref. [35]. Starting point is the exchange energy
+Ex = −1
+2
+� �
+drdr′
+|r − r′|Tr
+�
+γ(r, r′)γ(r′, r)
+�
+.
+(9)
+Here, γ denotes the one-particle spin-density matrix, a
+2 × 2 matrix in spin space whose elements are given
+by γσξ(r, r′) = �N
+j ψjσ(r)ψ∗
+jξ(r′), constructed from two-
+component spinor Kohn-Sham orbitals, where σ =↑, ↓
+and likewise for ξ; Tr is the trace over spin indices. The
+exact noncollinear exchange potential then follows by
+minimizing Ex with respect to the orbitals, under the
+constraint that the orbitals come from a single-particle
+equation with a local potential—this is the so-called op-
+timized effective potential (OEP) approach [48].
+This
+approach is system-independent, i.e., it can be defined in
+real space and for lattice models alike.
+The exact-exchange OEP requires solving an integral
+equation; we use here instead a simplification known as
+the Krieger-Li-Iafrate (KLI) approximation [49].
+The
+construction and numerical solution of the noncollinear
+KLI approximation have been discussed in detail in Refs.
+[32, 35]. KLI directly yields a scalar exchange potential
+and an exchange magnetic field with moderate numeri-
+cal effort and with very little loss of accuracy compared
+to the full OEP. In time-dependent SDFT, the exact-
+exchange OEP formally carries a memory [50]. The time-
+dependent KLI, on the other hand, is an adiabatic ap-
+proximation.
+KLI
+for
+noncollinear
+systems
+produces
+exchange
+torques in extended Hubbard systems [36]. For the pur-
+poses of the present study, we also define a projected KLI
+(KLIp) in which the exchange magnetic field Bx on each
+lattice site is projected along the local magnetization di-
+rection, and which therefore has no exchange torques.
+D.
+SDFT phase diagram
+In the SDFT modeling of the Hamiltonian (5), a simi-
+lar magnetic phase diagram is obtained as the exact one
+shown in Fig. 2a. The main difference is that the phase
+
+5
+boundaries at the critical angles θc are not as sharp as
+in the exact case but quite diffuse, as schematically de-
+picted in Fig. 2b. This is mainly due to the well-known
+tendency of SDFT to prefer symmetry breaking, unless
+highly accurate correlation functionals are used.
+The broadened phase boundary region has a tendency
+to exhibit “charge sloshing” [51] in the Kohn-Sham self-
+consistency iterations. Charge sloshing spoils the con-
+vergence behavior and must be overcome with special
+measures, e.g. charge preconditioning or imaginary time
+propagation [52]. A sufficiently strong external potential
+Vj can also be applied to one side of the model in or-
+der to prevent charge sloshing. A fairly strong external
+potential in the exchange-only SDFT modeling is also
+necessary in the vicinity of θc in order to match to the
+exact initial state because correlation effects tend to be
+stronger close to the phase boundaries (see Sec. III).
+For the simulations of section IV D, where the SDFT
+calculations are not tethered to an exact initial solu-
+tion, charge sloshing can arise in the stronger interaction
+regime, even far from the critical angle θc.
+We found
+that replacing the Kohn-Sham self-consistency loop with
+an imaginary time propagation algorithm [52] for com-
+puting the SDFT ground state was useful in mitigating
+charge sloshing.
+III.
+TIME PROPAGATION AND CHOICE OF
+INITIAL STATE
+In order to compare the dynamics of the exact and
+TD-SDFT solutions, we excite the system with a small,
+localized magnetic field burst along the y direction dur-
+ing a brief number of time steps. To propagate the full
+time-dependent many-body Schr¨odinger equation for our
+Hubbard bowtie we use a standard Crank-Nicolson algo-
+rithm.
+The time-dependent Kohn-Sham equations are
+also propagated using Crank-Nicolson, including a pre-
+dictor-corrector scheme (one corrector step suffices) [53].
+Since our interest is predominantly in the dynamical
+effects comparing KLI and KLIp, we start in both cases
+from the same ground state. This means that the ex-
+change torques must be included in the calculation of
+the KLIp initial state, as this is required in order to
+have KLIp start with the same initial conditions as the
+full KLI simulations; however, these torques are frozen
+in, effectively in the form of an external magnetic field.
+By contrast, in full KLI the exchange torques are time-
+dependent as the system evolves.
+Compared to the differences between exchange-only
+SDFT and exact many-body benchmarks, the differences
+between KLI and KLIp are small and can easily be over-
+shadowed. Since we are here interested in relatively sub-
+tle dynamical exchange torque effects, it is desirable to
+start from a KLI initial state with external scalar poten-
+tial Vj and magnetic field Bj chosen to reproduce the
+exact density and magnetization. With some effort, Vj
+and Bj can be numerically constructed by minimizing
+TABLE I. SOC angle θ and interaction strength U0 for the
+three ground states considered in Sec. IV, the total magnitude
+of the exact xc torque and the exchange-only torque, and the
+correlation and exchange energies.
+θ
+U0
+Σj|τxc|
+Σj|τ KLI
+x
+|
+Ec
+Ex
+30◦
+1
+4.2 × 10−2
+2.6 × 10−2
+-0.214
+-1.84
+30◦
+3
+4.8 × 10−2
+1.2 × 10−1
+-0.236
+-5.82
+60◦
+1
+1.3 × 10−4
+2.0 × 10−3
+-0.448
+-1.92
+the functional
+F(Vj, Bj) =
+�
+j
+�
+(nj − n(0)
+j )2 + |mj − m(0)
+j |2�
+,
+(10)
+where n(0)
+j
+and m(0)
+j
+are the target density and mag-
+netization, respectively.
+For each simulation matched
+to an exact initial state, we minimize F to an accu-
+racy of at least F = 10−25. The minimization is done
+via a conjugate gradient method with randomized resets
+when a local minimum of insufficient accuracy is reached.
+Searching over Vj and Bj of only the SDFT simulations
+to find the minimum of F is extremely computationally
+expensive due to the high dimensionality of the parame-
+ter space. In order to overcome this issue, we switch to
+minimizing F with respect to the external fields of the
+exact solution once F <∼ 10−4. Minimizing with respect
+to exact solution parameters is less computationally ex-
+pensive due to the much smoother response of the exact
+solution to small changes in the external fields.
+IV.
+RESULTS AND DISCUSSION
+The model system shown in Fig. 1 is simple yet ex-
+hibits quite a rich range of structural and dynamical be-
+havior. The parameter space to be explored comprises
+the hopping strength th, the SOC angle θ, and the inter-
+action strength U0 (fixing U1 = U0/2). In the following
+we set th = 1 and limit ourselves to three representative
+choices of (θ, U0) in the magnetic phase diagram. This
+will already be sufficient to gain insight into the signifi-
+cance of the xc torques.
+Table I gives an overview of the three parameter sets,
+the ground-state exchange and correlation energies Ex
+and Ec, and the magnitude of the exact xc torque τxc
+and of the exchange-only torque τx. These will be further
+discussed below.
+A.
+θ = 30◦, U0 = 1
+We first consider the case θ = 30◦, which is in the
+middle of the spin-frustrated region shown in yellow in
+the phase diagrams of Fig. 2, and for weak interaction
+strength U0 = 1. The magnetization dynamics compari-
+son of exact, KLI, and KLIp is shown in Fig. 3a, which
+
+6
+exact 
+KLI 
+KLIproj 
+ b 
+a
+Frequency 
+Y-axis Magentization (x10−5)
+Spectral Amplitude (a.u.)
+FIG. 3.
+Comparison of exact, KLI, and KLIp modeling for
+the case of θ = 30◦ and U0 = 1.
+(a) Dynamics of the y-
+component of the magnetization of a corner site exited by a
+small, short, local burst of magnetic field in the y direction.
+(b) Associated spectral amplitude (in arbitrary units), calcu-
+lated via Fourier transform of the data shown in part (a).
+depicts the magnetization along the y-direction of a cor-
+ner site. By construction (see Sec. III), all three methods
+start from the same initial value.
+KLI and KLIp stay fairly close to one another for
+much of the run time due to the relative smallness of the
+Hubbard interaction, which indicates that the exchange
+torques are not very important in the chosen regime. For
+the first few cycles of the precessional motion triggered by
+the short pulse, exchange-only SDFT is quite close to the
+exact result. In spite of that, both KLI and KLIp start
+to diverge significantly from the exact solution around
+t = 15, which shows that the correlation effects, although
+relatively small, eventually start playing a nonnegligible
+role in the time evolution of the system.
+To gain further insight, we perform a spectral analysis
+of the time-dependent data via Fourier transformation
+of the amplitude of the magnetization oscillations, which
+reveals the spectrum of magnetic excitations. As shown
+in Fig. 3b, KLI and KLIp agree well with the exact spec-
+trum at low frequencies (up to about a frequency ω = 3).
+At higher frequencies, the SDFT spectra differ from the
+ b 
+a
+Frequency 
+Y-axis Magentization (x10−5)
+Spectral Amplitude (a.u.)
+FIG. 4.
+Same as Fig. 3 but for U0 = 3.
+exact spectra, which may be due to the fact that we are
+using here an adiabatic approximation which does not
+produce double or higher excitations [53] and hence does
+not capture all peaks. However, KLI and KLIp remain
+very close to each other throughout, illustrating again
+that exchange torques are insignificant here.
+B.
+θ = 30◦, U0 = 3
+For the second case, we remain at θ = 30◦, away
+from the phase boundaries, but increase the interaction
+strength into the moderately strongly interacting regime,
+at U0 = 3. The real time magnetization dynamics and
+amplitude spectrum are shown in Fig. 4. Clearly, KLI
+and KLIp start to differ from each other almost right
+away, which points to the more important role of the
+exchange torques.
+At first glance, it is surprising to see that the projected
+KLI, which has no torques, agrees better with the exact
+magnetization oscillations, at least for the first few cycles.
+To explain this, it is helpful to consider the magnitudes
+of the initial τxc and τx given in Table I. For U0 = 1, the
+sum of the exchange torques is comparable to the sum of
+the xc torques (within a factor 1.6); at U0 = 3, on the
+other hand, the exchange torques are much larger than
+
+my[0]
+0.12826
+KS
+0.12828
+KSproj
+0.12830
+0.12832
+0.12834
+SIxe
+0.12836
+-0.12838
+0.0 
+2.5
+5.0
+7.5
+10.0
+12.5
+15.0
+17.5
+20.0
+Timemy[0]
+-0.12826
+exact
+KS
+-0.12828
+KSproj
+-0.12830
+-0.12832
+-0.12834
+-0.12836
+-0.12838
+0.0
+2.5
+5.0
+7.5
+10.0
+12.5
+15.0
+17.5
+20.0
+Timemy[O] FFT
+10-1
+exact
+KLI
+10-2
+KLIproj
+FFT amplitude (a.u.)
+10-4
+10-5
+10-6.
+10-7,
+0
+2
+4
+5
+6
+Excitation energy (in units of tg)my[o]
+le-5-1.567e.
+exact
+-5.2
+KLI
+KLIproj
+5.4
+5.6
+5.8
+6.0
+6.2
+-6.4
+0.0
+2.5
+5.0
+7.5
+10.0
+12.5
+15.0
+17.5
+20.0
+Timemy[O] FFT
+10-2
+amplitude (a.u.)
+10
+-4
+10-5
+FT
+10-6
+exact
+KLI
+KLIproj
+10-7
+0
+2
+3
+1
+4
+5
+6
+Excitation energy (in units of tg)my[0]
+1e-5-3.28e-1
+6.0
+6
+6.4
+6.6
+6.8
+7.0
+-7.2
+exact
+-7.4
+KLI
+KLIproj
+-7.6
+0.0
+2.5
+5.0
+7.5
+10.0
+12.5
+15.0
+17.5
+20.0
+Time7
+ b 
+a
+Frequency 
+Y-axis Magentization (x10−5)
+Spectral Amplitude (a.u.)
+FIG. 5.
+Same as Fig. 3 but for θ = 60◦.
+the xc torques, which suggests that the correlation con-
+tribution to the torques becomes relatively much more
+important. In other words, exchange-only overestimates
+the torques, and correlation compensates for it. KLIp
+avoids this overestimation (better no exchange torque at
+all, than too much of it), and brings the dynamics closer
+to the exact case. Notice that this could have not been
+anticipated just from looking at the exchange and cor-
+relation energies Ex and Ec of the initial state, which
+would have suggested that the exchange is dominant.
+The Fourier spectrum in Fig. 4b is less clear: while
+both KLI and KLIp seem to reproduce the rough trends
+of the exact spectrum, it is difficult to say which one of
+them agrees better. Neither of them captures the details
+of the exact spectrum particularly well.
+C.
+θ = 60◦, U0 = 1
+Lastly, we consider the case of θ = 60◦ and U0 = 1, see
+Fig. 5. This state is at a critical angle of the magnetic
+phase diagram where artificial charge density symmetry
+breaking in exchange-only SDFT is prevalent, indicat-
+ing that strong correlations are needed to reproduce the
+exact results. As shown in Table I, Ec is significantly en-
+hanced relative to Ex, compared to the case of θ = 30◦.
+∑j |τKLI
+x
+|
+0
+1
+2
+3
+4
+Fave
+U0
+FIG. 6.
+Red (right axis): Comparison between KLI and
+KLIp solutions as a function of interaction strength for the
+case of θ = 30◦ and U0, quantified by the time-averaged dis-
+tance measure Fave, Eq. (11). Blue (left axis): Σj|τ KLI
+x
+| of
+the Hubbard bowtie ground state versus U0.
+Correspondingly, the exchange torques are lower, due to
+the localization of the magnetization to one side of the
+system. The strong correlation effects at the transition
+angle result in both KLI and KLIp diverging from the
+exact solution fairly quickly. The magnetization oscilla-
+tions calculated with KLI and KLIp match each other
+fairly well, at least for the first few cycles, but then dif-
+ferences start to accumulate.
+The Fourier spectrum, see Fig. 5b, has well defined ex-
+citations, which are fairly well captured by both KLI and
+KLIp, but some inaccuracies are noticeable at both high
+and low frequencies.
+Notably, KLIp performs slightly
+better at estimating the gaps in the spectrum for mid-
+range frequency excitations. The better performance of
+KLIp occurs, similarly to Section IV B, due to the KLI
+exchange-only approximation substantially overestimat-
+ing the xc torques, with no correlation to compensate
+(see Table I).
+D.
+Distance between KLI and KLIp versus U0
+The effect of the exchange torques can be further quan-
+tified by introducing the time-averaged distance measure
+Fave = 1
+t
+� t
+0
+dt′ �
+j
+� �
+nKLI
+j
+− nKLIp
+j
+�2
++
+���mKLI
+j
+− mKLIp
+j
+���
+2 �
+,
+(11)
+where we calculate the time average over a short time
+(t = 2) after initial excitation.
+This provides an esti-
+mate of the degree of divergence between the solutions
+which can be compared with interaction strength and the
+magnitude of ground state KLI exchange torques.
+Figure 6 shows the time-averaged distance measure
+(11) between KLI and KLIp as a function of U0 at
+θ = 30◦, and, for the sake of comparison, the sum of
+
+my[O] FFT
+10-1
+exact
+KLI
+10-2
+KLIproj
+FFT amplitude (a.u.)
+10-4
+10-5
+10-6.
+10-7_
+0
+2
+3
+4
+5
+7
+6
+Excitation energy (in units of tg)my[o]
+1e-5-2.521e-1
+3.2
+Y-axis Magnetization Magnitude
+3.4
+3.6
+3.8
+4.0
+4.2
+exact
+KLI
+-4.4
+KLIproj
+0.0
+2.5
+5.0
+7.5
+10.0
+12.5
+15.0
+17.5
+20.0
+Time0.25
+0.12
+0.10
+0.20
+0.08
+0.15
+0.06
+0.10
+0.04
+0.05
+0.02
+0.00
+0.00
+0.0
+0.5
+1.0
+1.5
+2.08
+the magnitudes of the KLI exchange torques of the corre-
+sponding initial states. Both Fave and Σj|τ KLI
+x
+| start out
+linearly for small interaction strengths U0 and keep in-
+creasing well into the moderate interaction regime, where
+Fave appears to start leveling off around U0 = 2.
+A comparison with exact time-dependent xc torques
+is, unfortunately, not possible; even the construction of
+the exact Σj|τxc| over the whole range of U0 is numeri-
+cally too demanding, except for the three cases in Table
+I. Nevertheless, we can infer from the results presented
+in Fig.
+6 that both exchange and correlation torques
+must be accounted for even for relatively low interaction
+strengths in order to accurately describe the dynamics.
+V.
+CONCLUSION
+We have performed exact and approximate, exchange-
+only (TD)-SDFT calculations on a half-filled 5-site Hub-
+bard cluster with varying interaction and SOC strengths.
+The purpose of this study was to assess the significance of
+many-body magnetic torques for the description of spin
+dynamics. We considered three scenarios with weak and
+moderate interactions and close to and away from a tran-
+sition between different magnetic phases. While this is
+clearly not an exhaustive exploration of the parameter
+space, the examples studied here are good representa-
+tives and allow us to draw meaningful conclusions.
+We find that exchange torques become increasingly im-
+portant as non-local interactions become stronger, with
+an approximately linear dependence at low interactions
+(see Fig. 6), but the relationship becomes nonlinear for
+more general interaction strengths. Strong correlations in
+the vicinity of phase boundaries reduce the importance of
+exchange torques due to localization. When correlations
+are particularly strong, they appear to counteract the ex-
+change torques, leading to a net reduction of the total xc
+torques. This suggests that when lacking a sufficiently ac-
+curate correlation functional, completely projecting out
+the xc torques may improve the overall accuracy of TD-
+SDFT magnetic dynamics, at least for short times.
+The challenge for future work is clearly to construct
+correlation functionals that produce accurate torques,
+and test these against benchmarks. A good starting point
+will be to do this for similar finite Hubbard models, fol-
+lowed by tests for the magnetization dynamics in real
+magnetic materials in the linear and nonlinear regime.
+ACKNOWLEDGMENTS
+This work was supported by DOE Grant No.
+DE-
+SC0019109. The authors wish to thank Aurora Pribram-
+Jones for helpful discussion.
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+

GtE1T4oBgHgl3EQf_QYB/content/tmp_files/2301.03577v1.pdf.txt

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+arXiv:2301.03577v1  [nlin.CD]  9 Jan 2023
+Records and occupation time statistics for area-preserving maps⋆
+Roberto Artuso1,2,∗ Tulio M. de Oliveira3, and Cesar Manchein3†
+1Dipartimento di Scienza e Alta Tecnologia and Center for Nonlinear
+and Complex Systems, Via Valleggio 11, 22100 Como, Italy;
+2I.N.F.N, Sezione di Milano, Via Celoria 16, 20133 Milano, Italy; and
+3Departamento de Física, Universidade do Estado de Santa Catarina, 89219-710 Joinville, SC, Brazil
+⋆To Giulio Casati, celebrating his birthday and his achievements.
+(Dated: January 10, 2023)
+A relevant problem in dynamics is to characterize how deterministic systems may exhibit fea-
+tures typically associated to stochastic processes. A widely studied example is the study of (normal
+or anomalous) transport properties for deterministic systems on a non-compact phase space. We
+consider here two examples of area-preserving maps: the Chirikov-Taylor standard map and the
+Casati-Prosen triangle map, and we investigate transport properties, records’ statistics and occu-
+pation time statistics. While the standard map, when a chaotic sea is present, always reproduces
+results expected for simple random walks, the triangle map -whose analysis still displays many elu-
+sive points- behaves in a wildly different way, some of the features being compatible with a transient
+(non conservative) nature of the dynamics.
+Keywords: Area-preserving maps, record statistics, infinite ergodicity.
+I.
+INTRODUCTION
+One of the most remarkable advances in modern dy-
+namics lies in the recognition that deterministic sys-
+tems may exhibit statistical properties typical of purely
+stochastic processes: for instance such systems may dis-
+play diffusion properties similar to random walks [1–4].
+Area-preserving maps (see for instance [1]) represent a
+prominent example of Hamiltonian systems where subtle
+features of dynamics, as integrability vs chaotic proper-
+ties, may be studied. In this context one of the most out-
+standing example is represented by the Chirikov-Taylor
+standard map (SM) (see [1, 5] and references therein):
+we also mention the fundamental role of such a map in
+the development of quantum chaos, unveiling features
+like quantum dynamical localization [6]. Though the SM
+has been extensively explored by numerical simulations,
+very few rigorous results have been proven (see, for in-
+stance, the introduction in [7]): however it is generally
+believed that for large nonlinearity parameter this map
+typically exhibits good stochastic properties, and sensi-
+tive dependence upon initial conditions. Here a remark
+is due: such a map can be studied either on a 2-torus
+or on an (unbounded) cylinder: the latter representa-
+tion is naturally adopted when transport properties are
+concerned, and analogies with random walks are taken
+into account [1, 3, 8, 9]. While particular nonlinear pa-
+rameters in the standard map can be tuned to generate
+strong anomalous diffusion [10], here we will only deal
+with the case in which diffusion is normal.
+Our find-
+ings will be confronted with those obtained for another
+area-preserving map, characterized by the lack of expo-
+nential instability: the so called Casati-Prosen Triangle
+∗ roberto.artuso@uninsubria.it
+† cesar.manchein@udesc.br
+Map (TM) [11], introduced by considering, in an appro-
+priate limit, the Birkhoff dynamics of a triangular bil-
+liard: apart from its intrinsic interest, such a map is an
+ideal benchmark to test whether stochasticity properties,
+exhibited by strongly chaotic systems, are showcased also
+by systems lacking any exponential instability. It also
+turns out that many features about the TM are still de-
+bated, starting from basic properties like ergodicity and
+mixing (see for instance [12, 13]).
+More precisely we will compare different indicators
+for both map on the cylinder: though in principle fur-
+ther complications are added when one considers a non-
+compact phase space [14, 15], this is the appropriate sce-
+nario to discuss transport properties and record statis-
+tics, and to check whether tools from infinite ergodic
+theory may enrich our understanding of such systems.
+Our main findings are that the SM, in its typical
+chaotic regime, displays all stochastic properties of a
+purely stochastic system, while -as expected- results are
+far more complicated for the TM, even if we believe that
+some new insight is provided by our analysis, in partic-
+ular as regards persistence behaviour, occupation time
+statistics and the relationship between transport proper-
+ties and record statistics.
+The paper is organized as follows.
+In Sec. II,
+the Chirikov-Taylor standard map (1) and the triangle
+map (3) -our basic models- are presented and we also
+mention the main properties we analyze. Section III is
+dedicated to discuss transport properties, records’ statis-
+tics and occupation time statistics. We end with a dis-
+cussions in Sec. IV.
+
+2
+II.
+THE BASIC SETTING
+We recall the definition of the SM
+pn+1 = pn + K
+2π sin (2πxn),
+xn+1 = xn + pn+1
+mod 1;
+(1)
+K being the nonlinear parameter: when K is sufficiently
+big no KAM invariant circles bound the motion and one
+can study moments of the diffusing variable p ∈ R:
+⟨|pn − p0|q⟩ ∼ nqν(q).
+(2)
+The typical behaviour observed for the second moment in
+simulations is normal diffusion ν(2) = 1/2 [16, 17], while,
+for certain parameter values, the existence of stable run-
+ning orbits (accelerator modes) induces superdiffusion,
+ν(2) > 1/2) [18–20]. We point out that a finer descrip-
+tion of anomalous transport is obtained by considering
+the full spectrum ν(q): if ν(q) = α · q, for some α ̸= 1/2
+one speaks about weak anomalous diffusion whereas the
+case of a nontrivial ν(q) is dubbed strong anomalous dif-
+fusion [10]. As far as the SM is concerned we will consider
+the case where transport in the stochastic sea is normal
+(even if the phase space exhibits a mixture of chaotic and
+elliptic components (see Figure 1).
+Figure 1. Phase-space portrait for the standard map (1) on
+the 2-torus, for K = 2.6. Here 40 uniformly distributed initial
+conditions were used for x, while maintaining p0 = 0 fixed:
+each initial condition is iterated 104 times.
+On the other side the TM is defined (on the cylinder)
+as:
+pn+1 = pn + 2(xn− ⇂ xn ↿ −µ(−1)⇂xn↿),
+xn+1 = xn − 2pn+1
+mod 2,
+(3)
+where ⇂ · · · ↿ denotes the nearest integer. It was intro-
+duced in [11] (see also [21]) as a limit case for the Birkhoff
+map of irrational triangular billiards: systems lacking ex-
+ponential instability, whose ergodic properties are subtly
+related to irrationality properties of the angles [22–25]:
+we remark that polygonal billiards represent both a hard
+mathematical challenge [26–29], and a natural bench-
+mark when trying to assess which microscopic dynam-
+ical features lead to macroscopic transport laws [30–32]
+( see also [33, 34]): in this respect it is worth mentioning
+that anomalous transport has been associated to scaling
+exponents of the spectral measure [35], and that general-
+ized triangle maps have been investigated recently, both
+as connected to dynamical localization [36], and with
+respect to slow diffusion [37]. A typical phase portrait
+(on the torus) of the TM is shown in Figure 2. Before
+Figure 2. Phase-space dynamics for the triangle map (3), for
+µ =
+1+
+√
+5
+2
+(golden mean).
+Here 100 randomly distributed
+initial conditions were used for x and p: each initial condition
+is iterated 5×104 times. Notice the typical filament structure
+in the phase space [23, 24].
+mentioning the numerical experiments we performed, a
+crucial observation is in order. When looking at trans-
+port properties (and records statistics), considering maps
+on the cylinder is quite natural, while from the ergodic
+point of view this perspective is somehow delicate, since
+no renormalizable invariant density exists [14, 15], and
+the appropriate setting is infinite ergodic theory. When
+polygonal channels are considered, even establishing re-
+current properties of the dynamics is a demanding task
+[38].
+The first set of properties we investigated is more con-
+ventional, and a few results -as we will mention in the
+next section- have already been considered, especially as
+far as the SM is considered. We will look at transport
+properties, in particular through the first and the sec-
+ond moment of the diffusing variable We will also con-
+sider records’ statistics, which recently has turned very
+popular (see [39, 40] and references therein). Then we
+will study statistical properties like persistence probabil-
+ity and (generalized) arcsine law [41, 42]: while motion
+in the stochastic sea for the SM will exhibit typical prop-
+erties of a simple stochastic process like a random walk,
+
+1.0
+p
+0.0
+0.0
+2.00.5
+p
+-0.5
+0.03
+our findings are quite different in the case of the TM.
+III.
+RESULTS
+We start by considering properties associated to the
+spreading of trajectories over the phase space, then we
+will consider occupation time statistics.
+A.
+Diffusion
+This is a warm-up exercise, since transport properties
+have been studied both for the SM [1, 16, 17] and for the
+TM [25]. We observe normal transport for the case of
+the SM (see panels (c) and (d) in Figure 3), while for the
+TM are results indicate a superdiffusion, with
+⟨(pn − p0)2⟩ ∼ n1.83,
+(4)
+in agreement with [25]. We remark that by looking at the
+power-law exponents of the first two moments, we have
+that possibly anomalous diffusion is weak [10], namely if
+we consider the full spectrum of moments’ asymptotics:
+⟨|pn − p0|q⟩ ∼ nφ(q),
+(5)
+we have a single scaling, in the sense that
+φ(q) = α · q;
+(6)
+where normal diffusion is recovered when α = 1/2. This
+is reasonable since weak anomalous diffusion has been
+observed in polygonal billiards [43].
+B.
+Average number of records
+The statistics of records is very popular in the anal-
+ysis of correlated and uncorrelated stochastic time se-
+quences [39, 40]:
+since this subject has not been ex-
+plores thoroughly in the deterministic setting (with the
+remarkable exception of [44, 45]), we briefly review the
+basic concepts. First of all let us recall the (straightfor-
+ward) definition of a record: given a sequence of real data
+x0, x1, . . . , xk, . . . the element xm is a record if
+xm > xj
+j = 0, 1, . . . m − 1,
+(7)
+(we consider x0 to be the first record).
+To the se-
+quence of data points we associate the binary string
+σ0, σ1, . . . , σk, . . . , where
+σl =
+�
+1
+if xl is a record
+0
+otherwise
+(8)
+The number of records up to time N is then
+MN =
+N
+�
+j=0
+σj.
+(9)
+10−1
+100
+101
+102
+103
+104
+105
+106
+107
+108
+100
+101
+102
+103
+104
+105
+106
+107
+108
+(b)
+10−2
+10−1
+100
+101
+102
+103
+104
+105
+100
+101
+102
+103
+104
+105
+106
+(d)
+10−1
+100
+101
+102
+103
+104
+100
+101
+102
+103
+104
+105
+106
+107
+108
+(a)
+10−2
+10−1
+100
+101
+102
+100
+101
+102
+103
+104
+105
+106
+(c)
+σ2(n)
+⟨(pn − p0)2⟩
+n
+⟨M(n)⟩
+⟨pn − p0⟩
+n
+Figure 3.
+(a) Average number of records, (b) variance, (c)
+first, and (b) second moments of variable p for K = 2.6 in the
+standard map (1), as a function of time. These quantities were
+computed for 106 initial conditions for x0, arbitrarily chosen
+in the chaotic sea along the line p0 = 0. Black-continuous
+lines correspond to power-law asymptotics fit F(n) = anγ:
+the fitting parameters are, for (a) a = 0.86(0), γ = 0.50(9), for
+(b) a = 0.50(7), γ = 1.00(3), for (c) a = 0.77(7), γ = 0.50(1)
+and for (d) a = 0.02(1), γ = 0.99(1).
+The properties of the average number of records,
+⟨MN⟩, and the corresponding variance
+V ar(MN) = ⟨M 2
+N⟩ − ⟨MN⟩2
+(10)
+are important tools to access the nature of the data se-
+quence: as a matter of fact if the different xj are inde-
+pendent, identically distributed random variables, then,
+for large N we have [46, 47]:
+⟨MN⟩ = ln N + γE + O(N −1),
+(11)
+where γE = 0.5772 . . . is the Euler-Mascheroni constant,
+and
+V ar(MN) = σ2(N) = ln N + γE − π2
+6 + O(N −1). (12)
+We remark that both quantities are independent of the
+common distribution of the random variables: this uni-
+versality is an important feature of record statistics in
+different contexts.
+Results are quite different for a correlated sequence, as
+when xj denotes the position of a random walker at time
+j:
+xj+1 = xj + ξj+1,
+(13)
+where the jumps are taken from a common distribution
+℘(ξ). In this case the behaviour is [39, 40]:
+⟨MN⟩ ≈
+2
+√π
+√
+N,
+(14)
+
+4
+100
+102
+104
+106
+108
+1010
+1012
+100
+101
+102
+103
+104
+105
+106
+107
+(b)
+102
+104
+106
+108
+1010
+1012
+100
+101
+102
+103
+104
+105
+106
+107
+(d)
+10−1
+100
+101
+102
+103
+104
+105
+106
+100
+101
+102
+103
+104
+105
+106
+107
+(a)
+100
+101
+102
+103
+104
+105
+106
+107
+100
+101
+102
+103
+104
+105
+106
+107
+(c)
+σ2(n)
+⟨(pn − p0)2⟩
+n
+⟨M(n)⟩
+⟨pn − p0⟩
+n
+Figure 4. (a) Average number of records, (b) variance, (c)
+first, and (b) second moments of variable p for the golden
+mean µ =
+1+
+√
+5
+2
+in the triangular map (3) as a function of
+time.
+These quantities were computed for 106 initial con-
+ditions for x0, arbitrarily chosen in phase space along the
+line p0 = 0. Black-continuous lines correspond to power-law
+asymptotics F(n) = anγ: the fitting parameters are, for (a)
+a = 0.13(9), γ = 0.92(4), for (b) a = 0.04(0), γ = 1.84(9),
+for (c) a = 0.65(4), γ = 0.92(4) and for (c) a = 0.67(6),
+γ = 1.86(0) in (d).
+and
+V ar(MN) ≈ 2
+�
+1 − 2
+π
+�
+N,
+(15)
+so that the standard deviation is of the same order of
+magnitude as the average.
+Again this is a universal
+result, independent of the particular jump distribution
+℘(ξ), as long as the distribution is continuous and sym-
+metric. The crucial ingredient of the proof is that the
+process renews as soon as a new record is achieved and
+the appearance of the new record is related to the survival
+probability for the process, which is universal in view of
+Sparre-Andersen theorem [42, 48, 49] (see also [50]).
+Numerical results on records statistics are reported in
+Figures 3, 4, panels (a) and (b): for the SM our results
+are consistent with early investigations [44, 45], and with
+the asymptotic behaviour of a random walk, while for
+the TM we observe anomalous scaling w.r.t. (14,15): the
+behaviour is related to transport properties, in the sense
+that data are consistent with the growths:
+⟨MN⟩ ∼ N φ(1),
+V ar(MN) ∼ N φ(2).
+(16)
+A similar behaviour was observed in [44, 45], for the SM
+in the presence of accelerator modes. We remark that,
+though in the following we will fix our attention of a
+particular parameter value for the TM, we checked that
+reported experiments do not depend on the particular
+parameter choice, as exemplified in Figure 5, where the
+growth of the averaged number of records is reported for
+different parameters of the TM.
+While a general, quantitative relationship (if any) be-
+10−1
+100
+101
+102
+103
+104
+105
+106
+100
+101
+102
+103
+104
+105
+106
+107
+⟨Mn⟩
+n
+√
+7
+√
+2/2
+(
+√
+5 + e)/12
+Figure 5. (a) Average number of records for three additional
+parameters µ in the TM. Black-continuous line correspond to
+the power-law asymptotic fitting function F(n) = anγ, with
+γ = 0.92(4).
+tween transport exponents and statistical properties of
+records has not been fully developed, to the best of our
+knowledge, it is possible, in some cases, to connect φ(1)
+to the expected maximum of the walk [51, 52], that, for
+random walk with unit jumps, coincides with the num-
+ber of records. On the other side we mention that non-
+homogeneous random walks offer examples where such
+relationship does not hold [53–57].
+C.
+Occupation time statistics
+When we consider the evolution on the cylinder, both
+for the SM and the TM, we are in the presence of in-
+finitely ergodic systems [14, 15], since, while Lebesgue
+measure is preserved, due to area conservation, the (con-
+stant) phase space is unbounded, so the invariant density
+cannot be normalized. This has a series of remarkable
+consequences, which originally have been considered in
+the context of stochastic processes, and then explored in
+the deterministic evolution framework.
+One of the most striking property that has been in-
+vestigated is the generalized arcsine law (see [41] for the
+standard formulation for stochastic processes): we briefly
+recall the main result that lies at the basis of our analysis,
+namely Lamperti’s theorem [58]. The original formula-
+tion involves discrete stochastic processes, for which the
+infinite set of possible states can be separated into two
+sets A and B separated by a single site x0, such that a
+
+5
+transition from one set to the other can only be achieved
+by passing through x0, which can be taken as the start-
+ing site, and is supposed to be recurrent (namely the
+probability of returning to it is 1). For instance we can
+think of one dimensional random walk on an integer lat-
+tice, with x0 = 0 and A (B) consists of strictly positive
+(negative) lattice sites. We are interested in the limit-
+ing distribution of N(n)/n, the fraction of time spent in
+the positive semi-axis up to time n. The theorem states
+that such a distribution exists in the n → ∞ limit, and
+it is characterized by two parameters α and η. η is re-
+lated to symmetry properties of the process, being the
+expectation value of the fraction of time spent in R+:
+η = lim
+n→∞ E
+�N(n)
+n
+�
+:
+(17)
+for a symmetric process η = 1/2, and from now on we
+will only consider such a case.
+−1.0
+−0.5
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+(b)
+−0.4
+−0.2
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+1.4
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+(a)
+arccos(2(N(n)/n)−1)
+log10P(N(n)/n))
+arccos(2(N(n)/n)−1)
+Figure 6. Distribution of the fraction of time spent in the positive axis for the momentum p in the standard (1) (a) and triangle
+(3) (b) maps, in semi-logarithmic scale. To enhance readability of the border values, the transformation x → arccos(2x − 1) on
+the horizontal axis. The (light blue) points represent the simulation results, the (red) line the Lamperti distribution (20). Data
+are obtained by computing 106 initial conditions iterated 106 times for the standard map and 106 initial conditions iterated
+108 times for the triangle map. The fitting parameters are α = 0.49(9) for (a) and α = 0.42(0) for (b). In the case of the TM,
+data suggest a superposition of a (rescaled) Lamperti distribution and two Dirac’s δ centered of x = 0 and x = 1 (see text).
+The other parameter α is instead connected to the be-
+haviour of the generating function of first return prob-
+abilities to the starting site: it can be shown [59] that
+it can be related to the probability Pn of being at the
+starting site after n steps in the following way:
+Pn ∼ H(n)
+n1−α ,
+(18)
+where H(n) is a slowly varying function, namely
+lim
+n→∞
+H(yn)
+n
+= 1.
+(19)
+Under such conditions the density of ϕ = N(n)/n in the
+infinite time limit is given by Lamperti distribution:
+Gα(ϕ) = sin(πα)
+π
+ϕ1−α(1 − ϕ)1−α
+ϕ2α + 2ϕα(1 − ϕ)α cos(πα) + (1 − ϕ)2α ,
+(20)
+that reproduces the usual arcsine law
+P ((Nn/n) ≤ ξ) = 2
+π arcsin
+��
+ξ
+�
+(21)
+when α = 1/2, in the universality class of Sparre-
+Andersen theorem. Deviations from standard arcsine law
+have been reported for a number of cases, in the frame-
+work of deterministic dynamics [60–67], mainly in the
+context of intermittent maps. Numerical experiments for
+the SM confirm the validity of the arcsine law, α = 1/2,
+see panel (a) in Figure 6: to our knowledge this is the
+first time such an indicator has been considered in the
+analysis of area preserving maps.
+The results, as expected, are quite different for the TM,
+and they suggest novel features exhibited by this map. In
+particular (see panel (b) in Figure 6) numerical results are
+well fitted by a Lamperti distribution (with α ≈ 0.42),
+thus different from an ordinary random walk), except for
+the endpoints, that present enhanced peaks. Intuitively
+
+6
+such an additional contribution might be due to a frac-
+tion of orbits never returning to the origin: this would
+correspond, in stochastic language, to a transient random
+walk (we recall that, according to Pólya’s theorem [68] a
+simple symmetric random walk is recurrent -so the return
+to the starting site is sure- in one and two dimensions,
+and transient in higher dimensions). Such a possibility is
+indeed not excluded for infinite polygonal channels [38].
+Our last set of simulations concerns the survival proba-
+bility [61]:
+Pcum(n) = prob (pn ≥ 0 . . . p1 ≥ 0|p0 = 0) .
+(22)
+10−3
+10−2
+10−1
+100
+100
+101
+102
+103
+104
+105
+106
+(a)
+10−2
+10−1
+100
+100 101 102 103 104 105 106 107 108 109
+(b)
+Pcum(n)
+n
+n
+Figure 7. Cumulative distribution function for the survival times obtained for the variable p for (a) the standard map (1), and
+(b) the triangle map (3), in logarithmic scale. Data are obtained by simulating 106 and 105 initial conditions, respectively.
+Continuous-black lines correspond to power-law asymptotic functions F(n) = a + bn−α: the fitting parameters are a = 0, b =
+2.80(0), and α = 0.51(5) in (a) and a = 0.021(0), b = 1.62(6), and α = 0.42(0) in (b).
+When considering recurrent random walks, the asymp-
+totic behaviour of the survival probability is again ruled
+by Lamperti exponent [58, 59] (see also [69]):
+Pcum(n) ∼ n−α.
+(23)
+Once again SM simulations (see panel (a) in Figure 7)
+agree with expected behaviour for simple random walks
+(α = 1/2), while the situation is completely different for
+the TM, where the survival probability seems to tend to
+a finite limit for large n, see panel (b) in Figure 7. This
+is coherent with the transient nature of the TM, which
+we conjectured in the analysis of generalized arcsine law.
+IV.
+DISCUSSION
+We have performed a set of extensive numerical experi-
+ments on two paradigmatic area-preserving maps, the SM
+and the TM, focusing in the case where such maps are
+considered on a cylinder, namely a non compact phase
+space. Firstly we reproduced known results about nor-
+mal diffusion for typical (chaotic) parameters of the SM,
+and superdiffusion for the TM. Then we explored records’
+statistics: numerical simulations again confirm that the
+SM behave like a simple random walk, while anomalous
+growth is exhibited by the TM. The most interesting re-
+sults arise in the analysis of occupation times, like gen-
+eralized arcsine law and survival probability. While once
+again normal stochastic properties are displayed by the
+SM, the TM presents more surprising results, which we
+conjecture are possibly connected to lack of conserva-
+tivity [38] (or transient behaviour, in the language of
+random walks). This feature, which we think is worth
+of further investigations, might suggest new stochastic
+modeling of the TM (see [37]).
+AUTHORS’ CONTRIBUTIONS
+All authors have contributed substantially to the work.
+All authors have read and agreed to the published version
+of the manuscript.
+
+7
+ACKNOWLEDGEMENTS
+R.A. acknowledges partial support from PRIN Re-
+search Project No. 2017S35EHN “Regular and stochastic
+behavior in dynamical systems” of the Italian Ministry of
+Education, University and Research (MIUR). R.A. ac-
+knowledges an association to the GNFM group of IN-
+DAM. R.A thanks Gaia Pozzoli for discussions. C.M. ac-
+knowledges the National Council for Scientific and Tech-
+nological Development - CNPq (Brazilian agency) for
+partial financial support (Grant Number 310228/2020-4).
+T.M.O. acknowledges the Coordenação de Aperfeiçoa-
+mento de Pessoal de Nível Superior - CAPES (Brazilian
+agency ) - Finance Code 001, for partial financial sup-
+port. Additionally, T.M.O. and C.M. also acknowledges
+the Fundação de Amparo à Pesquisa e Inovação do Es-
+tado de Santa Catarina - FAPESC (Brazilian agency) for
+partial financial support.
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+

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+arXiv:2301.11245v1  [math.AP]  26 Jan 2023
+Exponential decay of the solutions to nonlinear Schr¨odinger systems
+Felipe Angeles∗, M´onica Clapp†, and Alberto Salda˜na (�)‡
+Abstract
+We show that the components of finite energy solutions to general nonlinear Schr¨odinger
+systems have exponential decay at infinity.
+Our results apply to positive or sign-changing
+components, and to cooperative, competitive, or mixed-interaction systems. As an application,
+we use the exponential decay to derive an upper bound for the least possible energy of a solution
+with a prescribed number of positive and nonradial sign-changing components.
+Keywords: Exponential decay; Schr¨odinger system; energy bounds; nodal solutions.
+MSC2010: 35B40; 35B45; 35J47; 35B06; 35J10;
+1
+Introduction
+Consider the nonlinear Schr¨odinger system
+
+
+
+
+
+
+
+−∆ui + Vi(x)ui =
+ℓ
+�
+j=1
+βij|uj|p|ui|p−2ui,
+ui ∈ H1(RN),
+i = 1, . . . , ℓ,
+(1.1)
+where N ≥ 1, Vi ∈ L∞(RN), βij ∈ R and 1 < p < 2∗
+2 . Here 2∗ is the usual critical Sobolev
+exponent, namely, 2∗ :=
+2N
+N−2 if N ≥ 3 and 2∗ := ∞ for N = 1, 2.
+Systems of this type occur as models for various natural phenomena. In physics, for example,
+they describe the behavior of standing waves for a mixture of Bose-Einstein condensates of different
+hyperfine states which overlap in space [13]. The coefficients βij determine the type of interaction
+between the states; if βij > 0, then there is an attractive force between ui and uj, similarly, if
+βij < 0, then the force is repulsive, and if βij = 0, then there is no direct interaction between
+these components. Whenever all the interaction coefficients are positive, we say that the system is
+cooperative. If βii > 0 and βij < 0 for all i ̸= j, then the system is called competitive. And if some
+βij are positive and others are negative for i ̸= j, then we say that the system has mixed couplings.
+All these regimes exhibit very different qualitative behaviors and have been studied extensively in
+recent years, see for instance [5,6,8–12,17,19–24,26] and the references therein.
+∗Instituto de Matem´aticas, Universidad Nacional Aut´onoma de M´exico, Circuito Exterior, Ciudad Universitaria,
+04510 Coyoac´an, Ciudad de M´exico, Mexico, felidaujal@im.unam.mx
+†Instituto de Matem´aticas, Universidad Nacional Aut´onoma de M´exico, Campus Juriquilla, Boulevard Juriquilla
+3001, 76230 Quer´etaro, Qro., Mexico, monica.clapp@im.unam.mx
+‡(Corresponding author �) Instituto de Matem´aticas, Universidad Nacional Aut´onoma de M´exico, Circuito Ex-
+terior, Ciudad Universitaria, 04510 Coyoac´an, Ciudad de M´exico, Mexico, alberto.saldana@im.unam.mx
+1
+
+System (1.1) has a variational structure, and therefore a natural strategy is to find weak solutions
+by minimizing an associated energy functional on a suitable set, under additional assumptions on
+the matrix (βij) and on the potentials Vi. Using this approach, several kinds of solutions have been
+found in terms of their signs and their symmetries. However, there seems to be no information
+available about the decay of these solutions at infinity. In this paper, we show that finite energy
+solutions must decay exponentially at infinity, and a rate can be found in terms of the potentials
+Vi. Our main result is the following one.
+Theorem 1.1. Assume that, for every i = 1, . . . , ℓ,
+(V1) Vi : RN → R is H¨older continuous and bounded,
+(V2) there exists ρ ≥ 0 such that
+σi :=
+inf
+RN∖Bρ(0) Vi > 0.
+Let (u1, . . . , uℓ) ∈
+�
+H1(RN)
+�ℓ be a solution of (1.1) and let µi ∈ (0, √σi). Then, there is C > 0
+such that
+|ui(x)| ≤ Ce−µi|x|
+for all x ∈ RN and i = 1, . . . , ℓ.
+(1.2)
+Furthermore, if Vi ≡ 1 for every i = 1, . . . , ℓ, then (1.2) holds true with µi = 1.
+We emphasize that each component may have a different decay depending on each potential Vi.
+The main obstacle to showing (1.2) is to handle the possibly sublinear term |ui|p−2ui for p ∈ (1, 2)
+(which is always the case for N ≥ 4). To explain this point in more detail, assume that (u1, . . . , uℓ)
+is a solution of (1.1) and write the i-th equation of the system as
+−∆ui +
+�
+ai(x) − ci(x)|ui(x)|p−2�
+ui = 0,
+ai := Vi − βii|ui|2p−2,
+ci :=
+ℓ
+�
+j̸=i
+βij|uj|p.
+(1.3)
+Since every uj ∈ H1(RN)∩C0(RN), we know that ai and ci are bounded in RN, but |ui|p−2 → ∞ as
+|x| → ∞ and it is also singular at the nodal set of a sign-changing solution. As a consequence, one
+cannot use directly previously known results about exponential decay for scalar equations, such as
+those in [1, 3, 18]. In fact, one can easily construct a one dimensional solution of a similar scalar
+equation that has a power-type decay. For instance, let w ∈ C2(R) be a positive function such that
+w(x) = |x|−2/3 for |x| > 1 and let
+c(x) := −w′′(x) + w(x)
+w(x)
+1
+2
+,
+x ∈ R.
+Then, w ∈ H1(R) is a solution of −w′′ + w = c w
+1
+2 in R, c(x) → 0 as |x| → ∞, and w decays as a
+power at infinity.
+This shows that the proof of the exponential estimate in Theorem 1.1 must rely on a careful
+study of the system structure. In other words, although the sublinear nonlinearity |ui|p−2ui appears
+in (1.1), the system is not sublinear. As a whole, it is always superlinear.
+With this in mind, we adapt some of the arguments in [1,18] preserving at each step the system
+structure of the problem. These arguments rely basically on elliptic regularity and comparison
+principles.
+2
+
+The exponential decay of solutions is a powerful tool in their qualitative study. As an application
+of Theorem 1.1, we derive energy bounds of solutions having prescribed positive and nonradial sign-
+changing components. For this, power type decay would not be enough.
+To be more precise, we consider the autonomous system
+
+
+
+
+
+
+
+−∆ui + ui =
+ℓ
+�
+j=1
+βij|uj|p|ui|p−2ui,
+ui ∈ H1(RN),
+i = 1, . . . , ℓ.
+(1.4)
+where the βij’s satisfy the following condition:
+(B1) The matrix (βij) is symmetric and admits a block decomposition as follows: For some 1 ≤
+q ≤ ℓ there exist 0 = ℓ0 < ℓ1 < · · · < ℓq−1 < ℓq = ℓ such that, if we set
+Ih := {i ∈ {1, . . . , ℓ} : ℓh−1 < i ≤ ℓh},
+h ∈ {1, . . . , q},
+then βii > 0, βij ≥ 0 if i, j ∈ Ih, and βij < 0 if i ∈ Ih, j ∈ Ik and h ̸= k.
+According to this decomposition, a solution u = (u1, . . . , uℓ) to (1.1) may be written in block-
+form as
+u = (u1, . . . , uq)
+with uh = (uℓh−1+1, . . . , uℓh),
+h = 1, . . . , q.
+We say that u is fully nontrivial if every component ui is different from zero.
+Set Q := {1, . . . , q}. Given a partition Q = Q+ ∪ Q− with Q+ ∩ Q− = ∅ we look for solutions
+such that every component of uh is positive if h ∈ Q+ and every component of uh is nonradial
+and changes sign if h ∈ Q−. To this end, we use variational methods in a space having suitable
+symmetries. As shown in [11, Section 3], to guarantee that the solutions obtained are fully nontrivial
+we need to assume the following two conditions:
+(B2) For each h ∈ Q, the graph whose set of vertices is Ih and whose set of edges is Eh := {{i, j} :
+i, j ∈ Ih, i ̸= j, βij > 0} is connected.
+(B3) If q ≥ 2 then, for every h ∈ {1, . . . , q} such that ℓh − ℓh−1 ≥ 2, the inequality
+�
+min
+{i,j}∈Eh
+βij
+�
+
+
+min
+h=1,...,q max
+i∈Ih βii
+�
+i,j∈Ih
+βij
+
+
+p
+p−1
+> C∗
+q
+�
+k=1
+k̸=h
+�
+i∈Ih
+j∈Ik
+|βij|
+holds true, where C∗ = C∗(N, p, q, Q+) > 0 is the explicit constant given in (3.7) below.
+In [11] it is shown that, for any q, the system (1.1) has a fully nontrivial solution satisfying the
+sign requirements described above. Furthermore, an upper bound for its energy is exhibited, but
+only for systems with at most 2 blocks, i.e., for q = 1, 2. Here we use Theorem 1.1 to obtain an
+energy bound for any number of blocks.
+For each h = 1, . . . , q, let RIh := {s = (sℓh−1+1, . . . , sℓh) : si ∈ R for all i ∈ Ih} and define
+µh := inf
+s∈RIh
+s̸=0
+
+
+
+�
+i∈Ih s2
+i
+� �
+i,j∈Ih βij|si|p|sj|p
+� 2
+2p
+
+
+
+p
+p−1
+.
+(1.5)
+3
+
+For any ℓ ∈ N, we write ∥u∥ for the usual norm of u = (u1, . . . , uℓ) in (H1(RN))ℓ, i.e.,
+∥u∥2 :=
+ℓ
+�
+i=1
+�
+RN (|∇ui|2 + |ui|2).
+We prove the following result.
+Theorem 1.2. Let N = 4 or N ≥ 6, and let Q = Q+ ∪ Q− with Q+ ∩ Q− = ∅. Assume (B1),
+(B2), and (B3). Then, there exists a fully nontrivial solution u = (u1, . . . , uq) to the system (1.4)
+with the following properties:
+(a) Every component of uh is positive if h ∈ Q+ and every component of uh is nonradial and
+changes sign if h ∈ Q−.
+(b) If q = 1, then
+∥u∥2 = µ1∥ω∥2 if Q = Q+
+and
+∥u∥2 < 10 µ1∥ω∥2 if Q = Q−.
+(c) If q ≥ 2 the following estimate holds true
+∥u∥2 <
+
+min
+k∈Q
+�
+akµk +
+�
+h∈Q∖{k}
+bhµh
+�
+
+ ∥ω∥2,
+(1.6)
+where ak := 1 if k ∈ Q+, ak := 12 if k ∈ Q−, bh := 6 if h ∈ Q+, bh := 12 if h ∈ Q−, and ω is the
+unique positive radial solution to the equation
+− ∆w + w = |w|2p−2w,
+w ∈ H1(RN).
+(1.7)
+To prove Theorem 1.2, we follow the approach in [11] and impose on the variational setting
+some carefully constructed symmetries which admit finite orbits. This approach immediately gives
+energy estimates but it requires showing a quantitative compactness condition which needs precise
+knowledge about the asymptotic decay of the components of the system. Here is where we use
+Theorem 1.1.
+The paper is organized as follows. Section 2 is devoted to the proof of the exponential decay
+stated in Theorem 1.1. The application of this result to derive energy bounds is contained in Section
+3, where we also give some concrete examples.
+Acknowledgments
+We thank Nils Ackermann for helpful comments and suggestions.
+F. Angeles and A. Salda˜na
+thank the Instituto de Matem´aticas - Campus Juriquilla for the kind hospitality. F. Angeles is
+supported by CONACYT (Mexico) through a postdoctoral fellowship under grant A1-S-10457. M.
+Clapp is supported by CONACYT (Mexico) through the research grant A1-S-10457. A. Salda˜na
+is supported by UNAM-DGAPA-PAPIIT (Mexico) grant IA100923 and by CONACYT (Mexico)
+grant A1-S-10457.
+4
+
+2
+Exponential decay
+This section is devoted to the proof of Theorem 1.1. As a first step, we extend the argument
+in [2, Lemma 5.3] to systems. Let Br denote the ball of radius r in RN centered at zero. Let σi
+and βij as in (V2) and (1.1), then we let σ := (σ1, . . . , σℓ) and β := (βij)ℓ
+i,j=1.
+Lemma 2.1. Let Vi ∈ L∞(RN) satisfy (V2) and let u = (u1, . . . , uℓ) be a solution of (1.1). Set
+ξi(r) :=
+�
+RN∖Br
+�
+|∇ui|2 + |ui|2�
+and
+ξ(r) := (ξ1(r), . . . , ξℓ(r)).
+Then, there are positive constants C = C(u, σ, β, N, ρ, p) and ϑ = ϑ(σ), with ρ and σi as in (V2),
+such that
+|ξ(r)|1 :=
+ℓ
+�
+i=1
+ξi(r) ≤ Ce−ϑr
+for every r ≥ 0.
+Proof. Let χ : RN → R be given by χ(r) := 0 if r ≤ 0, χ(r) := r if r ∈ (0, 1) and χ(r) := 1 if
+r ≥ 1. Let ur
+i (x) := χ(|x| − r)ui(x) for r ≥ 0, x ∈ RN, and i = 1, . . . , ℓ. Then ur
+i ∈ H1(RN) and
+ur
+i (x) = (|x| − r)ui(x),
+∇ur
+i (x) = (|x| − r)∇ui(x) + x
+|x|ui(x),
+if x ∈ Br+1 ∖ Br.
+Set δ := min{σ1, . . . , σℓ, 1}. Using that |ui x
+|x| · ∇ui| ≤ 1
+2(|∇ui|2 + |ui|2) we obtain
+�
+RN
+�
+∇ui · ∇ur
+i + Vi uiur
+i
+�
+≥ δξi(r + 1) +
+�
+Br+1∖Br
+�
+(|x| − r)
+�
+|∇ui|2 + Vi u2
+i
+�
++ ui
+x
+|x| · ∇ui
+�
+≥ δξi(r + 1) − 1
+2
+�
+Br+1∖Br
+�
+|∇ui|2 + |ui|2�
+≥ (δ + 1
+2)ξi(r + 1) − 1
+2ξi(r)
+if r + 1 ≥ ρ.
+(2.1)
+As u solves (1.1) we have that
+����
+�
+RN ∇ui · ∇ur
+i + Vi uiur
+i
+���� =
+������
+�
+RN
+ℓ
+�
+j=1
+βij|uj|p|ui|p−2uiur
+i
+������
+≤
+ℓ
+�
+j=1
+�
+RN\Br
+|βij||uj|p|ui|p−2|ui|2 =
+ℓ
+�
+j=1
+|βij|
+�
+RN∖Br
+|uj|p|ui|p
+and since |um|p ≤
+��ℓ
+k=1 |uk|2p�1/2
+for every m = 1, . . . , ℓ, we obtain
+����
+�
+RN ∇ui · ∇ur
+i + Vi uiur
+i
+���� ≤
+
+
+ℓ
+�
+j=1
+|βij|
+
+
+ℓ
+�
+k=1
+�
+RN∖Br
+|uk|2p.
+Given that uk ∈ H1(RN) for all k = 1, . . . , ℓ, Lemma A.1 implies the existence of a constant
+C1 = C1(N, p) > 0 such that
+����
+�
+RN ∇ui · ∇ur
+i + Vi uiur
+i
+���� ≤ C1
+
+
+ℓ
+�
+j=1
+|βij|
+
+
+ℓ
+�
+k=1
+��
+RN∖Br
+�
+|∇uk|2 + |uk|2��p
+(2.2)
+5
+
+for every r ≥ 1 and i = 1, . . . , ℓ. Set C2 := C1
+�ℓ
+i,j=1 |βij|. From (2.1) and (2.2), assuming without
+loss of generality that ρ ≥ 2 and adding over i, we get
+2δ + 1
+2
+|ξ(r + 1)|1 − 1
+2|ξ(r)|1 ≤ C2
+ℓ
+�
+k=1
+|ξk(r)|p =: C2 |ξ(r)|p
+p
+if r + 1 ≥ ρ.
+Therefore,
+|ξ(r + 1)|1
+|ξ(r)|1
+≤
+1
+2δ + 1
+�
+1 + 2C2
+|ξ(r)|p
+p
+|ξ(r)|1
+�
+≤
+1
+2δ + 1
+�
+1 + 2C2|ξ(r)|p−1
+1
+�
+=: γ(r)
+if r + 1 ≥ ρ. (2.3)
+Since |ξ(r)|1 → 0 as r → ∞, there is r0 = r0(u, p, β, ρ) ∈ N such that r0 ≥ ρ and γ(r) ≤ γ−1
+0
+for all
+r ≥ r0 with γ0 := 2δ+1
+δ+1 > 1. Then, for r > r0 + 1,
+|ξ(r)|1 ≤ |ξ(⌊r⌋)|1 = |ξ(r0)|1
+⌊r⌋−1
+�
+k=r0
+|ξ(k + 1)|1
+|ξ(k)|1
+≤ |ξ(r0)|1γr0−⌊r⌋
+0
+≤ ∥u∥2γr0−r+1
+0
+,
+where ⌊r⌋ denotes the floor of r. Since |ξ(r)|1 ≤ ∥u∥2 ≤ ∥u∥2γr0−r+1
+0
+for r ≤ r0 + 1 we have that
+|ξ(r)|1 ≤ ∥u∥2γr0−r+1
+0
+= ∥u∥2γr0+1
+0
+e− ln(γ0)r
+for every r ≥ 0,
+as claimed.
+Lemma 2.2. Assume (V1) and let u = (u1, . . . , uℓ) be a solution of (1.1). Then ui ∈ W 2,s(RN) ∩
+C2(RN) for every s ≥ 2 and i = 1, . . . , ℓ.
+Proof. Let N ≥ 3. The argument for N = 1, 2 is similar and easier. For each i = 1, . . . , ℓ set
+fi :=
+l
+�
+j=1
+βij|uj|p|ui|p−2ui.
+(2.4)
+Since |uk| ≤ |u| :=
+�
+u2
+1 + · · · + u2
+ℓ for every k = 1, . . . ℓ, we have that
+|fi| ≤
+ℓ
+�
+i,j=1
+|βij||uj|p|ui|p−1 ≤
+
+
+ℓ
+�
+j=1
+|βij|
+
+ |u|p|u|p−1 ≤
+
+
+ℓ
+�
+i,j=1
+|βij|
+
+ |u|2p−1.
+(2.5)
+Therefore, fi ∈ Ls1(RN) for s1 :=
+2∗
+2p−1 > 1 and, by the standard Lp-elliptic regularity theory,
+ui ∈ W 2,s1(RN) for all i = 1, . . . , ℓ (see, e.g., [14, Chapter 9] or [25, Section 3.2]).
+Using a
+bootstrapping argument, we conclude the existence of s > max{N
+2 , 2} such that ui ∈ W 2,s(RN) for
+all i = 1, . . . , ℓ and thus, by the Sobolev embedding theorem, ui ∈ C1,α(RN). Since Vi is H¨older
+continuous and bounded, applying the Schauder estimates repeatedly, we deduce that ui is of class
+C2 (see [15, Section 1.3]).
+In the rest of the paper, we write | · |t for the norm in Lt(RN), 1 ≤ t ≤ ∞. If u = (u1, . . . , uℓ) ∈
+[L∞(RN)]ℓ, then |u|∞ := �ℓ
+i=1 supRN |ui|. Moreover, for a proper open subset Ω of RN we denote
+the usual Sobolev norm in H1(Ω) by ∥ · ∥H1(Ω), i.e.,
+∥u∥2
+H1(Ω) :=
+�
+Ω
+(|∇u|2 + |u|2).
+6
+
+Lemma 2.3. Assume (V1). Let u = (u1, . . . , uℓ) be a solution of (1.1), s > max{2, N
+2 } and Λ > 0
+be such that |Vi|∞ ≤ Λ for i = 1, . . . , ℓ. Then there is a constant C = C(β, N, p, Λ, s) > 0 such
+that, for any x ∈ RN,
+∥ui∥W 2,s(B 1
+2 (x)) ≤ C
+
+|ui|
+s−2
+s
+∞ ∥ui∥
+2
+s
+H1(B1(x)) + |u|
+2ps−(s+2)
+s
+∞
+�
+ℓ
+�
+j=1
+∥uj∥2
+H1(B1(x))
+� p
+s
+
+ ,
+where |u| :=
+�
+u2
+1 + · · · + u2
+ℓ and BR(x) is the ball of radius R centered at x.
+Proof. Since ui ∈ W 2,s(RN) ⊂ L∞(RN), we have that
+|ui|s = |ui|s−2|ui|2 ≤ |ui|s−2
+∞ |ui|2.
+Set fi as in (2.4). By (2.5), there is a constant C2 = C2(β) such that
+|fi|s ≤ Cs
+2|u|(p−1)s|u|ps = Cs
+2|u|(p−1)s+p(s−2)(u2
+1 + · · · + u2
+ℓ)p
+≤ Cs
+2|u|2ps−(s+2)
+∞
+ℓp(u2p
+1 + · · · + u2p
+ℓ ),
+where (p − 1)s + p(s − 2) > 0. Then, by [14, Theorem 9.11], there is a positive constant C1 =
+C1(s, N, Λ) such that
+∥ui∥W 2,s(B 1
+2 (x)) ≤ C1
+�
+|ui|Ls(B1(x)) + |fi|Ls(B1(x))
+�
+for any x ∈ RN.
+From the previous inequalities we derive
+∥ui∥W 2,s(B 1
+2 (x)) ≤ C1
+
+|ui|
+s−2
+s
+∞ ∥ui∥
+2
+s
+H1(B1(x))) + C2ℓ
+p
+s C3|u|
+2ps−(s+2)
+s
+∞
+�
+ℓ
+�
+j=1
+∥uj∥2
+H1(B1(x))
+� p
+s
+
+ ,
+where C3 = C3(N, p) is the constant given by the Sobolev embedding H1(B1) ⊂ L2p(B1).
+Lemma 2.4. Assume (V1) − (V2), let u = (u1, . . . , uℓ) be a solution of (1.1) and let fi be as in
+(2.4). Then, there are constants η > 0, C1 > 0, and C2 > 0 such that
+|ui(x)| ≤ C1e−η|x|,
+|fi(x)| ≤ C2e−(2p−1)η|x|,
+for all x ∈ RN and i = 1, . . . , ℓ.
+Proof. For x ∈ RN with |x| ≥ 2, set r := 1
+2|x|. Then, B1(x) ⊂ RN ∖ Br and, by Lemma 2.1, there
+are positive constants K1 = K1(u, σ, β, N, ρ, p) and ϑ = ϑ(σ), with ρ and σi as in (V2), such that
+∥uj∥2
+H1(B1(x)) ≤ ∥uj∥2
+H1(RN∖Br) = ξj(r) ≤
+ℓ
+�
+i=1
+ξi(r) ≤ K1e−ϑr
+for every j = 1, . . . , ℓ.
+Fix s > max{N
+2 , 2}. By Lemma 2.3 there are positive constants K2 = K2(u, β, N, p, Λ, s) and
+K3 = K3(u, σ, β, ρ, N, p, s) such that
+∥ui∥W 2,s(B 1
+2 (x)) ≤ K2
+
+∥ui∥
+2
+s
+H1(B1(x))) +
+�
+ℓ
+�
+j=1
+∥uj∥2
+H1(B1(x))
+� p
+s
+
+ ≤ K2K3e− ϑ
+s r.
+7
+
+Therefore,
+|ui(x)| ≤ |ui|L∞(B 1
+2 (x)) ≤ K4∥ui∥W 2,s(B 1
+2 (x)) ≤ K2K3K4e− ϑ
+2s|x|
+for every x ∈ RN ∖ B2,
+where K4 is the positive constant given by the embedding W 2,s(B 1
+2) ⊂ L∞(B 1
+2).
+Since ui is
+continuous, we may choose C1 ≥ K2K3K4 such that |ui(x)| ≤ C1e− ϑ
+s for every x ∈ B2. So, setting
+η := ϑ
+2s, we obtain
+|ui(x)| ≤ C1e−η|x|
+for every x ∈ RN.
+The estimate for fi follows immediately from (2.5).
+The following result is a particular case of [18, Theorem 2.1]. We include a simplified proof for
+completeness.
+Lemma 2.5. Assume that V : RN → R satisfies σ := infRN∖Bρ(0) V > 0 for some ρ ≥ 0. Let w be
+a classical solution of −∆w + V w = f in RN such that
+|w(x)| ≤ Ce−η|x|
+and
+|f(x)| ≤ Ce−δ|x|
+for all x ∈ RN
+and for some constants C > 0, η ∈ (0, √σ) and δ ∈ (η, √σ]. Then, for any µ ∈ (η, δ), there is
+M = M(µ, δ, ρ, σ, C) > 0 such that
+|w(x)| ≤ Me−µ|x|
+for all x ∈ RN.
+Proof. Let ρ, σ, η, δ, µ, and C be as in the statement. Set v(x) := e−µ|x| for x ∈ RN. Then,
+∆v(x) = v(x)h(|x|)
+for x ∈ RN ∖ {0},
+where h(r) := µ2 − (N − 1)µ
+r .
+In particular, V (x) − h(|x|) ≥ σ − µ2 =: ε > 0 for |x| > ρ. Fix t ∈ R satisfying
+t > C
+ε e(µ−δ)ρ
+and
+w(x) < tv(x) for |x| = ρ.
+(2.6)
+We claim that w(x) ≤ tv(x) for all |x| > ρ. Indeed, let z := w − tv and assume, by contradiction,
+that m := sup|x|≥ρ z(x) > 0. Since lim|x|→∞ z(x) = 0, there is R > ρ such that z(x) ≤ m
+2 for
+|x| ≥ R. Let Ω := {x ∈ RN : ρ < |x| < R and z(x) > 0}. Then z ≤ m
+2 on ∂Ω and, by (2.6),
+−∆z(x) = −∆w(x) + t∆v(x) = f(x) − V (x)w(x) + tv(x)h(|x|)
+= f(x) − V (x)z(x) + tv(x)(h(|x|) − V (x))
+< Ce−δ|x| − εtv(x) = Ce−δ|x| − εte−µ|x| < 0
+for every x ∈ Ω.
+Then, by the maximum principle, m = maxΩ z = max∂Ω z ≤ m
+2 . This is a contradiction. Therefore
+m ≤ 0, namely, w(x) ≤ te−µ|x| for all |x| ≥ ρ. Arguing similarly for −w and using that w ∈ L∞(RN)
+we obtain that |w(x)| ≤ Me−µ|x| for all x ∈ RN, as claimed.
+We are ready to prove Theorem 1.1.
+Proof of Theorem 1.1. Iterating Lemmas 2.4 and 2.5, using that 2p − 1 > 1, one shows that, for
+any µi ∈ (0, √σi), there is C > 0 such that |ui(x)| ≤ Ce−µi|x| for all x ∈ RN and for all i = 1, . . . , ℓ.
+Now, assume that Vi ≡ 1 for every i = 1, . . . , ℓ and let µ ∈ (0, 1) be such that (2p − 1)µ > 1.
+By Lemma 2.4, we have that |fi(x)| ≤ C2e−(2p−1)µ|x| for all x ∈ RN.
+The claim now follows
+from [1, Theorem 2.3(c)].
+8
+
+3
+Energy estimates for seminodal solutions
+In this section we prove Theorem 1.2.
+Consider the autonomous system (1.4) where N ≥ 4,
+1 < p <
+N
+N−2 and βij satisfy the assumption (B1) stated in the Introduction. According to the
+decomposition given by (B1), a solution u = (u1, . . . , uℓ) to (1.4) may be written in block-form as
+u = (u1, . . . , uq)
+with uh = (uℓh−1+1, . . . , uℓh),
+h = 1, . . . , q.
+u is called fully nontrivial if every component ui is different from zero. We say that u is block-wise
+nontrivial if at least one component in each block uh is nontrivial.
+Following [11], we introduce suitable symmetries to produce a change of sign in some compo-
+nents. Let G be a finite subgroup of the group O(N) of linear isometries of RN and denote by
+Gx := {gx : g ∈ G} the G-orbit of x ∈ RN. Let φ : G → Z2 := {−1, 1} be a homomorphism of
+groups. A function u : RN → R is called G-invariant if it is constant on Gx for every x ∈ RN and
+it is called φ-equivariant if
+u(gx) = φ(g)u(x) for all g ∈ G, x ∈ RN.
+(3.1)
+Note that, if φ ≡ 1 is the trivial homomorphism and u satisfies (3.1), then u is G-invariant. On
+the other hand, if φ is surjective every nontrivial function satisfying (3.1) is nonradial and changes
+sign. Define
+H1(RN)φ := {u ∈ H1(RN) : u is φ-equivariant}.
+For each h = 1, . . . , q, fix a homomorphism φh : G → Z2. Take φi := φh for all i ∈ Ih and set
+φ = (φ1, . . . , φℓ). Denote by
+Hφ := H1(RN)φ1 × · · · × H1(RN)φℓ,
+and let J φ : Hφ → R be the functional given by
+J φ(u) := 1
+2
+ℓ
+�
+i=1
+∥ui∥2 − 1
+2p
+ℓ
+�
+i,j=1
+βij
+�
+RN |ui|p|uj|p.
+This functional is of class C1 and its critical points are the solutions to the system (1.4) satisfying
+(3.1). The block-wise nontrivial solutions belong to the Nehari set
+N φ := {u ∈ Hφ : ∥uh∥ ̸= 0 and ∂uhJ φ(u)uh = 0 for every h = 1, . . . , ℓ}.
+Note that
+∂uhJ φ|K(u)uh = ∥uh∥2 −
+ℓ
+�
+k=1
+�
+(i,j)∈Ih×Ik
+βij
+�
+RN |ui|p|uj|p,
+and that J φ(u) = p−1
+2p ∥u∥2 if u ∈ N φ. Let
+cφ :=
+inf
+u∈N φ J φ(u).
+If s = (s1, . . . , sq) ∈ Rq and u = (u1, . . . , uq) ∈ Hφ we write su := (s1u1, . . . , squq). The following
+facts were proved in [8].
+9
+
+Lemma 3.1.
+(i) cφ > 0.
+(ii) If the coordinates of u ∈ Hφ satisfy
+q
+�
+k=1
+�
+(i,j)∈Ih×Ik
+�
+RN βij|ui|p|uj|p > 0
+for every h = 1, . . . , q,
+(3.2)
+then there exists a unique su ∈ (0, ∞)q such that suu ∈ N φ. Furthermore,
+J φ(suu) =
+max
+s∈(0,∞)q J φ(su).
+Proof. See [8, Lemma 2.2] or [11, Lemma 2.2].
+Lemma 3.2. If cφ is attained, then the system (1.4) has a block-wise nontrivial solution u =
+(u1, . . . , uℓ) ∈ Hφ. Furthermore, if ui is nontrivial, then ui is positive if φi ≡ 1 and ui is nonradial
+and changes sign if φi is surjective.
+Proof. It is shown in [8, Lemma 2.4] that any minimizer of J φ on N φ is a block-wise nontrivial
+solution to (1.4). If ui ̸= 0 and φi is surjective, then ui is nonradial and changes sign. If φi ≡ 1 then
+|ui| is G-invariant and replacing ui with |ui| we obtain a solution with the required properties.
+Set Q := {1, . . . , q} and fix a decomposition Q = Q+ ∪ Q− with Q+ ∩ Q− = ∅. From now
+on, we consider the following symmetries. We write RN ≡ C × C × RN−4 and a point in RN as
+(z1, z2, y) ∈ C × C × RN−4.
+Definitions 3.3. Let i denote the imaginary unit. For each m ∈ N, let
+Km := {e2πij/m : j = 0, . . . , m − 1},
+Gm be the group generated by Km ∪{τ}∪O(N −4), acting on each point (z1, z2, y) ∈ C×C×RN−4
+as
+e2πij/m(z1, z2, y) := (e2πij/mz1, e2πij/mz2, y),
+τ(z1, z2, y) := (z2, z1, y),
+α(z1, z2, y) := (z1, z2, αy)
+if α ∈ O(N − 4),
+and θ : Gm → Z2 be the homomorphism satisfying
+θ(e2πij/m) = 1,
+θ(τ) = −1,
+and
+θ(α) = 1
+for every α ∈ O(N − 4).
+Define φh : Gm → Z2 by
+φh :=
+�
+1
+if h ∈ Q+,
+θ
+if h ∈ Q−.
+(3.3)
+Due to the lack of compactness, cφ is not always attained; see e.g. [11, Corollary 2.8(i)]. A
+sufficient condition for this to happen is given by the next lemma. We use the following notation.
+If Q′ ⊂ Q := {1, . . . , q} we consider the subsystem of (1.4) obtained by deleting all components of
+uh for every h /∈ Q′, and we denote by J φ
+Q′ and N φ
+Q′ the functional and the Nehari set associated
+to this subsystem. We write
+cφ
+Q′ :=
+inf
+u∈N φ
+Q′
+J φ
+Q′(u).
+If Q′ = {h} we omit the curly brackets and write, for instance, cφ
+h or J φ
+h .
+10
+
+Lemma 3.4 (Compactness). Let N ̸= 5, m ≥ 5 and φh : Gm → Z2 be as in (3.3). If, for each
+h ∈ Q := {1, . . . , q}, the strict inequality
+cφ <
+
+
+
+cφ
+Q∖{h} + mµh
+p−1
+2p ∥ω∥2,
+if h ∈ Q+,
+cφ
+Q∖{h} + 2mµh
+p−1
+2p ∥ω∥2,
+if h ∈ Q−,
+(3.4)
+holds true, then cφ is attained, where ω is the positive radial solution to (1.7) and µh is given by
+(1.5).
+Proof. This statement follows by combining [11, Corollary 2.8(ii)] with [11, Equation (5.1)].
+To verify condition (3.4) we introduce a suitable test function. Fix m ≥ 5 and let Km be as in
+Definitions 3.3. If h ∈ Q+, we take ζh := ( 1
+√
+2,
+1
+√
+2, 0) and, for each R > 1, we define
+�σhR(x) :=
+�
+g∈Km
+ω(x − Rgζh),
+x ∈ RN.
+If h ∈ Q− we take ζh := (1, 0, 0) and we define
+�σhR(x) :=
+�
+g∈G′m
+φh(g) ω(x − Rgζh),
+x ∈ RN,
+where ω is the positive radial solution to (1.7) and G′
+m is the subgroup of Gm generated by Km∪{τ}.
+Note that �σhR(gx) = φh(g)�σhR(x) for every g ∈ Gm, x ∈ RN. Let
+σhR := thR�σhR,
+(3.5)
+where thR > 0 is chosen so that ∥σhR∥2 =
+�
+RN |σhR|2p.
+Lemma 3.5. If m ≥ 5, then, for each h ∈ {1, . . . , q}, there exist th = (tℓh−1+1, . . . , tℓh) ∈
+(0, ∞)ℓh−ℓh−1 and C0, R0 > 0 such that thσhR := (tℓh−1+1σhR, . . . , tℓhσhR) ∈ N φ
+h and
+J φ
+h (thσhR) ≤ |Gmζh| µh
+p−1
+2p ∥ω∥2 − C0e−Rdm
+for every R ≥ R0,
+where |Gmζh| is the cardinality of the Gm-orbit of ζh, i.e., |Gmζh| = m if h ∈ Q+ and |Gmζh| = 2m
+if h ∈ Q−, and
+dm := |1 − e2πi/m|.
+(3.6)
+Proof. Take th = (tℓh−1+1, . . . , tℓh) ∈ (0, ∞)ℓh−ℓh−1 such that
+�
+i∈Ih
+t2
+i =
+�
+i,j∈Ih
+βijtp
+jtp
+i = µh
+and apply [11, Proposition 4.1(i) and Lemma 4.4].
+11
+
+Proof of Theorem 1.2. Assume (B1) and let φh : Gm → Z2 be given by (3.3). For q = 1 and m ≥ 5
+it is proved in [11, Corollary 4.2 and Proposition 4.5] that cφ is attained at u ∈ N φ satisfying
+∥u∥2 = µ1∥ω∥2 if Q+ = {1}
+and
+∥u∥2 < 2m µ1∥ω∥2 if Q− = {1}.
+Taking m = 5 gives statement (b).
+Fix m = 6. We claim that cφ is attained and that the estimate (c) holds true for every q ≥ 2.
+To prove this claim, we proceed by induction. Assume it is true for q − 1 with q ≥ 2.
+We will show that the compactness condition (3.4) holds true. Using a change of coordinates, it
+suffices to argue for h = q. By induction hypothesis there exists w = (w1, . . . , wq−1) ∈ N φ
+Q∖{q} such
+that J φ
+Q∖{q}(w) = cφ
+Q∖{q}. For each R > 1 let σqR be as in (3.5) and take tq ∈ (0, ∞)ℓ−ℓq−1 as in
+Lemma 3.5. Set whR = wh for h = 1, . . . , q−1 and wqR = tqσqR, and define wR = (w1R, . . . , wℓR) :=
+(w1R, . . . , wqR). Then, as w ∈ N φ
+Q∖{q} and the interaction between the components of w and σqR
+tends to 0 as R → ∞, we have that wR satisfies (3.2) for large enough R and, as a consequence,
+there exist R1 > 0 and (s1R, . . . , sqR) ∈ [1/2, 2]q such that (s1Rw1R, . . . , sqRwqR) ∈ N φ if R ≥ R1.
+Set uR = (u1R, . . . , uℓR) := (s1Rw1R, . . . , sqRwqR). Using that w ∈ N φ
+Q∖{q} and tqσqR ∈ N φ
+q , from
+the last statement in Lemma 3.1(ii) and Lemma 3.5 we derive
+J φ(uR) = 1
+2
+ℓ
+�
+i=1
+∥uiR∥2 − 1
+2p
+ℓ
+�
+i,j=1
+βij
+�
+RN |uiR|p|ujR|p
+≤ J φ
+Q∖{q}(w) + J φ
+q (tqσqR) − 1
+p
+q−1
+�
+h=1
+�
+(i,j)∈Ih×Iq
+βij
+�
+RN |shRwiR|p|sqRwjR|p
+≤ cφ
+Q∖{q} + |Gmζh| µq
+p−1
+2p ∥ω∥2 − C0e−Rdm + C1
+q−1
+�
+h=1
+�
+i∈Ih
+�
+RN |wiR|p|σqR|p,
+if R ≥ max{R0, R1}, where C0 and C1 are positive constants and dm is given in (3.6).
+It is well known that |ω(x)| ≤ Ce−|x| and, as w solves a subsystem of (1.4), Theorem 1.1 asserts
+that
+|wiR(x)| ≤ Ce−|x|
+for every i ∈ Ih with h = 1, . . . , q − 1.
+Therefore, for every g ∈ Gm,
+�
+RN |wiR|p|ω( · − Rgζh)|p ≤ C
+�
+RN e−p|x| e−p|x−Rgζh| dx ≤ Ce−Rp.
+So, if p > dm, we conclude that
+cφ < cφ
+Q∖{q} + |Gmζh| µq
+p−1
+2p ∥ω∥2
+and, by Lemmas 3.4 and 3.2, cφ is attained at a block-wise nontrivial solution u of (1.4) such
+that every component of uh is positive if h ∈ Q+ and every component of uh is nonradial and
+changes sign if h ∈ Q−. Furthermore, since we are assuming (B2) and (B3) with C∗ as in (3.7)
+below, [11, Theorem 3.3] asserts that u is fully nontrivial.
+Finally, note that p > 1 = dm because m = 6. As |Gmζh| = 6 if h ∈ Q+ and |Gmζh| = 12 if
+h ∈ Q−, the estimate in statement (c) follows by induction.
+12
+
+Remark 3.6. If m = 5 and p > dm we arrive to a similar conclusion, where, in this case, the
+constant bh in statement (b) is 5 if h ∈ Q+ and it is 10 if h ∈ Q−. Note, however, that numbers p
+satisfying d5 = 2 sin π
+5 < p <
+N
+N−2 exist only for N ≤ 13.
+Remark 3.7. For φh as in (3.3), the constant C∗ > 0 appearing in (B3) depends on N, p, q, and
+Q+. It is explicitly defined in [11, Equation (3.1)] as
+C∗ :=
+
+
+pdφ
+(p − 1)S
+p
+p−1
+φ
+
+
+p
+,
+(3.7)
+where
+dφ := p − 1
+2p
+inf
+(v1,...,vq)∈Uφ
+q
+�
+h=1
+∥vh∥2
+with Uφ := {(v1, . . . , vq) : vh ∈ H1(RN)φh ∖ {0}, ∥vh∥2 = |vh|2p
+2p, vhvk = 0 if h ̸= k}, and
+Sφ :=
+min
+h=1,...,q
+inf
+v∈H1(RN)φh∖{0}
+∥v∥2
+|v|2
+2p
+.
+Remark 3.8. In the proof of Theorem 1.2 we use [1, Theorem 2.3], which also characterizes the
+sharp decay rate for positive components by providing a bound from below. This kind of information
+can be useful to show uniqueness of positive solutions for some problems, see [4, Section 8.2].
+To conclude, we discuss some special cases.
+Examples 3.9. Assume (B1) and let p ∈ (1, 2∗
+2 ).
+(a) If q = 1 the system (1.4) is cooperative and more can be said. Indeed, it is shown in [11,
+Corollary 4.2 and Proposition 4.5] that, if (B2) is satisfied, then (1.4) has a synchronized
+solution u = (t1u, . . . , tℓu), where (t1, . . . , tℓ) ∈ (0, ∞)ℓ is a minimizer for (1.5) and u is a
+nontrivial φ-equivariant least energy solution of the equation
+−∆u + u = |u|2p−2u,
+u ∈ H1(RN)φ.
+(3.8)
+Here, if Q+ = {1}, then φ ≡ 1 (and therefore u = ω) and ∥u∥2 ≤ µ1∥ω∥2. On the other
+hand, if Q− = {1}, then φ : Gm → Z2 is the homomorphism θ given in Definitions 3.3 and
+∥u∥2 ≤ 10µ1∥ω∥2.
+(b) If q = ℓ ≥ 2 the system (1.4) is competitive, i.e., βii > 0 and βij < 0 if i ̸= j. Assumptions
+(B2) and (B3) are automatically satisfied and, as µi = β
+−
+1
+p−1
+ii
+, the estimate in Theorem 1.2(c)
+becomes
+∥u∥2 <
+
+min
+j∈Q
+�
+ajβ
+−
+1
+p−1
+jj
++
+�
+i∈Q∖{i}
+biβ
+−
+1
+p−1
+ii
+�
+
+ ∥ω∥2
+≤
+
+
+
+(6 |Q+| + 12 |Q−| − 5) β
+−
+1
+p−1
+0
+∥ω∥2
+if Q+ ̸= ∅,
+12 |Q−|β
+−
+1
+p−1
+0
+∥ω∥2
+if Q+ = ∅,
+where |Q±| denotes the cardinality of Q± and β0 := min{β11, . . . , βℓℓ}.
+13
+
+(c) Similarly, for any q ≥ 2, the estimate in Theorem 1.2(c) yields
+∥u∥2 ≤
+�
+(6 |Q+| + 12 |Q−| − 5) µ∗∥ω∥2
+if Q+ ̸= ∅,
+12 |Q−| µ∗∥ω∥2
+if Q+ = ∅.
+where µ∗ = max{µ1, . . . , µq}.
+Assumptions (B2) and (B3) guarantee that u is fully nontrivial. Note that the left-hand side of
+the inequality in (B3) depends only on the entries of the submatrices (βij)i,j∈Ih, h = 1, . . . , q,
+whereas the right-hand side only depends on the other entries. So, if the former are large
+enough with respect to the absolute values of the latter, (B3) is satisfied. For example, if we
+take ℓ = 2q and the matrix is
+
+
+
+
+
+
+
+
+
+
+
+λ
+λ
+β13
+β14
+β15
+. . .
+β1ℓ
+λ
+λ
+β23
+β24
+β25
+. . .
+β2ℓ
+β31
+β32
+λ
+λ
+β35
+. . .
+β3ℓ
+β41
+β42
+λ
+λ
+β45
+. . .
+β4ℓ
+...
+...
+...
+...
+βℓ−1 1
+. . .
+βℓ−1 ℓ−2
+λ
+λ
+βℓ1
+. . .
+βℓ ℓ−2
+λ
+λ
+
+
+
+
+
+
+
+
+
+
+
+.
+with λ > 0 and βji = βij < 0, then (B1) and (B2) are satisfied. If, additionally,
+λ > 4
+2p−1
+p−1 (q − 1)C∗
+and
+|βij| ≤ 1,
+then, for any h = 1, . . . , q,
+�
+min
+{i,j}∈Eh
+βij
+�
+
+
+min
+h=1,...,q max
+i∈Ih
+βii
+�
+i,j∈Ih
+βij
+
+
+p
+p−1
+= λ
+� λ
+4λ
+�
+p
+p−1
+> C∗4(q − 1) ≥ C∗
+q
+�
+k=1
+k̸=h
+�
+i∈Ih
+j∈Ik
+|βij|
+so (B3) is satisfied.
+A
+An auxiliary result
+Lemma A.1. For every r ≥ 1 there is a linear operator Er : H1(RN ∖ Br) → H1(RN) such that,
+for every u ∈ H1(RN ∖ Br),
+(i) Eru = u a.e. in RN ∖ Br,
+(ii) |Eru|2
+2 ≤ C1|u|2
+L2(RN∖Br)
+(iii) ∥Eru∥2 ≤ C1∥u∥2
+H1(RN∖Br)
+for some positive constant C1 depending only on N and not on r. As a consequence, given p ∈ (1, 2∗
+2 )
+there is a positive constant C depending only on N and p such that
+|u|L2p(RN∖Br) ≤ C∥u∥H1(RN∖Br)
+for every u ∈ H1(RN ∖ Br) and every r ≥ 1.
+14
+
+Proof. Fix a linear (extension) operator E1 : H1(RN ∖ B1) → H1(RN) and a positive constant C1
+satisfying (i), (ii) and (iii) for r = 1; see e.g. [16, Theorem 2.3.2]. For r > 1, set �u(x) := u(rx)
+and, for u ∈ H1(RN ∖ Br), define
+(Eru)(y) := (E1�u)
+�y
+r
+�
+.
+Then, �
+Eru = E1�u. Clearly, Er satisfies (i). Note that |�u|2
+L2(RN ∖B1) = r−N|u|2
+L2(RN∖Br) and that
+∥�u∥2
+H1(RN∖B1) = r−N
+��
+RN∖Br
+�
+r2|∇u|2 + |u|2��
+.
+Similar identities hold true when we replace RN ∖ B1 and RN ∖ Br with RN. Therefore,
+r−N|Eru|2
+2 = |�
+Eru|2
+2 = |E1�u|2
+2 ≤ C1∥�u∥2
+L2(RN∖B1) = r−NC1|u|2
+L2(RN ∖Br),
+which yields (ii). Furthermore,
+r−N
+��
+RN
+�
+r2|∇(Eru)|2 + |Eru|2��
+= ∥�
+Eru∥2 = ∥E1�u∥2
+≤ C1∥�u∥2
+H1(RN∖B1) = r−NC1
+��
+RN∖Br
+�
+r2|∇u|2 + |u|2��
+.
+This inequality, combined with (ii), yields
+r2∥Eru∥2 =
+�
+RN
+�
+r2|∇(Eru)|2 + |Eru|2�
++ (r2 − 1)
+�
+RN |Eru|2
+≤ C1
+�
+RN∖Br
+�
+r2|∇u|2 + |u|2�
++ C1(r2 − 1)
+�
+RN∖Br
+|u|2 = r2C1∥u∥2
+H1(RN∖Br),
+which gives (iii).
+For p ∈ (1,
+N
+N−2) let C2 = C2(N, p) be the constant for the Sobolev embedding H1(RN) ⊂
+L2p(RN). Then, for any u ∈ H1(RN ∖ Br), using statements (i) and (iii) we obtain
+|u|2
+L2p(RN∖Br) ≤ |Eru|2
+2p ≤ C2∥Eru∥2 ≤ C2C1∥u∥2
+H1(RN∖Br),
+as claimed.
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+17
+

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@@ -0,0 +1,1460 @@
+AE 
+Students’ Perceptions of Sustainable Universities in Hungary: An Importance-
+Performance Analysis 
+ 
+496 
+Amfiteatru Economic 
+STUDENTS’ PERCEPTIONS OF SUSTAINABLE UNIVERSITIES  
+IN HUNGARY: AN IMPORTANCE-PERFORMANCE ANALYSIS 
+ 
+Szabolcs Nagy1* and Mariann Veresné Somosi2 
+ 1)2) University of Miskolc, Miskolc, Hungary 
+ 
+ 
+ 
+Please cite this article as: 
+Nagy, S. and Somosi, M.V., 2020. Students’ 
+Perceptions of Sustainable Universities in Hungary: 
+An Importance-Performance Analysis. Amfiteatru 
+Economic, 22(54), pp. 496-515. 
+ 
+DOI: 10.24818/EA/2020/54/496 
+ 
+Article History 
+Received: 29 December 2019  
+Revised: 3 February 2020 
+Accepted: 30 March 2020 
+ 
+Abstract 
+In order to succeed, universities are forced to respond to the new challenges in the rapidly 
+changing world. The recently emerging fourth-generation universities should meet 
+sustainability objectives to better serve their students and their communities. It is essential 
+for universities to measure their sustainability performance to capitalise on their core 
+strengths and to overcome their weaknesses. In line with the stakeholder theory, the 
+objective of this study was to investigate students’ perceptions of university sustainability 
+including their expectations about and satisfaction with the efforts that universities make 
+towards sustainability. This paper proposes a new approach that combines the sustainable 
+university scale, developed by the authors, with the importance-performance analysis to 
+identify key areas of university sustainability. To collect data, an online survey was 
+conducted in Hungary in 2019. The sustainable university scale was found to be a reliable 
+construct to measure different aspects of university sustainability. Results of the 
+importance-performance analysis suggest that students consider Hungarian universities 
+unsustainable. Research findings indicate that Hungarian universities perform poorly in 
+sustainable purchasing and renewable energy use, but their location and their efforts 
+towards separate waste collection are their major competitive advantages. The main 
+domains of university sustainability were also discussed. This study provides university 
+decision-makers and researchers with insightful results supporting the transformation of 
+traditional universities into sustainable, fourth-generation higher education institutions. 
+ 
+Keywords: sustainable university, students’ perception, importance-performance analysis, 
+Hungary, student satisfaction, student expectation 
+ 
+JEL Classification: I23, Q56 
+ 
+ 
+                                                 
+* Corresponding author, Szabolcs Nagy – nagy.szabolcs@uni-miskolc.hu  
+ 
+
+Sustainable University  
+AE 
+ 
+Vol. 22 • No. 54 • May 2020 
+497 
+Introduction 
+We live in the age of rapid changes to which higher education institutions should adopt. A 
+university that wants to succeed needs to respond to the challenges of the new era. One of 
+them is the urgency to meet sustainability objectives (Filho, Manolas and Pace, 2015; Soini, 
+et al., 2018; Olalla and Merino, 2019). Universities are undergoing a rapid transformation 
+as they are not only traditionally engaged in education but are also playing an increasingly 
+important role in the society (Papp-Váry and Lukács, 2019). Nowadays, the emergence of 
+the so-called Fourth Generation universities, which actively shape their socio-economic 
+environment, can be seen (Pawłowski, 2009; Lukovics and Zuti, 2017).  
+The topic of sustainable development is increasingly present among the major concerns of 
+the international academic community (Grecu and Ipiña, 2014). Universities must take 
+steps to achieve the United Nations Sustainable Development Goals (Paletta, et al., 2019). 
+Target 4.7 declares that students have the right to acquire the knowledge and skills needed 
+to promote sustainable development (UN, 2019). Globally, the proliferation of the efforts to 
+assess universities’ responses to the challenges of sustainability can be seen (Li, Gu and 
+Liu, 2018). Adams, Martin, and Boom (2018) draw the attention to the importance of the 
+university sustainability culture. 
+Adaptation of the stakeholder theory is essential for higher education institutions 
+(Mainardes, et al., 2010) as stakeholders can create opportunities for or pose threats to an 
+organisation (Chapleo and Sims, 2017). Students as stakeholders have a serious impact on 
+the future development of universities (Degtjarjova, Lapina and Freidenfelds, 2018). 
+Commitment to sustainability of leaders and important stakeholders play a key role in the 
+effectiveness of sustainable development initiatives in higher education institutions 
+(Wright, 2010.)  
+The position of Hungarian higher education institutes in the world rankings is not very 
+favourable. The best Hungarian university can be found around the 500th place in global 
+rankings. There are only seven or eight Hungarian institutions that are ranked at all 
+(Polónyi and Kozma, 2019). The weak performance of the Hungarian higher education 
+institutions in sustainability rankings explains the need for a comprehensive analysis of 
+university sustainability in Hungary from the students as stakeholders’ perspective, which 
+is one of the main objectives of this study.  
+Students as stakeholders form expectations regarding university sustainability not only 
+generally, but also very specifically, and how those expectations are met determines the 
+level of their satisfaction. This study aims to investigate student expectations about and 
+satisfaction with the attributes of the sustainable university by using the sustainable 
+university scale (SUS) combined with the importance-performance analysis (IPA). SUS, 
+the items of which are the determinants of university sustainability, was developed by the 
+authors. IPA has been widely used to examine the relationship between importance, 
+performance, and satisfaction in many areas (Yuvinatileng, et al., 2013; Wyród-Wróbel and 
+Biesok, 2017, Kim, et al., 2018) However, no previous study has investigated it in the 
+context of university sustainability in spite of the fact that universities should use 
+managerial tools to develop their sustainability strategy. This study seeks to address this 
+research gap.  
+ 
+
+AE 
+Students’ Perceptions of Sustainable Universities in Hungary: An Importance-
+Performance Analysis 
+ 
+498 
+Amfiteatru Economic 
+1. Literature review 
+1.1. Perceptions of the sustainable university 
+In the UI GreenMetric World University Ranking 2019, which provides information about 
+the current conditions and policies related to Green Campus and Sustainability, only seven 
+Hungarian universities can be found. The University of Szeged is in the best position, 
+ranked first in Hungary, and 74th in the world. It is followed by the University of Pecs, 
+ranked 100th globally and the University of Debrecen, in the 202nd position in the world 
+ranking. The University of Miskolc, for which the authors work, can be found only in the 
+605th place in this ranking of 780 universities globally (Greenmetric, 2019). Students’ 
+perceptions of university sustainability were assumed to be in line with this poor ranking 
+performance. It is therefore hypothesized that students are not satisfied with the 
+sustainability performance of the Hungarian higher education institutions (H1). Mention 
+must be made of some of the shortcomings of the GreenMetric Ranking, i.e. non-
+compliance with the Berlin Principles (Ragazzi and Ghidini, 2017), however, it is still one 
+of the best tools to quantify university sustainability. 
+The perceptions of university students towards factors of a sustainable university was first 
+discussed by Nejati and Nejati (2013). The authors developed a reliable scale to assess the 
+university practices towards sustainability. They identified four main dimensions of the 
+sustainable university scale, which are respectively: 1) community outreach, 2) 
+sustainability commitment and monitoring, 3) waste and energy, and 4) land use and 
+planning. Their initial scale contained 28 items, which they reduced to a 12-item scale, 
+which could be a key instrument for university decision-makers and stakeholders to 
+measure the university’s performance regarding the implementation of the transition 
+strategy towards sustainability. Their construct measuring sustainability practices of 
+universities contains 1) community outreach programs; 2) green community centres; 3) 
+partnerships with government, non-governmental organizations, and industry working 
+toward sustainability; 4-5) written commitment to sustainability (university and department 
+level); 6-7) sustainability audits on the surrounding community and on campus; 8) reuse of 
+campus waste; 9) use of renewable and safe energy sources; 10) sustainable support 
+services (e.g. recycling bins on campus, efficient public transport throughout the 
+university); 11) sustainable campus building planning and 12) sustainable campus land-use. 
+Dagiliute, Liobikiene and Minelgaite (2018) were the first to investigate the differences in 
+the perceived sustainability performance between the ‘green’ and the ‘non-green’ 
+universities. They compared the students' attitudes towards sustainability in two Lithuanian 
+universities. They did not find any significant differences in sustainability aspects in 
+general, however, students of the green university sought more information about 
+sustainability and were more often involved in sustainability activities. They also found that 
+campus sustainability and environmental information have a significant impact on students’ 
+sustainable behaviour. In their study, they used a scale to measure perceptions made up of 
+16 items, grouped into four main constructs: 1) ‘campus sustainability’, 2) ‘environmental 
+information’, 3) ‘students’ sustainability involvement’, and 4) ‘university's role in 
+sustainable development. The item ‘university's self-representation as a green university’ 
+was also involved in their construct. Their 17-item scale involves 1) environmental student 
+organization(s); 2) use of public transport, bikes; 3) possibility to recycle waste at the 
+university; 4) use one's own non-disposable cup; 5) availability of strategic documents and 
+their implementation reports; 6) sustainability-related information during lectures;  
+
+Sustainable University  
+AE 
+ 
+Vol. 22 • No. 54 • May 2020 
+499 
+7) university website on environmental objectives; 8) participation in environmental, social 
+activities; 9) involvement in activities at the university; 10) energy and resource saving;  
+11) contribution to social well-being, tolerance; 12) environmental education; 13) 
+cooperation with other national and foreign universities and businesses; 14) inclusion of 
+sustainability aspects in study programmes; 15) sustainability research; 16) university's 
+self-representation as a green university; and 17) declared environmental objectives. They 
+found that students considered social aspects, i.e. social well-being, tolerance the most 
+important attribute of the sustainable university. However, students considered 
+environmental aspects, such as energy saving, environmental education, and actions less 
+important. 
+Li, Gu and Liu (2018) established a new scoring system for campus sustainability in 
+Australia. They suggest that sustainable campus performance indicators should be 
+identified from the different perspectives of the economy, environment and society. In 
+order to identify and prioritise the key sustainability indicators for university campuses, 
+they proposed a new approach combining the qualitative scoring method and an analytical 
+hierarchical process. After thorough literature review, they identified 54 indicators and 
+quantified the weight coefficients for the criteria, sub-criteria and elements, and proposed a 
+model that can be a flexible tool for university decision-makers. 
+It is hypothesized that combining the most relevant items of the constructs developed by 
+Nejati and Nejati (2013), Dagiliute, Liobikiene and Minelgaite (2018) and Li, Gu and Liu 
+(2018), a new, reliable scale to measure perceived university sustainability, i.e. the 
+sustainable university scale, can be developed (H2).  
+Shuqin, et al. (2019) aimed to assess and compare the sustainability performance of 
+different Chinese universities. The authors developed a campus sustainability evaluation 
+system that is made up of the five main domains of campus sustainability, which are 
+respectively: organization and management, energy and resource saving, friendly 
+environment, campus culture, and social outreach. Their evaluation system included 14 
+mandatory indicators and 69 optional indicators. They found that the most problematic 
+fields are organization management, resource saving and campus culture. For example, 
+there are issues with green education, green research and green humanities as they are not 
+so developed there. The assessment tool proposed by the authors can be used to guide the 
+green campus revolution in China and could be adopted by the rest of the world. 
+Wakkee, et al. (2019) demonstrated how (entrepreneurial) universities can drive regional 
+sustainable development in developing countries. They found that local campus leadership, 
+a holistic teaching and research programme, and student involvement can have significant 
+local effects.  
+1.2. Importance-Performance Analysis (IPA) 
+The importance-performance analysis (IPA) was developed by Martilla and James (1977). 
+The original version of IPA defines consumer satisfaction as the function of two 
+components that are respectively: the importance of an attribute of the product/service, and 
+the perceived performance of the company on this attribute. The mean of importance and 
+performance ratings of each attribute determines its position on the importance-
+performance matrix or grid, which is also often called the Cartesian diagram (Figure no. 1). 
+The overall mean of the performance/importance ratings is used as a delimiter of high and 
+low performance/importance (Yuvinatileng, Utomo and Latuperissa, 2013).  
+
+AE 
+Students’ Perceptions of Sustainable Universities in Hungary: An Importance-
+Performance Analysis 
+ 
+500 
+Amfiteatru Economic 
+The 2x2 IPA matrix can be divided into four quadrants. Each quadrant requires a different 
+approach and strategy (Wyród-Wróbel and Biesok, 2017):  
+ Quadrant 1: Keep up the good work. This is the best possible position for an attribute. 
+This quadrant contains the competitive advantages and major strengths of a company. The 
+organization must defend all of them to succeed. These are high importance/high 
+performance items. 
+ 
+  
+Figure no. 1: The modified Importance Performance Matrix 
+Source: Kim, Jeon, Cho and Kim, 2018. 
+ 
+ Quadrant 2: The territory of Possible overkill. Here low importance/high performance 
+attributes, i.e. items of overperformance, can be found. Organizations should deploy 
+business resources used here somewhere else (e.g. in Quadrant 1) or should increase the 
+importance of those attributes that can be found here to turn them into competitive 
+advantages. 
+ Quadrant 3: The area of Low priority. Low importance/low performance attributes can 
+be seen here. Those are minor weaknesses that require no additional resources. 
+Organizations are suggested to avoid investing in this quadrant. 
+ Quadrant 4: Concentrate here. High importance/low performance attributes can be 
+found here. Those are the major weaknesses of an organization that require immediate 
+corrective actions to increase consumer satisfaction and to avoid customer churn.  
+1.3. Stakeholder theory 
+The stakeholder theory originates from the 1980s. Freeman (1984) was the first to coin the 
+phrase as an opposite to the shareholder theory or Friedman’s doctrine, which suggests that a 
+company’s sole responsibility is to make money for its shareholders (Friedman, 1965). 
+
+High
+Quadrant 2
+Quadrant1
+Possible overkill
+Keep up the good work
+Performance
+Quadrant3
+Quadrant 4
+Low priority
+Concentrate here
+Low
+Low
+Importance
+HighSustainable University  
+AE 
+ 
+Vol. 22 • No. 54 • May 2020 
+501 
+According to the stakeholder theory, shareholders are only one of many stakeholders in a 
+company, and an organization’s key to market success is how it satisfies all the stakeholders, 
+not only its shareholders (Freeman, 2010). The stakeholder theory says that the stakeholder 
+ecosystem is made up of all parties that invested and involved in, or affected by, the company. 
+Therefore, companies must pay special attention to their employees, vendors, suppliers, 
+owners, community/neighbours, community groups, competitors, governmental bodies, 
+oversight organizations and the local ecology (Freeman, 2010).  
+The stakeholder theory is intertwined with the domains of ethics and sustainability. Carroll and 
+Buchholtz (2014) suggest that successful businesses in society adopt a stakeholder 
+management approach. The stakeholder theory is solid ground for corporate social 
+responsibility and business ethics inside the company (Kakabadse, Rozules and Davies, 2005). 
+The stakeholder ecosystem of a university comprises current, former (alumni) and potential 
+students, parents, municipalities, academics, faculties, management (Rector, the Senate, 
+Chancellor), administrative staff, governmental organisations, Academy of Sciences, 
+research partners and companies. In higher education institutions, students and employees 
+are always the major stakeholders in terms of their number. According to the stakeholder 
+theory, universities are service providers to students and students are one of the most 
+important stakeholders (Degtjarjova, Lapina, and Freidenfelds, 2018). The more satisfied 
+students are, the more likely it is that the university could succeed, also in the field of 
+sustainability. It is therefore assumed that IPA as a strategic tool should be used to 
+maximize student satisfaction with the efforts that universities make towards sustainability. 
+ 
+2. Methodology  
+2.1. Methodology and research questions 
+Based on the literature review presented above, and in line with the main objectives of the 
+research, this study aims to address the following research questions respectively: 
+ R1: What are the student expectations about university sustainability in Hungary? 
+(student expectations) 
+ R2: To what extent are students satisfied with the sustainability performance of 
+universities? (student satisfaction). H1 refers to this question. 
+ R3: Is combining sustainable university scale (SUS) with importance-performance 
+analysis (IPA) a powerful strategic tool for university decision-makers to identify key areas 
+of university sustainability? 
+ R4: What are the main components of the perceived university sustainability? 
+ R5: Is sustainable university scale (SUS) a reliable construct to measure students’ 
+perceptions of university sustainability? H2 refers to this question. 
+In line with the research questions, the following hypotheses were developed: 
+ H1: Students are not satisfied with the sustainability performance of the Hungarian 
+higher education institutions. 
+
+AE 
+Students’ Perceptions of Sustainable Universities in Hungary: An Importance-
+Performance Analysis 
+ 
+502 
+Amfiteatru Economic 
+ H2: Combining the most relevant items of the constructs developed by Nejati and 
+Nejati (2013), Dagiliute, Liobikiene and Minelgaite (2018) and Li, Gu and Liu (2018), a 
+new, reliable scale for measuring perceived university sustainability, i.e. the Sustainable 
+University Scale (SUS), can be developed. 
+To answer the research questions, and to thoroughly investigate students’ perceptions of the 
+sustainable university, a questionnaire made up of 47 questions grouped into three sections 
+were designed: 
+ Section 1: Importance of the sustainable university scale (SUS) items. It contains 21 
+statements measured on a five-point importance scale (1. not at all important … 5. very 
+important). Respondents were asked to answer the following question: “How important are 
+the followings to you regarding a sustainable university?”. SUS items can be seen in Table 
+no. 1. 
+ Section 2: Perceived performance of the sustainable university scale (SUS) items: The 
+very same 21 statements as in Section 1, measured on a five-point rating scale (1 – very 
+poor ... 5 – excellent), answering the question “How do you rate the sustainability 
+performance of your university?”. 
+ Section 3: Demographic variables. It contains 5 questions including gender, age, 
+study level, branch of sciences and the university where they study (Table no. 2).  
+The sustainable university scale (SUS), which contains 21 items, is a construct developed 
+by the authors. It is based on the domains of university sustainability discussed in the 
+literature review. More specifically, in our construct we combined 9 items (item 4, 5, 7, 9, 
+10, 15, 17, 18 and 20) from Dagiliute, Liobikiene and Minelgaite (2018) with 9 items (item 
+1, 3, 6, 7, 8, 11, 13, 14 and 16) used by Nejati and Nejati (2013), with 3 items (item 9, 11 
+and 16) from Li, Gu and Liu (2018). It must be noted that four items are overlapping. They 
+were found in not only one but two of the three reference studies (item 7, 9, 11 and 16). 
+Moreover, we added four new items to SUS (item 2, 12, 19 and 21). The newly added items 
+are 1) the awareness of the sustainability strategy of the university; 2) green location; 3) the 
+inclusion of sustainability information into normal courses and 4) the integration of 
+sustainability research results into the curricula. The sustainable university scale makes it 
+possible that university decision-makers could gain deep insight into how students perceive 
+their efforts towards sustainability.  
+Eight of 21 items were used without any modifications in its original form (referred as 
+‘original’), nine items were modified to be unambiguous (referred to as ‘revised’), and the 
+four new items that we added are labelled as ‘New’ (Table no 1.).  
+Table no. 1: The items of the sustainable university scale (SUS) 
+ 
+Sustainable university scale items 
+S* 
+Type 
+1 
+The university has a sustainability strategy 
+2 
+R 
+2 
+All the students, researchers, academic and non-academic staff are 
+aware of the sustainability strategy of the university 
+4 
+N 
+3 
+Regular sustainability audits are performed on campus 
+2 
+O 
+4 
+Sustainability information is readily available on the university's 
+website, newsletter, Neptun messages, etc. 
+1 
+R 
+
+Sustainable University  
+AE 
+ 
+Vol. 22 • No. 54 • May 2020 
+503 
+ 
+Sustainable university scale items 
+S* 
+Type 
+5 
+The university distinguishes itself as sustainable/green from other 
+higher education institutions. 
+1 
+R 
+6 
+The university established environmentally and socially responsible 
+purchasing practices  
+2 
+O 
+7 
+Separate waste collection is possible on campus, and the university 
+encourages everyone to do so. 
+1, 2 
+R 
+8 
+The university uses renewable energy sources (e.g. solar panels). 
+2 
+O 
+9 
+The university saves water and energy (e.g. LED lighting) 
+1, 3 
+R 
+10 
+The university encourages use of public transport, bikes. 
+1 
+O 
+11 
+The university buildings are designed / converted in an energy 
+efficient and sustainable way (e.g. windows, doors, insulation) 
+2, 3 
+R 
+12 
+The university buildings are located in a natural setting (quiet, 
+green area with many trees where the air quality is excellent) 
+4 
+N 
+13 
+The university engages in community outreach programs that 
+benefit the local environment. 
+2 
+O 
+14 
+The university has created partnerships with government, non-
+governmental organizations, and industry working toward 
+sustainability. 
+2 
+O 
+15 
+The university has active environmental student organization(s)  
+1 
+O 
+16 
+There are many green actions, projects running / available at the 
+university to support the achievement of sustainability goals 
+2, 3 
+R 
+17 
+The university offers a lot of study programmes related to 
+sustainability. 
+1 
+R 
+18 
+The university offers a lot of subjects/courses related to 
+sustainability. 
+1 
+R 
+19 
+There is also a lot of information about sustainability in normal 
+courses 
+4 
+N 
+20 
+The university promotes sustainability research  
+1 
+O 
+21 
+Sustainability research results are integrated into the curricula 
+4 
+N 
+Notes: S* (Source) = 1: Dagiliute, Liobikiene and Minelgaite (2018); 2: Nejati and Nejati (2013); 3: 
+Li, Gu and Liu (2018); 4: New variables added by the authors. 
+Type = N: new, O: original, R: revised. 
+An online survey, designed in Google Form, was conducted to collect data in October and 
+November 2019. Current student status (ongoing studies) was the one and only eligibility 
+criterion for students to participate in the study. Convenience sampling method was used. 
+Students of nine Hungarian universities located in different regions of Hungary were asked 
+to fill in the online questionnaire. The internal messaging systems of the universities were 
+used to reach their students. Due to the low response rate, the sample size is 297. 
+SPSS 24 was used for data analysis (Babbie, Wagner and Zaino, 2019), and MS Excel for 
+data visualisation (Walkenbach, 2016). Means were calculated to quantify the importance 
+(R1) and performance (R2) of each item of the sustainable university scale. Importance- 
+performance matrix was drawn to illustrate the position of SUS items to answer R3 (Kim et 
+al., 2018.). To answer R4, Principle Component Analysis was run to understand patterns in 
+
+AE 
+Students’ Perceptions of Sustainable Universities in Hungary: An Importance-
+Performance Analysis 
+ 
+504 
+Amfiteatru Economic 
+SUS items (Jolliffe, 2011). The reliability of the sustainable university scale (SUS) was 
+measured by Cronbach's alpha to answer R5 (DeVellis, 2017). Frequency tables of 
+demographic variables were also calculated (Babbie, Wagner and Zaino, 2019). 
+2.2. The sample 
+Of the sample of 297 respondents, 61.3% was female, 38.7% male (Table no. 2). Mostly 
+undergraduate students (77.1%) participated in this study, however some graduate students 
+(16.8%) and doctoral students (6.1%) contributed to the survey. The majority of the 
+respondents (54.2%) fell into the category ‘aged 18-24’. Most of the students in the sample 
+study social sciences (51.1%), engineering (23.9%) or humanities (13.87%). A significant 
+part of them study in Miskolc (75.4%), the rest (24.6%) in other Hungarian universities. 
+Therefore, this convenience sample is not representative, which is a limitation of this study. 
+Table no. 2: Distribution of demographic variables (N=297) 
+Demographic variables 
+Values 
+Frequency 
+Percent 
+Gender 
+male 
+115 
+38.7 
+ 
+female 
+182 
+61.3 
+Study level 
+bachelor 
+229 
+77.1 
+ 
+master 
+50 
+16.8 
+ 
+PhD 
+18 
+6.1 
+Age 
+18-24 
+161 
+54.2 
+ 
+25-31 
+60 
+20.2 
+ 
+32-38 
+29 
+9.8 
+ 
+39-45 
+27 
+9.1 
+ 
+46- 
+20 
+6.7 
+Branch of science 
+agricultural sciences 
+2 
+0.7 
+ 
+arts 
+4 
+1.3 
+ 
+engineering 
+71 
+23.9 
+ 
+humanities 
+41 
+13.8 
+ 
+medicine 
+19 
+6.4 
+ 
+natural sciences 
+7 
+2.4 
+ 
+social sciences 
+153 
+51.5 
+University  
+Corvinus University 
+2 
+0.7 
+ 
+Eszterhazy Uni. Eger 
+14 
+4.7 
+ 
+METU Budapest 
+3 
+1.0 
+ 
+National Uni. of Public Service 
+2 
+0.7 
+ 
+Szechenyi Uni. Gyor 
+7 
+2.4 
+ 
+University of Miskolc 
+224 
+75.4 
+ 
+University of Pannonia 
+27 
+9.1 
+ 
+University of Pecs 
+15 
+5.1 
+ 
+University of Szeged 
+3 
+1.0 
+ 
+
+Sustainable University  
+AE 
+ 
+Vol. 22 • No. 54 • May 2020 
+505 
+ 
+3. Results and discussion 
+This chapter is divided into five main sections, and each of them discusses the results 
+related to one of the research questions. 
+3.1. Importance of the sustainable university scale items (student expectations) 
+In order to answer the first research question (R1), and to investigate student expectations 
+about university sustainability in Hungary, the items of SUS were measured on a five-point 
+importance scale. The lowest value (1) means the item is not at all important, whereas the 
+highest value (5) indicates the item is very important. The importance of SUS items refers 
+to the students’ expectations regarding university sustainability. It expresses their opinion 
+on what a university should do in order to be sustainable. 
+It was found that the opportunity for separate waste collection on campus and 
+encouragement of this activity by the university is the most important attribute of university 
+sustainability (4.54), whereas regular sustainability audits performed on campus is the least 
+important for university students (3.51). They consider water and energy savings (e.g. the 
+use of LEDs) as well as sustainable university buildings that are designed or converted in 
+an energy efficient and sustainable way (e.g. windows, doors, insulation) extremely 
+important (4.43).  
+If a university intends to be more sustainable, it must make efforts to provide the necessary 
+infrastructure for sperate waste collection and promote this activity. The sustainable 
+university should save water and energy and invest in sustainable, energy efficient 
+buildings on campus. These findings are not fully consistent with those of Dagiliute, 
+Liobikiene and Minelgaite (2018), who found recycling is less important for students. 
+It is also crucial for the students that university buildings must be located in a natural 
+setting, e.g. in a quiet, green area with many trees where the air quality is excellent (4.39). 
+Students, therefore, expect sustainable universities not only to be green, but to be located in 
+a green environment. For the most important stakeholders, it is also essential that the 
+sustainable university should use renewable energy sources, e.g. solar panels (4.35), it has a 
+sustainability strategy (4.1) and promotes sustainability research (4.07).  
+It was also found that students think it important that the sustainable university carries out 
+environmentally and socially responsible purchasing practices (4.0) and encourages the use 
+of public transport, bikes (4.0). In a sustainable university, it is important that all the 
+students, researchers, academic and non-academic staff should be aware of the 
+sustainability strategy of the university (3.95) and sustainability information should be 
+readily available on the university’s website, newsletters, etc. (3.94), as well as the 
+university should create partnerships with government, non-governmental organizations, 
+and industry working toward sustainability (3.94). Green actions and projects (3.9) and 
+community outreach programs (3.89) were found to be even less important. 
+The existence of environmental student organization(s) (3.76), the integration of 
+sustainability research results into the curricula (3.72) as well as sustainability-focused 
+positioning, when the university distinguishes itself as sustainable/green from other higher 
+education institutions (3.71) are even less central for the students. There is only moderate 
+demand for subjects/courses about sustainability (3.71) Students do not require that a lot of 
+
+AE 
+Students’ Perceptions of Sustainable Universities in Hungary: An Importance-
+Performance Analysis 
+ 
+506 
+Amfiteatru Economic 
+information about sustainability should be integrated into normal courses (3.61) or the 
+university should offer a lot of study programmes related to sustainability issues (3.6). 
+These results match those observed in earlier studies (Dagiliute, Liobikiene and Minelgaite, 
+2018). The overall mean of the importance items is 3.98. (Table no. 3.) 
+3.2. Perceived performance of the sustainable university items (student satisfaction) 
+In order to answer the second research question (R2), and to find out to what extent 
+students are satisfied with the performance of the Hungarian universities towards 
+sustainability, students were asked to rate the performance of the universities on a five-
+point rating scale. The lowest score (1) indicates very poor rating (dissatisfaction), whereas 
+the highest score (5) means excellent rating (very high satisfaction). The rating scores of the 
+sustainable university scale items refer to how students are satisfied with the sustainability 
+performance of the university where they study. It expresses their opinion on how 
+sustainable the university is perceived regarding each attribute (item) of the sustainable 
+university scale. It allows decision-makers to get more insight into how their efforts 
+towards sustainability are seen by their students, their most important stakeholders. 
+As far as the perceived sustainability performance of the Hungarian universities is 
+concerned, their overall performance rating is only 3.23, which means that that students are 
+not satisfied with it and consider Hungarian universities unsustainable (Table no. 3). These 
+results provide support for the first hypothesis (H1), therefore it has been accepted.  
+Students are most satisfied with the location of the university buildings, the rating of which 
+is very good (4.17). It suggests that Hungarian universities have preferred locations that are 
+mostly situated in quiet, green areas with many trees where the air quality is excellent. This 
+could be a strength they capitalise on. Student are also satisfied with the separate waste 
+collection opportunities on campus (3.7), community outreach programs benefiting the 
+local environment (3.5) and the promotion of sustainability research (3.47). Students are 
+somewhat satisfied with the efforts made towards sustainability strategy (3.35), 
+partnerships with government, non-governmental organizations, and industry working 
+towards sustainability (3.34), as well as sustainable university buildings (3.33), water and 
+energy savings in the university (3.29) and the use of public transport and bikes (3.27). 
+However, students are not really satisfied with how much information about sustainability 
+is integrated into normal courses (3.14) and the mostly unsustainable purchasing practices 
+of universities (3.13). They are not convinced by the green actions/projects (3.12) and the 
+integration of sustainability research results into the curricula (3.11).  
+Moreover, students think only limited information on sustainability is available for them on 
+the website or in the newsletters of the universities (3.08). This is a serious problem as the 
+lack of information is usually one of the greatest barriers towards sustainability (Avila et al, 
+2017). Also, students think that they and other important stakeholders (researchers, 
+academic and non-academic staff) are not aware of the sustainability strategy of the 
+university (3.06), however their participation would be essential in the implementation. 
+Students do not think that universities position themselves as sustainable/green (3.03) or 
+use solar panels or other renewable energy sources (3.02). They are not content with the 
+number of subjects/courses about sustainability (3.0), green/environmental student 
+organizations (2.99) and the number of study programmes related to sustainability (2.93). 
+Students were found to be the least satisfied with the sustainability audits on campus (2.82).  
+
+Sustainable University  
+AE 
+ 
+Vol. 22 • No. 54 • May 2020 
+507 
+3.3. Importance-performance analysis (IPA) 
+In this section, in line with research question 3 (R3), it is discussed whether combining the 
+sustainable university scale (SUS) with importance-performance analysis (IPA) could be a 
+useful strategic tool for university decision-makers to identify key areas of university 
+sustainability. In order to determine the position of each item of the sustainable university 
+scale in the quadrants of the importance-performance matrix, deviations of the means from 
+the overall mean of importance (Δ IMP) and performance (Δ PER) were calculated. Table 
+no. 3 shows the results and the position of each item in the quadrants of IPA. 
+Seven attributes of the sustainable university scale including location, separate waste 
+collection, strategy, energy and water savings, public transport, research and sustainable 
+buildings fall into the ‘Keep up the good work” quadrant (Q1), which contains the 
+competitive advantages (strengths) of the Hungarian universities with regard to 
+sustainability. It is suggested that universities should use all of them in communication 
+campaigns targeted at students who are concerned about sustainability.  
+Table no. 3: Importance and performance of the sustainable university scale items  
+ 
+ 
+IMP 
+means 
+PER 
+means 
+Δ IMP 
+Δ PER 
+Quad- 
+rant 
+1 
+Sustainability strategy 
+4.10 
+3.35 
+0.12 
+0.12 
+Q1 
+2 
+Awareness of the sust. strategy 
+3.95 
+3.06 
+-0.03 
+-0.17 
+Q3 
+3 
+Sustainability audits 
+3.51 
+2.82 
+-0.47 
+-0.41 
+Q3 
+4 
+Sustainability information 
+3.94 
+3.08 
+-0.04 
+-0.15 
+Q3 
+5 
+Green positioning 
+3.71 
+3.03 
+-0.27 
+-0.20 
+Q3 
+6 
+Green purchasing 
+4.00 
+3.13 
+0.02 
+-0.10 
+Q4 
+7 
+Separate waste collection 
+4.54 
+3.70 
+0.56 
+0.47 
+Q1 
+8 
+Renewable energy sources 
+4.35 
+3.02 
+0.37 
+-0.21 
+Q4 
+9 
+Water and energy savings 
+4.43 
+3.29 
+0.45 
+0.06 
+Q1 
+10 
+Public transport, bikes 
+4.00 
+3.27 
+0.02 
+0.04 
+Q1 
+11 
+Sustainable buildings 
+4.43 
+3.33 
+0.45 
+0.10 
+Q1 
+12 
+Green location 
+4.39 
+4.17 
+0.41 
+0.94 
+Q1 
+13 
+Community outreach programs 
+3.89 
+3.50 
+-0.09 
+0.27 
+Q2 
+14 
+Sustainability partnerships 
+3.94 
+3.34 
+-0.04 
+0.11 
+Q2 
+15 
+Green student organization(s) 
+3.76 
+2.99 
+-0.22 
+-0.24 
+Q3 
+16 
+Green actions, projects 
+3.90 
+3.12 
+-0.08 
+-0.11 
+Q3 
+17 
+Green study programmes 
+3.60 
+2.93 
+-0.38 
+-0.30 
+Q3 
+18 
+Green subjects/courses 
+3.71 
+3.00 
+-0.27 
+-0.23 
+Q3 
+19 
+Greening normal courses 
+3.61 
+3.14 
+-0.37 
+-0.09 
+Q3 
+20 
+Sustainability research 
+4.07 
+3.47 
+0.09 
+0.24 
+Q1 
+21 
+Sustainability research integration 
+3.72 
+3.11 
+-0.26 
+-0.12 
+Q3 
+ 
+Total 
+3.98 
+3.23 
+ 
+ 
+ 
+  Notes: IMP: importance; PER: performance; Quadrants: (1) Keep up the good work  (2) Possible 
+overkill (3) Low priority (4) Concentrate here 
+ 
+
+AE 
+Students’ Perceptions of Sustainable Universities in Hungary: An Importance-
+Performance Analysis 
+ 
+508 
+Amfiteatru Economic 
+Campus location is found to be the biggest strength. The favourable location is very 
+important for the students. They require that university buildings should be situated in a 
+quiet, green environment, and for most of them, this expectation is fully met. Separate 
+waste collection, which is the most important aspect of the sustainable university from the 
+students’ perspective, is also a major strength as students are quite satisfied with it. 
+Hungarian universities must communicate that they provide the infrastructure for separate 
+waste collection and promote this activity. 
+Based on our findings, it is advisable for universities to emphasize that their students are 
+satisfied with their efforts towards energy and water savings and appreciate their 
+endeavours to increase energy efficiency on campus. Also, students are content with how 
+sustainable the design of the university buildings is. It can also be suggested that Hungarian 
+higher education institutions should communicate that they promote sustainability research, 
+encourages the use of public transport, bikes and they have a written sustainability strategy. 
+Two items can be found in Q2, which is the possible overkill quadrant. It contains items 
+that are not important for the students, however they, the most important stakeholders of the 
+universities are satisfied with it (performance ratings are better than the overall average). 
+The performance of universities concerning community outreach programs, partnership 
+with governmental, non-governmental organizations, and industry is better than required. In 
+this case, it is suggested that universities should make a communication campaign to 
+increase the importance of their community outreach programs and sustainability 
+partnership to turn those activities into competitive advantages. 
+Ten items – nearly the half of the sustainable university scale items – can be found in Q3, 
+which represents “Low priority” attributes having low importance and low perceived 
+performance. The items that fall into this quadrant are respectively: 1) awareness of the 
+sustainability strategy; 2) regular sustainability audits; 3) information regarding 
+sustainability (website, newsletters, etc.); 4) sustainability-focused positioning of the 
+universities; 5) active green student organizations; 6) sustainability-related projects/actions; 
+7) study programs related to sustainability; 8) subjects/courses related to sustainability; 9) 
+integration of sustainability into normal/traditional courses; and finally, 10) integration of 
+sustainability research results into the curricula. Hungarian universities are strongly advised 
+to avoid any investments in those activities.  
+Last but not at least, two sustainable university items can be found in Q4. This is the 
+“Concentrate here” quadrant representing attributes that universities should immediately 
+improve to achieve higher student satisfaction with regard to their attempts to be more 
+sustainable. The items listed here have high importance and low perceived performance 
+suggesting that students are really dissatisfied with them in spite of the fact that those items 
+are really important for them. On the one hand, they do not believe that universities have 
+environmentally and socially responsible purchasing practices, on the other hand they are 
+disappointed with the use renewable energy sources (e.g. solar panels) on campus. It is 
+therefore suggested that universities should concentrate more on green/socially responsible 
+procurement and should increase the use of renewable energy sources to make students 
+who are concerned about sustainability more satisfied. Universities should consider more 
+the sustainability performance of their suppliers. They should be greening their tenders, 
+prefer local suppliers, and install more solar panels, etc. (Figure no. 2). 
+ 
+
+Sustainable University  
+AE 
+ 
+Vol. 22 • No. 54 • May 2020 
+509 
+ 
+Figure no. 2: Importance-performance of the sustainable university scale items 
+As no research has been found that surveyed the perceived importance and performance of 
+the attributes of university sustainability, it is therefore not possible to compare the results 
+discussed here to the findings of previous works. However, this study fills this gap in the 
+literature and propose a new methodology to investigate the attributes of university 
+sustainability. As an answer to R3, it can be concluded that importance performance 
+analysis (IPA) is a strong strategic tool for university decision-makers to identify key areas 
+of university sustainability when combined with the sustainable university scale (SUS). 
+Using the results of IPA, universities could implement corrective actions to make students 
+as stakeholders more satisfied with their efforts to be more sustainable. 
+3.4. Factor analysis of the sustainable university scale items 
+In order to investigate patterns in perceived university sustainability, and answer R4, factor 
+analysis was used. The dataset of the importance of SUS items were analysed as it refers to 
+the students’ expectation. The very high Kaiser-Meyer-Olkin Measure of Sampling 
+Adequacy value (KMO=0.938) indicates that a factor analysis is a useful method with our 
+data. The Bartlett's Test of Sphericity (Approx. Chi-Square = 4400.484; df = 210; Sig.= 
+0.000) also reconfirms it (Jolliffe, 2011).  
+The extraction communalities are acceptable, although the lower values of Green Location 
+and Green Positioning show that they don't fit as well as the others. Only three factors in 
+the initial solution have eigenvalues greater than 1. Together, they account for almost 65% 
+of the variability in the original variables (Table no. 4). This suggests that three latent 
+influences are associated with sustainable university perceptions, but there remains room 
+for a lot of unexplained variation (Babbie, Wagner and Zaino, 2019). The scree plot also 
+confirmed the choice of three components. 
+Table no. 4: Total variance explained 
+
+greenlocation
+4.1
+separatewaste
+collection
+3.6
+Performance
+communityoutreach
+programs
+sustainabilityresearch
+sustainabilitypartnerships
+sustainablebuildings
+sustainabilitystrategy
+publictransport,bikes
+water andenergy savings
+greeningnormal courses
+greenactions,projects
+sustainability
+green purchasing
+3.1
+researchintegration
+sustainabilityinformation
+green subjects/courses
+green positioning
+awarenessofthesust.strategy
+renewableenergysources
+green studentorganization(s)
+greenstudyprogranmimes
+sustainabilityaudits
+2.6
+3.4
+3.6
+3.8
+4
+4.2
+4.4
+4.6
+ImportanceAE 
+Students’ Perceptions of Sustainable Universities in Hungary: An Importance-
+Performance Analysis 
+ 
+510 
+Amfiteatru Economic 
+Compo-
+nent 
+Initial Eigenvalues 
+Extraction Sums of 
+Squared Loadings 
+Rotation Sums of Squared 
+Loadings 
+Total 
+% of 
+Variance 
+Cumulative 
+% 
+Total 
+% of 
+Variance 
+Cumulative 
+% 
+Total % of 
+Variance 
+Cumulative 
+% 
+1 
+10.648 50.706 
+50.706 
+10.648 50.706 
+50.706 
+5.782 27.535 
+27.535 
+2 
+1.650 7.856 
+58.563 
+1.650 7.856 
+58.563 
+3.937 18.746 
+46.281 
+3 
+1.333 6.346 
+64.909 
+1.333 6.346 
+64.909 
+3.912 18.628 
+64.909 
+4 
+.916 4.360 
+69.269 
+ 
+ 
+ 
+ 
+ 
+ 
+21 
+.139 .663 
+100.000 
+ 
+ 
+ 
+ 
+ 
+ 
+Notes: Extraction Method: Principal Component Analysis. 
+To rotate the factor components, Varimax rotation with Kaiser normalization was used. The 
+first rotated factor component is most highly correlated with community outreach 
+programs, sustainability partnerships, green study programmes, green subjects/courses, 
+greening normal courses, sustainability research and sustainability research integration 
+items (Table no. 5). These variables are not particularly correlated with the other two factor 
+components, and each of them refers to actions towards meeting sustainability objectives, 
+or related to education or research, it is therefore the first component called as Sustainable 
+Actions, Education & Research (SAER).  
+Table no. 5: Rotated component matrix 
+Items 
+1. Sust. Actions, 
+Education & 
+Research 
+2. Sust. 
+Operation/ 
+Infrastructure 
+3. Sust. 
+Strategy 
+Type 
+Sustainability strategy 
+0.20 
+0.23 
+0.76 
+ST1 
+Awareness of the sust. strategy 
+0.24 
+0.24 
+0.80 
+ST2 
+Sustainability audits  
+0.39 
+0.06 
+0.73 
+ST3 
+Sustainability information 
+0.25 
+0.32 
+0.74 
+ST4 
+Green positioning 
+0.31 
+0.29 
+0.48 
+ST5 
+Green purchasing 
+0.51 
+0.37 
+0.44 
+PU1 
+Separate waste collection  
+0.15 
+0.78 
+0.24 
+WE1 
+Renewable energy sources 
+0.26 
+0.75 
+0.30 
+WE2 
+Water and energy savings 
+0.13 
+0.82 
+0.31 
+WE3 
+Public transport, bikes 
+0.50 
+0.56 
+-0.01 
+WE4 
+Sustainable buildings 
+0.28 
+0.73 
+0.26 
+WE5 
+Green location 
+0.39 
+0.49 
+0.08 
+LO1 
+Community outreach programs  
+0.60 
+0.31 
+0.33 
+SA1 
+Sustainability partnerships 
+0.68 
+0.25 
+0.33 
+SA2 
+Green student organization(s) 
+0.57 
+0.33 
+0.41 
+SA3 
+Green actions, projects  
+0.63 
+0.28 
+0.45 
+SA4 
+Green study programmes 
+0.83 
+0.16 
+0.19 
+SER1 
+Green subjects/courses 
+0.82 
+0.17 
+0.21 
+SER2 
+Greening normal courses 
+0.71 
+0.16 
+0.27 
+SER3 
+Sustainability research 
+0.71 
+0.34 
+0.30 
+SER4 
+Sustainability research 
+0.78 
+0.22 
+0.26 
+SER5 
+
+Sustainable University  
+AE 
+ 
+Vol. 22 • No. 54 • May 2020 
+511 
+Items 
+1. Sust. Actions, 
+Education & 
+Research 
+2. Sust. 
+Operation/ 
+Infrastructure 
+3. Sust. 
+Strategy 
+Type 
+integration 
+Notes: (1) Extraction Method: Principal Component Analysis. Rotation Method: Varimax with Kaiser 
+Normalization. Rotation converged in 6 iterations. (2) ST: Strategy, commitment & monitoring; PU: 
+Purchasing; WE: Waste & energy; LO: Location; SA: Sustainability actions; SER: Sustainable 
+education & research 
+The second factor component, which is called Sustainable Operation/Infrastructure, are 
+made up of separate waste collection, renewable energy sources, water and energy savings 
+and sustainable buildings. All those items are related to the domain of waste and energy. 
+The third component, Sustainable Strategy, has been named after the items that correlated 
+with it the most. All of them are related to the sustainability strategy including the written 
+sustainability strategy, and its awareness, regular sustainability audits and sustainability 
+information. Because of their moderately large correlations with the first and the third 
+components, green student organizations and green action/projects bridges Sustainable 
+Actions, Education & Research and Sustainable Strategy. Public transport, bikes and green 
+location variables bridge the first and the second components, whereas green positioning 
+and green purchasing are highly correlated with all the three factor components.  
+These results suggest that students form expectations about the three main domains of 
+university sustainability: 1) sustainable strategy, 2) sustainable operations/infrastructure, 
+and 3) sustainable actions/education/research. These are the main topics of the university 
+sustainability in the mind of the most important stakeholder. These findings are not in line 
+with those of previous studies (Nejati and Nejati, 2013; Dagiliute, Liobikiene and 
+Minelgaite, 2018). In both of the earlier studies, the number of factor components was 
+higher, and the structure of the component was different from our results. 
+It is proposed that universities should deal with all the three components separately, and it 
+would be beneficial for them to assign managers in charge to each domain to fully meet 
+student expectations.  
+3.5. Reliability of the sustainable university scale 
+In order to test H2 and to investigate whether all the 21 items of the sustainable university 
+scale reliably measure the same latent variable, a Cronbach's alpha was run on both SUS 
+importance and SUS performance datasets.  
+In the reliability statistics table of SUS importance, Cronbach's alpha was 0.95, which 
+indicates a very high level of internal consistency for our scale with this specific sample 
+(DeVellis, 2017.). The "Cronbach's Alpha “If Item Deleted" column showed that removal 
+of any item would result in a lower Cronbach's alpha, so no items were removed from the 
+21 item-scale. In the reliability statistics table of SUS performance dataset, Cronbach's 
+alpha was even higher (0.985), which indicates an even higher level of internal consistency. 
+Here also no items were removed as the “If Item Deleted" column showed that removal of 
+any item would result in a lower Cronbach's alpha.  
+Also, a reliability analysis was run in order to ensure internal consistency of the identified 
+constructs after the principle component analysis. The high Cronbach’s alpha values 
+confirmed the reliability of the constructs (α Sustainable Strategy = 0.850, no. of items = 5; 
+
+AE 
+Students’ Perceptions of Sustainable Universities in Hungary: An Importance-
+Performance Analysis 
+ 
+512 
+Amfiteatru Economic 
+α Sustainable Operation & Infrastructure = 0.861, no. of items = 6; and α Sustainable 
+Actions, Education & Research= 0.938, no. of items =10). 
+All these findings support H2. It is therefore accepted that the sustainable university scale 
+(SUS) is a reliable construct to measure perceived university sustainability. 
+  
+Conclusions 
+The stakeholder theory suggests that organizations should fully meet stakeholders’ 
+expectations to be successful (Freeman, 2010). Students are one of the biggest and most 
+important stakeholders of universities (Degtjarjova, Lapina and Freidenfelds, 2018), and 
+could have a significant impact on the environment (Emanuel and Adams, 2011). 
+Nowadays, the public demand for more sustainable universities is growing (Md Shahbudin,  
+et al., 2011.). More and more students want to study about sustainability, expect the 
+integration of sustainability research into curricula and prefer universities that make efforts 
+to operate in a more sustainable manner (Dagiliute, Liobikiene and Minelgaite, 2018). 
+University decision-makers (Rector, Chancellor, Deans and the Senate) should consider 
+sustainability issues to a greater extent when developing organizational strategy. This study 
+extends the knowledge of the above decision-makers regarding students’ perception of 
+university sustainability in many aspects. 
+The current study found that separate waste collection on campus is the most important 
+student expectation about sustainability. However, it is not in line with the result of 
+previous studies. Dagiliute, Liobikiene and Minelgaite (2018) found recycling less 
+important for students. Nonetheless, our findings are consistent with those of other studies 
+suggesting that students expect water and energy savings and energy efficient, sustainable 
+university buildings in a sustainable university. Also, it is important for the students that the 
+buildings should be located in a green environment. Universities are therefore advised to 
+promote separate waste collection, save water and energy, and maintain sustainable, energy 
+efficient buildings that are situated in green parks (R1).  
+In the current study, the low value of general satisfaction with the performance of 
+universities towards sustainability (3.23) confirmed H1 and suggests that students are not 
+satisfied with it and consider Hungarian universities rather unsustainable. Students’ 
+perceptions of university sustainability are in line with the weak positions of the Hungarian 
+higher education institution in green rankings (Greenmetric, 2019). Our findings show that 
+students are most satisfied with the location of the university buildings, which suggests that 
+Hungarian universities have preferred locations. Also, students are content with the 
+opportunity to collect waste separately on campus, the community outreach programs that 
+universities offer, and the promotion of research on sustainability (R2). 
+The findings of this research confirmed H3. By combining the importance-performance 
+analysis (IPA) with the sustainable university scale (SUS), a simple but powerful strategic 
+managerial tool can be developed. It could be widely used by university decision-makers to 
+investigate the key areas of university sustainability. IPA helps to identify competitive 
+advantages and major weaknesses in the domains of sustainability and make it possible for 
+decision-makers to implement corrective actions to make students as stakeholders more 
+satisfied with the university's efforts to address sustainability. Two major weaknesses were 
+found in our study. Hungarian universities perform poorly in sustainable purchasing and 
+
+Sustainable University  
+AE 
+ 
+Vol. 22 • No. 54 • May 2020 
+513 
+use less renewable energy (e.g. solar panels) on campus than it is expected by their 
+students. It is therefore suggested that universities should immediately make both their 
+energy use and purchasing process more sustainable. On the other hand, it was also found 
+that campus location and separate waste collection are the major competitive advantages. It 
+is suggested that the major strengths are used in the marketing campaigns of universities to 
+make their green positioning more effective and to build the sustainable university brand 
+image. Strategy, energy and water savings, public transport, sustainable buildings and 
+research are also strengths of the Hungarian universities that should be communicated (R3). 
+The three main domains of university sustainability were also identified. These are the 
+strategy towards sustainability, actions to promote sustainability including education and 
+research, and the sustainable infrastructure/operations. This is a unique structure and 
+different from those presented in earlier studies (Nejati and Nejati, 2013; Dagiliute, 
+Liobikiene and Minelgaite, 2018), which suggests that Hungarian universities should use a 
+nation-specific approach to university sustainability. Future studies on this topic are 
+therefore recommended to investigate it in different cultural and national contexts (R4). 
+The sustainable university scale (SUS) was found to be a reliable construct to measure 
+perceived university sustainability (H2 accepted). The adaptation of this construct is 
+therefore proposed to both researchers and university decision-makers to investigate how 
+students do perceive the efforts that universities make towards sustainability. Combined 
+with IPA, it could be a powerful benchmarking tool, which is an important practical 
+implication (R5). 
+Further research should be done to compare the perceived university sustainability of green 
+and non-green universities in different cultural settings. 
+ 
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+

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+1 
+ 
+Phonon-mediated room-temperature quantum Hall transport in graphene 
+Daniel Vaquero1,†, Vito Clericò1,† , Michael Schmitz2,3, Juan Antonio Delgado-Notario1,4, Adrian Martín-Ramos1, 
+Juan Salvador-Sánchez1, Claudius S. A. Müller5,6, Km Rubi5,6, Kenji Watanabe7, Takashi Taniguchi8, Bernd 
+Beschoten2, Christoph Stampfer2,3, Enrique Diez1, Mikhail I. Katsnelson6, Uli Zeitler5,6, Steffen Wiedmann5,6, 
+Sergio Pezzini9,* 
+1Nanotechnology Group, USAL–Nanolab, Universidad de Salamanca, E-37008 Salamanca, Spain. 
+2JARA-FIT and 2nd Institute of Physics, RWTH Aachen University, 52074 Aachen, Germany. 
+3Peter Grünberg Institute (PGI-9), Forschungszentrum Jülich, 52425 Jülich, Germany. 
+4CENTERA Laboratories, Institute of High Pressure Physics, Polish Academy of Sciences, 29/37 Sokołowska Str, Warsaw, 
+Poland. 
+5High Field Magnet Laboratory (HFML-EMFL), Radboud University, Toernooiveld 7, 6525 ED Nijmegen, The Netherlands. 
+6Radboud University, Institute for Molecules and Materials, Heyendaalseweg 135, 6525 AJ Nijmegen, The Netherlands. 
+7Research Center for Functional Materials, National Institute for Materials Science, 1-1 Namiki Tsukuba, Ibaraki 305-0044, 
+Japan. 
+8International Center for Materials Nanoarchitectonics, National Institute for Materials Science, 1-1 Namiki Tsukuba, 
+Ibaraki 305-0044, Japan. 
+9NEST, Istituto Nanoscienze-CNR and Scuola Normale Superiore, Piazza San Silvestro 12, 56127 Pisa, Italy. 
+ 
+ 
+†These authors contributed equally to this work 
+*email: sergio.pezzini@nano.cnr.it 
+ 
+Abstract 
+The quantum Hall (QH) effect in two-dimensional electron systems (2DESs) is conventionally observed at liquid-
+helium temperatures, where lattice vibrations are strongly suppressed and bulk carrier scattering is dominated 
+by disorder. However, due to large Landau level (LL) separation (~2000 K at B = 30 T), graphene can support the 
+QH effect up to room temperature (RT), concomitant with a non-negligible population of acoustic phonons 
+with a wave-vector commensurate to the inverse electronic magnetic length. Here, we demonstrate that 
+graphene encapsulated in hexagonal boron nitride (hBN) realizes a novel transport regime, where dissipation in 
+the QH phase is governed predominantly by electron-phonon scattering. Investigating thermally-activated 
+transport at filling factor 2 up to RT in an ensemble of back-gated devices, we show that the high B-field 
+
+2 
+ 
+behaviour correlates with their zero B-field transport mobility. By this means, we extend the well-accepted 
+notion of phonon-limited resistivity in ultra-clean graphene to a hitherto unexplored high-field realm. 
+ 
+Introduction 
+Van der Waals heterostructures of graphene and hBN have recently granted experimental access to novel 
+phenomena in condensed matter [1]. The use of hBN as atomically-flat encapsulating dielectric, in particular, 
+permits a drastic reduction of extrinsic disorder in graphene devices [2], leading to the observation of zero-field 
+transport regimes dominated by either electron-electron [3], electron-hole [4] or electron-phonon (e-ph) 
+interaction [5], which manifest over different carrier density and temperature ranges. Toward RT (T ~ 300 K), 
+the scattering of electrons with acoustic phonons was theoretically identified as the main intrinsic contribution 
+to the electrical resistivity in graphene [6–8], implying a carrier mobility exceeding 105 cm2V-1s-1 at low carrier 
+concentration (n < 1012 cm-2). While such figures could already be inferred from early data on disordered SiO2-
+supported graphene (~104 cm2V-1s-1 mobility) [9, 10], at present, the reach of the zero-field acoustic-phonon-
+limit is firmly established as a generic property of high-quality graphene devices [5], also when encapsulated in 
+hBN crystals from different sources [11] or engineered to high doping levels (n > 1013 cm-2) [12]. Notable 
+exceptions to the cleanness-implies-high-RT-mobility scenario are suspended graphene samples, where flexural 
+phonons dramatically contribute to carrier scattering leading to a T2 behaviour of the resistivity [13], and 
+rotationally faulted graphene bilayers close to magic-angle, showing strong phonon-driven T-linear resistivity 
+[14]. The difference between freely suspended graphene and graphene encapsulated in hBN is due to the fact 
+that in the latter case van der Waals interaction between graphene and substrate makes flexural phonons 
+harder, suppressing an intrinsic rippling instability [15]. 
+In this work, we address the fundamental question whether the e-ph mechanism in clean graphene could also 
+govern the electrical transport in the QH regime [16] at temperatures close to RT. In this sense, we note that 
+
+3 
+ 
+previous literature on the RT-QH effect in graphene [17–20] exclusively includes experiments on SiO2-
+supported devices, precluding such investigation.  
+ 
+Results 
+The QH effect in 2DESs manifests when the Fermi level (EF) lies on the localised states between two LLs, formed 
+in a perpendicular magnetic field and separated by an energy gap ΔLL. The interplay between this energy scale 
+and the thermal energy kT governs the basic phenomenology of the electrical transport in the QH regime. 
+When 𝑘𝑇 ≪ 𝛥𝐿𝐿, no conduction takes place in the 2D bulk, while 1D chiral edge states carry the electrical 
+current ballistically, leading to zero longitudinal resistivity (ρxx) when measured in four-probe configuration 
+(Figure 1a, upper panel). As the temperature increases and 𝑘𝑇~𝛥𝐿𝐿, thermal excitation of extended bulk states 
+(close to the LLs centre) exponentially restores bulk conduction and carrier scattering (Figure 1a, lower panel), 
+resulting in a finite value of the longitudinal resistivity minimum according to 𝜌𝑥𝑥 = 𝜌0 exp (− 𝛥𝐿𝐿 2𝑘𝑇
+⁄
+). This 
+relation is vastly employed to estimate the inter-LL separation via T-dependent measurements of the local 
+resistivity minimum (under the precaution that the activation energy underestimates ΔLL due to disorder-
+broadening of the LLs [21]). The pre-factor to the exponential term, ρ0, which is often not considered explicitly, 
+determines the magnitude of the T-activated resistivity (shaded yellow area Figure 1a, lower) and contains 
+information regarding the disorder potential [22, 23]. In perpendicular magnetic fields, e-ph scattering requires 
+lattice vibrations with a wave-vector in the order of the inverse of the magnetic length (𝑙𝐵~ 25 nm √𝐵[T]
+⁄
+) 
+[24], which defines a third energy scale relevant to our problem 𝐸𝑝ℎ = ℏv𝑠 𝑙𝐵
+⁄
+ (where vs is the sound velocity in 
+the material). In conventional 2DESs, the small ΔLL leads to a complete suppression of the QH effect within a 
+few K [25], where the Eph-controlled phonon population can be considered negligible. Although the low 
+electronic mass in 2DESs such as InSb [26] and HgTe [27–29] enables the observation of the QHE up to liquid-
+nitrogen temperature, this is insufficient to ensure 𝑘𝑇 ≫ 𝐸𝑝ℎ and therefore insufficient to realize a 
+predominance of e-ph interaction. This condition, as sketched in Figure 1b, is instead fulfilled by graphene in 
+
+4 
+ 
+the RT-QH regime (the field dependence of Eph and the corresponding T-dependent excitation probability for 
+acoustic phonons in graphene at B = 30 T are shown in SI Figure S1). Under this circumstance, the T-activated 
+resistivity (shaded dark cyan area in Figure 1b) should directly relate to e-ph scattering [24]. 
+Figure 1c shows a representative measurement of the RT-QH effect, acquired at B = 30 T in a 
+hBN/graphene/hBN back-gated Hall bar (sample D2). The Hall conductivity (σxy) presents weak slope changes 
+around filling factors ν = ±2 (Vg ~ ±20 V), while the shelves-like features at low carrier concentration originate 
+from the onset of electron-hole coexistence in the highly-degenerate N = 0 LL [30]. ρxx, in addition to the 
+pronounced maximum around the charge neutrality point (CNP), shows two sizable minima (Figure 1c, inset), 
+indicative of T-activated QH states. Notably, the overall robustness of the RT-QH signatures dramatically differs 
+in high-mobility graphene with respect to SiO2-supported samples [17]; we thoroughly address this striking 
+observation in a separate work, where we study the suppression of the σxy plateaus in ultra-high-quality 
+devices at temperatures significantly lower than RT. In the following, we will focus on the magnitude of ρxx in 
+the RT-QH regime and identify the underlying mechanism employing a collection of dry-assembled hBN/ 
+graphene/hBN heterostructures. 
+ 
+In Figure 2 we present the main transport characteristics of our devices (details on the fabrication are given in 
+Methods), measured at zero magnetic field and at elevated temperatures. Figure 2a shows the RT mobility of 
+three hBN-encapsulated devices, calculated according to the Drude model ( 𝜇 = 1 (𝑛𝑒𝜌𝑥𝑥)
+⁄
+ ), as a function of 
+the carrier density n. All the mobility curves are well above the typical values for SiO2-supported graphene 
+(grey shaded area) over the whole n range. Importantly, sample D3 shows a μ(n) dependence comparable to 
+the data of Ref. [5] (dash-dotted line), demonstrating the standard fingerprint of phonon-limited RT mobility in 
+zero magnetic field [11, 12] (as confirmed by temperature-dependent resistivity data shown in SI Figure S2). 
+We note that, although Wang et al. employed a 15 μm-wide van der Pauw device, e-ph scattering imposes a ~1 
+μm upper bound to the electronic mean free path at B = 0 and RT [5]. Therefore, the zero-field e-ph limit can 
+also be realized using narrow Hall bars, provided that their channel width exceeds 1 μm (1.5 μm to 2.3 μm in 
+
+5 
+ 
+our devices). The overall high quality of the samples is also supported by the observation of fractional QH 
+states at liquid-helium temperature (see data for sample D2 in SI Figure S3, and Ref. [31] for sample D4, 
+fabricated using CVD-grown graphene). In Figure 2b we explore the correlation between the carrier mobility 
+(calculated using the field-effect formula [32]) and the charge inhomogeneity in the CNP region, estimated as 
+the usual n* parameter [33] (see Figure 2b inset for an example of the extraction). We consider data at T = 220 
+K, where clear thermal activation is observed in the RT-QH regime. n* values above the intrinsic CNP thermal 
+broadening (~2.6 × 1010 cm-2 at 220 K, beginning of the x-axis in Figure 2b) quantify the residual disorder, 
+which, in our devices, remains well below the typical observations for graphene on SiO2 (n* in the few-1011 cm-2 
+range).  In addition, as for Refs. [33, 34], the linear μ-1(n*) dependence (see shaded area in Figure 2b) indicates 
+scattering from long-range potentials, attributed to random strain variations generic to graphene on substrates 
+[35]. We can therefore conclude that the devices at disposal (i) span a low-disorder range unexplored in 
+previous RT-QH experiments, and (ii) present a well-defined disorder type, with increasing impact along the D4-
+to-D1 sequence. 
+ 
+We then employ the sample temperature as an experimental knob to control the excitation of both phonons 
+(see SI Figure S1) and bulk-extended electronic states in strong magnetic fields. In Figure 3a we sketch the 
+effect of increasing T on the Landau-quantized electrons in graphene at B = 30 T. Toward RT, the broadening of 
+the Fermi-Dirac distribution around EF (experimentally set by Vg) ensures excited charge carriers from both the 
+N = 0 and N = 1 LLs, across the giant gap ΔLL. Accordingly, the local resistivity minimum at filling factor ν = 2 
+leaves zero and displays increasing finite values, as shown in the experimental curves of Figure 3b. In Figure 3c, 
+we present a complete picture of the T-dependence of ρxx (ν = 2) for samples D1-4, at selected magnetic fields 
+(30 T and 25 T in the main panel and inset, respectively; data at ν = -2 are shown in SI Figure S4). In addition to 
+our data, we show reference points from Ref. [20] (black diamonds, ρxx (ν = 2) in graphene on SiO2), and two 
+theoretical calculations defining different dissipation limits (continuous lines). In both cases we take an 
+activation energy equal to ΔLL/2: this was shown to be accurate for high B-fields in Ref. [20] and should hold 
+
+6 
+ 
+true for clean graphene with reduced LL broadening. The upper line (yellow) assumes the universal 
+conductivity pre-factor due to long-range disorder (2e2/h) [23], multiplied by a factor 4 to take into account the 
+LL degeneracy of graphene. The lower line (dark cyan) is based on the work by Alexeev et al. [24], who 
+calculated the conductivity mediated by two-phonon scattering for graphene in the RT-QH regime. The 
+relevant e-ph process conserves the LL number, but modifies the in-plane electronic momentum. We note that 
+this phenomenology is fundamentally different from that of magneto-phonons oscillations, recently discovered 
+in extra-wide graphene devices [36], which rely on resonant inter-LL scattering at T < 200 K. Here, two-phonon 
+scattering within each LL contributes with a conductivity pre-factor σ0 = σN(T/300 K)(B/10 T)1/2 , which depends 
+both on temperature and magnetic field (in contrast to the constant pre-factor commonly assumed in QH 
+studies). In the ν = 2 state, the predominant contribution to the σN terms comes from the N = 0 LL (0.65 e2/h, 
+one order of magnitude larger with respect to N = 1, 0.06 e2/h) [24]. Strikingly, the resulting activated 
+behaviour, not including any free parameter, is well approximated by our devices, while the reference data 
+from graphene on SiO2 follow the long-range disorder limit. The qualitative agreement between theoretical 
+calculations and experimental data, together with the contrasting behaviour with respect to previous reports 
+[20], indicate that graphene/hBN heterostructures support an e-ph-dominated transport in the RT-QH regime. 
+Arrhenius-type fits to the conductivity [37], shown in SI Figure S5, confirm the contrasting magnitude of the 
+pre-factor for the two generations of graphene devices (as well as the correctness of the assumed gap size). 
+  
+Despite the presence of long-range potentials (Figure 2b), our data clearly indicate that the e-ph pre-factor 
+does not simply add up to the standard long-range disorder term. To elucidate this point, we quantitatively 
+analyse the deviation from the phonon-mediated limit in the different devices. We proceed by fitting the data 
+from samples D1-3 (only two high T curves are acquired for D4 due to experimental limitations) with a 
+generalized relation (Figure 4, inset), which adds to the theoretical e-ph dependence from Ref. [24] an 
+activation part with a constant pre-factor (ρD). This term is intended to account for the effect of residual 
+disorder, and it is the only free parameter in the fits. In Figure 4 we plot the extracted ρD for the three samples 
+
+7 
+ 
+at different magnetic fields, as a function of the n* parameter (averaged between the electron and hole-side). 
+The linear ρD(n*) behaviour observed here (shaded area in Figure 4) indicates that the random strain variations 
+inducing the CNP broadening are also responsible for ρxx exceeding the e-ph limit in the RT-QH regime. Notably, 
+the only device to display an exact e-ph-type dependence (D3, ρD ~ 0), is also the one to show a Drude mobility 
+comparable to the zero-field e-ph limit [5]. Taking into account the sample-dependent correction due to 
+residual disorder, in SI (Figure S6) we proceed to a quantitative investigation of the field and temperature 
+dependence of the conductivity pre-factor in our samples, revealing the expected B1/2 behaviour of the e-ph 
+term. However, we note that the simplified pre-factor proposed in Ref. [24] is the result of several 
+approximations and, more importantly, it neglects the effect of disorder. To better understand the interplay 
+between the different scattering mechanisms underlying the activated resistivity, in SI (Figures S7 and S8) we 
+discuss additional data at lower temperature (down to 50 K) and magnetic field (down to 1 T). We find that ρD 
+drastically increases toward low T, with the activated resistivity exceeding the e-ph limit by more than one 
+order of magnitude in a clean sample. However, as the temperature and magnetic field are increased, ρD 
+progressively drops (i. e., the activated resistivity tends toward the e-ph limit), suggesting a temperature-driven 
+crossover between regimes dominated by either disorder or e-ph interaction (the latter being realized only 
+close to RT). While it is not surprising that the e-ph limit works as a lower bound to the activated resistivity of 
+real samples, the non-universality (i.e., the sample and temperature dependence) of the disorder contribution 
+deserves particular attention in future theoretical treatments of the RT-QH in graphene.  
+ 
+Discussion 
+The physics of graphene is essentially determined by its deviations from flatness (that is, ripples), due to either 
+thermal fluctuations associated to flexural phonons for freely suspended samples or to roughness of substrate 
+like for graphene on SiO2 [15]. In both cases, ripples induce inhomogeneity of electron density with electron 
+and hole puddles in the vicinity of the CNP [38, 39]. In particular, for the case of graphene on SiO2 the 
+
+8 
+ 
+amplitude of induced inhomogeneity of charge-carrier density is estimated as 3×1011 cm-2 [39], in agreement 
+with the above cited experimental values of n*. This makes the system strongly disordered, and any intrinsic 
+scattering mechanisms become irrelevant. Oppositely, the hBN substrate is atomically flat [1] and at the same 
+time suppresses intrinsic ripples which increases the RT carrier mobility by an order of magnitude and makes 
+intrinsic scattering mechanisms dominant [15]. Indeed, experimentally measured n* for our samples is an 
+order-of-magnitude smaller than what is supposed to be induced by ripples at RT. This results in an essentially 
+different picture of QH physics at high enough temperatures. 
+ 
+In conclusion, we showed experimental evidence of predominant e-ph scattering in the QH regime. This is 
+realized by uniquely combining strong magnetic fields, high temperatures and hBN-encapsulation of graphene. 
+Although the RT-QH in graphene has long been known, we showed that mitigation of disorder via van der 
+Waals engineering provides novel insights on the transport mechanisms in this phenomenon. 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+
+9 
+ 
+Methods 
+Graphene-hBN van der Waals assembly and device fabrication 
+hBN/graphene/hBN samples D1-3 are assembled using the standard van der Waals dry pick-up [5], starting 
+from micromechanically exfoliated graphene flakes previously identified by optical and Raman microscopy. 
+Sample D4 is obtained by CVD growth on Cu foil and direct hBN-mediated pick-up after controlled decoupling 
+via Cu surface oxidation [31]. All the devices are fabricated making use of electron beam lithography, reactive 
+ion etching and e-beam evaporation of Cr/Au 1D edge contacts [5]. 
+Magnetotransport measurements 
+We use standard lock-in acquisition at low frequency (13 Hz), with simultaneous ρxx and ρxy measurements in 
+four-probe configuration, either under a constant current excitation (12.5 nA, sample D1-D3) or a constant 
+voltage bias (300 µV, sample D4). The devices are mounted in a VTI system with low-pressure 4He serving as 
+exchange gas, coupling the samples to a liquid-N2 reservoir. The cryogenic system is accommodated in the 
+access bore of a resistive Bitter magnet at HFML-EMFL, with a maximum field of 33 T. 
+ 
+Data Availability 
+The data presented in this study are available at https://doi.org/10.5281/zenodo.7352031 . 
+ 
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+ 
+ 
+Acknowledgements 
+We acknowledge technical support from Y. Lechaux and J. Quereda. This work has been supported by 
+Ministerio de Ciencia e Innovación (Grant PID2019-106820RB-C2-2) and Junta de Castilla y León (Grants 
+SA256P18 and SA121P20, including EU/FEDER funds). This work was supported by HFML-RU/NWO-I, member 
+of the European Magnetic Field Laboratory (EMFL). This work was also supported by CENTERA Laboratories in 
+the frame of the International Research Agendas Program for the Foundation for Polish Sciences co-financed by 
+
+13 
+ 
+the European Union under the European Regional Development Fund (no. MAB/2018/9). D.V. acknowledges 
+financial support from the Ministry of Universities (Spain) (Ph.D. contract FPU19/04224). J.A.D-N thanks the 
+support from the Universidad de Salamanca for the María Zambrano postdoctoral grant funded by the Next 
+Generation EU Funding for the Requalification of the Spanish University System 2021–23, Spanish Ministry of 
+Universities. K.W. and T.T. acknowledge support from the Elemental Strategy Initiative conducted by the MEXT, 
+Japan (Grant Number JPMXP0112101001) and JSPS KAKENHI (Grant Numbers 19H05790, 20H00354 and 
+21H05233). 
+ 
+This version of the article has been accepted for publication, after peer review, but is not the Version of Record 
+and does not reflect post-acceptance improvements, or any corrections. The Version of Record is available 
+online at: https://doi.org/10.1038/s41467-023-35986-3 . 
+ 
+Author Contributions Statement 
+U.Z., S.W. and S.P. conceived the experiment and coordinated the collaboration. D.V., V.C. and M.S. fabricated 
+the graphene devices and performed the transport measurements. J.A.D.-N., A.M.-R. and J.S.-S. provided 
+technical assistance in the cleanroom processing. C.S.A.M. and K.R. provided technical assistance during the 
+high-field experiments. K.W. and T.T. provided single crystals of hBN. B.B., C.S. and E.D. supervised the 
+experimental work. D.V., V.C., M.S., and S.P. performed the data analysis. M.I.K. provided theoretical input for 
+the interpretation of the results. S.P. wrote the manuscript with input from all the co-authors. 
+ 
+Competing Interests Statement 
+The authors declare no competing interests. 
+ 
+ 
+
+14 
+ 
+Figures and Captions 
+ 
+Figure 1 | Dissipation regimes in the quantum Hall phase: high-quality graphene at RT. a, Schematics of 
+temperature-dependent transport in conventional quantum Hall systems, such as 2DESs in semiconductors. At 
+low T (relative to the LL separation, upper part), the electrical current is carried by chiral edge states, leading to 
+zero longitudinal resistance. At higher T (lower part), thermally-excited bulk states give a finite resistivity due to 
+disorder scattering (yellow shading), with negligible contribution from lattice vibrations. b, At RT, graphene 
+supports both the QH effect (due to large inter-LL spacing) and predominant e-ph scattering in high-mobility 
+samples, enabling the realization of a different transport regime, with phonon-mediated dissipation at high 
+magnetic fields (dark cyan shading). c, ρxx (black) and σxy (red) as a function of the back-gate voltage (corrected 
+by a 5.2 V offset from the CNP), measured in hBN-encapsulated sample D2 at B = 30 T and T = 295 K. Inset: 
+zoom-in of ρxx in the vicinity of filling factor ν = 2 (the dark cyan shading indicates the finite value of the 
+resistivity minimum). 
+ 
+
+a
+kT << △LL
+c
+kT<Eph
+ballistic edge transport
+0.4
+25
+conventional quantum
+2
+Hall systems 
+(Uy)
+0.2
+Pxx
+20
+Pxx
+kT ~ △LL
+1
+disorder-mediated
+kT<Eph
+0.0!
+dissipation
+2V
+0xy
+(Uy)
+15
+0
+ (e2/h)
+Pxx
+0
+Pxx
+10
+B = 30 T
+-1
+b
+T= 295 K
+kT~ △LL
+phonon-mediated
+ clean graphene
+dissipatior
+5
+D2
+at RT
+Pxx
+-2
+V
+0
+Pxx
+-30
+-20
+-10
+0
+10
+20
+30
+Vg (V)15 
+ 
+ 
+ 
+Figure 2 | Phonon-limited transport and residual disorder at zero magnetic field. a, RT carrier mobility 
+(calculated according to the Drude model) as a function of the carrier concentration, for three hBN-
+encapsulated devices. The reference dash-dotted line are data from Ref. [5], indicating a carrier mobility 
+limited by electron-acoustic phonon scattering. The grey-shaded area shows the typical mobility for SiO2-
+supported graphene devices, 1-2 × 104 cm2V-1s-1. b, Inverse of the high-temperature (220 K) field-effect mobility 
+as a function of charge inhomogeneity n*, for hBN/graphene/hBN devices D1-4. The shaded area covers a 
+linear fit to the data, as in Ref. [33], ± one standard error on the best-fit intercept and slope. Inset: Log-Log plot 
+of the longitudinal conductivity of sample D1 as a function of the carrier density, exemplifying the extraction of 
+n* (black arrow). 
+
+a
+b
+1.5
+2.0
+· Ref. [5]
+T = 295 K
+D1
+100
+Oxx
+D1
+e'
+D2
+(e2/h)
+D3
+1.5
+ (105 cm²V-1s-1)
+1.0
+: 10
+1
+10
+100
+n (1010 cm2)
+D1
+1.0
+D2
+≥0.5
+h+
+D3
+D4 
+A
+口
+D1
+D2
+0.5
+T = 220 K
+graphene on SiO,/Si
+e'
+△
+D3
+D4
+0.0
+0.0
+0.5
+1.0
+1.5
+2.0
+4
+6
+8
+n (1012 cm-2)
+n* (1010 cm2)16 
+ 
+ 
+Figure 3 | Temperature-activated resistivity and phonon-mediated dissipation in the quantum Hall effect. a, 
+Density of states (DOS) of graphene as a function of energy, at B = 30 T (with a realistic value of LL broadening 
+of 15 K). On top of the DOS we show the Fermi-Dirac distribution, with EF positioned in the middle of the N = 0 
+and N = 1 LL, at two different temperatures, representative of the experimental range considered. b, 
+Temperature-activated longitudinal resistivity in the vicinity of ν = 2, measured in sample D1. c, Minimum of ρxx 
+at ν = 2 as a function of temperature, for the hBN-encapsulated devices. The reference data (black diamonds) 
+are from Ref. [20]. The yellow and dark cyan continuous line are theoretical calculations based on Ref. [23] and 
+Ref. [24], respectively (the shading covers resistivity values within the two theoretical calculations). The 
+magnetic field is 30 T (25 T) in the main panel (inset). 
+ 
+ 
+ 
+
+a
+c
+1.5
+EF
+B= 30 T
+B = 30 T
+B=25 T
+(arb. units)
+Pxx
+DOS
+ALL
+295K
+1.0
+disorder-
+1.0
+((v= 2) (kΩ)
+mediated
+125K
+-2
+0
+2
+0.5
+E (103 K)
+() (Z =4) d
+b
+B = 30 T
+0.0
+0.6
+0.5
+300
+250
+200
+150
+T (K)
+0.4
+283 K
+(Uy) d
+Ref. [20]
+0.2
+口
+D1
+D2
+0.0
+phonon-
+125 K
+D3
+mediated
+D1
+D4
+0.0
+18
+20
+22
+24
+300
+250
+200
+150
+Vg (V)
+T (K)17 
+ 
+ 
+Figure 4 | Sample-dependent disorder contribution to the activated resistivity. Correlation between the T-
+independent pre-factor to the activated resistivity and n*(220 K) for devices D1-3. The shaded area is defined 
+as in Figure 2b. The error bars correspond to ± one standard error from the fits shown in the inset. Inset: fit to 
+the minimum resistivity as a function of temperature (continuous lines), using the generalized formula 
+including both e-ph and disorder contributions, at B = 25 T. 
+
+0.4
+B = 25 T
+0.3
+Pxx (v= 2) (k2)
+0.2
+0.1
+0.0
+300
+250
+200
+150
+Pp (h/e2)
+0.2
+T (K)
+口
+D1
+20 T
+0
+D2
+D3
+D1
+25 T
+D2
+D3
+ATAI
+0.0
+D1
+30 T
+D2
+4
+6
+8
+n* (1010 cm2)

NdFLT4oBgHgl3EQfOC8n/content/tmp_files/2301.12022v1.pdf.txt

@@ -0,0 +1,1399 @@
+arXiv:2301.12022v1  [cs.AI]  27 Jan 2023
+ǫ-Identifiability of Causal Quantities
+Ang Li , Scott Mueller and Judea Pearl
+Cognitive Systems Laboratory, Department of Computer Science,
+University of California, Los Angeles,
+Los Angeles, California, USA.
+{angli, scott, judea}@cs.ucla.edu
+Abstract
+Identifying the effects of causes and causes of ef-
+fects is vital in virtually every scientific field. Of-
+ten, however, the needed probabilities may not be
+fully identifiable from the data sources available.
+This paper shows how partial identifiability is still
+possible for several probabilities of causation. We
+term this ǫ-identifiability and demonstrate its use-
+fulness in cases where the behavior of certain sub-
+populations can be restricted to within some nar-
+row bounds. In particular, we show how unidentifi-
+able causal effects and counterfactual probabilities
+can be narrowly bounded when such allowances are
+made. Often those allowances are easily measured
+and reasonably assumed. Finally, ǫ-identifiability
+is applied to the unit selection problem.
+1
+Introduction
+Both Effects of Causes (EoC) and Causes of Effects (CoE)
+play an important role in several fields, such as health
+science, social science, and business.
+For example, the
+causal effects identified by the adjustment [Pearl, 1993]
+formula helps decision-maker avoid randomized controlled
+trial using purely observational data.
+For another exam-
+ple, probabilities of causation have been proven critical in
+personalized decision-making [Mueller and Pearl, 2022]. Be-
+sides, a linear combination of probabilities of causation
+has been used to solve the unit selection problem defined
+by Li and Pearl [Li and Pearl, 2022b; Li and Pearl, 2019;
+Li and Pearl, 2022d]. Causal quantities can also increase the
+accuracy of machine learning models by combining causal
+quantities with the model’s label [Li et al., 2020].
+The causal quantities have been studied for decades.
+Pearl first defined the causal quantities such as causal
+effects [Pearl, 1993], probability of necessity and suffi-
+ciency (PNS), probability of sufficiency (PS), and prob-
+ability of necessity (PN) [Pearl, 1999] and their identi-
+fiability [Pearl, 2009] using the structural causal model
+(SCM) [Galles and Pearl, 1998; Halpern, 2000]. Pearl also
+proposed the identification conditions of the causal ef-
+fects (i.e., back-door and front-door criteria) [Pearl, 1993].
+Pearl, Bareinboim, etc.
+have studied more conditions for
+identifying the causal effects [Bareinboim and Pearl, 2012;
+Shpitser and Pearl, 2009]. If the causal effects are not iden-
+tifiable, the informative bounds are given by Li and Pearl
+using non-linear programming [Li and Pearl, 2022c]. Then,
+Tian and Pearl proposed the identification conditions of
+the binary probabilities of causation (i.e., monotonicity)
+[Tian and Pearl, 2000]. If the probabilities of causation are
+not identifiable, Tian and Pearl [Tian and Pearl, 2000] also
+have informative tight bounds for them using Balke’s Linear
+programming [Balke and Pearl, 1997]. Mueller, Li, and Pearl
+[Mueller et al., 2021], as well as Dawid [Dawid et al., 2017],
+increased those bounds using additional covariate informa-
+tion and the corresponding causal structure. Recently, Li and
+Pearl also proposed the theoretical work for non-binary prob-
+abilities of causation [Li and Pearl, 2022a].
+In real-world applications, decision-makers are more likely
+to have identifiable cases (i.e., the causal quantities have point
+estimations) because the bounds under unidentifiable cases
+may be less informative (e.g., 0.1 ≤ PNS ≤ 0.9). Besides,
+estimating the bounds often requires a combination of exper-
+imental and observational data. So we wonder if something
+is sitting between the identifiable and the bounds. Inspired by
+the idea of the confidence interval, in this paper, we proposed
+the definition of ǫ-identifiability, in which more conditions
+of ǫ-identifiability can be found while the estimations of the
+causal quantities are still near point estimations.
+2
+Preliminaries
+Here, we review the definition of PNS, PS, and PN de-
+fined by Pearl [Pearl, 1999], as well as the definition of
+identifiable and the conditions for identifying PNS, PS, and
+PN [Tian and Pearl, 2000].
+Besides, we review the tight
+bounds of PNS, PS, and PN when they are unidentifiable
+[Tian and Pearl, 2000]. Readers who are familiar with the
+above knowledge may skip this section.
+Similarly to any works mentioned above,
+we used
+the causal language of the SCM [Galles and Pearl, 1998;
+Halpern, 2000].
+The introductory counterfactual sentence
+“Variable Y would have the value y, had X been x” in this
+language is denoted by Yx = y, and shorted as yx. We have
+two types of data: experimental data, which is in the form
+of causal effects (denoted as P(yx)), and observational data,
+which is in the form of a joint probability function (denoted
+as P(x, y)).
+
+First, the definition of identifiable for any causal quantities
+defined using SCM is as follows:
+Definition 1 (Identifiability). Let Q(M) be any computable
+quantity of a class of SCM M that is compatible with graph
+G. We say that Q is identifiable in M if, for any pairs of
+models M1 and M2 from M, Q(M1) = Q(M2) whenever
+PM1(v) = PM2(v), where P(v) is the statistical data over
+the set V of observed variables. If our observations are lim-
+ited and permit only a partial set FM of features (of PM(v))
+to be estimated, we define Q to be identifiable from FM if
+Q(M1) = Q(M2) whenever FM1 = FM2. [Pearl, 2009]
+Second, the definitions of three binary probabilities of cau-
+sation defined using SCM are as follow [Pearl, 1999]:
+Definition 2 (Probability of necessity (PN)). Let X and Y
+be two binary variables in a causal model M, let x and y
+stand for the propositions X = true and Y = true, respec-
+tively, and x′ and y′ for their complements. The probability
+of necessity is defined as the expression
+PN
+=
+∆
+P(Yx′ = false|X = true, Y = true)
+=
+∆
+P(y′
+x′|x, y)
+Definition 3 (Probability of sufficiency (PS)). Let X and Y
+be two binary variables in a causal model M, let x and y
+stand for the propositions X = true and Y = true, respec-
+tively, and x′ and y′ for their complements. The probability
+of sufficiency is defined as the expression
+PS =
+∆ P(yx|y′, x′)
+Definition 4 (Probability of necessity and sufficiency (PNS)).
+Let X and Y be two binary variables in a causal model M, let
+x and y stand for the propositions X = true and Y = true,
+respectively, and x′ and y′ for their complements. The proba-
+bility of necessity and sufficiency is defined as the expression
+PNS =
+∆ P(yx, y′
+x′)
+Third, we review the identification conditions for causal
+effects [Pearl, 1993; Pearl, 1995].
+Definition 5 (Back-door criterion). Given an ordered pair of
+variables (X, Y ) in a directed acyclic graph G, a set of vari-
+ables Z satisfies the back-door criterion relative to (X, Y ), if
+no node in Z is a descendant of X, and Z blocks every path
+between X and Y that contains an arrow into X.
+If a set of variables Z satisfies the back-door criterion for
+X and Y , the causal effects of X on Y are identifiable and
+given by the adjustment formula:
+P(yx) =
+�
+z
+P(y|x, z)P(z).
+(1)
+Definition 6 (Front-door criterion). A set of variables Z is
+said to satisfy the front-door criterion relative to an ordered
+pair of variables (X, Y ) if:
+• Z intercepts all directed paths from X to Y ;
+• there is no back-door path from X to Z; and
+• all back-door paths from Z to Y are blocked by X.
+If a set of variables Z satisfies the front-door criterion for
+X and Y , and P(x, Z) > 0, then the causal effects of X on
+Y are identifiable and given by the adjustment formula:
+P(yx) =
+�
+z
+P(z|x)
+�
+x′
+P(y|x′, z)P(x′).
+If causal effects are not identifiable, Tian and Pearl
+[Tian and Pearl, 2000] provided the following bounds for the
+causal effects.
+P(x, y) ≤ P(yx) ≤ 1 − P(x, y′).
+(2)
+Finally, we review the identification conditions for PNS,
+PS, and PN [Tian and Pearl, 2000].
+Definition 7. (Monotonicity) A Variable Y is said to be mono-
+tonic relative to variable X in a causal model M iff
+y′
+x ∧ yx′ = false.
+Theorem 8. If Y is monotonic relative to X, then PNS, PN,
+and PS are all identifiable, and
+PNS = P(yx) − P(yx′),
+PN = P(y) − P(yx′)
+P(x, y)
+,
+PS = P(yx) − P(y)
+P(x′, y′)
+.
+If PNS, PN, and PS are not identifiable, informative bounds
+are given by Tian and Pearl [Tian and Pearl, 2000].
+max
+
+
+
+
+
+0,
+P(yx) − P(yx′),
+P(y) − P(yx′),
+P(yx) − P(y)
+
+
+
+
+
+≤ PNS
+(3)
+min
+
+
+
+
+
+
+
+
+
+P(yx),
+P(y′
+x′),
+P(x, y) + P(x′, y′),
+P(yx) − P(yx′)+
+P(x, y′) + P(x′, y)
+
+
+
+
+
+
+
+
+
+≥ PNS
+(4)
+max
+�
+0,
+P (y)−P (yx′)
+P (x,y)
+�
+≤ PN
+(5)
+min
+�
+1,
+P (y′
+x′)−P (x′,y′)
+P (x,y)
+�
+≥ PN
+(6)
+max
+�
+0,
+P (y′)−P (y′
+x)
+P (x′,y′)
+�
+≤ PS
+(7)
+min
+�
+1,
+P (yx)−P (x,y)
+P (x′,y′)
+�
+≥ PS
+(8)
+The identification conditions mentioned above (i.e., back-
+door and front-door criteria and monotonicity) are robust.
+However, it may still be hard to achieve in real-world appli-
+cations. In this work, we extend the definition of identifia-
+bility, in which a sufficiently small interval is allowed. By
+the new definition, the estimates of causal quantities are still
+near point estimations, and more conditions for identifiability
+could be discovered. If nothing is specified, the discussion
+in this paper will be restricted to binary treatment and effect
+(i.e., X and Y are binary).
+
+3
+Main Results
+First, we extend the definition of identifiability, which we call
+ǫ-identifiability.
+Definition 9 (ǫ-Identifiability). Let Q(M) be any computable
+quantity of a class of SCM M that is compatible with graph
+G. We say that Q is ǫ-identifiable in M (and ǫ-identified to
+q) if, there exists q s.t. for any model m from M, Q(m) ∈
+[q − ǫ, q + ǫ] with statistical data PM(v), where P(v) is the
+statistical data over the set V of observed variables. If our
+observations are limited and permit only a partial set FM
+of features (of PM(v)) to be estimated, we define Q to be ǫ-
+identifiable from FM if Q(m) ∈ [q − ǫ, q + ǫ] with statistical
+data FM.
+With the above definition, the causal quantity is at a max-
+imum distance of ǫ from its true value. We will use the in-
+fix operator symbol ≈ǫ to represent its left-hand side being
+within ǫ of its right-hand side:
+r ≈ǫ q ⇐⇒ r ∈ [q − ǫ, q + ǫ].
+(9)
+The
+following
+sections
+explicate
+conditions
+for
+ǫ-
+identifiability of causal effects, PNS, PS, and PN.
+3.1
+ǫ-Identifiability of Causal Effects
+The causal effect P(YX) can be ǫ-identified with information
+about the observational joint distribution P(X, Y ). This can
+be seen by rewriting Equation (2) as:
+P(x, y) ⩽ P(yx) ⩽ P(x, y) + P(x′).
+(10)
+Here, P(yx) is ǫ-identified to P(x, y) + ǫ when P(x′) ⩽ 2ǫ.
+This ǫ-identification indicates a lower bound of P(x, y) and
+an upper bound of P(x, y) + 2ǫ. Since P(x′) ⩽ 2ǫ, these
+bounds are equivalent to (10). Notably, only P(x, y) and an
+upper bound on P(x′) are necessary to ǫ-identify P(yx). This
+is generalized in Theorem 10, without any assumptions of the
+causal structure.
+Theorem 10. The causal effect P(YX) is ǫ-identified as fol-
+lows:
+P(yx) ≈ǫ P(x, y) + ǫ
+if P(x′) ⩽ 2ǫ,
+(11)
+P(y′
+x) ≈ǫ P(x, y′) + ǫ
+if P(x′) ⩽ 2ǫ,
+(12)
+P(yx′) ≈ǫ P(x′, y) + ǫ
+if P(x) ⩽ 2ǫ,
+(13)
+P(y′
+x′) ≈ǫ P(x′, y′) + ǫ
+if P(x) ⩽ 2ǫ.
+(14)
+Proof. See Appendix 8.1.
+When the complete distribution P(X, Y ) is known, The-
+orem 10 provides no extra precision over Equation (10). Its
+power comes from when only part of the distribution is known
+and only an upper bound on P(X) is available or able to be
+assumed.
+Knowledge of a causal structure can aid ǫ-identification. In
+Figure 1, there is a binary confounder U. If the full joint dis-
+tribution P(X, Y, U) was available, the causal effect P(YX)
+could be computed simply through the backdoor adjustment
+formula. In the absence of the full joint distribution, Theo-
+rem 11 allows ǫ-identification of P(yx) with only knowledge
+of P(x) and the conditional probability P(y|x) as well as an
+upper bound on P(u).
+U
+X
+Y
+Figure 1: The causal graph, where X is a binary treatment, Y is a
+binary effect, and U is a binary confounder.
+Theorem 11. Given the causal graph in Figure 1 and
+P(u) ≤ P(x) − c for some constant c, where 0 < c ⩽ P(x),
+P(yx) ≈ǫ P(y|x) +
+P(x) − c
+2cP(x) + P(x) + c · ǫ
+if P(u) ≤
+2cP(x)
+2cP(x) + P(x) + c · ǫ.
+(15)
+Specifically, if P(x) ≥ 0.5, then the causal effect P(yx) is
+ǫ-identified to P(y|x) +
+ǫ
+13 if P(u) <
+4
+13ǫ.
+Proof. See Appendix 8.2.
+Note that x ∈ {x, x′}, y ∈ {y, y′}, and u ∈ {u, u′} in
+Theorem 11. The constant c should be maximized satisfying
+both c ⩽ P(x) − P(u) and the condition in Equation (15) for
+a given ǫ. The larger c is, the closer P(yx) is ǫ-identified to
+P(y|x). This needs to be balanced with minimizing ǫ.
+As an example, if P(x) ≥ 0.5 and P(u) ⩽ 0.1, then the
+causal effect P(yx) is ǫ-identified to P(y|x) +
+ǫ
+13 if P(u) ⩽
+4
+13ǫ.
+Essentially, P(yx) is ǫ-identified to P(y|x) plus some frac-
+tion of ǫ when P(u) is sufficiently small.
+Therefore, the
+causal effect P(yx) is near P(y|x) if P(U) is specific (i.e.,
+P(u) or P(u′) is minimal). In this case, Theorem 11 can be
+advantageous over the backdoor adjustment formula to com-
+pute P(yx), even when data on X, Y , and U are available,
+because P(Y |X, U), required for the adjustment formula, is
+impractical to estimate with P(U) close to 0.
+3.2
+ǫ-Identifiability of PNS
+Even though Tian and Pearl derived tight bounds on PNS
+[Tian and Pearl, 2000], the PNS can be potentially further
+narrowed when taking into account particular upper bound
+assumptions on causal effects or observational probabilities.
+This can be seen by analyzing the bounds of PNS in Equa-
+tions (3) and (4). Picking any of the arguments to the max
+function of the lower bound and any of the arguments to the
+min function of the upper bound, we can make a condition
+that the range of those two values is less than 2ǫ. For ex-
+ample, let us pick the second argument of the max function,
+P(yx) − P(yx′), and the first argument of the min function,
+P(yx):
+P(yx) − [P(yx) − P(yx′)] ⩽ 2ǫ,
+P(yx′) ⩽ 2ǫ.
+(16)
+Equation (16) is the assumption and the PNS is the
+ǫ-identified to ǫ above the lower bound or ǫ below the upper
+bound:
+PNS ≈ǫ P(yx) − P(yx′) + ǫ, or
+(17)
+PNS ≈ǫ P(yx) − ǫ.
+(18)
+
+Since it is assumed that P(yx′) ⩽ 2ǫ, Equation (17) is equiv-
+alent to Equation (18). The complete set of ǫ-identifications
+and associated conditions are stated in Theorem 12.
+Theorem 12. The PNS is ǫ-identified as follows:
+PNS ≈ǫ ǫ
+if P(yx) ⩽ 2ǫ,
+(19)
+PNS ≈ǫ ǫ
+if P(y′
+x′) ⩽ 2ǫ,
+(20)
+PNS ≈ǫ ǫ
+if P(x, y) + P(x′, y′) ⩽ 2ǫ,
+(21)
+PNS ≈ǫ ǫ
+if P(yx) − P(yx′)+
+P(x, y′) + P(x′, y) ⩽ 2ǫ,
+(22)
+PNS ≈ǫ P(yx) − ǫ
+if P(yx′) ⩽ 2ǫ,
+(23)
+PNS ≈ǫ P(y′
+x′) − ǫ
+if P(y′
+x) ⩽ 2ǫ,
+(24)
+PNS ≈ǫ P(yx)−
+P(yx′) + ǫ
+if P(x, y′) + P(x′, y) ⩽ 2ǫ,
+(25)
+PNS ≈ǫ P(yx)−
+P(yx′) + ǫ
+if P(yx′) − P(yx)+
+P(x, y) + P(x′, y′) ⩽ 2ǫ,
+(26)
+PNS ≈ǫ P(x, y)−
+P(x′, y′) − ǫ
+if P(yx′) − P(yx)+
+P(x, y) + P(x′, y′) ⩽ 2ǫ,
+(27)
+PNS ≈ǫ P(y′
+x′) − ǫ
+if P(y′) ⩽ 2ǫ,
+(28)
+PNS ≈ǫ P(yx) − ǫ
+if P(yx) + P(yx′)−
+P(y) ⩽ 2ǫ,
+(29)
+PNS ≈ǫ P(y) − P(yx′) + ǫ
+if P(yx) + P(yx′)−
+P(y) ⩽ 2ǫ,
+(30)
+PNS ≈ǫ P(x, y)+
+P(x′, y′) − ǫ
+if P(x′, y′) + P(yx′)−
+P(x′, y) ⩽ 2ǫ,
+(31)
+PNS ≈ǫ P(y) − P(yx′) + ǫ
+if P(x′, y′) + P(yx′)−
+P(x′, y) ⩽ 2ǫ,
+(32)
+PNS ≈ǫ P(y) − P(yx′) + ǫ
+if P(x′, y) + P(y′
+x′)−
+P(x′, y′) ⩽ 2ǫ,
+(33)
+PNS ≈ǫ P(yx) − ǫ
+if P(y) ⩽ 2ǫ,
+(34)
+PNS ≈ǫ P(y′
+x′) − ǫ
+if P(y′
+x′) − P(yx)+
+P(y) ⩽ 2ǫ,
+(35)
+PNS ≈ǫ P(y) − P(yx′) + ǫ
+if P(y′
+x′) − P(yx)+
+P(y) ⩽ 2ǫ,
+(36)
+PNS ≈ǫ P(x, y)+
+P(x′, y′) − ǫ
+if P(x, y) + P(y′
+x)−
+P(x, y′) ⩽ 2ǫ,
+(37)
+PNS ≈ǫ P(yx) − P(y) + ǫ
+if P(x, y) + P(y′
+x)−
+P(x, y′) ⩽ 2ǫ,
+(38)
+PNS ≈ǫ P(yx) − P(y) + ǫ
+if P(x′, y) + P(y′
+x′)−
+P(x′, y′) ⩽ 2ǫ.
+(39)
+Proof. See Appendix 8.3.
+Note that in the above theorem, eight conditions consist
+solely of experimental probabilities or solely of observational
+probabilities. This potentially eliminates the need for some
+types of studies, at least partially, even when estimating
+a counterfactual quantity such as PNS. For example, if a
+decision-maker knows that P(y) is large (P(y) ⩾ 0.95), they
+can immediately conclude PNS ≈0.05 P(y′
+x′) − 0.05 with-
+out knowing the specific value of P(y). Thus, only a control
+group study would be sufficient.
+3.3
+ǫ-Identifiability of PN and PS
+Tian and Pearl derived tight bounds on PN and PS in addi-
+tion to PNS. Similar to the derivation of Theorem 12, we can
+potentially narrow those bounds by taking into account upper
+bound assumptions on causal effects or observational proba-
+bilities. The set of ǫ-identifications and associated conditions
+are stated in Theorems 13 and 14.
+Theorem 13. The PN is ǫ-identified as follows:
+PN ≈ǫ ǫ
+if P(y′
+x′) − P(x′, y′)
+⩽ 2ǫP(x, y),
+(40)
+PN ≈ǫ 1 − ǫ
+if P(yx′) − P(x′, y)
+⩽ 2ǫP(x, y),
+(41)
+PN ≈ǫ
+P(y) − P(yx′)
+P(x, y)
++ ǫ
+if P(yx′) − P(x′, y)
+⩽ 2ǫP(x, y),
+(42)
+PN ≈ǫ
+P(y′
+x′) − P(x′, y′)
+P(x, y)
+− ǫ
+if P(x, y′)
+⩽ 2ǫP(x, y),
+(43)
+PN ≈ǫ
+P(y) − P(yx′)
+P(x, y)
++ ǫ
+if P(x, y′)
+⩽ 2ǫP(x, y).
+(44)
+Proof. See Appendix 8.4.
+
+Table 1: Results of an observational study with 1500 individuals
+who have access to the medicine, where 1260 individuals chose to
+receive the medicine and 240 individuals chose not to.
+Take the medicine
+Take no medicine
+Recovered
+780
+210
+Not recovered
+480
+30
+Theorem 14. The PS is ǫ-identified as follows:
+PS ≈ǫ ǫ
+if P(yx) − P(x, y)
+⩽ 2ǫP(x′, y′),
+(45)
+PS ≈ǫ 1 − ǫ
+if P(y′
+x) − P(x, y′)
+⩽ 2ǫP(x′, y′),
+(46)
+PS ≈ǫ
+P(y′) − P(y′
+x)
+P(x′, y′)
++ ǫ
+if P(y′
+x) − P(x, y′)
+⩽ 2ǫP(x′, y′),
+(47)
+PS ≈ǫ
+P(yx) − P(x, y)
+P(x′, y′)
+− ǫ
+if P(x′, y)
+⩽ 2ǫP(x′, y′),
+(48)
+PS ≈ǫ
+P(y′) − P(y′
+x)
+P(x′, y′)
++ ǫ
+if P(x′, y)
+⩽ 2ǫP(x′, y′).
+(49)
+Proof. See Appendix 8.5.
+4
+Examples
+Here, we illustrate how to apply ǫ-Identifiability in real appli-
+cations by two simulated examples.
+4.1
+Causal Effects of Medicine
+Consider a medicine manufacturer who wants to know the
+causal effect of a new medicine on a disease. They conducted
+an observational study where 1500 patients were given access
+to the medicine; the results of the study are summarized in Ta-
+ble 1. In addition, the expert from the medicine manufacturer
+acknowledged that family history is the only confounder of
+taking medicine and recovery, and the family history of the
+disease is extremely rare; only 1% of the people have the fam-
+ily history.
+Let X = x denote that a patient chose to take the medicine,
+and X = x′ denote that a patient chose not to take the
+medicine. Let Y = y denote that a patient recovered, and
+Y = y′ denote that a patient did not recover. Let U = u de-
+note that a patient has the family history, and U = u′ denote
+that a patient has no family history.
+To obtain the causal effect of the medicine (i.e., using ad-
+justment formula (1)), we have to know the observational data
+associated with family history, which is difficult to obtain.
+Fortunately, from Table 1, we obtain that P(x) = 0.84 and
+P(y|x) = 0.62. We also have the prior that P(u) = 0.01.
+Since 0.01 = P(u) ≤ P(x) − 0.8 (let c = 0.8) and
+0.01 = P(u) <
+2c∗0.025P (x)
+2cP (x)+P (x)+c = 0.0113, we can ap-
+ply Theorem 11 to obtain that P(yx) is 0.025-identified to
+P(y|x)+
+P (x)−c
+2cP (x)+P (x)+c0.025 = 0.62. This means the causal
+effect of the medicine is very close to 0.62 (i.e., 0.025 close),
+which can not be 0.025 far from 0.62. Then the medicine man-
+ufacturer can conclude that the causal effect of the medicine
+is roughly 0.62 without knowing the observational data asso-
+ciated with the family history.
+Or even simpler, note that P(x) = 0.84 > 0.5 and P(u) =
+0.01 < 0.1, P(u) = 0.01 <
+4
+13 ∗ 0.035 = 0.0108. We obtain
+that P(yx) is 0.035-identified to P(y|x) + 0.035
+13
+= 0.62. The
+decision-maker can make the same conclusion as above.
+4.2
+PNS of Flu Shot
+Consider a newly invented flu shot. After a vaccination com-
+pany introduced a new flu shot, the number of people infected
+by flu reached the lowest point in 20 years (i.e., less than 5%
+of people infected by flu). The government concluded that
+the new flu shot is the key to success. However, some anti-
+vaccination associations believe it is because people’s physi-
+cal quality increases yearly. Therefore, they all want to know
+how many percentages of people are uninfected because of
+the flu shot. The PNS of the flu shot (i.e., the percentage of
+individuals who would not infect by the flu if they had taken
+the flu shot and would infect otherwise) is indeed what they
+want.
+Let X = x denote that an individual has taken the flu shot
+and X = x′ denote that an individual has not taken the flu
+shot. Let Y = y denote an individual infected by the flu and
+Y = y′ denote an individual not infected by the flu.
+If they want to apply the bounds of PNS in Equations (3)
+and (4), they must conduct both experimental and observa-
+tional studies. However, note that P(y) < 0.05, one could
+apply Equation (34) in Theorem 12, which PNS is 0.025-
+identified to P(yx)− 0.025 (i.e., PNS is very close to P(yx)).
+Thus, according to [Li et al., 2022], only an experimental
+study for the treated group with a sample size of 385 is ad-
+equate for estimating PNS.
+5
+ǫ-Identifiability in Unit Selection Problem
+One utility of the causal quantities is the unit selection prob-
+lem [Li and Pearl, 2022b; Li and Pearl, 2019], in which Li
+and Pearl defined an objective causal function to select a set
+of individuals that have the desired mode of behavior.
+Let X denote the binary treatment and Y denote the bi-
+nary effect. According to Li and Pearl, individuals were di-
+vided into four response types: Complier (i.e., P(yx, y′
+x′)),
+always-taker (i.e., P(yx, yx′)), never-taker (i.e., P(y′
+x, y′
+x′)),
+and defier (i.e., P(y′
+x, yx′)). Suppose the payoff of selecting
+a complier, always-taker, never-taker, and defier is β, γ, θ, δ,
+respectively (i.e., benefit vector). The objective function (i.e.,
+benefit function) that optimizes the composition of the four
+types over the selected set of individuals c is as follows:
+f(c) = βP(yx, y′
+x′|c) + γP(yx, yx′|c) +
+θP(y′
+x, y′
+x′|c) + δP(y′
+x, yx′|c).
+Li and Pearl provided two types of identifiability condi-
+tions for the benefit function. One is about the response type
+such that there is no defier in the population (i.e., monotonic-
+ity). Another is about the benefits vector’s relations, such that
+β + δ = γ + θ (i.e., gain equality). These two conditions
+
+Table 2: Results of an experimental study with 1500 randomly se-
+lected customers were forced to apply the discount, and 1500 ran-
+domly selected customers were forced not to.
+Discount
+No discount
+Bought the purchase
+900
+750
+No purchase
+600
+750
+are helpful but still too specific and challenging to satisfy in
+real-world applications. If the benefit function is not identifi-
+able, it can be bounded using experimental and observational
+data. Here in this paper, we extend the gain equality to the
+ǫ-identifiability as stated in the following theorem.
+Theorem 15. Given a causal diagram G and distribution
+compatible with G, let C be a set of variables that does not
+contain any descendant of X in G, then the benefit function
+f(c) = βP(yx, y′
+x′|c) + γP(yx, yx′|c) + θP(y′
+x, y′
+x′|c) +
+δP(yx′, y′
+x|c) is |β−γ−θ+δ|
+2
+-identified to (γ − δ)P(yx|c) +
+δP(yx′|c) + θP(y′
+x′|c) + β−γ−θ+δ
+2
+.
+One critical use case of the above theorem is that decision-
+makers usually only care about the sign (gain or lose) of the
+benefit function. Decision-makers can apply the above theo-
+rem before conducting any observational study to see if the
+sign of the benefit function can be determined, as we will il-
+lustrate in the next section.
+5.1
+Example: Non-immediate Profit
+Consider the most common example in [Li and Pearl, 2019].
+A sale company proposed a discount on a purchase in
+order to increase the total non-immediate profit.
+The
+company assessed that the profit of offering the dis-
+count to complier, always-taker, never-taker, and defier is
+$100, −$60, $0, −$140, respectively. Let X = x denote that
+a customer applied the discount, and X = x denote that a
+customer did not apply the discount. Let Y = y denote that a
+customer bought the purchase and Y = y′ denote that a cus-
+tomer did not. The benefit function is then (here c denote all
+customers)
+f(c) = 100P(yx, y′
+x′|c) − 60P(yx, yx′|c) +
+0P(y′
+x, y′
+x′|c) − 140P(y′
+x, yx′|c).
+The company conducted an experimental study where 1500
+randomly selected customers were forced to apply the dis-
+count, and 1500 randomly selected customers were forced not
+to. The results are summarized in Table 2. The experimental
+data reads P(yx|c) = 0.6 and P(yx′|c) = 0.5.
+Before conducting any observational study, one can con-
+clude that the benefit function is 10-identified to −12 using
+Theorem 15. This result indicates that the benefit function is
+at most 10 away from −12; thus, the benefit function is nega-
+tive regardless of the observational data. The decision-maker
+then can easily conclude that the discount should not offer to
+the customers.
+6
+Discussion
+We have defined the ǫ-identifiability of causal quantities and
+provided a list of ǫ-identifiable conditions for causal effects,
+PNS, PN, and PS. We still have some further discussions
+about the topic.
+First, all conditions except Theorem 11 are conditions from
+observational or experimental data. In other words, if some of
+the observational or experimental distributions satisfied a par-
+ticular condition, then the causal quantities are ǫ-identifiable.
+These conditions are advantageous in real-world applications
+as no specific causal graph is needed.
+However, we still
+love to discover more graphical conditions of ǫ-identifiability,
+such as back-door or front-door criterion.
+Second, the bounds of PNS, PS, PN, and the benefit func-
+tion can be narrowed by covariates information with their
+causal structure [Dawid et al., 2017; Li and Pearl, 2022d;
+Mueller et al., 2021].
+The ǫ-identifiability can also be ex-
+tended if covariates information and their causal structure are
+available, which should be an exciting direction in the future.
+Third, monotonicity is defined using a causal quantity, and
+in the meantime, monotonicity is also an identifiable condi-
+tion for other causal quantities (e.g., PNS). Thus, another
+charming direction is how the ǫ-identifiability of monotonic-
+ity affects the ǫ-identifiability of other causal quantities.
+7
+Conclusion
+In this paper, we defined the ǫ-identifiability of causal quan-
+tities, which is easier to satisfy in real-world applications.
+We provided the ǫ-identifiability conditions for causal effects,
+PNS, PS, and PN. We further illustrated the use cases of the
+proposed conditions by simulated examples.
+References
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+
+8
+Appendix
+8.1
+Proof of Theorem 10
+Proof. From Equation (2) we have,
+P(x, y) ≤ P(yx) ≤ 1 − P(x, y′).
+Let 1 − P(x, y′) − P(x, y) ≤ 2ǫ, we obtain P(x′) ≤ 2ǫ.
+Therefore, P(yx) is ǫ-identified to P(x, y) + ǫ if P(x′) ≤ 2ǫ,
+Equation (11) holds. Similarily, we can substitute x, y with
+x′, y′, respectively. Equations (12) to (14) hold.
+8.2
+Proof of Theorem 11
+Proof. First, by adjustment formula in Equation (1), we have,
+P(yx) = P(y|x, u)P(u) + P(y|x, u′)P(u′).
+Thus,
+P(yx)
+≥
+P(y|x, u′)P(u′)
+=
+P(y|x, u′)(1 − P(u))
+=
+P(x, y, u′)
+P(x, u′) (1 − P(u))
+≥
+P(x, y) − P(u)
+P(x)
+(1 − P(u))
+=
+P(y|x) − P(y|x)P(u) − P(u)
+P(x) + P 2(u)
+P(x)
+≥
+P(y|x) − P(u) − P(u)
+P(x)
+=
+P(y|x) − (1 +
+1
+P(x))P(u).
+Also if P(x) ≥ P(u) + c for some constant c > 0, we have,
+P(yx)
+≤
+P(u) + P(y|x, u′)(1 − P(u))
+≤
+P(u) + P(x, y, u′)
+P(x, u′) (1 − P(u))
+≤
+P(u) +
+P(x, y)
+P(x) − P(u)(1 − P(u))
+≤
+P(u) +
+P(x, y)
+P(x) − P(u)
+=
+P(u) +
+P(x, y)
+P(x)(1 − P (u)
+P (x))
+=
+P(u) +
+P(x, y)(1 − P (u)
+P (x)) + P(y|x)P(u)
+P(x)(1 − P (u)
+P (x))
+=
+P(u) + P(y|x) + P(y|x)P(u)
+P(x) − P(u)
+≤
+P(y|x) + P(u) +
+P(u)
+P(x) − P(u)
+≤
+P(y|x) + P(u) + P(u)
+c
+=
+P(y|x) + P(u)(1 + 1
+c )
+Therefore, we have,
+P(y|x) − (1 +
+1
+P(x))P(u) ≤ P(yx) ≤ P(y|x) + (1 + 1
+c)P(u).
+Let
+(1 + 1
+c)P(u) + (1 +
+1
+P(x))P(u) ≤ 2ǫ.
+We have,
+P(u)
+≤
+2
+2 + 1
+c +
+1
+P (x)
+ǫ
+=
+2cP(x)
+2cP(x) + P(x) + cǫ.
+Then we know that if P(u) ≤
+2cP (x)
+2cP (x)+P (x)+cǫ,
+P(y|x) − (1 +
+1
+P(x))
+2cP(x)
+2cP(x) + P(x) + cǫ ≤
+P(yx),
+P(y|x) + (1 + 1
+c )
+2cP(x)
+2cP(x) + P(x) + cǫ ≥
+P(yx),
+P(y|x) −
+2cP(x) + 2c
+2cP(x) + P(x) + cǫ ≤
+P(yx),
+P(y|x) +
+2cP(x) + 2P(x)
+2cP(x) + P(x) + cǫ ≥
+P(yx).
+Therefore, P(yx) is ǫ-identified to P(y|x)−
+2cP (x)+2c
+2cP (x)+P (x)+cǫ+
+ǫ = P(y|x) +
+P (x)−c
+2cP (x)+P (x)+cǫ.
+Besides, if P(x) ≥ 0.5 and P(u) ≤ 0.1, let c = 0.4, we have
+P(y|x) − (1 +
+1
+P(x))P(u) ≤ P(yx),
+P(y|x) + (1 + 1
+c )P(u) ≥ P(yx).
+P(y|x) − (1 + 1
+0.5)P(u) ≤ P(yx),
+P(y|x) + (1 + 1
+0.4)P(u) ≥ P(yx).
+P(y|x) − 3P(u) ≤ P(yx) ≤ P(y|x) + 3.5P(u).
+Let 3.5P(u) + 3P(u) ≤ 2ǫ, we have P(u) ≤
+4
+13ǫ, and
+P(y|x) − 12
+13ǫ ≤
+P(yx)
+≤ P(y|x) + 14
+13ǫ.
+Therefore, P(yx) is ǫ-identified to P(y|x) − 12
+13ǫ + ǫ =
+P(y|x) +
+ǫ
+13.
+8.3
+Proof of Theorem 12
+Proof. From the bounds of PNS in Equations (3) and (4) is
+as follows:
+max
+
+
+
+
+
+0,
+P(yx) − P(yx′),
+P(y) − P(yx′),
+P(yx) − P(y)
+
+
+
+
+
+≤ PNS
+min
+
+
+
+
+
+
+
+
+
+P(yx),
+P(y′
+x′),
+P(x, y) + P(x′, y′),
+P(yx) − P(yx′)+
++P(x, y′) + P(x′, y)
+
+
+
+
+
+
+
+
+
+≥ PNS.
+
+Let P(yx) − 0 ≤ 2ǫ, we obtain that PNS is ǫ-identified to ǫ if
+P(yx) ≤ 2ǫ, Equation (19) holds.
+Similarly, the rest of 20 equations can be obtained by letting
+P(y′
+x′) − 0
+≤
+2ǫ,
+P(x, y) + P(x′, y′) − 0
+≤
+2ǫ,
+P(yx) − P(yx′) + P(x, y′) + P(x′, y) − 0
+≤
+2ǫ,
+P(yx) − (P(yx) − P(yx′))
+≤
+2ǫ,
+P(y′
+x′) − (P(yx) − P(yx′))
+≤
+2ǫ,
+P(x, y) + P(x′, y′) − (P(yx) − P(yx′))
+≤
+2ǫ,
+P(yx) − P(yx′) + P(x, y′) + P(x′, y)−
+(P(yx) − P(yx′))
+≤
+2ǫ,
+P(yx) − (P(y) − P(yx′))
+≤
+2ǫ,
+P(y′
+x′) − (P(y) − P(yx′))
+≤
+2ǫ,
+P(x, y) + P(x′, y′) − (P(y) − P(yx′))
+≤
+2ǫ,
+P(yx) − P(yx′) + P(x, y′) + P(x′, y)−
+(P(y) − P(yx′))
+≤
+2ǫ,
+P(yx) − (P(yx) − P(y))
+≤
+2ǫ,
+P(y′
+x′) − (P(yx) − P(y))
+≤
+2ǫ,
+P(x, y) + P(x′, y′) − (P(yx) − P(y))
+≤
+2ǫ,
+P(yx) − P(yx′) + P(x, y′) + P(x′, y)−
+(P(yx) − P(y))
+≤
+2ǫ.
+8.4
+Proof of Theorem 13
+Proof. From the bounds of PN in Equations (5) and (6) is as
+follows:
+max
+�
+0,
+P (y)−P (yx′)
+P (x,y)
+�
+≤ PN ≤ min
+�
+1,
+P (y′
+x′)−P (x′,y′)
+P (x,y)
+�
+Let
+P (y′
+x′)−P (x′,y′)
+P (x,y)
+−0 ≤ 2ǫ, we obtain that PN is ǫ-identified
+to ǫ if P(y′
+x′) − P(x′, y′) ≤ 2P(x, y)ǫ, Equation (40) holds.
+Similarly, the rest of 4 equations can be obtained by letting
+1 − P(y) − P(yx′)
+P(x, y)
+≤
+2ǫ,
+P(y′
+x′) − P(x′, y′)
+P(x, y)
+− P(y) − P(yx′)
+P(x, y)
+≤
+2ǫ.
+8.5
+Proof of Theorem 14
+Proof. From the bounds of PS in Equations (7) and (8) is as
+follows:
+max
+�
+0,
+P (y′)−P (y′
+x)
+P (x′,y′)
+�
+≤ PS ≤ min
+�
+1,
+P (yx)−P (x,y)
+P (x′,y′)
+�
+Let P (yx)−P (x,y)
+P (x′,y′)
+− 0 ≤ 2ǫ, we obtain that PS is ǫ-identified
+to ǫ if P(yx) − P(x, y) ≤ 2P(x′, y′)ǫ, Equation (45).
+Similarly, the rest of 4 conditions can be obtained by letting
+1 − P(y′) − P(y′
+x)
+P(x′, y′)
+≤
+2ǫ,
+P(yx) − P(x, y)
+P(x′, y′)
+− P(y′) − P(y′
+x)
+P(x′, y′)
+≤
+2ǫ.
+8.6
+Proof of Theorem 15
+Proof.
+f(c)
+=
+βP(yx, y′
+x′|c) + γP(yx, yx′|c) +
+θP(y′
+x, y′
+x′|c) + δP(y′
+x, yx′|c)
+=
+βP(yx, y′
+x′|c) + γ[P(yx|c) − P(yx, y′
+x′|c)] +
+θ[P(y′
+x′) − P(yx, y′
+x′|c)] + δP(y′
+x, yx′|c)
+=
+γP(yx|c) + θP(y′
+x′|c) + (β − γ − θ)P(yx, y′
+x′|c) +
+δP(y′
+x, yx′|c).
+(50)
+Note that, we have,
+P(y′
+x, yx′|c) = P(yx, y′
+x′|c) − P(yx|c) + P(yx′|c).
+(51)
+Substituting Equation (51) into Equation (50), we have,
+f(c)
+=
+γP(yx|c) + θP(y′
+x′|c) + (β − γ − θ)P(yx, y′
+x′|c) +
+δP(y′
+x, yx′|c)
+=
+γP(yx|c) + θP(y′
+x′|c) + (β − γ − θ)P(yx, y′
+x′|c) +
+δ[P(yx, y′
+x′|c) − P(yx|c) + P(yx′|c)]
+=
+(γ − δ)P(yx|c) + δP(yx′|c) + θP(y′
+x′|c) +
+(β − γ − θ + δ)P(yx, y′
+x′|c).
+Case 1: If β − γ − θ + δ ≥ 0,
+f(c)
+≤
+(γ − δ)P(yx|c) + δP(yx′|c) + θP(y′
+x′|c) +
+β − γ − θ + δ
+2
++ |β − γ − θ + δ|
+2
+=
+(γ − δ)P(yx|c) + δP(yx′|c) + θP(y′
+x′|c) +
+β − γ − θ + δ.
+and,
+f(c)
+≥
+(γ − δ)P(yx|c) + δP(yx′|c) + θP(y′
+x′|c) +
+β − γ − θ + δ
+2
+− |β − γ − θ + δ|
+2
+=
+(γ − δ)P(yx|c) + δP(yx′|c) + θP(y′
+x′|c).
+Therefore, f(c) is |β−γ−θ+δ|
+2
+-identified to (γ − δ)P(yx|c) +
+δP(yx′|c) + θP(y′
+x′|c) + β−γ−θ+δ
+2
+.
+Case 2: If β − γ − θ + δ < 0,
+f(c)
+≤
+(γ − δ)P(yx|c) + δP(yx′|c) + θP(y′
+x′|c) +
+β − γ − θ + δ
+2
++ |β − γ − θ + δ|
+2
+=
+(γ − δ)P(yx|c) + δP(yx′|c) + θP(y′
+x′|c).
+and,
+f(c)
+≥
+(γ − δ)P(yx|c) + δP(yx′|c) + θP(y′
+x′|c) +
+β − γ − θ + δ
+2
+− |β − γ − θ + δ|
+2
+=
+(γ − δ)P(yx|c) + δP(yx′|c) + θP(y′
+x′|c) +
+β − γ − θ + δ.
+
+Therefore, f(c) is |β−γ−θ+δ|
+2
+-identified to (γ − δ)P(yx|c) +
+δP(yx′|c) + θP(y′
+x′|c) + β−γ−θ+δ
+2
+.
+

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@@ -0,0 +1,1074 @@
+arXiv:2301.04728v1  [cs.LO]  11 Jan 2023
+Submitted to MFPS 22
+Patch Locale of a Spectral Locale
+in
+Univalent Type Theory
+Ayberk Tosuna,1
+Mart´ın H. Escard´oa,2
+a School of Computer Science
+University of Birmingham
+Birmingham, United Kingdom
+Abstract
+Stone locales together with continuous maps form a coreflective subcategory of spectral locales and perfect maps. A proof
+in the internal language of an elementary topos was previously given by the second-named author. This proof can be easily
+translated to univalent type theory using resizing axioms. In this work, we show how to achieve such a translation without
+resizing axioms, by working with large, locally small, and small complete frames with small bases. This turns out to be
+nontrivial and involves predicative reformulations of several fundamental concepts of locale theory.
+Keywords:
+locale theory, pointfree topology, patch locale, spectral locale, stone space, univalent type theory
+1
+Introduction
+The category Stone of Stone locales together with continuous maps forms a coreflective subcategory of
+the category Spec of spectral locales and perfect maps i.e. maps preserving compact opens. A proof in the
+internal language of an elementary topos was previously constructed in [8,10], defining the patch frame as
+the frame of Scott continuous nuclei on a given frame.
+The objective of this paper is to carry out this construction in predicative, constructive univalent
+foundations. In the presence of Voevodsky’s resizing axioms [15], it is straightforward to translate the
+above proof to univalent type theory. However, at the time of writing, there is no known constructive
+interpretation of the resizing axioms. In such a predicative situation, the usual approach to locale theory
+is to work with presentations of locales, known as formal topologies [2, 3, 13]. However, we show that
+it is possible to work with locales directly, if we adopt large, locally small, and small complete frames
+with small bases [6]. This requires a number of substantial modifications to the proofs and constructions
+of [8,10]:
+(i) The patch is defined as the frame of Scott continuous nuclei in [8,10]. In order to prove that this is
+indeed a frame, one starts with the frame of all nuclei, and then exhibits the Scott continuous nuclei
+as a subframe. However, this procedure does not seem to be possible in our predicative setting as
+1 Email: a.tosun@pgr.bham.ac.uk
+2 Email: m.escardo@cs.bham.ac.uk
+MFPS 22 Proceedings will appear in Electronic Notes in Theoretical Informatics and Computer Science
+
+Tosun and Escard´o
+it is not clear whether all nuclei form a frame; so we construct the frame of Scott continuous nuclei
+directly, which requires reformulations of all proofs about it inherited from the frame of all nuclei.
+(ii) In the impredicative setting, any frame has all Heyting implications, which is needed to construct
+open nuclei. Again, this does not seem to be the case in the predicative setting. We show, however,
+that it is possible to construct Heyting implications in locally small frames with small bases, by an
+application of the Adjoint Functor Theorem for posets.
+(iii) Similar to (ii), we use the Adjoint Functor Theorem for posets to define the right adjoint of a frame
+homomorphism, using which we define the notion of a perfect map, namely, a map whose defining
+frame homomorphism’s right adjoint is Scott continuous. This notion is used in [8,10].
+For the purposes of this work, a spectral locale is a locale in which the compact opens form a small
+basis closed under finite meets.
+A continuous map of spectral locales is spectral if its defining frame
+homomorphism preserves compact opens. A Stone locale is one that is compact and zero-dimensional (i.e.
+whose clopens form a basis). Every Stone locale is spectral since the clopens coincide with the compact
+opens in Stone locales. The patch frame construction is the right adjoint to the inclusion Stone ֒→ Spec.
+The main contribution of our work is the construction of this right adjoint in the predicative context
+of univalent type theory.
+We have also formalised the development of this paper in the Agda proof
+assistant [1], though our presentation here is self-contained and can be followed independently of the
+formalisation. Although we have omitted some proofs for lack of space, we have included all the crucial
+differences from [8,10] in full.
+The organisation of this paper is as follows. In Section 2, we present the type-theoretical context in
+which we work. In Section 3, we present our definitions of spectral and Stone locales that provide a suitable
+basis for a predicative development. In Section 4, we present a predicative version of the Adjoint Functor
+Theorem for the simplified context of locales that is central to our development. In Section 5, we define
+the meet-semilattice of perfect nuclei as preparation for the complete lattice of perfect nuclei, which we
+then construct in Section 6. Finally in Section 7, we prove the desired universal property, namely, that
+the patch locale exhibits the category Stone as a coreflective subcategory of Spec.
+2
+Foundations
+In this section, we present the type-theoretical setting in which we work and then provide the type-
+theoretical formulations of some of the preliminary notions that form the basis of our work. Our type-
+theoretical conventions follow those of de Jong and Escard´o [5] and the Univalent Foundations Programme
+[14].
+We work in Martin-L¨of Type Theory with binary sums −+−, dependent products �, dependent sums
+�, the identity type − = −, and inductive types including the empty type 0, the unit type 1, and the
+type List(A) of lists over any type A. We adhere to the convention of [14] of using − ≡ − for judgemental
+equality and − = − for the identity type.
+We work explicitly with universes, for which we adopt the convention of using the variables U, V, W,
+and T . The ground universe is denoted U0 and the successor of a given universe U is denoted U+. The
+least upper bound of two universes is given by the operator − ⊔ − which is assumed to be associative,
+commutative, and idempotent. Furthermore, (−)+ is assumed to distribute over − ⊔ −. Universes are
+computed for the given type formers as follows:
+• Given types X : U and Y : V, the type X + Y inhabits universe U ⊔ V.
+• Given a type X : U and an X-indexed family, Y : X → V, both �
+x:X Y (x) and �
+x:X Y (x) inhabit the
+universe U ⊔ V.
+• Given a type X : U and inhabitants x, y : X, the identity type x = y inhabits universe U.
+• The type N of natural numbers inhabits U0.
+• The empty type 0 and the unit type 1 have copies in every universe U, which we occasionally make
+explicit using the notations 0U and 1U.
+• Given a type A : U, the type List(A) inhabits U.
+We assume only function extensionality, propositional extensionality and quotients, and do not need
+2
+
+Tosun and Escard´o
+full univalence for our development. We always maintain a distinction between structure and property,
+and reserve logical connectives for propositional types i.e. types A satisfying isProp (A) := �
+x,y:A x = y.
+We denote by ΩU the type of propositional types in universe U i.e. ΩU := ΣA:UisProp (A).
+We assume the existence of propositional truncation, given by a type former ∥−∥ : U → U and a unit
+operation |−| : A → ∥A∥. The existential quantification operator is defined using propositional truncation
+as:
+∃
+x:A
+B(x)
+:=
+�����
+�
+x:A
+B(x)
+����� .
+When presenting proofs informally, we adopt the following conventions for avoiding ambiguity between
+propositional and non-propositional types:
+• For the anonymous inhabitation |A| of a type, we say that A is inhabited.
+• For truncated Σ types, we use the terminologies there is and there exists;
+2.1
+Directed families
+We now proceed to define some preliminary notions in the type-theoretical setting that we have just
+presented.
+Definition 2.1 (Family) A U-family on a type A is a pair (I, f) where I : U and f : I → A. We denote
+the type of U-families on type A by FamU(A) i.e. FamU (A) := �
+(I:U) I → A.
+We often use the shorthand {xi}i:I for families. In other words, instead of writing (I, f) for a family,
+we write {xi}i:I where xi denotes the application f(i).
+Definition 2.2 (Subfamily) By a subfamily of some U-family (I, f) we mean a family (J, f ◦ g) where
+(J, g) is itself a U-family on I.
+When considering a subfamily J of some family {xi}i:I, we often use the abbreviation {xj | j ∈ J}.
+As mentioned in the introduction, Scott continuity plays a central role in our development. To define
+Scott continuity, we define the notion of a directed family. The definition that we work with (also used by
+de Jong and Escard´o [5]) is the following:
+Definition 2.3 (Directed family) Let {xi}i:I be a family in some type A that is equipped with a preorder
+− ≤ −. The family {xi}i:I is called directed if (1) I is inhabited, and (2) for every i, j : I, there exists
+some k : I such that xk is the upper bound of {xi, xj}.
+2.2
+Definition of locale
+A locale is a notion of space characterised solely by its frame of opens.
+Our definition of a frame is
+parameterised by three universes: (1) for the carrier set, (2) for the order, and (3) for the index type of
+families on which the join operation is defined. We adopt the convention of using the universe variables
+U, V, and W for these respectively. We often omit universe levels in contexts where they are not relevant
+to the discussion. In cases where only the index universe W is relevant, we speak of a W-locale for the
+sake of brevity and omit universes U and V.
+Definition 2.4 (Frame) A (U, V, W)-frame L consists of:
+• a set |L| : U,
+• a partial order − ≤ − : |L| → |L| → ΩV,
+• a top element ⊤ : |L|,
+• an operation − ∧ − : |L| → |L| → |L| giving the greatest lower bound U ∧ V of any two U, V : |L|,
+• an operation �
+: FamW (|L|) → |L| giving the least upper bound �
+i:I Ui of any W-family {Ui}i:I,
+such that binary meets distribute over arbitrary joins, i.e.
+U ∧
+�
+i:I
+Vi =
+�
+i:I
+U ∧ Vi
+3
+
+Tosun and Escard´o
+for every U : |L| and W-family {Vi}i:I in |L|.
+It follows automatically from the antisymmetry condition for partial orders that the underlying type
+of a frame is a set. Finally, we note that most of our results are restricted to (U+, U, U)-frames for a fixed
+universe U, which we refer to as large, locally small, and small complete frames. Even though some of our
+results apply to frames of a more general form, we refrain from presenting the specific level of generality
+for the sake of brevity. For the precise universe levels, we refer the reader to the formalisation.
+Definition 2.5 (Frame homomorphism) Let K and L be a (U, V, W)-frame and a (U′, V′, W)-frame
+respectively. A function h : |K| → |L| is called a frame homomorphism if it preserves the top element,
+binary meets, and joins of W-families. We denote the category of frames and their homomorphisms by
+Frm.
+We adopt the notational conventions of [12]. A locale is a frame considered in the opposite category
+called Loc := Frmop. To highlight this, we adopt the standard convention of using the letters X, Y, Z, . . .
+(or sometimes A, B, C, . . .) for locales and denoting by O(X) the frame corresponding to a locale X. For
+variables that range over the frame of opens of a locale X, we use the letters U, V, W, . . . We use the letters
+f and g for continuous maps X → Y of locales. A continuous map f : X → Y is given by a frame
+homomorphism f ∗ : O(Y ) → O(X).
+Definition 2.6 (Nucleus) A nucleus on a locale X is an endofunction j : O(X) → O(X) that is infla-
+tionary, idempotent, and preserves binary meets.
+In Section 6, we will work with inflationary and binary-meet-preserving functions that are not neces-
+sarily idempotent. Such functions are called prenuclei. We also note that, to show a prenucleus j to be
+idempotent, it suffices to show j(j(U)) ≤ j(U) as the other direction follows from inflationarity. In fact,
+the notion of a nucleus could be defined as a prenucleus satisfying the inequality j(j(U)) ≤ j(U), but we
+define it as in Definition 2.6 for the sake of simplicity and make implicit use of this fact in our proofs of
+idempotency.
+3
+Spectral and Stone locales
+We start by defining the notion of a small basis for a frame. This is crucial not just for the definitions of
+spectral and Stone locales that we use in our development, but also for the Adjoint Functor Theorem that
+we present in Section 4.
+Definition 3.1 (Small basis) Given a W-locale X, a W-family {Bi}i:I of opens of X is said to form a
+basis for O(X) if
+�
+U:O(X)
+∃
+J:FamW(I)
+U =
+�
+{Bj | j ∈ J}.
+A W-locale X is then said to have a small basis if there exists a W-family {Bi}i:I in O(X) that forms a
+basis for O(X).
+Given an open U : O(X) with a small basis, we refer to the family {Bj | j ∈ J} giving U as its join as
+the basic covering family for U.
+It is important to note here that we use propositional truncation when defining the notion of a locale
+having a basis. So even though we often speak of a “locale with some small basis {Bi}i:I”, the existence of
+this basis is a property meaning we have access to it only in contexts where the goal is itself a proposition.
+We often need covering families given by a basis to be directed. This is easy to achieve if we work with
+bases closed under finite joins, which we can do without loss of generality, as this closure produces another
+basis.
+The standard impredicative definition of a spectral locale is as one in which the compact opens form
+a basis closed under binary meets. To talk about compactness, we define the way below relation:
+Definition 3.2 (Way below) Given a W-locale X and opens U, V : O(X), U is said to be way below
+V , written U ≪ V , if �
+(I,f):FamW(O(X))(I, f) directed → V ≤ �(I, f) → ∃i:I U ≤ f(i).
+Proposition 3.3 Given any two opens U and V of a locale, the type U ≪ V is a proposition.
+4
+
+Tosun and Escard´o
+The statement U ≪ V is thought of as expressing that U is compact relative to V . An open is said to
+be compact if it is compact relative to itself:
+Definition 3.4 (Compact open of a locale) An open U : O(X) is called compact if U ≪ U.
+We denote the type of compact opens of a locale X by K(X). We adopt the convention of using letters
+C, D, . . . : K(X) for compact opens.
+Definition 3.5 (Compact locale) A locale X is called compact if its top element ⊤ : O(X) is compact.
+The standard definition of a spectral locale as one in which the compact opens form a basis closed under
+finite meets is problematic in our predicative setting, as it is not always the case that the type of compact
+opens of a (U, V, W)-locale lives in W. In particular, the type of compact opens of a (U+, U, U)-locale
+lives in U+ and it is accordingly said to be large. To address this problem, we restrict attention to locales
+with small bases and express the notion of spectrality by imposing the conditions of interest on the basic
+elements instead.
+Definition 3.6 (Spectral locale) A locale X is said to be spectral if there exists a small basis {Bi}i:I
+such that:
+(i) every Bi is compact, and
+(ii) {Bi}i:I is closed under finite meets i.e. there is t : I with Bt = ⊤ and for any two i, j : I, there is
+k : I such that Bk = Bi ∧ Bj.
+We have previously remarked that we can assume without loss of generality that bases of locales are
+closed under finite joins. Note here that this assumption can also be made for bases of spectral locales as
+compact opens are also closed under finite joins.
+Spectral locales together with spectral maps constitute the category Spec. We now define the notion
+of a spectral map.
+Definition 3.7 (Spectral map) A continuous map f : X → Y between spectral locales X and Y is
+called spectral if f ∗(V ) : O(X) is a compact open of X whenever V is a compact open of Y .
+A natural question to ask about our definition of spectral space is whether it corresponds to the previous
+informal definition: can there be compact opens that do not fall in the basis?
+Proposition 3.8 For any spectral locale X, every compact open of X falls in the basis.
+Proof. Let X be a spectral locale and denote by {Bi}i:I its basis closed under finite joins. Let C : O(X)
+be a compact open and let {Bj}j∈J be the covering family for C. Because the basis is closed under finite
+joins, this family is directed. As C ≤ �
+i:I Bi there must be some k : I by the compactness of C such that
+C ≤ Bk. It is also clearly the case that Bk ≤ C and so Bk = C, meaning C falls in the basis.
+✷
+3.1
+Zero-dimensional and regular locales
+Clopenness is central to the notion of a zero-dimensional locale, similar to the fundamental role played by
+the notion of a compact open in the definition of a spectral space. To define the clopens, we first define
+the well inside relation.
+Definition 3.9 (Well inside relation) Given a locale X and opens U, V : O(X), U is said to be well
+inside V (written U ⪕ V ) if
+∃
+W :O(X)
+(U ∧ W = ⊥) × (V ∨ W = ⊤) .
+Definition 3.10 (Clopen) An open U is called a clopen if it is well inside itself, which amounts to
+saying that it has a Boolean complement.
+Before we proceed to defining zero-dimensionality, we record the following important fact about the
+well inside relation:
+Proposition 3.11 Given opens U, V, W : O(X) of a locale X,
+(i) if U ⪕ V and V ≤ W then U ⪕ W; and
+5
+
+Tosun and Escard´o
+(ii) if U ≤ V and V ⪕ W then U ⪕ W.
+Our definition of zero-dimensionality is analogous to the definition of a spectral locale where conditions
+of interest apply only to basic opens.
+Definition 3.12 (Zero-dimensional frame) A locale is called zero-dimensional if it has a small basis
+{Bi}i:I with each Bi clopen.
+Zero-dimensionality can in fact be viewed as a special case of regularity. For purposes of our develop-
+ment, we need the result that U ≪ V implies U ⪕ V in any zero-dimensional locale [11, Lemma VII.3.5,
+pg. 303]. As this can be strengthened to apply to the more general case of regular locales, we now define
+the notion of regularity, using which we obtain a result slightly more general than needed.
+Definition 3.13 (Regular locale) A locale is called regular if it has some basis {Bi}i:I such that for
+any open U, every Bj in the covering family for U is well inside U.
+Similar to the case of spectral locales, the basis of a regular locale can be assumed to be closed under
+finite joins without loss of generality as every basis can be closed under finite joins to obtain another basis
+satisfying the regularity condition of Definition 3.13.
+Proposition 3.14 Every zero-dimensional locale is regular.
+Proof. Let X be a zero-dimensional locale and call its basis {Bi}i:I. Consider some U : O(X). There
+must be a basic covering U = �
+i∈J Bj such that each Bj is clopen for every j ∈ J. Clearly, Bj ≤ U so we
+have Bj ⪕ Bj ≤ U which implies Bj ⪕ U (by Proposition 3.11.(i)).
+✷
+The following two propositions are needed to prove that compact opens and clopens coincide in Stone
+locales, which we will need later. They are adaptations of standard proofs [11, pg. 303, Lemma VII.3.5]
+into our predicative setting.
+Proposition 3.15 In any regular locale, U ≪ V implies U ⪕ V for any two opens U, V .
+Proof. Let {Bi}i:I be the basis, closed under finite joins, of a regular locale X, let U, V : O(X) such
+that U ≪ V , and let {Bj}j∈J be the basic family covering V . As V ≤ �
+j∈J Bj there must exist some
+k ∈ J such that U ≤ Bk by the fact that U ≪ V . We then have U ≤ Bk ⪕ V which implies U ⪕ V by
+Proposition 3.11.
+✷
+Proposition 3.16 In any compact locale, U ⪕ V implies U ≪ V for any two opens U, V .
+The proof of Proposition 3.16 is omitted as it is exactly the same as in [11, pg. 303].
+Definition 3.17 (Stone locale) A Stone locale is one that is compact and zero-dimensional.
+Proposition 3.18 In any Stone locale, an open is compact iff it is clopen.
+Proof. By propositions 3.15 and 3.16 and the fact that every zero-dimensional locale is regular (Propo-
+sition 3.14).
+✷
+4
+Adjoint Functor Theorem for frames with small bases
+We start with the definition of the notion of an adjunction in the simplified context of posetal categories.
+Definition 4.1 Let P and Q be two posets. An adjunction between P and Q consists of a pair of monotonic
+maps f : P → Q and g : Q → P satisfying f ⊣ g := �
+x:P
+�
+y:Q f(x) ≤ y ↔ x ≤ g(y).
+In locale theory, it is standard convention to denote by f∗ : O(X) → O(Y ) the right adjoint of a
+frame homomorphism f ∗ : O(Y ) → O(X) corresponding to a continuous map of locales f : X → Y . The
+right adjoint of a frame homomorphism is defined using the Adjoint Functor Theorem which amounts
+to the definition: f∗ := U �→ �{V : O(Y ) | f ∗(V ) ≤ U}.
+In the predicative setting of type theory
+however, it is not clear how the right adjoint of a frame homomorphism would be defined as the family
+{V : O(Y ) | f ∗(V ) ≤ U} might be too big in general, meaning it is not clear a priori that its join in O(X)
+exists. To resolve this problem, we restrict attention once again to frames with small bases in which we
+circumvent this problem by quantifying over the basic elements.
+6
+
+Tosun and Escard´o
+Theorem 4.2 (AFT) Let X and Y be two large, locally small, and small complete locales and let f ∗ :
+O(Y ) → O(X) be a monotone map. Assume that Y has a small basis {Bi}i:I. The map f ∗ has a right
+adjoint iff f ∗(�
+i Ui) = �
+i f ∗(Ui) for any small family {Ui}i:I in O(Y ).
+Proof. Let f ∗ : O(Y ) → O(X) be a monotone map from frame O(Y ) to frame O(Y ) and assume that Y
+has a small basis {Bi}i:I.
+The forward direction is easy: suppose f ∗ : O(Y ) → O(X) has a right adjoint f∗ : O(X) → O(Y ).
+Let {Ui}i:I be a family in O(Y ). By the uniqueness of joins, it is sufficient to show that f ∗(�
+i Ui) is the
+join of {f ∗(Ui)}i:I. It is clearly an upper bound by the fact that f ∗ is monotone. Given any other upper
+bound V of {f ∗(Ui)}i:I, we have that f ∗(�
+i Ui) ≤ V since f ∗(�
+i Ui) ≤ V ↔ (�
+i Ui) ≤ f∗(V ) meaning it
+is sufficient to show Ui ≤ f∗(V ) for each Ui. Since Ui ≤ f∗(V ) iff f ∗(Ui) ≤ V , we are done as the latter
+can be seen to hold directly from the fact that V is an upper bound of {f ∗(Ui)}i:I.
+For the converse, suppose f ∗(�
+i Ui) = �
+i:I f ∗(Ui) for every family {Ui}i:I. We define the right adjoint
+of f ∗ as:
+f∗(V )
+:=
+�
+{Bi | i : I, f ∗(Bi) ≤ V } .
+We need to show that f∗ is the right adjoint of f ∗ i.e. that f ∗(U) ≤ V ↔ U ≤ f∗(V ) for any two
+U, V : O(X). For the forward direction, assume f ∗(U) ≤ V . We know that there exists a covering family
+{Bj}j∈J for U with U = �
+j∈J Bj so it suffices to show that Bj ≤ f∗(V ) for every j ∈ J. It remains to
+show that f ∗(Bj) ≤ V . This follows from the fact that f ∗(Bj) ≤ f ∗(�
+j∈J Bj) ≤ f ∗(U) ≤ V . For the
+backward direction, let U ≤ f∗(V ). We have:
+f ∗(U)
+≤
+f ∗(f∗(V ))
+≡
+f ∗ ��
+{Bi | f ∗(Bi) ≤ V }
+�
+≤
+�
+{f ∗(Bi) | f ∗(Bi) ≤ V }
+[since f ∗ preserves joins]
+≤
+V.
+✷
+Our primary use case for the Adjoint Functor Theorem is the construction of Heyting implications in
+locally small frames with small bases.
+Definition 4.3 (Heyting implication) Let X be a large, locally small, and small complete locale with a
+small basis and let U : O(X). As the map −∧U : O(X) → O(X) preserves joins by the frame distributivity
+law, it must have a right adjoint h : O(X) → O(X), by Theorem 4.2, that satisfies W ∧U ≤ V ↔ W ≤ h(V )
+for all W, V : O(X). We then define the Heyting implication as: U ⇒ V := h(V ).
+The Adjoint Functor Theorem also allows us to define the notion of a perfect frame homomorphism.
+Definition 4.4 (Perfect frame homomorphism) Let X and Y be two large, locally small, and small
+complete locales and assume that Y has a small basis. A continuous map f : X → Y is said to be perfect
+if the right adjoint f∗ of its defining frame homomorphism f ∗ is Scott continuous.
+Proposition 4.5 Let f : X → Y be a perfect map where Y is a locale with small basis.
+The frame
+homomorphism f ∗ respects the way below relation, that is, U ≪ V implies f ∗(U) ≪ f ∗(V ), for any two
+U, V : O(Y ).
+A proof of Proposition 4.5 can be found in [8]. Our proof is mostly the same, once it is ensured that
+the Heyting implication exists through the small basis assumption. We thus omit the proof.
+Corollary 4.6 Perfect maps are spectral as they preserve compact opens.
+In fact, the converse is also true in the case of spectral locales so Corollary 4.6 can be strengthened to
+an equivalence in this case.
+Proposition 4.7 Let X and Y be two large, locally small, and small complete spectral locales and assume
+that Y has a small basis. A continuous map f : X → Y is perfect iff it is spectral.
+7
+
+Tosun and Escard´o
+Proof. The forward direction is given by Corollary 4.6. For the backward direction, assume f : X →
+Y to be a spectral map.
+We have to show that the right adjoint f∗ : O(X) → O(Y ) of its defining
+frame homomorphism is Scott continuous. Let {Ui}i:I be a directed family in O(X). We have to show
+f∗(�
+i:I Ui) = �
+i:I f∗(Ui). The �
+i:I f∗(Ui) ≤ f∗(�
+i:I Ui) direction is immediate. For the f∗(�
+i:I Ui) ≤
+�
+i:I f∗(Ui) direction, let C be a compact open such that C ≤ f∗(�
+i:I Ui). By the fact that f ∗ ⊣ f∗, it must
+be the case that f ∗(C) ≤ �
+i:I Ui and since f ∗(C) is compact, by the spectrality assumption of f ∗, there
+must exist some l : I such that f ∗(C) ≤ Ul. Again by adjointness, C ≤ f∗(Ul) so clearly C ≤ �
+i:I f∗(Ui).✷
+5
+Meet-semilattice of Scott continuous nuclei
+In this section, we take the first step towards constructing the defining frame of the patch locale on a
+spectral locale i.e. the frame of Scott continuous nuclei. We construct the meet-semilattice of all nuclei on
+a frame.
+Proposition 5.1 The type of nuclei on a given frame O(X) forms a meet-semilattice under the pointwise
+order.
+Proof. We need to show that the type O(X) has all finite meets. The top nucleus is defined as − �→ ⊤
+and the meet of two nuclei as j ∧ k := U �→ j(U) ∧ k(U). It is easy to see that j ∧ k is the greatest lower
+bound of j and k so it remains to show that j ∧ k satisfies the nucleus laws.
+The inflation property can be seen to be satisfied from the inflation properties of j and k combined
+with the fact that j(U)∧k(U) is the greatest lower bound of j(U) and k(U). To see that meet preservation
+holds, let U, V : O(X); we have:
+(j ∧ k)(U ∧ V )
+≡
+j(U ∧ V ) ∧ k(U ∧ V )
+=
+j(U) ∧ j(V ) ∧ k(U) ∧ k(V )
+=
+(j(U) ∧ k(U)) ∧ (j(V ) ∧ k(V ))
+≡
+(j ∧ k)(U) ∧ (j ∧ k)(V ).
+For idempotency, let U : O(X). We have:
+(j ∧ k)((j ∧ k)(U))
+≡
+j(j(U) ∧ k(U)) ∧ k(j(U) ∧ k(U))
+=
+j(j(U)) ∧ j(k(U)) ∧ k(j(U)) ∧ k(k(U))
+≤
+j(j(U)) ∧ k(k(U))
+=
+j(U) ∧ k(U)
+≡
+(j ∧ k)(U).
+✷
+We now show that this meet-semilattice can be refined to only those nuclei that are Scott continuous
+(i.e. the perfect nuclei).
+Proposition 5.2 The Scott continuous nuclei on any locale form a meet-semilattice.
+Proof. Let X be a locale. The construction is the same as the one from Proposition 5.1; the top element
+is − �→ ⊤ which is trivially Scott continuous so it remains to show that the meet of two Scott continuous
+nuclei is Scott continuous. Consider two Scott continuous nuclei j and k on O(X) and a directed small
+8
+
+Tosun and Escard´o
+family {Ui}i:I. We then have:
+(j ∧ k)
+��
+i:I
+Ui
+�
+≡
+j
+��
+i:I
+Ui
+�
+∧ k
+
+�
+j:I
+Uj
+
+
+=
+��
+i:I
+j(Ui)
+�
+∧
+
+�
+j:I
+k(Uj)
+
+
+[Scott continuity of j and k]
+=
+�
+(i,j):I×I
+j(Ui) ∧ k(Uj)
+[distributivity]
+=
+�
+i:I
+j(Ui) ∧ k(Ui)
+[†]
+≡
+�
+i:I
+(j ∧ k)(Ui)
+[meet preservation]
+where, for the † step, we use antisymmetry. The backwards direction is immediate. For the forwards
+direction, we need to show that �
+(i,j):I×I j(Ui) ∧ k(Uj) ≤ �
+i:I j(Ui) ∧ k(Ui), for which it suffices to show
+that �
+i:I j(Ui)∧k(Ui) is an upper bound of {j(Ui)∧k(Uj)}(i,j):I×I. Let m, n : I be two indices. As {Ui}i:I
+is directed, there must exist some o such that Uo is an upper bound of {Um, Un}. Using the monotonicity
+of j and k, we get j(Um) ∧ k(Un) ≤ j(Uo) ∧ k(Uo) ≤ �
+i:I j(Ui) ∧ k(Ui) as desired.
+✷
+6
+Joins in the frame of Scott continuous nuclei
+The nontrivial component of the patch frame construction is the join of a family {ki}i:I of perfect nuclei, as
+the pointwise join fails to be idempotent in general, and not inflationary when the family in consideration
+is empty.
+A construction of the join, given in [9], is based on the idea that, if we start with a family {ki}i:I of
+nuclei, we can consider the family
+{ki0 ◦ · · · ◦ kin}(i0,··· ,in):List(I) ,
+whose index type is the type of lists of indices in I, that will always be directed. We will use the following
+notation for lists over a type X:
+• ε denotes the empty list,
+• x :: s, with x : X and s : List(X), denotes the list with first element x followed by the elements of s,
+• s t denotes the concatentation of lists s and t.
+To define the join operation, we will use the iterated composition function o that we define as follows:
+Definition 6.1 (Iterated composition of nuclei) Given a small family K := {ki}i:I of nuclei on a
+given locale X, we denote by K∗ the family (List(I), o) where o is defined as follows:
+o(ε)
+:=
+id;
+o(i :: s)
+:=
+o(s) ◦ ki.
+By an easy proof by induction, we have the following.
+Proposition 6.2 For any family K := {ki}i:I of prenuclei on a locale and any s, t : List(I), we have that
+o(s t) = o(s) ◦ o(t).
+Proposition 6.3 Given a family K := {ki}i:I of nuclei on a locale, every α ∈ K∗ is a prenucleus, that
+is, for every s : List(I), the function o(s) is a prenucleus.
+9
+
+Tosun and Escard´o
+Proof. If s = ε, we are done as it is immediate that the identity function id is a prenucleus. If s = i :: s′,
+we need to show that o(s′) ◦ ki is a prenucleus. For meet preservation, let U, V : O(X). We have that:
+(o(s′) ◦ ki)(U ∧ V )
+≡
+o(s′)(ki(U ∧ V ))
+=
+o(s′)(ki(U) ∧ ki(V ))
+[ki is a nucleus]
+=
+o(s′)(ki(U)) ∧ o(s′)(ki(V ))
+[inductive hypothesis]
+≡
+(o(s′) ◦ ki)(U) ∧ (o(s′) ◦ ki)(V ).
+For the inflation property, consider some U : O(X). We have that U ≤ ki(U) ≤ o(s′)(ki(U)), by the
+inflation property of ki and the inductive hypothesis.
+✷
+Proposition 6.4 Given a nucleus j and a family K := {ki}i:I of nuclei on a locale, if j is an upper bound
+of K then it is also an upper bound of K∗.
+Proof. Let j and K := {ki}i:I be, respectively, a nucleus and a family of nuclei on a locale. Let s : List(I).
+We denote by {αs}s:List(S) the family K∗. We proceed by induction on s. If s = ε, we have that id(U) ≡
+U ≤ j(U). If s = i :: s′, we then have:
+αs′(ki(U))
+≤
+αs′(j(U))
+[monotonicity of αs′ (Prop. 6.3 and monotonicity of prenuclei)]
+≤
+j(j(U))
+[inductive hypothesis]
+≤
+j(U)
+[idempotency of j].
+✷
+Proposition 6.5 Given a family {ki}i:I of Scott continuous nuclei on a locale, every prenucleus α ∈ K∗
+is Scott continuous.
+Proof. Any composition of finitely many Scott continuous functions is Scott continuous.
+✷
+Proposition 6.6 Given a family K :≡ {ki}i:I of nuclei on a locale, the family K∗ is directed.
+Proof. K∗ is indeed always inhabited by the identity nucleus. The upper bound of nuclei o(s) and o(t) is
+given by o(s t), which is o(s) ◦o(t) by Proposition 6.2. The fact that this is an upper bound of {o(s), o(t)}
+follows from monotonicity and inflationarity.
+✷
+Proposition 6.7 Let j be a nucleus and K := {ki}i:I a family of nuclei on a locale. Denote by {αs}s:List(I)
+the family K∗ and by {βs}s:List(I) the family {j ∧ k | k ∈ K}∗. We have that βs is a lower bound of {αs, j}
+for every s : List(I).
+We are now ready to construct the join operation in the meet-semilattice of Scott continuous nuclei
+hence defining the patch frame O(Patch(X)) of the frame of a locale X.
+Theorem 6.8 (Join of Scott continuous nuclei) Let K := {ki}i:I be a family of Scott continuous
+nuclei. The join of K can be calculated as �N K := U �→ �
+α∈K∗ α(U).
+Proof. It must be checked that this is (1) indeed the join, (2) is a Scott continuous nucleus i.e. it is
+inflationary, binary-meet-preserving, idempotent, and Scott continuous. The inflation property is direct.
+10
+
+Tosun and Escard´o
+For meet preservation, consider some U, V : O(X). We have:
+� N
+�
+i:I
+ki
+�
+(U ∧ V )
+≡
+�
+α∈K∗
+α(U ∧ V )
+=
+�
+α∈K∗
+α(U) ∧ α(V )
+[Proposition 6.5]
+=
+�
+β,γ∈K∗
+β(U) ∧ γ(V )
+[†]
+=
+
+ �
+β∈K∗
+β(U)
+
+ ∧
+
+ �
+γ∈K∗
+γ(V )
+
+
+[distributivity]
+≡
+� N
+�
+i:I
+ki
+�
+(U) ∧
+� N
+�
+i:I
+ki
+�
+(V ),
+where the step (†) uses antisymmetry. The �
+α∈K∗ α(U) ∧ α(V ) ≤ �
+β,γ∈K∗ β(U) ∧ γ(V ) direction is direct
+whereas for the �
+β,γ∈K∗ β(U) ∧ γ(V ) ≤ �
+α∈K∗ α(U) ∧ α(V ) direction we show that �
+α∈K∗ α(U) ∧ α(V ) is
+an upper bound of the set {β(U)∧γ(V ) | β, γ ∈ K∗}. Consider arbitrary β, γ ∈ K∗. By the directedness of
+K∗ we know that there exists some δ ∈ K∗ that is an upper bound of {β, γ}. We then have: β(U)∧γ(V ) ≤
+δ(U) ∧ δ(V ) ≤ �
+α∈K∗ α(U) ∧ α(V ). For idempotency, let U : O(X). We have that:
+� N
+�
+i
+ki
+� �� N
+�
+i
+ki
+�
+(U)
+�
+≡
+�
+α∈K∗
+α
+
+ �
+β∈K∗
+β(U)
+
+
+=
+�
+α∈K∗
+�
+β∈K∗
+α(β(U))
+[Proposition 6.5]
+≤
+�
+α,β∈K∗
+α(β(U))
+[flattening joins]
+≤
+�
+α∈K∗
+α(U)
+[†]
+≡
+� N
+�
+i
+ki
+�
+(U),
+where for the step (†) it suffices to show that �
+α∈K∗ α(U) is an upper bound of the family
+{α(β(U)) | (α, β) ∈ K∗ × K∗}.
+Consider arbitrary α, β ∈ K∗.
+There must be lists s and t of indices
+of K such that α ≡ o(s) and β ≡ o(t). Picking δ := o(s t) then gives a δ ∈ K∗, is then an upper bound
+for o(s) and o(t) (as in Proposition 6.6). By Proposition 6.2, we have that o(s)(o(t)(U)) ≡ o(s t)(U) ≡
+δ(U) ≤ �
+α∈K∗ α(U).
+11
+
+Tosun and Escard´o
+For Scott continuity, let {Uj}j:J be a directed family over O(X). The result then follows as:
+� N
+�
+K
+� 
+�
+j:J
+Uj
+
+
+≡
+�
+α∈K∗
+α
+
+�
+j:J
+Uj
+
+
+=
+�
+α∈K∗
+�
+j:J
+α(Uj)
+[Proposition 6.5]
+=
+�
+j:J
+�
+α∈K∗
+α(Uj)
+[joins commute]
+≡
+�
+j:J
+� N
+�
+K
+�
+(Uj)
+as required.
+The fact that �N
+i ki is an upper bound of K is easy to verify: given some ki and U : O(X), ki(U) ∈
+{α(U) | α ∈ K∗} since ki ∈ K∗. To see that it is the least upper bound, consider a Scott continuous
+nucleus j that is an upper bound of K. Let U : O(X). We need to show that (�N
+i ki)(U) ≤ j(U). We
+know by Proposition 6.4 that j is an upper bound of K∗, since it is an upper bound of K, which is to
+say K∗
+s(U) ≤ j(U) for every s : List(I) i.e. j(U) is an upper bound of the family {α(U) | α ∈ K∗}. Since
+(�N
+i ki)(U) is the least upper bound of this family by definition, it follows that it is below j(U).
+✷
+We use Proposition 6.7 to prove the following.
+Proposition 6.9 (Distributivity) For any Scott continuous nucleus j and any family {ki}i:I of Scott
+continuous nuclei, we have that
+j ∧
+��
+i:I
+ki
+�
+=
+�
+i:I
+j ∧ ki.
+It follows that the Scott continuous nuclei form a frame.
+Definition 6.10 (Patch locale of a spectral locale) Let X be a large, locally small, and small com-
+plete spectral locale. The patch locale of X, written Patch(X), is given by the frame of Scott continuous
+nuclei on X.
+Note that we do not assume that locale X is spectral in Definition 6.10. This is to highlight the fact
+that the construction of the patch frame does not rely on this assumption in a crucial way. However,
+the reader is reminded that the patch construction is meaningful only on spectral locales, as its universal
+property can be proved only for spectral locales.
+Definition 6.10 gives rise to a problem that we need to address: the patch of a locally small locale does
+not yield a locally small locale. Starting with a (U+, U, U)-locale X, Patch(X) is a (U+, U+, U)-locale since
+the pointwise ordering of nuclei (defined in Proposition 5.1) quantifies over arbitrary opens. In most of
+our development, we have restricted attention to only locally small frames meaning we run into problems
+if Patch(X) is not locally small (e.g. applying the Adjoint Functor Theorem). We circumvent this by using
+the following small version of the same relation:
+Definition 6.11 (Basic nuclei ordering on spectral locales) Let X be a spectral locale and denote
+its basis by {Bi}i:I. Let j, k : O(X) → O(X) be two nuclei. We define the basic nuclei ordering − ≤K −
+as
+j ≤K k
+:=
+�
+i:I
+j(Bi) ≤ k(Bi).
+Given two nuclei j and k on a (U, V, W)-locale, the relation j ≤K k lives in universe V ∨ W meaning,
+in the case of a (U+, U, U)-locale, it lives in U as desired.
+Proposition 6.12 The basic nuclei ordering defined in Definition 6.11 is logically equivalent to the point-
+wise ordering of nuclei.
+12
+
+Tosun and Escard´o
+Proof. The usual pointwise ordering obviously implies the basic ordering so we address the other direction.
+Let j and k be two Scott continuous nuclei on a spectral locale X and assume that j ≤K k. We need to
+show that j(U) ≤ k(U) for every open U. It must be the case that U = �
+l∈L Bl where {Bl}l∈L is the basic
+covering family of compact opens covering U that is directed. We then have j(�
+l∈L Bl) = �
+l∈L j(Bl) by
+Scott continuity and �
+l∈L j(Bl) ≤ �
+l∈L k(Bl) since j(Bl) ≤ k(Bl) for every l : L.
+✷
+Thanks to Proposition 6.12 our theorems that have the local smallness assumption apply to the patch
+frame as we know that Patch(X) always has an equivalent version that is locally small. We also note that
+we will not be precise in distinguishing between the basic order and the regular order on nuclei and will
+freely switch between the two, making implicit use of Proposition 6.12.
+7
+The coreflection property of Patch
+We prove in this section that our construction of Patch has the desired universal property: it exhibits
+Stone as a coreflective subcategory of Spec. We also note that when we talk about Stone and spectral
+locales in this section, we implicitly assume them to be large, locally small, and small complete, and refrain
+from explicitly stating this assumption.
+The notions of closed and open nuclei are crucial for proving the universal property. We start with the
+definitions of these. Let U be an open of a locale X;
+(i) The closed nucleus induced by U is the map V �→ U ∨ V ;
+(ii) The open nucleus induced by U is the map V �→ U ⇒ V .
+We denote the closed nucleus by ‘U’ and, because the open nucleus is the Boolean complement of the closed
+nucleus, we denote it by ¬‘U’. This follows the notation of [8,10]. We now prove the Scott continuity of
+these nuclei.
+Lemma 7.1 For any spectral locale X and any monotone map h : O(X) → O(X), if for every U : O(X)
+and compact C : O(X) with C ≤ h(U), there is some compact D ≤ U such that C ≤ h(D), then h is Scott
+continuous
+Lemma 7.2 Let X be a spectral locale. The closed nucleus ‘U’ on X is Scott continuous for any open U,
+whereas the open nucleus is Scott continuous if the open U is compact.
+Proof.
+Closed nucleus. Let U be an open of a locale and let {Vi}i:I be a directed family of opens. We need to
+show that ‘U’(�
+i:I Vi) = �
+i:I ‘U’(Vi). It is clear that U ∨(�
+i:I Vi) is an upper bound of {U ∨Vi}i:I. Let W
+be an arbitrary upper bound of {U ∨ Vi}i:I. It suffices to show that W is an upper bound of {U, (�
+i:I Vi)}.
+For the case of �
+i:I Vi, we have that �
+i:I Vi ≤ �
+i:I U ∨ Vi ≤ W. For the case of U, we use the fact that Vi
+directed. Since Vi is directed it must be inhabited by some Vk. We then have U ≤ U ∨ Vk ≤ W as W is
+an upper bound of {U ∨ Vi}i:I.
+Open nucleus. Let U be a compact open of a locale. By Lemma 7.1, it is sufficient to show that
+for any open V and any compact open C1 with C1 ≤ U ⇒ V , there exists some compact C2 ≤ U such
+that C1 ≤ U ⇒ C2. Let V and C1 be two opens with C1 compact and satisfying C1 ≤ U ⇒ V . Pick
+C2 := U ∧ C1. We know that this is compact by spectrality. It remains to check (1) C2 ≤ V and (2)
+C1 ≤ U ⇒ C2, both of which are direct.
+✷
+In Lemma 7.5, we prove that the map whose inverse image sends an open U to the closed nucleus ‘U’
+is perfect. Before Lemma 7.5, we record two auxiliary lemmas that are needed in the proof.
+Lemma 7.3 Let X be a spectral locale with a small basis. The right adjoint ε∗ : O(Patch(X)) → O(X)
+of ‘−’ is given by the assignment j �→ j(⊥) i.e. ε∗(j) = j(⊥) for every Scott continuous nucleus j on X.
+Lemma 7.4 Given a directed family {ki}i:I of Scott continuous nuclei, their join is computed pointwise,
+that is, (�
+i:I ki) (U) = �
+i:I ki(U).
+Proofs of Lemma 7.3 and Lemma 7.4 can be found in [8]. They are omitted here as they are mostly
+unchanged in our type-theoretical setting.
+13
+
+Tosun and Escard´o
+Lemma 7.5 The function that sends an open U to the closed nucleus ‘U’ is a perfect frame homomorphism
+O(X) → O(Patch(X)).
+Proof. We have to show that the right adjoint ε∗ of ‘−’ is Scott continuous. Let {ki}i:I be a directed family
+of Scott continuous nuclei. By Lemma 7.3, it suffices to show (�
+i:I ki) (⊥) = �
+i:I ε∗(ki). By Lemma 7.4,
+we have that (�
+i:I ki) (⊥) = �
+i:I ki(⊥). The desired result of �
+i:I ki(⊥) = �
+i:I ε∗(ki) is then immediate
+by Lemma 7.3.
+✷
+This function defines a continuous map ε : Patch(X) → X, which we we will show to be the counit of
+the coreflection in consideration.
+7.1
+Patch is Stone
+Before we proceed to showing that the Patch locale has the desired universal property, we first need to
+show that Patch(X) is Stone (as given in Definition 3.17) for any spectral locale X. We start by addressing
+the question of zero-dimensionality.
+To show that Patch(X) is zero-dimensional, we need to construct a basis consisting of clopens. We will
+use the following fact, which was already mentioned above:
+Proposition 7.6 The open nucleus ¬‘U’ is the Boolean complement of the closed nucleus ‘U’.
+Lemma 7.7 The patch of any spectral locale X with a basis {Bi}i:I of compact opens is zero-dimensional,
+with a basis of clopens of the form �
+(m,n)∈M×N ‘Bm’ ∧ ¬‘Bn’ with M and N finite, which is clearly closed
+under finite joins.
+More precisely, if the given basis of X is the family B : I → O(X), then the constructed basis of Patch(X)
+is the family C : List(I × I) → O(Patch(X)) defined by
+C([(n0, m0), . . . , (nk−1, mk−1)]) :=
+�
+0≤i<k
+‘Bmi’ ∧ ¬‘Bni’.
+That is, the index set of the basis consists of formal expressions for finite joins.
+Proof. We need to show that this (1) consists of clopens, and (2) indeed forms a basis. For (1), ‘B1’∧¬‘B2’
+has complement ¬‘B1’∨‘B2’, by Proposition 7.6, and finite unions of complemented sets are complemented.
+For (2), let j : O(X) → O(X) be a perfect nucleus on O(X). We need to show that there exists a subfamily
+of B that yields j as its join. For this we pick the subfamily Bj := {‘Bm’ ∧ ¬‘Bn’ | m, n : I, Bm ≤ j(Bn)}.
+The fact that j is the least upper bound of Bj follows from Lemma 7.8 and Lemma 7.9:
+j
+=
+�
+n:I
+‘j(Bn)’ ∧ ¬‘Bn’
+[Lemma 7.8]
+=
+�
+{‘Bm’ ∧ ¬‘Bn’ | m, n : I, Bm ≤ j(Bn)}
+[Lemma 7.9]
+✷
+The following is adapted from Johnstone [11, Proposition II.2.7].
+Lemma 7.8 Given any perfect nucleus j : Patch(X), we have that j = � {‘j(Bn)’ ∧ ¬‘Bn’ | n : I}.
+Lemma 7.9 Let X be a spectral locale.
+Given any perfect nucleus j
+: Patch(X), we have that
+� {‘j(Bn)’ ∧ ¬‘Bn’ | n : I} = � {‘Bm’ ∧ ¬‘Bn’ | m, n : I, Bm ≤ j(Bn)}.
+Theorem 7.10 Given any spectral locale X, we have that Patch(X) is a Stone locale.
+Proof. Zero-dimensionality is given by Lemma 7.7 so it only remains to show compactness. Recall that
+the top element ⊤ of Patch(X) is defined as ⊤ := − �→ ⊤X. Because ε is a frame homomorphism, it must
+be the case that ⊤ = ε(⊤X) meaning what we want to show is ε(⊤X) ≪ ε(⊤X). By Proposition 4.5, it
+suffices to show ⊤X ≪ ⊤X which is immediate as spectral locales are compact.
+✷
+14
+
+Tosun and Escard´o
+7.2
+The universal property of the patch construction
+We now prove the universal property of Patch corresponding to the fact that it is the right adjoint to the
+inclusion Stone ֒→ Spec. For this purpose, we use the following lemma, which is not needed in [8, 10]
+thanks to the existence of the frame of all nuclei in the impredicative setting.
+Lemma 7.11 Let L, L′ be two spectral frames and B a small Boolean algebra embedded in L such that
+(i) L is generated by A, and
+(ii) B contains all compact opens of L.
+Then for any lattice homomorphism h : B → L′, there is a unique frame homomorphism ¯h : L → L′
+satisfying h = ¯h ◦ η, where η : B ֒→ L denotes the embedding of B into L, as illustrated in the following
+diagram:
+B
+L
+L′.
+h
+η
+¯h
+(†)
+Proof. Define ¯h(x) := � {h(b) | η(b) ≤ x, b : B}. We need to show that (1) ¯h is a frame homomorphism,
+and (2) is the unique map satisfying h = ¯h ◦ η.
+(1) It is clear that ¯h preserves ⊥, ⊤, and joins of directed families. To show that it preserves binary
+joins, we make use of the fact that for any b ≤ x ∨ y with b compact (in any spectral locale), there exist
+compact opens c ≤ x and d ≤ y such that b ≤ c ∨ d. As it preserves both binary joins and directed joins,
+it must preserve arbitrary joins.
+(2) It is easy to see that ¯h satisfies the equation h = ¯h ◦ η. Uniqueness follows from the fact that η is
+injective.
+✷
+We can now present the universal property.
+Theorem 7.12 Given any spectral map f : X → A from a Stone locale into a spectral locale, there exists
+a unique spectral map ¯f : X → Patch(A) satisfying ε ◦ ¯f = f, as illustrated in the following diagram in the
+category of spectral locales:
+X
+A
+Patch(A)
+f
+¯f
+ε
+Proof. We apply Lemma 7.11 with L := O(Patch(A)), L′ := O(X), B := K(Patch(A)) and h defined by
+h
+
+
+�
+(j,k)∈J×K
+‘Bj’ ∧ ¬‘Bk’
+
+
+:=
+�
+(j,k)∈J×K
+f ∗(Bj) ∧ ¬f ∗(Bk).
+It is easy to see that h is well-defined, in the sense that if the same clopen is expressed in two different
+ways as a finite join of binary meets, then h gives the same value for them. It is easy to check that the
+embedding K(Patch(A)) ֒→ O(Patch(A)) satisfies the premise of the lemma. We then take ¯f ∗ to be ¯h as
+constructed in the lemma. We need to show that this satisfies ¯f ∗(‘U’) = f ∗(U) for all U : O(A). It suffices
+to consider the case where U is a compact open C, as the compact opens form a basis. Because C can be
+written as �{‘C’ ∧ ¬‘⊥’}, we have that
+¯f ∗(‘C’) = h
+��
+{‘C’ ∧ ¬‘⊥’}
+�
+=
+�
+{f ∗(C) ∧ ¬f ∗(⊥)} =
+�
+{f ∗(C) ∧ ⊤} = f ∗(C),
+as required.
+✷
+15
+
+Tosun and Escard´o
+8
+Summary and discussion
+We have constructed the patch locale of a spectral locale in the predicative and constructive setting of
+univalent type theory, using only propositional and functional extensionality. Furthermore, we have shown
+that the patch construction Patch : Spec → Stone is the right adjoint to the inclusion Stone ֒→ Spec
+which is to say that patch exhibits the category Stone as a coreflective subcategory of Spec.
+As we have elaborated in Section 3, answering this question in a predicative setting has involved the
+reformulation of several fundamental concepts of locale theory. In particular, we have reformulated notions
+of spectrality, zero-dimensionality, and regularity, and have shown that crucial facts about these notions
+remain valid in the predicative setting.
+We have also formalised almost all of our development, most importantly Theorem 7.10 and
+Lemma 7.11. The formalisation has been carried out by the first-named author as part 3 of the second-
+named author’s TypeTopology library [7]. Almost all of the presented results have already been imple-
+mented, including:
+(i) All of Section 3 in the module Locales.CompactRegular;
+(ii) The Adjoint Functor Theorem and its application to define Heyting implications in frames (Sec-
+tion 4) in modules Locales.GaloisConnection, Locales.AdjointFunctorTheoremForFrames, and
+Locales.HeytingImplication;
+(iii) All of Section 5 and Section 6 in module Locales.PatchLocale; and
+(iv) The extension lemma (Lemma 7.11) from Section 7.2 in Locales.BooleanAlgebra.
+The only result that remains to be formalised is the universal property which we have proved using
+Lemma 7.11. The formalisation of this result is work in progress and is soon to be completed.
+In previous work [8,10], that forms the basis of the present work, the patch construction was used to
+(i) exhibit Stone as a coreflective subcategory of Spec, which we have addressed here, and
+(ii) exhibit the category of compact regular locales and continuous maps as a coreflective subcategory of
+of stably compact locales and perfect maps, which we leave for future work.
+Coquand and Zhang [4] tackled (ii) using formal topology. We conjecture that it should be possible to
+instead use the approach we have presented here, namely, working with locales with small bases and
+constructing the patch as the frame of Scott continuous nuclei.
+References
+[1] Agda development team, The Agda Proof Assistant (version 2.6.2).
+URL https://agda.readthedocs.io/en/v2.6.2/team.html
+[2] Coquand, T., G. Sambin, J. Smith and S. Valentini, Inductively generated formal topologies 124, pp. 71–106.
+[3] Coquand, T. and A. Tosun, Formal Topology and Univalent Foundations, in: Proof and Computation II, WORLD
+SCIENTIFIC pp. 255–266.
+[4] Coquand, T. and G.-Q. Zhang, A representation of stably compact spaces, and patch topology 305, pp. 77–84.
+[5] de Jong, T. and M. H. Escard´o, Domain theory in constructive and predicative univalent foundations, in: C. Baier
+and J. Goubault-Larrecq, editors, 29th EACSL Annual Conference on Computer Science Logic (CSL 2021), Leibniz
+International Proceedings in Informatics (LIPIcs) 183, pp. 28:1–28:18.
+[6] de Jong, T. and M. H. Escard´o, Predicative Aspects of Order Theory in Univalent Foundations, in: N. Kobayashi, editor,
+6th International Conference on Formal Structures for Computation and Deduction (FSCD 2021), Leibniz International
+Proceedings in Informatics (LIPIcs) 195, pp. 8:1–8:18.
+[7] Escard´o, M., and contributors, TypeTopology, Agda library.
+URL https://github.com/martinescardo/TypeTopology
+[8] Escard´o, M. H., On the Compact-regular Coreflection of a Stably Compact Locale 20, pp. 213–228.
+3 The HTML rendering of the Agda code can be browsed here https://www.cs.bham.ac.uk/∼mhe/TypeTopology/Locales.index.html
+16
+
+Tosun and Escard´o
+[9] Escard´o, M. H., Properly injective spaces and function spaces 89, pp. 75–120.
+[10] Escard´o, M. H., The regular locally compact coreflection of a stably locally compact locale 157, pp. 41–55.
+[11] Johnstone, P. T., “Stone Spaces,” Cambridge Univ. Press.
+[12] Mac Lane, S. and I. Moerdijk, “Sheaves in Geometry and Logic: A First Introduction to Topos Theory,” Universitext,
+Springer-Verlag.
+[13] Sambin, G., Intuitionistic Formal Spaces — A First Communication, in: D. G. Skordev, editor, Mathematical Logic and
+Its Applications, Springer US pp. 187–204.
+[14] UFP, “Homotopy Type Theory: Univalent Foundations of Mathematics,” .
+URL https://homotopytypetheory.org/book
+[15] Voevodsky, V., Resizing Rules — their use and semantic justification, invited talk at TYPES 2011, Bergen, Norway.
+17
+

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+arXiv:2301.11481v1  [cs.GT]  27 Jan 2023
+Are Equivariant Equilibrium Approximators Beneficial?
+Zhijian Duan1, Yunxuan Ma1, Xiaotie Deng1,2
+1Center on Frontiers of Computing Studies, Peking University
+2Center for Multi-Agent Research, Institute for AI, Peking University
+{zjduan,charmingmyx,xiaotie}@pku.edu.cn
+Abstract
+Recently, remarkable progress has been made by approximating Nash equilibrium (NE), corre-
+lated equilibrium (CE), and coarse correlated equilibrium (CCE) through function approximation
+that trains a neural network to predict equilibria from game representations. Furthermore, equiv-
+ariant architectures are widely adopted in designing such equilibrium approximators in normal-
+form games. In this paper, we theoretically characterize benefits and limitations of equivariant
+equilibrium approximators. For the benefits, we show that they enjoy better generalizability than
+general ones and can achieve better approximations when the payoff distribution is permutation-
+invariant. For the limitations, we discuss their drawbacks in terms of equilibrium selection and
+social welfare. Together, our results help to understand the role of equivariance in equilibrium
+approximators.
+1
+Introduction
+The equivariant equilibrium property states that, given a Nash Equilibrium (NE) solution of a
+game, the permuted solution is also an NE for the game whose actions of representation are permuted
+in the same way. The same property also holds in correlated equilibrium (CE) and coarse correlated
+equilibrium (CCE), as well as the approximate solutions for all three solution concepts.
+In this paper, we are interested in understanding the equivariant equilibrium property in designing
+neural networks that predict equilibria from game payoffs, following such recent approaches in de-
+signing equivariant equilibrium approximators [Feng et al., 2021, Marris et al., 2022] in normal-form
+games. Informally, such equivariant approximators keep the same permutation of the output strate-
+gies (represented as vectors or tensors) when the input game representations (e.g., the game payoff
+tensors) are permuted 1. While equivariant approximators achieved empirical success, little work has
+theoretically discussed whether they are beneficial.
+We theoretically characterize benefits and limitations of equivariant NE, CE and CCE approx-
+imators. For the benefits, we first show that equivariant approximators enjoy better generalizabil-
+ity, where we evaluate the approximators using the maximum exploitability [Lockhart et al., 2019,
+Goktas and Greenwald, 2022] over all players. To get such a result, we derive the generalization bounds
+and the sample complexities of the NE, CE, and CCE approximators: The generalization bounds offer
+confidence intervals on the expected testing approximations based on the empirical training approxi-
+mations; The sample complexities describe how many training samples the equilibrium approximators
+need to achieve desirable generalizability. The generalization bounds and sample complexities include
+the covering numbers [Shalev-Shwartz and Ben-David, 2014], which measure the representativeness of
+the approximators’ function classes. Afterward, we prove that the equivariant approximators have
+lower covering numbers than the general models, therefore have lower generalization bounds and sam-
+ple complexities. We then show that the equivariant approximators can achieve better approximation
+when the payoff distribution is permutation-invariant.
+As for the limitations, we find the equivariant approximators unable to find all the equilibria of
+some normal-form games. Such a result is caused by the limited representativeness of the equivariant
+approximators’ function class. Besides, we find that the equivariant NE approximator may lose social
+welfare. Specifically, in an example we constructed, while the maximum NE social welfare is large, the
+maximum social welfare of NEs that the equivariant NE approximators could find can be arbitrary
+1We will provide a formal definition of equivariance equilibrium approximators in Section 3
+1
+
+close to zero. Such a negative result inspires us to balance generalizability and social welfare when we
+design the approximators’ architectures.
+1.1
+Further Related Work
+Solving (approximate) NE, CE, and CCE for a single game are well studied [Fudenberg et al., 1998,
+Cesa-Bianchi and Lugosi, 2006]. However, many similar games often need to be solved [Harris et al.,
+2022] , both in practice and in some multi-agent learning algorithms [Marris et al., 2021, Liu et al.,
+2022]. For instance, in repeated traffic routing games [Sessa et al., 2020], the payoffs of games de-
+pend on the capacity of the underlying network, which can vary with time and weather condi-
+tions.
+In repeated sponsored search auctions, advertisers value different keywords based on the
+current marketing environment [Nekipelov et al., 2015].
+In many multi-agent learning algorithms
+such as Nash Q-learning [Hu and Wellman, 2003], Correlated-Q learning [Greenwald et al., 2003], V-
+learning [Jin et al., 2022] and PSRO [Lanctot et al., 2017], an NE, CE or CCE of a normal-form game
+need to be solved in every update step.
+In these settings, it is preferred to accelerate the speed of game solving by function approximation:
+Marris et al. [2022] introduces a neural equilibrium approximator to approximate CE and CCE for n-
+player normal-form games; Feng et al. [2021] proposes a neural NE approximator in PSRO [Lanctot et al.,
+2017]; Wu and Lisser [2022] designs a CNN-based NE approximator for zero-sum bimatrix games. Dif-
+ferentiable approximators have also been developed to learn QREs [Ling et al., 2018], NE in chance-
+constrained games [Wu and Lisser, 2023], and opponent’s strategy [Hartford et al., 2016].
+Equivariance is an ideal property of the equilibrium approximator. We will discuss the literates of
+equivariant approximators after formally defining them in Section 3.
+2
+Preliminary
+In this section, we introduce the preliminary and notations of our paper. We also provide a brief
+introduction to equilibrium approximators.
+2.1
+Game Theory
+Normal-Form Game
+Let a normal-form game with joint payoff u be Γu = (n, A, u), in which
+• n ∈ N≥2 is the number of players. Each player is represented by the index i ∈ [n].
+• A = ×i∈[n]Ai is the product action space of all players, where Ai = {1, 2, . . ., mi}. For player
+i ∈ [n], let ai ∈ Ai be a specific action of i (An action is also referred to as a pure strategy). A
+joint action a = (a1, a2, . . . , an) ∈ A represents one play of the game in which the player i takes
+action ai. The action space A is a Cartesian product that contains all possible joint actions. We
+have |A| = �
+i∈[n] |Ai| = �
+i∈[n] mi.
+• u = (ui)i∈[n] is the joint payoff or utility of the game. ui is an n-dimensional tensor (or matrix
+if n = 2) describing player i’s payoff on each joint action.
+In our paper, following previous
+literatures [Tsaknakis and Spirakis, 2007, Deligkas et al., 2022], we normalize all the elements of
+payoff into [0, 1].
+A joint (mixed) strategy is a distribution over A. Let σ = ×i∈[n]σi be a product strategy and
+π ∈ ∆A be a joint (possibly correlated) strategy. Denote πi as the marginal strategy of player i in π.
+The expected utility of player i under π is
+ui(π) = Ea∼π[ui(a)] =
+�
+a∈A
+π(a)ui(a).
+Besides, on behalf of player i, the other players’ joint strategy is denoted as π−i, so as a−i and σ−i.
+2
+
+Nash Equilibrium (NE)
+We say a product strategy σ∗ = (σ∗
+1, σ∗
+2, . . . , σ∗
+n) is a NE if each player’s
+strategy is the best response given the strategies of others, i.e.,
+ui(σi, σ∗
+−i) ≤ ui(σ∗
+i , σ∗
+−i), ∀i ∈ [n], σi ∈ ∆Ai.
+(NE)
+Computing NE for even general 2-player or 3-player games is PPAD-hard [Chen et al., 2009, Daskalakis et al.,
+2009], which leads to research on approximate solutions. For arbitrary ǫ > 0, we say a product strat-
+egy ˆσ is an ǫ-approximate Nash equilibrium (ǫ-NE) if no one can achieve more than ǫ utility gain by
+deviating from her current strategy. Formally,
+ui(σi, ˆσ−i) ≤ ui(ˆσi, ˆσ−i) + ǫ, ∀i ∈ [n], σi ∈ ∆Ai.
+(ǫ-NE)
+The definition of ǫ-NE reflects the idea that players might not be willing to deviate from their strategies
+when the amount of utility they could gain by doing so is tiny (not more than ǫ).
+Coarse Correlated Equilibrium (CCE)
+We say a joint (possibly correlated) strategy π∗ is a CCE
+if no player can receive a higher payoff by acting independently, i.e.,
+ui(σi, π∗
+−i) ≤ ui(π∗), ∀i ∈ [n], σi ∈ ∆Ai,
+(CCE)
+and we say ˆπ is an ǫ-approximate coarse correlated equilibrium (ǫ-CCE) for ǫ > 0 if
+ui(σi, ˆπ−i) ≤ ui(ˆπ) + ǫ, ∀i ∈ [n], σi ∈ ∆Ai,
+(ǫ-CCE)
+The difference between NE and CCE is that in an NE, players execute their strategy individu-
+ally in a decentralized way, while in a CCE, players’ strategies are possibly correlated.
+A stan-
+dard technique to correlate the strategy is sending each player a signal from a centralized controller
+[Shoham and Leyton-Brown, 2008].
+Correlated Equilibrium (CE)
+CE is similar to CCE, except that in a CE, each player can observe
+her recommended action before she acts. Thus, player i deviates her strategy through strategy mod-
+ification φi : Ai → Ai. φi maps actions in Ai to possibly different actions in Ai. Based on strategy
+modification, we say a joint (possibly correlated) strategy π∗ is a CE if
+�
+a∈A
+π∗(a)ui(φi(ai), a−i) ≤ ui(π∗), ∀i, ∀φi,
+(CE)
+and a joint strategy ˆπ is an ǫ-approximate correlated equilibrium (ǫ-CE) for ǫ > 0 if
+�
+a∈A
+ˆπ(a)ui(φi(ai), a−i) ≤ ui(ˆπ) + ǫ, ∀i, ∀φi,
+(ǫ-CE)
+Note that for a finite n-player normal-form game, at least one NE, CE, and CCE must exist. This
+is because NE always exists [Nash et al., 1950] and NE ⊆ CE ⊆ CCE.
+Equilibrium Approximation
+To evaluate the quality of a joint strategy to approximate an equilib-
+rium, we define approximation based on exploitability [Lockhart et al., 2019, Goktas and Greenwald,
+2022].
+Definition 2.1 (Exploitability and Approximation). Given a joint strategy π, the exploitability (or
+regret) Ei(π, u) of player i is the maximum payoff gain of i by deviating from her current strategy, i.e.,
+Ei(π, u) := max
+σ′
+i
+ui(σ′
+i, π−i) − ui(π) = max
+a′
+i
+ui(a′
+i, π−i) − ui(π)
+and the exploitability under strategy modification ECE
+i
+(π, u) of player i is the maximum payoff gain of
+i by deviating through strategy modification, i.e.,
+ECE
+i
+(π, u) := max
+φi
+�
+a∈A
+π(a)ui(φi(ai), a−i) − ui(π).
+3
+
+Algorithm 1 Example: learning NE/CCE approximator via minibatch SGD
+1: Input: Training set S
+2: Parameters: Number of iterations T > 0, batch size B > 0, learning rate η > 0, initial parameters
+w0 ∈ Rd of the approximator model.
+3: for t = 0 to T do
+4:
+Receive minibatch St = {u(1), . . . , u(B)} ⊂ S
+5:
+Compute the empirical average approximation of St:
+6:
+LSt(f wt) ← 1
+B
+�B
+i=1 E(f wt(u(i)), u(i))
+7:
+Update model parameters:
+8:
+wt+1 ← wt − η∇wtLSt(f wt)
+9: end for
+The equilibrium approximation is defined as the maximum exploitability over all players 2, i.e.,
+E(π, u) :=
+�
+maxi∈[n] Ei(π, u)
+, for NE and CCE
+maxi∈[n] ECE
+i
+(π, u)
+, for CE
+Based on approximation, we can restate the definition of solution concepts. A product strategy σ
+is an NE of game Γu if E(σ, u) = 0 and is an ǫ-NE if E(σ, u) ≤ ǫ. A joint strategy π is a (C)CE of Γu
+if E(π, u) = 0 and is an ǫ-(C)CE if E(π, u) ≤ ǫ.
+2.2
+Equilibrium Approximator
+The equilibrium approximators, including NE, CE, and CCE approximators, aim to predict solution
+concepts from game representations. In our paper, we fix the number of players n and action space A.
+We denote by U the set of all the possible game payoffs. The NE approximator f NE : U → ×i∈[n]∆Ai
+maps a game payoff to a product strategy, where f NE(u)i ∈ ∆Ai is player i’s predicted strategy. We
+define FNE as the function class of the NE approximator. Similarly, we define (C)CE approximator
+as f (C)CE : U → ∆A and (C)CE approximator class as F(C)CE.
+An equilibrium approximator can be learned through machine learning paradigms based on empir-
+ical data. For instance, Feng et al. [2021] learn the NE approximator using the game payoffs generated
+in the learning process of PSRO, and optimize the approximator by gradient descent in exploitability.
+Marris et al. [2022] learn the CE and CCE approximators using the games i.i.d. generated from a
+manually designed distribution, and optimize the approximators using maximum welfare minimum
+relative entropy loss. Such a loss balances the equilibrium approximation, the social welfare, and the
+relative entropy of the joint strategy. Additionally, another simple way to learn NE or CCE equilibrium
+approximator is to apply minibatch stochastic gradient descent (SGD) on approximation. Specifically,
+we denote w ∈ Rd as the d-dimensional parameter of the approximator, such as the weights of the
+neural network. We can optimize w by the standard minibatch SGD algorithm on approximation (See
+Algorithm 1).
+3
+Equivariant Equilibrium Approximator
+In this section, we introduce the equivariance of the equilibrium approximators and the way how
+we apply orbit averaging [Elesedy and Zaidi, 2021] to construct equivariant approximators. Equiv-
+ariant approximator has been developed in many literatures [Hartford et al., 2016, Feng et al., 2021,
+Marris et al., 2022, Wu and Lisser, 2022], which we will discuss latter.
+We first define the permutation of a game. Let ρi : Ai → Ai be a permutation function of player i,
+which is a bijection from Ai to Ai itself. Define Gi ∋ ρi as the class of permutation function for player
+i, which forms an Abelian group.
+Definition 3.1 (Permutation of a game). For a normal-form game Γu = (n, u, A), we define the
+ρi-permutation of payoff tensor u as ρiu = (ρiuj)j∈[n], in which
+(ρiuj)(ai, a−i) = uj(ρ−1
+i
+(ai), a−i), ∀a ∈ A.
+2A similar metric of equilibrium approximation is called Nikaido-Isoda function [Nikaidˆo and Isoda, 1955] or Nash-
+Conv [Lockhart et al., 2019], which is the sum of exploitability over all players, i.e., �
+i∈[n] Ei(π, u).
+4
+
+We also define the ρi-permutation of joint strategy π as ρiπ, where
+(ρiπ)(ai, a−i) = π(ρ−1
+i
+(ai), a−i), ∀a ∈ A,
+and the ρi-permutation of product strategy σ as ρiσ = (ρiσj)j∈[n], where
+∀aj ∈ Aj, ρiσj(aj) =
+�
+σj(aj)
+, if j ̸= i
+σi(ρ−1
+i ai)
+, if j = i
+Equivariant NE Approximator
+Considering the relationship of ρi-permuted game and ρi-permuted
+product strategy, we have the following result for NE:
+Lemma 3.2. In a normal-form game Γu = (n, u, A), for arbitrary player i ∈ [n] and any (ǫ-)NE
+strategy σ = (σi, σ−i), ρiσ = (ρiσi, σ−i) is also an (ǫ-)NE for the ρi-permuted game Γρiu.
+Lemma 3.2 tells the unimportance of action order with respect to NE approximation. Based on it,
+we can summarize two ideal properties for NE approximators, which are defined as follows:
+Definition 3.3 (Player-Permutation-Equivariance, PPE). We say an NE approximator f NE satisfies
+player i permutation-equivariant (i-PE) if for arbitrary permutation function ρi ∈ Gi we have
+f NE(ρiu)i = ρif NE(u)i,
+(i-PE)
+Moreover, we say f NE is player-permutation-equivariant (PPE) if f NE satisfies i-PE for all player
+i ∈ [n].
+Definition 3.4 (Opponent-Permutation-Invariance, OPI). We say an NE approximator f NE is oppo-
+nent i permutation-invariant (i-PI) if for any other player j ∈ [n] − {i} and arbitrary permutation
+function ρi ∈ Gi we have
+f NE(ρiu)j = f NE(u)j, ∀j ̸= i
+(i-PI)
+and we say f NE is opponent-permutation-invariant (OPI) if f NE satisfies i-PI for all player i ∈ [n].
+Equivariant (C)CE approximator
+Considering the relationship of ρi-permuted game and ρi-
+permuted joint strategy, we have a similar result for CE and CCE:
+Lemma 3.5. In a normal-form game Γu = (n, u, A), for an arbitrary player i ∈ [n] and any (ε-)CE
+or (ǫ-)CCE strategy π, ρiπ is also an (ε-)CE or (ǫ-)CCE for the ρi-permuted game Γρiu.
+Inspired by Lemma 3.5, we can also summarize an ideal property for CE and CCE approximators
+defined as follows.
+Definition 3.6 (Permutation-Equivariance,PE). We say an (C)CE approximator f (C)CE is player i
+permutation-equivariant (i-PE) if for permutation function ρi ∈ Gi we have
+f (C)CE(ρiu) = ρif (C)CE(u),
+and we say f (C)CE is permutation-equivariant (PE) if f (C)CE satisfies i-PE for all player i ∈ [n].
+Equivariant Approximators in Literature
+For two-player games, Feng et al. [2021] propose an
+MLP-based NE approximator that satisfies both PPE and OPI for zero-sum games. Additionally, they
+also design a Conv1d-based NE approximator that satisfies PPE only. Hartford et al. [2016] give a PPE
+approximator to predict players’ strategies. The traditional algorithms Tsaknakis and Spirakis [2007]
+and Deligkas et al. [2022], which approximate NE by optimization, are also PPE and OPI to payoff
+and the initial strategies. For n-player general games, Marris et al. [2022] provide a permutation-
+equivariant approximator to approximate CE and CCE. Equivariant architectures are also adopted
+in optimal auction design [Rahme et al., 2021, Duan et al., 2022, Ivanov et al., 2022], and Qin et al.
+[2022] theoretically characterize the benefits of permutation-equivariant in auction mechanisms. We
+follow the rough idea of Qin et al. [2022] when we analyze the benefits of equivariant equilibrium
+approximators.
+5
+
+3.1
+Orbit Averaging
+Orbit averaging is a well-known method to enforce equivariance or invariance for a function [Schulz-Mirbach,
+1994]. It averages the inputs of a function over the orbit of a group (e.g., the permutation group in
+our paper).
+Orbit Averaging for NE Approximator
+For an NE approximator f NE and any player i ∈ [n],
+we can construct a i-PI or i-PE NE approximator by averaging with respect to all the permutations
+of player i. Specifically, we construct an i-PI NE approximator by operator Oi with
+(Oif NE)(u)j =
+�
+f NE(u)i
+, if j = i
+1
+|Ai|!
+�
+ρi∈Gi f NE(ρiu)j
+, otherwise
+and we construct an i-PE NE approximator by operator Pi with:
+(Pif NE)(u)j =
+�
+1
+|Ai|!
+�
+ρi∈Gi ρ−1
+i f NE(ρiu)i
+, if j = i
+f NE(u)j
+, otherwise
+Lemma 3.7. Oif NE is i-PI and Pif NE is i-PE. Specially, if f NE is already i-PI or i-PE, then we
+have Oif NE = f NE or Pif NE = f NE, respectively.
+To construct a PPE or OPI NE approximator, we composite the operators with respect to all
+players. Let O = O1 ◦ O2 ◦ · · · ◦ On and P = P1 ◦ P2 ◦ · · · ◦ Pn, we get the following corollary:
+Lemma 3.8. Of NE is OPI and Pf NE is PPE. If f NE is already OPI or PPE, we have Of NE = f NE
+or Pf NE = f NE, respectively.
+Furthermore, we can also compose P ◦O to construct a NE approximator with both PPE and OPI.
+Orbit Averaging for (C)CE Approximator
+For CE or CCE approximator f, we define Qi-
+project for player i ∈ [n] to construct an i-PE approximator, which averages with respect to all the
+permutations of player i.
+(Qif (C)CE)(u) =
+1
+|Ai|!
+�
+ρi∈Gi
+ρ−1
+i f (C)CE(ρiu)
+Similarly, we define Q = Q1 ◦ Q2 ◦ · · · ◦ Qn as the composite operator.
+Lemma 3.9. Qif (C)CE is i-PE and Qf (C)CE is PE. Specifically, If f (C)CE is already i-PE or PE,
+then we have Qif (C)CE = f (C)CE or Qf (C)CE = f (C)CE, respectively.
+Combined with Lemma 3.7, Lemma 3.8 and Lemma 3.9, we can have the following corollary directly.
+Corollary 3.10. O2 = O, P2 = P, Q2 = Q.
+The benefit of using orbit averaging is shown in the following lemma:
+Lemma 3.11. Denote X as an idempotent operator, i.e. X 2 = X (e.g. O, P or Q). For function
+class F of NE, CE or CCE approximator, let FX be any subset of F that is closed under X, then XFX
+is the largest subset of FX that is invariant under X.
+According to Lemma 3.8, Lemma 3.9 and Lemma 3.11, OFNE(or PFNE/QF(C)CE) is the largest
+subset of FNE(or FNE/F(C)CE) with the corresponding property (OPI, PPE, or PE) if FNE(or
+FNE/F(C)CE) is closed operator under O(or P/Q). The result tells that the orbit averaging oper-
+ators, while enforcing the operated function to be equivariance or invariance, keep as large capacity
+of the function class as possible. Therefore, we believe that orbit averaging is an ideal approach to
+constructing equivariant or invariant functions.
+6
+
+4
+Theoretical Analysis of Benefits
+In this section, we theoretically analyze the benefits of equivariant approximators with respect to
+generalizability and approximation.
+4.1
+Benefits for Generalization
+We first derive the generalization bound and sample complexity for general approximator classes,
+and then we show the benefits of equivariant approximators by applying orbit averaging to the ap-
+proximators.
+The representativeness of an approximator class is measured by the covering numbers [Shalev-Shwartz and Ben-David,
+2014] under ℓ∞-distance, which are defined as follows:
+Definition 4.1 (ℓ∞-distance). The ℓ∞-distance between two equilibrium approximators f, g is:
+ℓ∞(f, g) = max
+u∈U ∥f(u) − g(u)∥,
+where we define the distance of two product strategies σ and σ′ as
+∥σ1 − σ2∥ = max
+i∈[n]
+�
+ai∈Ai
+|σ1
+i (ai) − σ2
+i (ai)|
+and the distance of two joint strategy π and π′ as
+∥π1 − π2∥ =
+�
+a∈A
+|π1(a) − π2(a)|
+Definition 4.2 (r-covering number). For r > 0, we say function class Fr r-covers another function
+class F under ℓ∞-distance if for all function f ∈ F, there exists fr ∈ Fr such that ∥f − fr∥∞ ≤ r. The
+r-covering number N∞(F, r) of F is the cardinality of the smallest function class Fr that r-covers F
+under ℓ∞-distance.
+Based on covering numbers, we provide the generalization bounds of NE, CE and CCE approxima-
+tors. The bounds describe the difference between the expected testing approximation and empirical
+training approximation.
+Theorem 4.3 (Generalization bound). For function class F of NE, CE or CCE approximator, with
+probability at least 1 − δ over draw of the training set S (with size m) from payoff distribution D, for
+all approximator f ∈ F we have
+Eu∼D[E(f(u), u)] − 1
+m
+�
+u∈S
+E(f(u), u) ≤ 2 · inf
+r>0{
+�
+2 ln N∞(F, r)
+m
++ Lr} + 4
+�
+2 ln(4/δ)
+m
+,
+where L = 2n for NE approximator, and L = 2 for CE and CCE approximators.
+To get the theorem, we first show that all three equilibrium approximations are Lipschitz continuous
+with respect to strategies. Afterward, we derive the Rademacher complexity [Bartlett and Mendelson,
+2002] of the expected approximation based on the Lipschitz continuity and covering numbers. See
+Appendix A.6 for the detailed proof.
+We can see from Theorem 4.3 that, with a large enough training set, the generalization gaps of
+equilibrium approximators go to zero if the covering number N∞(F, r) is bounded. As a result, we
+can estimate the expected testing performance through the empirical training performance.
+We can also derive the sample complexities of equilibrium approximators to achieve the desirable
+generalizability.
+Theorem 4.4 (Sample complexity). For ǫ, δ ∈ (0, 1), function class F of NE, CE or CCE approxi-
+mator and distribution D, with probability at least 1 − δ over draw of the training set S with
+m ≥
+9
+2ǫ2
+�
+ln 2
+δ + ln N∞(F, ǫ
+3L)
+�
+7
+
+games sampled from D, ∀f ∈ F we have
+Eu∼D[E(f(u), u)] ≤ 1
+m
+�
+u∈S
+E(f(u), u) + ǫ,
+where L = 2n for NE approximators, and L = 2 for CE and CCE approximators.
+The proof is based on the Lipschitz continuity of approximation, uniform bound, and concentration
+inequality. See Appendix A.7 for details. Theorem 4.4 is also called the uniform convergence of function
+class F, which is a sufficient condition for agnostic PAC learnable [Shalev-Shwartz and Ben-David,
+2014].
+As for the benefits of equivariant approximators for generalizability, the following result indicates
+that the projected equilibrium approximators have smaller covering numbers.
+Theorem 4.5. The O-projected, P-projected and Q-projected approximator classes have smaller cov-
+ering numbers, i.e., ∀r > 0 we have
+N∞(OFNE, r) ≤ N∞(FNE, r),
+N∞(PFNE, r) ≤ N∞(FNE, r),
+N∞(QF(C)CE, r) ≤ N∞(F(C)CE, r)
+The proof is done by showing all the operators are contraction mappings. See Appendix A.8 for
+details.
+Both the generalization bounds in Theorem 4.3 and the sample complexities in Theorem 4.4 decrease
+with the decrease of covering numbers N∞(F, r). Thus, we can see from Theorem 4.5 that both PPE
+and OPI can improve the generalizability of NE approximators, and PE can improve the generalizability
+of CE and CCE approximators.
+4.2
+Benefits for Approximation
+We then show the benefits of equivariance for approximation when the payoff distribution is invari-
+ant under permutation. The permutation-invariant distribution holds when the action is anonymous
+or indifferent or when we pre-train the equilibrium approximators using a manually designed distribu-
+tion [Marris et al., 2022].
+(C)CE Approximator
+The following theorem tells the benefit of permutation-equivariance in de-
+creasing the exploitability of (C)CE approximators.
+Theorem 4.6. When the payoff distribution D is invariant under the permutation of payoffs, the
+Q-projected (C)CE approximator has a smaller expected equilibrium approximation. Formally, for all
+f (C)CE ∈ F(C)CE and permutation-invariant distribution D, we have
+Eu∼D[E(Qf (C)CE(u), u)] ≤ Eu∼D[E(f (C)CE(u), u)],
+The proof is done by using the convexity of approximation. See Appendix A.10 for details. We can
+see from Theorem 4.6 that, when payoff distribution is invariant under permutation, it is beneficial to
+use equivariant architecture as the CE or CCE approximators.
+NE Approximator
+As for NE approximator, we have similar results.
+Theorem 4.7. For bimatrix constant-sum games, when the payoff distribution D is invariant under the
+permutation of payoffs, then the X-projected (X ∈ {O, P}) NE approximator has a smaller expected
+exploitability. Formally, for all f NE ∈ FNE and permutation-invariant distribution D for bimatrix
+constant-sum games, we have
+Eu∼D[
+�
+i
+Ei((Xf NE)(u), u)] ≤ Eu∼D[
+�
+i
+Ei(f NE(u), u)]
+8
+
+Theorem 4.8. When the payoff distribution D is invariant under the permutation of payoffs, and
+f NE satisfies OPI, then the P-projected NE approximator has a smaller expected NE approximation.
+Formally, for all f NE ∈ FNE that is OPI and permutation-invariant distribution D, we have
+Eu∼D[E((Pf NE)(u), u)] ≤ Eu∼D[E(f NE(u), u)].
+Theorem 4.9. For bimatrix games, when the payoff distribution D is invariant under the permutation
+of payoffs, and f NE satisfies PPE, then the O-projected NE approximator has a smaller expected NE
+approximation. Formally, for all f NE ∈ FNE that is PPE and permutation-invariant distribution D of
+bimatrix games, we have
+Eu∼D[E((Of NE)(u), u)] ≤ Eu∼D[E(f NE(u), u)].
+Theorem 4.8 and Theorem 4.9 tell that PPE and OPI approximators can achieve better approxi-
+mation than ones with only PPE or OPI. Meanwhile, we can see from Theorem 4.7 that for bimatrix
+constant-sum games (such as zero-sum games), it can be preferred to introduce PPE or OPI to the
+architectures.
+5
+Theoretical Analysis of Limitations
+As we discussed in Section 4, equivariant approximators enjoy better generalizability and better
+approximation sometimes. However, as we will show, they have some limitations regarding equilibrium
+selection and social welfare. Such limitations attribute to the limited representativeness caused by
+equivariance.
+5.1
+Equilibrium Selection
+We first show that there may be equilibria points that equivariant approximators will never find.
+We illustrate such limitation in permutation-invariant games, which is defined as follows:
+Definition 5.1 (Permutation-ρ-Invariant Game). We say a game Γu is permutation-ρ-invariant, where
+ρ = ◦i∈[n]ρi, if the payoff u is permutation-invariant with respect to ρ. That is, ρu = u.
+Permutation-ρ-invariance indicates that one cannot distinguish joint action a from ρa using only
+the payoff u. We’d like to provide an example to show more insight of permutation-ρ-invariant games:
+Example 5.2. For a 2-player game Γu = (2, u = (u1, u2), A = ([m1], [m2])) , Let ρi = (mi, mi −
+1, . . . , 1) and ρ = ρ1 ◦ ρ2. If one of the following conditions holds, then u is permutation-ρ-invariant:
+1. u1 and u2 are symmetric and persymmetric (i.e., symmetric with respect to the northeast-to-
+southwest diagonal) squares.
+2. Both u1 and u2 are centrosymmetric, i.e., ui(x, y) = ui(m1 +1−x, m2 +1−y) for i ∈ {1, 2}, x ∈
+[m1] and y ∈ [m2].
+For permutation ρ = (◦i∈[n]ρi) and player k ∈ [n], we denote the set of non-fixed actions of player
+k under ρk as
+V (ρk) := {ak|ak ∈ Ak, ρk(ak) ̸= ak}.
+Based on V (ρk), we find some equilibria points of permutation-ρ-invariant games that any equivariant
+approximators will never find.
+Theorem 5.3. For a permutation-ρ-invariant game Γu. if there is a pure NE a∗ = (a∗
+i )i∈[n] and at
+least one player k ∈ [n] such that a∗
+k ∈ V (ρk), then a∗ will never be found by any NE approximator
+with both PPE and OPI. Besides, a∗ (as a pure CE or CCE) will also never be found by any CE or
+CCE approximator with PE.
+We illustrate Theorem 5.3 by the following example:
+9
+
+Example 5.4. Consider a bimatrix game with identity utility
+u =
+�
+1, 1
+0, 0
+0, 0
+1, 1
+�
+There are two pure NE (bolded in the above matrix) and one mixed NE of σ1 = (0.5, 0.5) and σ2 =
+(0.5, 0.5). Let ρi be the unique permute function (except for identity function) of player i ∈ [2], and
+ρ = ρ1 ◦ ρ2. The game is permutation-ρ-invariant.
+Case 1: Let f be a permutation-equivariant CE or CCE approximator, and denote π = f(u). We
+have
+π = f(u)
+(a)
+= f(ρu)
+(b)
+= ρf(u),
+where (a) holds by permutation-ρ-invariance of u, and (b) holds by PE of f. Thus, we have π1,1 =
+π2,2 ∈ [0, 1
+2] and π1,2 = π2,1 ∈ [0, 1
+2]. As a result, the two pure (C)CEs cannot be found.
+Case 2: Let f be a NE approximator that holds PPE and OPI. Denote f(u) = (σ1, σ2), where
+σ1 = (p1, 1 − p1) and σ2 = (p2, 1 − p2). By PPE and OPI of f, we have
+f(u)1 = (p1, 1 − p1)
+(a)
+= f(ρ1ρ2u)1
+(b)
+= ρ1f(ρ2u)1
+(c)
+= ρ1f(u)1 = (1 − p1, p1),
+where (a) holds by permutaion-ρ-invariance of u, (b) holds by PPE of f, and (c) holds by OPI of f.
+As a result, the only NE that f could find is the mixed NE.
+As we can see from the example and Theorem 5.3, the equivariance, while introducing inductive bias
+to the approximator architecture, is also a strong constraint. Such a constraint is why the equivariant
+approximators cannot find all the equilibria points.
+5.2
+Social Welfare
+The social welfare of a joint strategy π is defined as the sum of all players’ utilities, i.e.,
+SW(π, u) =
+�
+i∈[n]
+ui(π).
+The equilibrium with higher social welfare is usually preferred [Marris et al., 2022].
+To analyze the social welfare of equivariant approximators, we define the worst social welfare ratio
+as follows:
+Definition 5.5. For any N, M ≥ 2 and two NE (or CE/CCE) approximator classes F1, F2 that target
+on games with number of players n ≤ N and |Ai| ≤ M, we define the worst social welfare ratio of F1
+over F2 as:
+SWRN,M(F1, F2) := inf
+D
+maxf1∈F1 Eu∼DSW(f1(u), u)
+maxf2∈F2 Eu∼DSW(f2(u), u)
+SWRN,M(F1, F2) measures the relative representativeness of F1 over F2 in terms of social welfare.
+Based on that, we have the following result for equivariant CE and CCE approximator classes:
+Theorem 5.6. Given N, M ≥ 2, let F(C)CE
+PE
+be the function class (target on games with number of
+players n ≤ N and |Ai| ≤ M) of all the (C)CE approximators with PE. Denote by F(C)CE
+general the function
+class of all the (C)CE approximators. Then we have
+SWRN,M(F(C)CE
+PE
+, F(C)CE
+general) = 1.
+Theorem 5.6 tells that, while the permutation-equivariant (C)CE approximator class may not be
+able to find all the (C)CE in a game, it can keep the social welfare of the output solutions.
+However, when considering equivariant NE approximators, we have the following negative result:
+10
+
+Theorem 5.7. Given N, M ≥ 2, let FNE
+OPI, FNE
+PPE and FNE
+both be the function classes (target on games
+with number of players n ≤ N and |Ai| ≤ M) of all the NE approximators with OPI, PPE and both.
+Denote the function class of all the NE approximators as FNE
+general. Then we have
+SWRN,M(FNE
+OPI, FNE
+general) =
+1
+M N−1 ,
+(1)
+SWRN,M(FNE
+PPE, FNE
+general) ≤ 1
+M ,
+(2)
+SWRN,M(FNE
+both, FNE
+general) =
+1
+M N−1 .
+(3)
+Additionally, when M ≥ 3, denote by �FNE
+both the function class of all the NE oracles (functions that
+always output exact NE solutions of the input games) with both PPE and OPI, and by �
+FNE
+general the
+function class of all the NE oracles. Then we have
+SWRN,M( �FNE
+both, �FNE
+general) = 0.
+(4)
+The proof is done by construction (See Appendix A.15 for details). As an illustration of Equa-
+tion (4), consider a bimatrix game with the following payoff:
+u =
+
+
+1, 1
+0, 0
+0, 1
+2 + ε
+0, 0
+1, 1
+0, 1
+2 + ε
+1
+2 + ε, 0
+1
+2 + ε, 0
+ε, ε
+
+
+for ǫ ∈ (0, 1
+2). The maximum NE (the upper-left corner of u) social welfare is 2, which can be found
+by at least one NE oracle in �FNE
+general. However, the only NE (the lower-right corner of u) that the NE
+oracles in �FNE
+both could find only has a social welfare of 2ǫ. As a result,
+SWR2,3( �FNE
+both, �FNE
+general) ≤ 2ǫ
+2 = ǫ,
+which goes to zero as ǫ → 0. Recall that we always have SWRN,M ≥ 0, thus Equation (4) holds when
+N = 2 and M = 3.
+Theorem 5.7 tells that equivariant NE approximators may lose some social welfare while enjoying
+better generalizability. Such a result inspires us to balance generalizability and social welfare when
+designing the NE approximator architecture.
+6
+Conclusion and Future Work
+In this paper, we theoretically analyze the benefits and limitations of equivariant equilibrium
+approximators, including player-permutation-equivariant (PPE) and opponent-permutation-invariant
+(OPI) NE approximator, and permutation-equivariant (PE) CE and CCE approximators. For the
+benefits, we first show that these equivariant approximators enjoy better generalizability. To get the
+result, we derive the generalization bounds and sample complexities based on covering numbers, and
+then we prove that the symmetric approximators have lower covering numbers. We then show that
+the equivariant approximators can decrease the exploitability when the payoff distribution is invariant
+under permutation. For the limitations, we find the equivariant approximators may fail to find some
+equilibria points due to their limited representativeness caused by equivariance. Besides, while equiv-
+ariant (C)CE approximators can keep the social welfare, the equivariant NE approximators reach a
+small worst social welfare ratio comparing to the general approximators. Such a result indicates that
+equivariance may reduce social welfare; therefore, we’d better balance the generalizability and social
+welfare when we design the architectures of NE approximators.
+As for future work, since in our paper we assume the training and testing payoff distribution are
+the same, an interesting topic is to study the benefits of equivariant approximators under the payoff
+distribution shift. Moreover, since we consider fixed and discrete action space, another interesting
+future direction is to analyze the benefits of equivariant approximators in varying or continuous action
+space.
+11
+
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+13
+
+A
+Omitted Proof
+A.1
+Useful Lemma
+We first introduce a lemma, which will be frequently used in the following proofs.
+Lemma A.1. ∀i, j ∈ [n], ρi ∈ Gi we have (ρiu)j(σi, σ−i) = uj(ρ−1
+i σi, σ−i) and (ρiu)j(π) = uj(ρ−1
+i π)
+Proof. Define �ai := ρ−1
+i ai. For product strategy σ = (σi)i∈[n],
+(ρiu)j(σi, σ−i) =
+�
+ai∈Ai
+�
+a−i∈A−i
+(ρiu)j(ai, a−i) · σi(ai) · σ−i(a−i)
+=
+�
+ai∈Ai
+�
+a−i∈A−i
+uj(ρ−1
+i ai, a−i) · σi(ai) · σ−i(a−i)
+=
+�
+ai∈Ai
+�
+a−i∈A−i
+uj(ρ−1
+i ai, a−i) · (ρ−1
+i
+σi)(ρ−1
+i ai) · σ−i(a−i)
+=
+�
+�ai∈Ai
+�
+a−i∈A−i
+uj(�ai, a−i) · (ρ−1
+i
+σi)(�ai) · σ−i(a−i)
+=uj(ρ−1
+i σi, σ−i)
+For joint strategy π,
+(ρiu)j(π) =
+�
+ai∈Ai
+�
+a−i∈A−i
+(ρiuj)(ai, a−i) · π(ai, a−i)
+=
+�
+ai∈Ai
+�
+a−i∈A−i
+uj(ρ−1
+i ai, a−i) · π(ai, a−i)
+=
+�
+ai∈Ai
+�
+a−i∈A−i
+uj(ρ−1
+i ai, a−i) · (ρ−1
+i
+π)(ρ−1
+i
+ai, a−i)
+=
+�
+�ai∈Ai
+�
+a−i∈A−i
+uj(�ai, a−i) · (ρ−1
+i
+π)(�ai, a−i)
+=uj(ρ−1
+i π)
+A.2
+Proof of Lemma 3.2
+Proof. For player i, we have
+Ei(ρiσ, ρiu) = max
+ai∈Ai ρiui(ai, ρiσ−i) − ρiui(ρiσ) = max
+ai∈Ai ρiui(ai, σ−i) − ρiui(ρiσi, σ−i)
+= max
+ai∈Ai ui(ρ−1
+i ai, σ−i) − ui(ρ−1
+i ρiσi, σ−i)
+(a)
+= max
+ai∈Ai ui(ai, σ−i) − ui(σi, σ−i) = Ei(σ, u),
+where (a) holds since ρi is a bijection on Ai. For player j ̸= i, we have
+Ej(ρiσ, ρiu) = max
+aj∈A ρiuj(aj, ρiσ−j) − ρiuj(ρiσ) = max
+aj∈Aj uj(aj, ρ−1
+i ρiσ−j) − uj(ρ−1
+i ρiσ)
+= max
+aj∈Aj uj(aj, σ−j) − uj(σ) = Ej(σ, u)
+From above, we have E(ρiσ, ρiu) = E(σ, u), thus if σ is a ε-NE of Γu, then ρiσ must be a ε-NE of
+Γρiu.
+14
+
+A.3
+Proof of Lemma 3.5
+CCE
+For player i, we have
+Ei(ρiπ, ρiu) = max
+ai∈Ai(ρiui)(ai, (ρiπ)−i) − (ρiui)(ρiπi)
+= max
+ai∈Ai(ρiui)(ai, (ρiπ)−i) − ui(ρ−1
+i ρiπi)
+= max
+ai∈Ai(ρiui)(ai, (ρiπ)−i) − ui(πi)
+= max
+ai∈Ai
+�
+b∈A
+(ρiui)(ai, b−i) · (ρiπ)(b) − ui(πi)
+= max
+ai∈Ai
+�
+bi∈Ai,b−i∈A−i
+ui(ρ−1
+i
+ai, b−i) · π(ρ−1
+i
+bi, b−i) − ui(πi)
+= max
+ai∈Ai
+�
+bi∈Ai,b−i∈A−i
+ui(ai, b−i) · π(bi, b−i) − ui(πi)
+, ρi is a bijection on Ai
+=Ei(π, u)
+For player j ̸= i, we have
+Ej(ρiπ, ρiu) = max
+aj∈Aj(ρiuj)(aj, (ρiπ)−j) − (ρiuj)(ρiπj)
+= max
+aj∈Aj(ρiuj)(aj, (ρiπ)−j) − uj(ρ−1
+i ρiπj)
+= max
+aj∈Aj(ρiuj)(aj, (ρiπ)−j) − uj(πj)
+= max
+aj∈Aj
+�
+b∈A
+(ρiuj)(aj, b−j) · (ρiπ)(b) − uj(πj)
+= max
+aj∈Aj
+�
+bi∈Ai,b−i∈A−i
+uj(aj, (b−j)−i, ρ−1
+i bi) · π(ρ−1
+i
+bi, b−i) − uj(πj)
+= max
+aj∈Aj
+�
+bi∈Ai,b−i∈A−i
+uj(aj, (b−j)−i, bi) · π(bi, b−i) − uj(πj)
+, ρi is a bijection on Ai
+=Ej(π, u)
+Thus, we have E(ρiπ, ρiu) = E(π, u). Thus, if π is a ε-CCE of Γu, then ρiπ must be a ε-CCE of Γρiu.
+CE
+For player j ̸= i, we have
+ECE
+j
+(ρiπ, ρiu) =
+max
+φj:Aj→Aj
+�
+a∈A
+(ρiπ)(a) · (ρiuj)(φj(aj), a−j) − (ρiuj)(ρiπ)
+=
+max
+φj:Aj→Aj
+�
+a∈A
+π(ρ−1
+i
+ai, a−i) · uj(φj(aj), a−i,j, ρ−1
+i ai) − uj(π)
+=
+max
+φj:Aj→Aj
+�
+a∈A
+π(ai, a−i) · uj(φj(aj), a−i,j, ai) − uj(π)
+, ρi is a bijection on Ai
+=ECE
+j
+(π, u)
+For player i, we define operator ¯ρi as (¯ρiφi)(ai) = ρ−1
+i φi(ρiai). We can verify that ¯ρi is a bijection
+on {φi : Ai → Ai}, because ¯· is a homomorphism in the sense that ρ1
+i ◦ ρ2
+i = ρ2
+i ρ1
+i and ¯· maps the
+identity mapping of Ai to the identity mapping of {Ai → Ai}. Specifically,
+ρ1
+i ◦ ρ2
+i φi(ai) = (ρ1
+i )−1(ρ2
+i φi)(ρ1
+i ai) = (ρ1
+i )−1(ρ2
+i )−1φi(ρ2
+i ρ1
+i ai) = ρ2
+i ρ1
+i φi(ai),
+and
+eiφi(ai) = e−1
+i φi(eiai) = φi(ai).
+15
+
+Based on ¯ρi, we have
+ECE
+i
+(ρiπ, ρiu)
+=
+max
+φi:Ai→Ai
+�
+a∈A
+(ρiπ)(a) · (ρiui)(φi(ai), a−i) − ui(π)
+=
+max
+φi:Ai→Ai
+�
+a∈A
+π(ρ−1
+i ai, a−i)ui(ρ−1
+i φi(ai), a−i) − ui(π)
+=
+max
+φi:Ai→Ai
+�
+a∈A
+π(ρ−1
+i ai, a−i)ui(ρ−1
+i φi(ρi(ρ−1
+i ai)), a−i) − ui(π)
+=
+max
+φi:Ai→Ai
+�
+a∈A
+π(ai, a−i)ui(ρ−1
+i φi(ρiai), a−i) − ui(π)
+, ρi is a bijection on Ai
+=
+max
+φi:Ai→Ai
+�
+a∈A
+π(ai, a−i)ui((¯ρiφi)(ai), a−i) − ui(π)
+=
+max
+φi:Ai→Ai
+�
+a∈A
+π(ai, a−i)ui(φi(ai), a−i) − ui(π)
+, ¯ρi is a bijection on {Ai → Ai}
+=ECE
+i
+(π, u)
+Thus, we have E(ρiπ, ρiu) = E(π, u), thus if π is a ε-CE of Γu, then ρiπ must be a ε-CE of Γρiu.
+A.4
+Proof of Lemma 3.7 to Lemma 3.9
+Proof of Lemma 3.7. ∀j ̸= i, ρ0 ∈ Gi, for operator Oi we have
+(Oif NE)(ρ0u)j =
+1
+|Ai|!
+�
+ρi∈Gi
+f NE(ρiρ0u)j
+(a)
+=
+1
+|Ai|!
+�
+�ρi∈Gi
+f NE(�ρiu)j = (Oif NE)(u)j
+where in (a) we define �ρi = ρiρ0, and (a) holds since ρ0 is a bijection on Gi. As a result, Oif NE is i-PI.
+For operator Pi we have
+(Pif NE)(ρ0u)i =
+1
+|Ai|!
+�
+ρi∈Gi
+ρ−1
+i f NE(ρiρ0u)j = ρ0
+1
+|Ai|!
+�
+ρi∈Gi
+ρ−1
+0 ρ−1
+i f NE(ρiρ0u)j
+=ρ0
+1
+|Ai|!
+�
+�ρi∈Gi
+�ρ−1
+i f NE(�ρiu)j = ρ0(Pif NE)(u)i,
+therefore Pif NE is i-PE.
+If f NE is already i-PI, ∀j ̸= i we have
+Oif NE(u)j =
+1
+|Ai|!
+�
+ρi∈Gi
+f NE(ρiu)j =
+1
+|Ai|!
+�
+ρi∈Gi
+f NE(u)j = f NE(u)j,
+and Oif NE(u)i = f NE(u)i according to definition of Oi. Therefore, Oif NE = f NE for i-PI f NE.
+If f NE is already i-PE, we have
+Pif NE(u)i =
+1
+|Ai|!
+�
+ρi∈Gi
+ρ−1
+i f NE(ρiu)i =
+1
+|Ai|!
+�
+ρi∈Gi
+ρ−1
+i ρif NE(u)i =
+1
+|Ai|!
+�
+ρi∈Gi
+f NE(u)i = f NE(u)i,
+and ∀j ̸= i, Pif NE(u)j = f NE(u)j according to definition of Pi. Therefore, Pif NE = f NE for i-PE
+f NE.
+Proof of Lemma 3.8. A direct inference from Lemma 3.7
+Proof of Lemma 3.9. ∀ρ0 ∈ Gi, we have
+16
+
+(Qif (C)CE)(ρ0u) =
+1
+|Ai|!
+�
+ρi∈Gi
+ρ−1
+i f (C)CE(ρiρ0u) = ρ0
+1
+|Ai|!
+�
+ρi∈Gi
+ρ−1
+0 ρ−1
+i
+f (C)CE(ρiρ0u)
+=ρ0
+1
+|Ai|!
+�
+�ρi∈Gi
+�ρ−1
+i f (C)CE(�ρiu) = ρ0(Qif (C)CE)(u)
+If f (C)CE is already i-PE, we have
+Qif (C)CE(u) =
+1
+|Ai|!
+�
+ρi∈Gi
+ρ−1
+i f (C)CE(ρiu) =
+1
+|Ai|!
+�
+ρi∈Gi
+ρ−1
+i ρif (C)CE(u) =
+1
+|Ai|!
+�
+ρi∈Gi
+f (C)CE(u) = f (C)CE(u)
+A.5
+Proof of Lemma 3.11
+We prove the three claims below.
+1. XFX ⊆ FX .
+2. X 2FX = XFX .
+3. If XY = Y ⊆ FX , then Y ⊆ XFX
+The first claim holds because FX is closed under X, and the second claim holds because X is
+idempotent. For the third claim, from Y ⊆ FX we know XY ⊆ XFX , then Y = XY ⊆ XFX .
+We immediately know XFX is the largest subset of FX that is invariant under X.
+A.6
+Proof of Theorem 4.3
+Some of the techniques come from D¨utting et al. [2019] and Duan et al. [2021]. We first introduce
+some useful lemmas. Denote ℓ : F × U → R as the loss function (such as ℓ(f, u) := E(f(u), u)). We
+measure the capacity of the composite function class ℓ ◦ F using the empirical Rademacher complex-
+ity [Bartlett and Mendelson, 2002] on the training set S, which is defined as:
+RS(ℓ ◦ F) := 1
+mEx∼{+1,−1}m
+�
+sup
+f∈F
+m
+�
+i=1
+xi · ℓ(f, u(i))
+�
+,
+where x is distributed i.i.d. according to uniform distribution in {+1, −1}. We have
+Lemma A.2 (Shalev-Shwartz and Ben-David [2014]). Let S be a training set of size m drawn i.i.d.
+from distribution D over U. Then with probability at least 1 − δ over draw of S from D, for all f ∈ F,
+Eu∼D[ℓ(f, u)] − 1
+m
+�
+u∈S
+ℓ(l, u) ≤ 2RS(ℓ ◦ F) + 4
+�
+2 ln(4/δ)
+m
+Lemma A.3. If |ℓ(·)| ≤ c for constant c > 0 and ∀f, f ′ ∈ F, |ℓ(f, u) − ℓ(f ′, u)| ≤ L∥f − f ′∥∞, then
+we have
+Eu∼D[ℓ(f, u)] − 1
+m
+�
+u∈S
+ℓ(l, u) ≤ 2 inf
+r>0
+�
+c
+�
+2 ln N∞(F, r)
+m
++ Lr
+�
++ 4
+�
+2 ln(4/δ)
+m
+Proof. For function class F, let Fr with |Fr| = N∞(F, r) be the function class that r-covers F for
+17
+
+some r > 0. Similarly, ∀f ∈ F, denote fr ∈ Fr be the function that r-covers f. We have
+RS(ℓ ◦ F) = 1
+mEx
+�
+sup
+f∈F
+m
+�
+i=1
+xi · ℓ(f, u(i))
+�
+= 1
+mEx
+�
+sup
+f∈F
+m
+�
+i=1
+xi ·
+�
+ℓ(fr, u(i)) + ℓ(f, u(i)) − ℓ(fr, u(i))
+��
+≤ 1
+mEx
+�
+sup
+fr∈Fr
+m
+�
+i=1
+xi · ℓ(fr, u(i))
+�
++ 1
+mEx
+�
+sup
+f∈F
+m
+�
+i=1
+|xi · Lr|
+�
+, |ℓ(f, u) − ℓ(fr, u)| ≤ L∥f − fr∥∞ = Lr
+≤ sup
+fr∈Fr
+�
+�
+�
+�
+m
+�
+i=1
+ℓ2(fr, u(i)) ·
+�
+2 ln N∞(F, r)
+m
++ Lr
+m Ex∥x∥
+, the first term holds by Massart’s lemma
+≤
+√
+c2m ·
+�
+2 ln N∞(F, r)
+m
++ Lr
+m Ex∥x∥
+≤c
+�
+2 ln N∞(F, r)
+m
++ Lr,
+(5)
+Combining Lemma A.2 and Equation (5), we get
+Eu∼D[ℓ(f, u)] − 1
+m
+�
+u∈S
+ℓ(l, u) ≤ 2 inf
+r>0
+�
+c
+�
+2 ln N∞(F, r)
+m
++ Lr
+�
++ 4
+�
+2 ln(4/δ)
+m
+NE Approximator
+Lemma A.4. For arbitrary product mixed strategy σ and σ′, we have
+|E(σ, u) − E(σ′, u)| ≤ 2n∥σ − σ′∥,
+Proof. ∀σ, σ′, we define y−j := (σ1, . . . , σj−1, σ′
+j+1, . . . , σ′
+n). Then, ∀i ∈ [n] we have
+|ui(σ) − ui(σ′)| =|ui(σ1, σ2, . . . , σn) − ui(σ′, σ′
+2, . . . , σ′
+n)|
+=
+���
+n
+�
+j=1
+�
+ui(σ1, . . . , σj, σ′
+j+1, . . . , σ′
+n) − ui(σ1, . . . , σ′
+j, σ′
+j+1, . . . , σ′
+n)
+����
+=
+���
+n
+�
+j=1
+�
+ui(σj, y−j) − ui(σ′
+j, y−j)
+����
+=
+���
+n
+�
+j=1
+�
+aj
+(σj(aj) − σ′
+j(aj))
+�
+a−j
+ui(aj, a−j)y−j(a−j)
+���
+≤
+n
+�
+j=1
+�
+aj
+���σj(aj) − σ′
+j(aj)
+���
+�
+a−j
+ui(aj, a−j)y−j(a−j)
+≤
+n
+�
+j=1
+�
+aj
+���σj(aj) − σ′
+j(aj)
+���
+�
+a−j
+y−j(a−j)
+, ui(·) ∈ [0, 1]
+≤
+n
+�
+j=1
+�
+aj∈Aj
+���σj(aj) − σ′
+j(aj)
+��� ≤ n max
+j∈[n]
+�
+aj∈Aj
+���σj(aj) − σ′
+j(aj)
+���
+=n∥σ − σ′∥,
+Therefore, ∀ai ∈ Ai,
+ui(ai, σ−i) − ui(σ) =ui(ai, σ−i) − ui(ai, σ′
+−i) + ui(ai, σ′
+−i) − ui(σ′) + ui(σ′) − ui(σ)
+≤n∥σ − σ′∥ + E(σ′, u) + n∥σ − σ′∥
+=E(σ′, u) + 2n∥σ − σ′∥.
+18
+
+Based on that, we get
+E(σ, u) =
+max
+i∈N,ai∈Ai[ui(ai, σ−i) − ui(σ)] ≤ E(σ′, u) + 2n∥σ − σ′∥
+Similarly, we also have
+E(σ′, u) ≤ E(σ, u) + 2n∥σ − σ′∥
+Based on Lemma A.4, ∀f, f ′ ∈ FNE, we have
+E(f(u), u) − E(f ′(u), u) ≤ 2∥f(u) − f ′(u)∥ ≤ 2∥f − f ′∥∞
+Considering that |E(·)| ≤ 1, according to Lemma A.3, we have:
+Eu∼D[E(f NE(u), u)] − 1
+m
+�
+u∈S
+E(f NE(u), u) ≤ 2 · inf
+r>0
+��
+2 ln N∞(FNE, r)
+m
++ 2nr
+�
++ 4
+�
+2 ln(4/δ)
+m
+CCE Approximator
+Lemma A.5. For arbitrary joint mixed strategy π and π′, we have
+|E(π, u) − E(π′, u)| ≤ 2∥π − π′∥,
+Proof. ∀π, π′, ∀i ∈ [n] we have
+|ui(π) − ui(π′)| =
+�
+a∈A
+(π(a) − π′(a))ui(a)
+(a)
+≤
+�
+a∈A
+|π(a) − π′(a)| = ∥π − π′∥
+(6)
+where (a) holds since ui(·) ∈ [0, 1]. Therefore, ∀ai ∈ Ai,
+ui(ai, π−i) − ui(π) =ui(ai, π−i) − ui(ai, π′
+−i) + ui(ai, π′
+−i) − ui(π′) + ui(π′) − ui(π)
+≤∥π − π′∥ + E(π′, u) + ∥π − π′∥
+=E(π′, u) + 2∥π − π′∥.
+Based on that, we get
+E(π, u) =
+max
+i∈N,ai∈Ai[ui(ai, π−i) − ui(π)] ≤ E(π′, u) + 2∥π − π′∥
+Similarly, we also have
+E(π′, u) ≤ E(π, u) + 2∥π − π′∥
+Based on Lemma A.5, ∀f, f ′ ∈ FCCE, we have
+E(f(u), u) − E(f ′(u), u) ≤ 2∥f(u) − f ′(u)∥ ≤ 2∥f − f ′∥∞
+Considering that |E(·)| ≤ 1, according to Lemma A.3, we have:
+Eu∼D[E(f CCE(u), u)] − 1
+m
+�
+u∈S
+E(f CCE(u), u) ≤ 2 · inf
+r>0
+��
+2 ln N∞(FCCE, r)
+m
++ 2r
+�
++ 4
+�
+2 ln(4/δ)
+m
+19
+
+CE Approximator
+Lemma A.6. For arbitrary joint mixed strategy π and π′, we have
+|ECE(π, u) − ECE(π′, u)| ≤ 2∥π − π′∥,
+Proof. ∀ai ∈ Ai, ∀φi, we have
+�
+a∈A
+π(a)ui(φ(ai), a−i) − ui(π) =
+�
+a∈A
+π(a)ui(φ(ai), a−i) −
+�
+a∈A
+π′(a)ui(φ(ai), a−i)
++
+�
+a∈A
+π′(a)ui(φ(ai), a−i) − ui(π′) + ui(π′) − ui(π)
+≤∥π − π′∥ + ECE(π′, u) + ∥π − π′∥
+=ECE(π′, u) + 2∥π − π′∥.
+Based on that, we get
+ECE(π, u) = max
+i∈N max
+φi
+�
+a∈A
+π(a)ui(φ(ai), a−i) − ui(π) ≤ ECE(π′, u) + 2∥π − π′∥
+Similarly, we also have
+ECE(π′, u) ≤ ECE(π, u) + 2∥π − π′∥
+Based on Lemma A.5, ∀f, f ′ ∈ FCE, we have
+ECE(f(u), u) − ECE(f ′(u), u) ≤ 2∥f(u) − f ′(u)∥ ≤ 2∥f − f ′∥∞
+Considering that |E(·)| ≤ 1, according to Lemma A.3, we have:
+Eu∼D[ECE(f CE(u), u)] − 1
+m
+�
+u∈S
+ECE(f CE(u), u) ≤ 2 · inf
+r>0
+��
+2 ln N∞(FCE, r)
+m
++ 2r
+�
++ 4
+�
+2 ln(4/δ)
+m
+A.7
+Proof of Theorem 4.4
+For function class F of NE, CE or CCE approximators, according to Lemma A.4, Lemma A.5 and
+Lemma A.6, ∀f, g ∈ F we have
+E(CE)(f(u), u) − E(CE)(g(u), u) ≤ L∥f(u) − g(u)∥ ≤ L∥f − g∥∞,
+(7)
+where L = 2n for NE approximators, and L = 2 for CE and CCE approximators.
+For simplicity, we denote LS(f) =
+1
+m
+�
+u∈S E(CE)(f(u), u) and LD(f) = Eu∼D[E(CE)(f(u), u)]. let
+Fr with |Fr| = N∞(F, r) be the function class that r-covers F for some r > 0. ∀ǫ ∈ (0, 1), by setting
+r =
+ǫ
+3L we have
+PS∼Dm
+�
+∃f ∈ F,
+��LS(f) − LD(f)
+�� > ǫ
+�
+≤PS∼Dm
+�
+∃f ∈ F,
+��LS(f) − LS(fr)
+�� +
+��LS(fr) − LD(fr)
+�� +
+��LD(fr) − LD(f)
+�� > ǫ
+�
+(a)
+≤PS∼Dm
+�
+∃f ∈ F, Lr +
+��LS(fr) − LD(fr)
+�� + Lr > ǫ
+�
+≤PS∼Dm
+�
+∃fr ∈ Fr,
+��LS(fr) − LD(fr)
+�� > ǫ − 2Lr
+�
+(b)
+≤N∞(F, r)PS∼Dm
+���LS(f) − LD(f)
+�� > ǫ − 2Lr
+�
+(c)
+≤2N∞(F, r) exp(−2m(ǫ − 2Lr)2),
+=2N∞(F, ǫ
+3L) exp(−2
+9mǫ2)
+where (a) holds by Equation (7), (b) holds by union bound, and (c) holds by Hoeffding inequality. As
+a result, when m ≥
+9
+2ǫ2
+�
+ln 2
+δ + ln N∞(F,
+ǫ
+3L)
+�
+, we have PS∼Dm
+�
+∃f ∈ F,
+���LS(f) − LD(f)
+��� > ǫ
+�
+< δ.
+20
+
+A.8
+Proof of Theorem 4.5
+We first provide an auxiliary lemma.
+Lemma A.7. For function class F and orbit averaging operator X, if ∀f, g ∈ F, ℓ∞(Xf, Xg) ≤
+ℓ∞(f, g), then N∞(XF, r) ≤ N∞(F, r) for any r > 0.
+Proof. ∀r > 0, Denote Fr as the smallest r-covering set that covers F with size N∞(F, r). ∀f ∈ F,
+let fr ∈ Fr be the function that r-covers f. We have ℓ∞(Xfr, Xf) ≤ ℓ∞(fr, f) ≤ r. Therefore, XFr
+is a r-covering set of XF, and we have N∞(XF, r) ≤ |XFr| ≤ |Fr| = N∞.
+For player i ∈ [n] and ∀f NE, gNE ∈ FNE, assuming U is closed under any ρi ∈ Gi. For Oi,
+l∞(Oif NE, OigNE) = max
+u∈U ∥Oif NE(u) − OigNE(u)∥
+= max
+j∈[n] max
+u∈U ∥(Oif NE)(u)j − (OigNE)(u)j∥
+= max
+�
+max
+u∈U ∥f NE(u)i − gNE(u)i∥, max
+j̸=i max
+u∈U ∥
+1
+|Ai|!
+�
+ρi∈Gi
+(f NE(ρiu)j − gNE(ρiu)j)∥
+�
+≤ max
+�
+max
+u∈U ∥f NE(u)i − gNE(u)i∥, max
+j̸=i
+1
+|Ai|!
+�
+ρi∈Gi
+max
+u∈U ∥f NE(ρiu)j − gNE(ρiu)j∥
+�
+= max
+�
+max
+u∈U ∥f NE(u)i − gNE(u)i∥, max
+j̸=i
+1
+|Ai|!
+�
+ρi∈Gi
+max
+u∈U ∥f NE(u)j − gNE(u)j∥
+�
+= max
+�
+max
+u∈U ∥f NE(u)i − gNE(u)i∥, max
+j̸=i max
+u
+∥f NE(u)j − gNE(u)j∥
+�
+=l∞(f NE, gNE)
+Since O = O1 ◦ · · · ◦ On, we have
+ℓ∞(Of NE, OgNE) ≤ ℓ∞(f NE, gNE).
+(8)
+For Pi,
+l∞(Pif NE, PigNE) = max
+u∈U max
+j∈[n] ∥(Pif NE)(u)j − (PigNE)(u)j∥
+= max
+�
+max
+j̸=i max
+u
+∥f NE(u)j − gNE(u)j∥, max
+u
+∥
+1
+|Ai|!
+�
+ρi∈Gi
+ρ−1
+i (f NE(ρiu)i − gNE(ρiu)i)∥
+�
+= max
+�
+max
+j̸=i max
+u
+∥f NE(u)j − gNE(u)j∥, max
+u
+∥
+1
+|Ai|!
+�
+ρi∈Gi
+(f NE(ρiu)i − gNE(ρiu)i)∥
+�
+≤ max
+�
+max
+j̸=i max
+u
+∥f NE(u)j − gNE(u)j∥,
+1
+|Ai|!
+�
+ρi∈Gi
+max
+u
+∥f NE(ρiu)i − gNE(ρiu)i∥
+�
+= max
+�
+max
+j̸=i max
+u
+∥f NE(u)j − gNE(u)j∥,
+1
+|Ai|!
+�
+ρi∈Gi
+max
+u
+∥f NE(u)i − gNE(u)i∥
+�
+= max
+�
+max
+j̸=i max
+u
+∥f NE(u)j − gNE(u)j∥, max
+u
+∥f NE(u)i − gNE(u)i∥
+�
+=l∞(f NE, gNE)
+Since P = P1 ◦ · · · ◦ Pn, we have
+ℓ∞(Pf NE, PgNE) ≤ ℓ∞(f NE, gNE).
+(9)
+21
+
+For CE or CCE approximator f (C)CE ∈ F(C)CE and Qi, we have
+l∞(Qif (C)CE, Qig(C)CE) = max
+u∈U ∥(Qif (C)CE)(u) − (Qig(C)CE)(u)∥
+= max
+u
+∥
+1
+|Ai|!
+�
+ρi∈Gi
+ρ−1
+i (f (C)CE(ρiu) − g(C)CE(ρiu))∥
+≤ max
+u
+1
+|Ai|!
+�
+ρi∈Gi
+∥ρ−1
+i (f (C)CE(ρiu) − g(C)CE(ρiu))∥
+≤
+1
+|Ai|!
+�
+ρi∈Gi
+max
+u
+∥ρ−1
+i (f (C)CE(ρiu) − g(C)CE(ρiu))∥
+=
+1
+|Ai|!
+�
+ρi∈Gi
+max
+u
+∥f (C)CE(ρiu) − g(C)CE(ρiu)∥
+=
+1
+|Ai|!
+�
+ρi∈Gi
+max
+u
+∥f (C)CE(u) − g(C)CE(u)∥
+=l∞(f (C)CE, g(C)CE)
+Since Q = Q1 ◦ · · · ◦ Qn, we have
+ℓ∞(Qf (C)CE, Qg(C)CE) ≤ ℓ∞(f (C)CE, g(C)CE).
+(10)
+Combing Lemma A.7, Equation (8), Equation (9) and Equation (10), we finish the proof.
+A.9
+Proof of Theorem 4.8
+We first introduce a useful lemma. It is about the property of Ei(σ, u)
+Lemma A.8. Ei(σ, u) is
+1. Linear on σi, i.e.
+pEi((σ1
+i , σ−i), u) + (1 − p)Ei((σ2
+i , σ−i), u) = Ei((pσ1
+i + (1 − p)σ2
+i , σ−i), u), ∀p ∈ [0, 1]
+2. Convex on σj, i.e.
+pEi((σ1
+j , σ−j), u) + (1 − p)Ei((σ2
+j , σ−j), u) ≥ Ei((pσ1
+j + (1 − p)σ2
+j , σ−j), u), ∀p ∈ [0, 1], j ̸= i
+Proof. We recall the definition Ei(σ, u) = maxai∈Ai ui(ai, σ−i) − ui(σ). Notice that ui(σ) is linear on
+σk for all k ∈ [n], thus both ui(ai, σ−i) and ui(σ) are linear on σk for any k ∈ [n]. Moreover, the
+maximum operator on a set of linear functions will induce a convex function.
+Proof of Theorem 4.8. We prove the theorem in two steps.
+Step 1
+First, we show that
+Eu∼D[Ei((Pif NE)(u), u)] = Eu∼D[Ei(f NE(u), u)],
+∀f NE ∈ FNE
+22
+
+By definition,
+Eu∼D[Ei(Pif NE(u), u)]
+=Eu∼D[Ei((
+1
+|Ai|!
+�
+ρi∈Gi
+ρ−1
+i f(ρiu)i, f(u)−i), u)]
+=
+1
+|Ai|!
+�
+ρi∈Gi
+Eu∼D[Ei((ρ−1
+i
+f(ρiu)i, f(u)−i), u)]
+, by linearity of Ei(σ, u) on σi
+=
+1
+|Ai|!
+�
+ρi∈Gi
+Ev∼D[Ei((ρ−1
+i
+f(v)i, f(ρ−1
+i v)−i), ρ−1
+i v)]
+, let v = ρiu and use the invariance of D
+=
+1
+|Ai|!
+�
+ρi∈Gi
+Ev∼D[Ei((ρ−1
+i
+f(v)i, f(v)−i), ρ−1
+i v)]
+, OPI of f
+=
+1
+|Ai|!
+�
+ρi∈Gi
+Eu∼D[Ei((f(u)i, f(u)−i), u)]
+, invariance of Ei(σ, u) under ρ−1
+i
+∈ Gi
+=Eu∼D[Ei(f NE(u), u)]
+Step 2
+Then we show that
+Eu∼D[Ej((Pif NE)(u), u)] ≤ Eu∼D[Ej(f NE(u), u)],
+∀f NE ∈ FNE, j ̸= i
+Eu∼D[Ej((Pif NE)(u), u)]
+=Eu∼D[Ej((
+1
+|Ai|!
+�
+ρi∈Gi
+ρ−1
+i f(ρiu)i, f(u)−i), u)]
+≤
+1
+|Ai|!
+�
+ρi∈Gi
+Eu∼D[Ej((ρ−1
+i
+f(ρiu)i, f(u)−i), u)]
+, by convexity of Ej(σ, u) on σi
+=
+1
+|Ai|!
+�
+ρi∈Gi
+Ev∼D[Ej((ρ−1
+i
+f(v)i, f(ρ−1
+i v)−i), ρ−1
+i v)]
+, let v = ρiu and use the invariance of D
+=
+1
+|Ai|!
+�
+ρi∈Gi
+Ev∼D[Ej((ρ−1
+i
+f(v)i, f(v)−i), ρ−1
+i v)]
+, OPI of f
+=
+1
+|Ai|!
+�
+ρi∈Gi
+Eu∼D[Ej((f(u)i, f(u)−i), u)]
+, invariance of Ej(σ, u) under ρ−1
+i
+∈ Gi
+=Eu∼D[Ej(f NE(u), u)]
+Since P = ◦iPi and E = maxi Ei, we have
+Eu∼D[E((Pf NE)(u), u)] ≤ Eu∼D[E(f NE(u), u)]
+A.10
+Proof of Theorem 4.6
+Similar to the proof of Theorem 4.8, we first prove a lemma about the property of Ei(π, u) and
+ECE
+i
+(π, u).
+Lemma A.9. Ei(π, u) and ECE
+i
+(π, u) are convex on π, i.e.
+pE(CE)
+i
+(π1, u) + (1 − p)E(CE)
+i
+(π2, u) ≥ E(CE)
+i
+(pπ1 + (1 − p)π2, u),
+∀p ∈ [0, 1]
+23
+
+Proof. We recall the definition Ei(π, u) = maxai∈Ai ui(ai, π−i) − ui(π) for CCE approximator and
+ECE
+i
+(π, u) = maxφi∈Ai→Ai
+�
+a π(a)ui(φi(ai), a−i) − ui(π) for CE approximator. ui(ai, π−i) is linear
+on π.
+Given φ, �
+a π(a)ui(φi(ai), a−i) is also linear on π. Moreover, the maximum operator on a set of
+linear functions will induce a convex function.
+Proof of Theorem 4.6. For f ∈ F(C)CE and ∀i, j ∈ [n],
+Eu∼D[E(CE)
+i
+(Qjf(u), u)] =Eu∼D[E(CE)
+i
+(
+1
+|Aj|!
+�
+ρj∈Gj
+ρ−1
+j f(ρju), u)]
+, by definition
+≤
+1
+|Aj|!
+�
+ρj∈Gj
+Eu∼D[E(CE)
+i
+(ρ−1
+j f(ρju), u)]
+, by convexity
+=
+1
+|Aj|!
+�
+ρj∈Gj
+Ev∼D[E(CE)
+i
+(ρ−1
+j f(v), ρ−1
+j v)]
+, let v = ρju
+=
+1
+|Aj|!
+�
+ρj∈Gj
+Ev∼D[E(CE)
+i
+(f(v), v)]
+, invariance of E(CE)
+i
+(π, u) under ρ−1
+j
+∈ Gj
+=Eu∼D[E(CE)
+i
+(f(u), u)]
+Since Q = ◦iQi and E = maxi Ei, we have
+Eu∼D[E(Qf(u), u)] ≤ Eu∼D[E(f(u), u)]
+A.11
+Proof of Theorem 4.9
+We prove the theorem in two steps, similar to the proof of Theorem 4.8.
+Step 1
+First we show that for player i ∈ {1, 2}, let {j} = {1, 2}\{i},
+Eu∼D[Ei((Oif NE)(u), u)] ≤ Eu∼D[Ei(f NE(u), u)]
+This is because
+Eu∼D[Ei((Oif NE)(u), u)] =Eu∼D[Ei((f NE(u)i,
+1
+|Ai|!
+�
+ρi∈Gi
+f NE(ρiu)j), u)]
+≤
+1
+|Ai|!
+�
+ρi∈Gi
+Eu∼D[Ei((f NE(u)i, f NE(ρiu)j), u)]
+, by convexity of Ei on σj
+=
+1
+|Ai|!
+�
+ρi∈Gi
+Ev∼D[Ei((f NE(ρ−1
+i v)i, f NE(v)j), ρ−1
+i
+v)]
+, let v = ρiu
+=
+1
+|Ai|!
+�
+ρi∈Gi
+Ev∼D[Ei((ρ−1
+i
+f NE(v)i, f NE(v)j), ρ−1
+i
+v)]
+, by PPE of f NE
+=
+1
+|Ai|!
+�
+ρi∈Gi
+Ev∼D[Ei((f NE(v)i, f NE(v)j), v)]
+, invariance of Ei(σ, u) under ρ−1
+i
+∈ G
+=Eu∼D[Ei((f NE)(u), u)]
+Step 2
+Then we show that if j ̸= i and {i, j} = {1, 2}
+Eu∼D[Ej((Oif NE)(u), u)] = Eu∼D[Ej(f NE(u), u)]
+24
+
+This is because
+Eu∼D[Ej((Oif NE)(u), u)] =Eu∼D[Ej((f NE(u)i,
+1
+|Ai|!
+�
+ρi∈Gi
+f NE(ρiu)j), u)]
+=
+1
+|Ai|!
+�
+ρi∈Gi
+Eu∼D[Ej((f NE(u)i, f NE(ρiu)j), u)]
+, by linearity of Ej on σj
+=
+1
+|Ai|!
+�
+ρi∈Gi
+Ev∼D[Ej((f NE(ρ−1
+i v)i, f NE(v)j), ρ−1
+i v)]
+, let v = ρiu
+=
+1
+|Ai|!
+�
+ρi∈Gi
+Ev∼D[Ej((ρ−1
+i f NE(v)i, f NE(v)j), ρ−1
+i v)]
+, by PPE of f NE
+=
+1
+|Ai|!
+�
+ρi∈Gi
+Ev∼D[Ej((f NE(v)i, f NE(v)j), v)]
+, invariance of Ej(σ, u) under ρ−1
+i
+∈ Gi
+=Eu∼D[Ej(f NE(u), u)]
+Since O = ◦iOi and E = maxi Ei, we have
+Eu∼D[E(Of NE(u), u)] ≤ Eu∼D[E(f NE(u), u)]
+A.12
+Proof of Theorem 4.7
+We only prove for the P-projected case; the proof for O-projected case is similar and therefore
+omitted.
+Recall
+Ei(σ, u) = max
+ai∈Ai ui(ai, σ−i) − ui(σ)
+Denote u1(σ) + u2(σ) ≡ c, then
+�
+i
+Ei(σ, u) =
+max
+a1∈A1,a2∈A2 u1(a1, σ2) + u2(a2, σ1) − c
+Then we have
+Eu∼D[
+�
+i
+Ei((Pf NE)(u), u)] =Eu∼D[max
+a1,a2 u1(a1, (Pf NE)(u)2) + u2(a2, (Pf NE)(u)1) − c]
+=Eu∼D[max
+a1 u1(a1, (Pf NE)(u)2)] + Eu∼D[max
+a2 u2(a2, (Pf NE)(u)1)] − c
+For the first term,
+Eu∼D[max
+a1 u1(a1, (Pf NE)(u)2)] =Eu∼D[max
+a1 u1(a1,
+1
+|A2|!
+�
+ρ2∈G2
+ρ��1
+2 f NE(ρ2u)2)]
+≤
+1
+|A2|!
+�
+ρ2∈G2
+Eu∼D[max
+a1 u1(a1, ρ−1
+2 f NE(ρ2u)2)]
+=
+1
+|A2|!
+�
+ρ2∈G2
+Ev∼D[max
+a1 (ρ−1
+2 v)1(a1, ρ−1
+2 f NE(v)2)]
+=
+1
+|A2|!
+�
+ρ2∈G2
+Ev∼D[max
+a1 v1(a1, f NE(v)2)]
+=Eu∼D[max
+a1 u1(a1, f NE(u)2)]
+Similarly, for the second term,
+Eu∼D[max
+a2 u2(a2, (Pf NE)(u)1)] ≤ Eu∼D[max
+a2 u2(a2, f NE(u)1)]
+25
+
+Above all,
+Eu∼D[
+�
+i
+Ei((Pf NE)(u), u)] =Eu∼D[max
+a1 u1(a1, (Pf NE)(u)2)] + Eu∼D[max
+a2 u2(a2, (Pf NE)(u)1)] − c
+≤Eu∼D[max
+a1 u1(a1, f NE(u)2)] + Eu∼D[max
+a2 u2(a2, f NE(u)1)] − c
+=Eu∼D[
+�
+i
+Ei(f NE(u), u)]
+A.13
+Proof of Theorem 5.3
+Let f be a PPE and OPI NE approximator. Denote f(u) = (σi)i∈[n]. For player k that a∗
+k ∈ V (ρk),
+we get
+σk = f(u)k
+(a)
+= f(ρu)k
+(b)
+= f(ρku)k
+(c)
+= ρkf(u)k = ρkσk,
+(11)
+where (a) holds since u is permutable w.r.t. ρ, (b) holds by OPI of f, and (c) holds by PPE of f.
+If a∗ can be found by f, we will have 1 = σk(a∗
+k)
+(d)
+= ρkσk(a∗
+k) = σk(ρ−1
+k (a∗
+k)), where (d) holds by
+Equation (11). However, such result leads to a contradiction, because a∗
+k ̸= ρ−1
+k (ak) but σk(a∗
+k) =
+σ(ρ−1
+k (a∗
+k)) = 1.
+Let f be a PE (C)CE approximator. Denote f(u) = π, we have
+π = f(u)
+(a)
+= f(ρu)
+(b)
+= ρf(u) = ρπ
+(12)
+where (a) holds since u is permutable w.r.t. ρ, (b) holds by PE of f. If a∗ can be found by f, we
+will have 1 = π(a∗)
+(c)
+= ρπ(a∗) = π(ρ−1a∗) = π(ρ−1
+1 a∗
+1, · · · , ρ−1
+n a∗
+n), where (c) holds by Equation (12).
+However, from a∗
+k ∈ V (ρk) we know ρ−1
+k (a∗
+k) ̸= a∗
+k, then ρ−1a∗ ̸= a∗, but π(a∗) = π(ρ−1a∗) = 1.
+A.14
+Proof of Theorem 5.6
+Proof. Assume f ∈ F(C)CE
+general is an (C)CE approximator that always finds the strategy that maximizes
+the social welfare. Afterward, we construct another f0 that satisfies PE and always finds the strategy
+that maximizes social welfare. f0 is constructed by orbit averaging:
+f0(u) = Qf(u),
+thus f0 is PE.
+Denote D as an arbitrary payoff distribution of u such that D is invariant under permutation and
+the cardinality of its support is finite. We have
+Eu∼DSW(Qif(u), u) =Eu∼DSW(
+1
+|Ai|!
+�
+ρi∈Gi
+ρ−1
+i
+f(ρiu), u)
+=Eu∼D
+n
+�
+i=1
+ui(
+1
+|Ai|!
+�
+ρi∈Gi
+ρ−1
+i f(ρiu))
+=
+1
+|Ai|!
+�
+ρi∈Gi
+Eu∼D
+n
+�
+i=1
+ui(ρ−1
+i
+f(ρiu))
+=
+1
+|Ai|!
+�
+ρi∈Gi
+Ev∼D
+n
+�
+i=1
+(ρ−1
+i
+v)i(ρ−1
+i f(v))
+, let v = ρiu
+=
+1
+|Ai|!
+�
+ρi∈Gi
+Ev∼D
+n
+�
+i=1
+vi(f(v))
+=Eu∼D
+n
+�
+i=1
+ui(f(u))
+=Eu∼DSW(f(u), u)
+26
+
+Due to that Q = Q1 ◦ · · · ◦ Qn, we have
+Eu∼DSW(f0(u), u) = Eu∼DSW(f(u), u)
+Due to the arbitrariness of D, we know that f0 maximizes the social welfare w.r.t. any u.
+From above, we immediately know
+SWRN,M(F(C)CE
+PE
+, F(C)CE
+general) = 1
+A.15
+Proof of Theorem 5.7
+A.15.1
+Proof of Equation (1) and Equation (3)
+We first prove the theorem with respect to FNE
+OPI and FNE
+both
+Step 1
+On the one part, we prove
+SWRN,M(FNE
+OPI, FNE
+general)
+SWRN,M(FNE
+both, FNE
+general)
+�
+≤
+1
+M N−1
+We prove this by construction.
+Consider a game with N player and Ai = [M] for i ∈ [N]. ∀a ∈ A, i ∈ [N], define the payoff ¯u as
+follows:
+¯ui(a) =
+�
+1
+, if a1 = a2 = · · · = aN
+0
+, otherwise
+Define U = {u′|u′ = ◦iρi¯u, ρi ∈ Gi} and D as a uniform distribution on U. Easy to certify that D is a
+permutation-invariant distribution.
+Let ˜f ∈ ˜FNE
+general be the NE oracle that ˜f(¯u)i = 1 and for any u′ = ◦iρi¯u ∈ U, ˜f(u′)i = ρi(1).
+Intuitively, the oracle will choose the action that will provide all players with revenue 1, leading to a
+social welfare of N. Since each player has got her maximum possible utility, we have
+max
+f∈F NE
+general
+Eu∼DSW(f(u), u) =
+max
+˜f∈ �
+F NE
+general
+Eu∼DSW( ˜f(u), u) = N.
+(13)
+For any j1, j2 ∈ [M] and j1 < j2, let ρ(j1,j2)
+i
+= (1, . . . , j2, . . . , j1, . . . , M) for all player i ∈ [N] be
+the swap permutation that swaps actions j1 and j2 and keeps other actions still. Then ◦i̸=jρ(j1,j2)
+i
+¯u =
+ρ(j1,j2)
+j
+¯u for player j. For f ∈ FNE
+OPI, we have f(¯u)j = f(◦i̸=jρ(j1,j2)
+i
+¯u)j = f(ρ(j1,j2)
+j
+¯u)j for arbitrary swap
+permutation ρ(j1,j2)
+j
+. Since any permutation can be achieved by composition of swap permutations,
+we have ∀ρj ∈ Gj, f(¯u)j = f(ρj ¯u)j.
+Based on that, and by OPI of f, ∀ρ = ◦i∈[N]ρi we have
+f(¯u)j = f(ρ¯u)j, i.e. f is a constant function on U. Without loss of generality, we denote f(u) ≡ σ for
+all u ∈ U. Then
+Eu∼DSW(f(u), u) =
+1
+|U|
+�
+u′∈U
+SW(σ, u′) =
+1
+(M!)N−1 SW(σ,
+�
+u′∈U
+u′).
+Additionally, we have (�
+u′∈U u′)(a) = ((M − 1)!)N−1 for any a ∈ A. Based on that, we have
+max
+f∈F NE
+OPI
+Eu∼DSW(f(u), u) =
+1
+(M!)N−1 · N((M − 1)!)N−1 =
+N
+M N−1 .
+(14)
+Combining Equation (13) and Equation (14), we have
+SWRN,M(FNE
+OPI, FNE
+general) ≤
+1
+M N−1 .
+Due to FNE
+both ⊆ FNE
+OPI, we immediately know
+SWRN,M(FNE
+both, FNE
+general) ≤
+1
+M N−1
+27
+
+Step 2
+On the other part, we prove
+SWRN,M(FNE
+OPI, FNE
+general)
+SWRN,M(FNE
+both, FNE
+general)
+�
+≥ 1/M N−1
+Define the maximum possible utility (MPU) for player i with respect to utility ui and action ai as
+MPU(ui, ai) :=
+max
+a−i∈A−i ui(ai, a−i)
+(15)
+Define the set of maximum possible utility best response for player i w.r.t. ui as
+Bi(ui) := {ai ∈ Ai : MPU(ui, ai) = max
+a′
+i∈Ai MPU(ui, a′
+i)}
+We first conduct some simplification to the target.
+SWRN,M(FNE
+both, FNE
+general) = inf
+D
+maxf∈F NE
+both Eu∼DSW(f(u), u)
+maxf∈F NE
+general Eu∼DSW(f(u), u) ≥ inf
+D
+maxf∈F NE
+both Eu∼DSW(f(u), u)
+Eu∼D maxσ SW(σ, u)
+Then we constrain u to be a cooperation game. For a normal form game Γu, we define ˜u = (˜ui)i∈[n]
+in which ˜ui = 1
+n
+�n
+i=1 ui. Then we have SW(σ, u) = SW(σ, ˜u), which means that constraining u to be
+a cooperation game will induce the same social welfare. Then
+inf
+D
+maxf∈F NE
+both Eu∼DSW(f(u), u)
+Eu∼D maxσ SW(σ, u)
+= inf
+D
+maxf∈F NE
+both Eu∼DSW(f(u), ˜u)
+Eu∼D maxσ SW(σ, ˜u)
+Denote f0 be the approximator that always outputs uniform strategy on Bi(˜ui) for player i. It’s
+obvious that f0 is both OPI and PPE because the operations from u to f0(u) are all permutation-
+equivariant. Then,
+inf
+D
+maxf∈F NE
+both Eu∼DSW(f(u), ˜u)
+Eu∼D maxσ SW(σ, ˜u)
+≥ inf
+D
+Eu∼DSW(f0(u), ˜u)
+Eu∼D maxσ SW(σ, ˜u)
+Ignore the infimum and the expectation operator, consider
+SW(f0(u),˜u)
+maxσ SW(σ,˜u) for arbitrary ˜u, denote b
+be the maximum element appeared in ˜u, then the denominator equals Nb. But for the numerator,
+for player i, no matter what action ai ∈ Bi(˜ui) she chooses, she always has probability at least
+�
+j̸=i
+1
+|Bj| ≥
+1
+MN−1 to achieve revenue b, therefore inducing SW(f0(u), ˜u) ≥
+Nb
+MN−1 .
+Then,
+SW(f0(u),˜u)
+maxσ SW(σ,˜u) ≥
+1
+MN−1 , so as infD
+Eu∼DSW(f0(u),˜u)
+Eu∼D maxσ SW(σ,˜u), SWRN,M(FNE
+both) and SWRN,M(FNE
+OPI).
+Above all,
+SWRN,M(FNE
+OPI, FNE
+general)
+SWRN,M(FNE
+both, FNE
+general)
+�
+=
+1
+M N−1
+A.15.2
+Proof of Equation (2)
+We next prove the theorem with respect to FNE
+PPEthat
+SWRN,M(FNE
+PPE, FNE
+general) ≤ 1
+M
+Consider a bimatrix game and Ai = [M] for i ∈ [2]. ∀a ∈ A, i ∈ [2], define the payoff ¯u as follows:
+¯ui(a) =
+�
+1
+, if a1 = a2
+0
+, otherwise
+Define U := {u′|u′ = ρ1ρ2¯u, ρi ∈ Gi} and D as a uniform distribution on U. Easy to certify that
+U = {u′|u′ = ρ1¯u, ρ1 ∈ G1} = {u′|u′ = ρ2¯u, ρ2 ∈ G2} and D is a permutation-invariant distribution.
+28
+
+Let ˜f ∈ ˜FNE
+general be the NE oracle that ˜f(¯u)i = 1 and for any u′ = ◦iρi¯u ∈ U, ˜f(u′)i = ρi(1).
+Intuitively, the oracle will choose the action that will provide all players with revenue of 1, leading to
+a social welfare of 2.
+For a permutation ̺ on [M], let P̺ ∈ {0, 1}M×M be the corresponding permutation matrix. Denote
+P as the set of all permutation matrice. As a result, ∀u ∈ U, ∀ρ1 ∈ G1, ρ1u = (Pρ1u1, Pρ1u2) =: Pρ1u
+and ∀ρ2 ∈ G2, ρ2u = (u1P T
+ρ2, u2P T
+ρ2) =: uP T
+ρ2. Specially, we have P̺¯uP T
+̺ = ¯u. For f ∈ FNE
+PPE, Denote
+f(¯u) = σ = (σ1, σ2). For permutation ̺ in [M] and payoff u′ = P̺¯u = ¯u(P T
+̺ )−1, by PPE of f, we have
+f(u′)1 = f(P̺¯u)1 = P̺σ1 = ̺σ1, and f(u′)2 = f(¯u(P T
+̺ )−1)2 = (P̺)−1σ2 = ̺−1σ2. Then we have
+SW(f(u′), u′) =
+�
+i
+(P̺¯u)i(̺σ1, ̺−1σ2) =
+�
+i
+¯ui(σ1, ̺−1σ2) =
+�
+i
+(¯uP T
+̺ )i(σ1, σ2) = SW(f(¯u), ¯uP T
+̺ )
+Therefore
+Eu∼DSW(f(u), u) =
+1
+|U|
+�
+u′∈U
+SW(f(u′), u′)
+=
+1
+|U|
+�
+P̺∈P
+SW(f(¯u), ¯uP T
+̺ )
+=
+1
+|U|
+�
+u=¯u(P T
+̺ )∈U
+SW(f(¯u), u)
+=
+1
+|U|SW(σ,
+�
+u′∈U
+u′).
+Since |U| =
+1
+M! and �
+u′∈U u′ is a tensor with all elements equal to (M −1)!. Thus Eu∼DSW(f(u), u) =
+2
+M and
+SWRN,M(FNE
+PPE, FNE
+general) ≤ 1
+M
+A.15.3
+Proof of Equation (4)
+Consider a 3 × 3 game as follows, where ǫ ∈ (0, 1
+2):
+u =
+
+
+1, 1
+0, 0
+0, 1
+2 + ε
+0, 0
+1, 1
+0, 1
+2 + ε
+1
+2 + ε, 0
+1
+2 + ε, 0
+ε, ε
+
+
+It is obvious that maxσ∗⊆NE(Γu) SW(σ∗, u) = 2, and the corresponding strategy has been bolded.
+However, for NE oracles with both PPE and OPI, it can only output a unique NE with a pure strategy
+that induces utility (ε, ε).
+Let ρ1 = ρ2 = (2, 1, 3), we have ρ1ρ2u = u. From the analysis above we know if f NE ∈ �
+FNE
+both and
+f NE(u) = (σ1, σ2), then σ1(1) = σ1(2), σ2(1) = σ2(2). We integrate the first two actions of player 1
+and player 2 into a new action that will choose randomly between the first two actions, then we form
+the utility matrix below:
+u =
+�
+1
+2, 1
+2
+0, 1
+2 + ε
+1
+2 + ε, 0
+ε, ε
+�
+There is a unique NE in this Prisoner’s Dilemma, which has been bolded. The game u is the
+same with the game u under the assumption that σ1(1) = σ1(2) and σ2(1) = σ2(2) in u.
+Then
+maxf∈ �
+F NE
+both SW(f(u), u) = 2ε. Since ε can be arbitrarily small, we have SWR2,3( �
+FNE
+both, �FNE
+general) = 0.
+As a result, we have SWRN,M( �FNE
+both, �FNE
+general) = 0 for all N ≥ 2 and M ≥ 3.
+29
+

VtE4T4oBgHgl3EQfMgzk/content/tmp_files/2301.04949v1.pdf.txt

@@ -0,0 +1,3642 @@
+arXiv:2301.04949v1  [math.OC]  12 Jan 2023
+A FORMAL POWER SERIES APPROACH TO MULTIPLICATIVE
+DYNAMIC FEEDBACK CONNECTION
+VENKATESH G. S. AND KURUSCH EBRAHIMI-FARD
+Abstract. The goal of the paper is multi-fold. The first of which is to derive an explicit
+formula to compute the generating series of a closed-loop system when a plant, given in a
+Chen–Fliess series description is in multiplicative output feedback connection with another
+system given in Chen–Fliess series description. Further, the objective extends in showing
+that the multiplicative dynamic output feedback connection has a natural interpretation as
+a transformation group acting on the plant. A computational framework for computing the
+generating series for multiplicative dynamic output feedback is devised utilizing the dual
+Hopf algebras corresponding to the shuffle group and the multiplicative feedback group.
+Contents
+1.
+Introduction
+2
+2.
+Preliminaries: Formal Power Series
+2
+2.1.
+Shuffle Product
+3
+3.
+Bialgebra and Hopf algebra: Preliminaries
+4
+3.1.
+Algebra
+4
+3.2.
+Coalgebra
+5
+3.3.
+Bialgebra
+6
+3.4.
+Hopf Algebra
+7
+4.
+Unshuffle Hopf algebra and its Coaction
+8
+4.1.
+Unshuffle Hopf Algebra
+8
+4.2.
+Gradation of Bialgebra H
+10
+4.3.
+Coaction of H
+11
+5.
+Chen–Fliess Series and its Interconnections
+13
+5.1.
+Chen–Fliess Series
+13
+5.2.
+Interconnections of Chen–Fliess Series: Parallel and Cascade Connections
+14
+5.3.
+Cascading of Chen–Fliess with Multiplicative Feedforward of Input
+15
+5.4.
+Multiplicative Dynamic Output Feedback Group
+16
+6.
+Chen–Fliess Series Under Multiplicative Dynamic Output Feedback
+18
+7.
+Invariance of Class and Relative Degree under multiplicative dynamic feedback
+connection
+20
+8.
+Computational Framework for Multiplicative Mixed Composition & Dynamic
+Feedback Product
+24
+8.1.
+Hopf Algebra Corresponding to the Multiplicative Dynamic Feedback Subgroup 24
+8.2.
+Coaction of Hopf algebra H on Algebra of Coordinate Map
+25
+8.3.
+Coaction of Hopf algbera H on the Hopf algebra H
+27
+8.4.
+Coproduct, Antipode Computations and Grading of Hopf algebra H
+29
+9.
+Conclusions and Future work
+35
+References
+35
+
+2
+VENKATESH G. S. AND KURUSCH EBRAHIMI-FARD
+1. Introduction
+The objective of the document is two fold and works with the Chen–Fliess functional
+series [Fliess(1981)]. There is no need that these input-output systems have a state space
+realization and thus, the results presented here are independent of any state space embed-
+ding when a realization is possible [Fliess(1983)]. Firstly, let Fc and Fd be two nonlinear
+input-output systems represented by Chen–Fliess series. It was shown in [Gray & Li(2005)]
+that the additive feedback interconnection of two such systems result in a Chen–Fliess se-
+ries description for the closed-loop system. The convergence of the closed-loop system was
+characterized in [Thitsa & Gray(2012)]. An efficient computation of the generating series for
+closed-loop system is facilitated through a combinatorial Hopf algebra [Gray, et al.(2014a),
+Foissy(2015), Duffaut Espinosa, et al.(2016)]. The feedback product formula and its com-
+putation were used to solve system inversion problems [Gray, et al.(2014b)] and trajectory
+generation problems [Duffaut Espinosa & Gray(2017)].
+However, when the nature of interconnection becomes multiplicative feedback, the similar
+set of questions persist in general. It is known that, in single-input single-output (SISO)
+setting, the closed-loop system in the affine feedback case (of which multiplicative feedback
+is a special case) has a Chen–Fliess series description and the computation of feedback for-
+mula is facilitated through a combinatorial Hopf algebra [Gray & Ebrahimi-Fard(2017)]. The
+present document, in one part, shows that even in multi-input multi-output (MIMO) setting
+the closed-loop system under multiplicative feedback has a Chen–Fliess series representation
+and provides an explicit expression of the closed-loop generating series which will be called
+as multiplicative dynamic feedback product . Furthermore, it will be shown that this feedback
+product has a natural interpretation as a transformation group acting on the plant. The
+algorithmic framework for the computation of the multiplicative dynamic feedback product
+formula for a general MIMO case is devised using the dual Hopf algebras corresponding to
+the shuffle product and to the multiplicative dynamic output feedback group. The charac-
+terization of convergence of the Chen–Fliess series for the closed-loop system is deferred for
+future work.
+The paper is organized as follows. The next section provides a summary of the concepts
+related to non-commutative formal power series, Hopf algebra, Chen–Fliess series and their
+interconnections. The Section 5.4 builds the pivotal multiplicative dynamic output feedback
+group. The Hopf algebra construction corresponding to the shuffle group is drafted in Sec-
+tion 4. Section 6 is where the multiplicative dynamic feedback connection is analyzed. The
+invariance of relative degree under multiplicative output feedback is asserted in Section 7.
+The framework for computing the feedback product is devised in Section 8 and is demon-
+strated using examples. The conclusions of the paper and directions for future work is given
+in the last section.
+2. Preliminaries: Formal Power Series
+A finite nonempty set of noncommuting symbols X = {x0, x1, . . . , xm} is called an alphabet.
+Each element of X is called a letter. Any finite sequence, η = xi1 · · · xik, of letters from X
+is called a word over X and its length is |η| = k. The set X∗ of all words includes the
+empty word, denoted ∅ ∈ X∗ and X+ := X∗\∅, and forms a monoid under catenation.
+Any mapping c : X∗ → Rℓ is called a formal power series. The value of c at η ∈ X∗ is
+denoted by (c, η) and called the coefficient of η in c. Normally, c is written as a formal sum
+c = �
+η∈X∗(c, η)η. A series c is proper when the coefficient (c, ∅) = 0 else it is a non-proper
+series. The support of c is the set supp(c) containing all words having nonzero coefficients.
+The order of c, denoted ord(c), is the length of the minimal length word in its support. The
+
+FORMAL SERIES APPROACH TO MULTIPLICATIVE DYNAMIC FEEDBACK CONNECTION
+3
+collection of all formal power series over X is denoted by Rℓ⟨⟨X⟩⟩. The ith component of a
+vector v ∈ Rℓ is denoted by vi and consequently the ith component of a series c ∈ Rℓ⟨⟨X⟩⟩
+is denoted by ci viz. (ci, η) = (c, η)i.
+A series c′ ∈ Rℓ⟨⟨X⟩⟩ is called a subseries of c ∈ Rℓ⟨⟨X⟩⟩ if there exists another series
+c′′ ∈ Rℓ⟨⟨X⟩⟩ such that the intersection supp (c′) ∩ supp (c′′) is empty and the series c can
+be decomposed as c = c′ + c′′.
+Definition 2.1. Let c ∈ Rℓ⟨⟨X⟩⟩, then the natural part of the series c is the subseries
+denoted by cN such that c = cN + cF and supp (cF) ⊆ X∗ \ {xk
+0 : k ∈ N0}. The subseries cF
+is called as forced part of the series c.
+Definition 2.1 asserts that the forced part cF of a series c should not contain any word
+formed by the letter x0 alone, including the empty word ∅. For the remainder of the docu-
+ment, Rℓ is given the structure of a unital commutative ring under Hadamard or pointwise
+product viz. (xy)i = xiyi with ll = [1 1 · · ·1]t ∈ Rℓ being the multiplicative unit. Formal
+power series Rℓ⟨⟨X⟩⟩ form a Rℓ-module and the submodule of all proper series in Rℓ⟨⟨X⟩⟩
+is denoted by Rℓ
+p ⟨⟨X⟩⟩, while the subset of non-proper series is denoted by Rℓ
+np ⟨⟨X⟩⟩.
+Definition 2.2. A series c ∈ Rℓ⟨⟨X⟩⟩ is called purely improper if ci is non-proper ∀i =
+1, . . . , ℓ. The subset of all purely improper series in Rℓ⟨⟨X⟩⟩ is denoted by Rℓ
+pi ⟨⟨X⟩⟩.
+Observe that Rℓ
+pi ⟨⟨X⟩⟩ ⊊ Rℓ
+np ⟨⟨X⟩⟩ if ℓ > 1, otherwise Rpi ⟨⟨X⟩⟩ = Rnp ⟨⟨X⟩⟩.
+2.1. Shuffle Product. The shuffle product α
+β of two words is a bilinear product on the
+linear span of words, which can be uniquely specified iteratively
+(xiη)
+(xjξ) := xi(η
+(xjξ)) + xj((xiη)
+ξ),
+where η, ξ ∈ X∗ and xi, xj ∈ X. See for instance [Fliess(1981)]. The shuffle product of two
+series, (c, d) �→ c
+d is defined as
+(c
+d, η) =
+�
+ζ,ν∈X∗
+η∈supp(ζ
+ν)
+(c, ζ) (d, ν) .
+We define for any xi, xj ∈ X and any word η ∈ X∗
+x−1
+i (xjη) :=
+�η,
+i = j
+0,
+else
+The following proposition is vital in understanding the bialgebra and Hopf algebra devised
+in Sections 4.1 and 4.3.
+Proposition 2.1. If c, d ∈ Rℓ⟨⟨X⟩⟩, then ∀xi ∈ X
+x−1
+i
+(c
+d) =
+�
+x−1
+i
+(c)
+d
+�
++
+�
+c
+x−1
+i
+(d)
+�
+.
+Note that Rℓ⟨⟨X⟩⟩ forms an associative and commutative Rℓ-algebra under the shuffle
+product. If d ∈ Rℓ
+pi ⟨⟨X⟩⟩, then shuffle inverse of d, denoted by d
+−1 is defined as
+d
+−1
+i
+= (di, ∅)−1
+��
+k∈N0
+(d′
+i)
+k
+�
+,
+where d′
+i = 1 − (di/ (di, ∅)). Hence, Rℓ
+pi ⟨⟨X⟩⟩ forms an Abelian group under the shuffle
+product with ll as the identity element.
+
+4
+VENKATESH G. S. AND KURUSCH EBRAHIMI-FARD
+Example 2.1. Let X = {x0, x1} and c ∈ R⟨⟨X⟩⟩ described as c = 1 − x1. Then the shuffle
+inverse is computed as:
+c
+−1 =
+�
+k∈N0
+(1 − (1 − x1))
+k
+=
+�
+k∈N0
+x
+k
+1
+=
+�
+k∈N0
+k!xk
+1.
+Therefore, c
+−1 = 1 + x1 + 2x2
+1 + 6x3
+1 + · · · + n!xn
+1 + · · · .
+Observe that (c
+d, ∅) = (c, ∅) (d, ∅). Hence, the set
+M
+= { ll + c : c ∈ Rn
+p ⟨⟨X⟩⟩},
+where c is a proper series in Rn⟨⟨X⟩⟩, forms a subgroup of the shuffle group. The group
+M
+is vital in the design of a computational framework of multiplicative dynamic feedback
+product as explained in Section 8.
+The set Rℓ⟨⟨X⟩⟩ is endowed with ultrametric structure where the metric κ is defined as
+κ(c, d) = σord(c−d),
+for c, d ∈ Rℓ⟨⟨X⟩⟩ and σ ∈]0, 1[. For brevity, κ(c, 0) is written as κ(c), and κ(c, d) = κ(c−d).
+The ultrametric space (Rℓ⟨⟨X⟩⟩, κ) is Cauchy complete [Berstel & Reutenauer(1988)]. The
+following definition of contraction maps between metric spaces will be useful.
+Definition 2.3. Given metric spaces (E, d) and (E′, d′), a map f : E −→ E′ is said to be a
+strong contraction map if ∀s, t ∈ E, it satisfies the condition d′(f(s), f(t)) ≤ αd(s, t) where
+α ∈ [0, 1[. If α = 1, then the map f is said to be a weak contraction map or a non-expansive
+map.
+3. Bialgebra and Hopf algebra: Preliminaries
+The goal is to provide the definitions of algebraic structures such as algebra, coalgebra,
+bialgebra and Hopf algebra [Abe(2004), Sweedler(1969)]. We let K be a commutative ring
+with identity 1K.
+3.1. Algebra. The definition of an algebra can be facilitated through the category of mod-
+ules. It allows to define the concept of a coalgebra (the dual notion) with ease.
+Definition 3.1. An algebra over K is a K-module A along with the morphisms of K-
+modules m : A ⊗ A −→ A , called the multiplication or product map, and η : K −→ A ,
+called the unit map, such that the following diagrams are commutative.
+(1)
+A ⊗ A ⊗ A
+m⊗idA
+�
+idA ⊗m
+�
+A ⊗ A
+m
+�
+A ⊗ A
+m
+� A
+K ⊗ A
+η⊗idA
+�
+∼
+=
+�▲
+▲
+▲
+▲
+▲
+▲
+▲
+▲
+▲
+▲
+▲
+▲
+▲
+▲
+▲
+▲
+▲
+▲
+▲
+▲
+A ⊗ A
+m
+�
+A
+A ⊗ K
+∼
+=
+�r
+r
+r
+r
+r
+r
+r
+r
+r
+r
+r
+r
+r
+r
+r
+r
+r
+r
+r
+idA ⊗η
+� A ⊗ A
+m
+�
+
+FORMAL SERIES APPROACH TO MULTIPLICATIVE DYNAMIC FEEDBACK CONNECTION
+5
+The tuple (A , m, η) is called a K-algebra.
+The commutative diagrams (1) mean that a K-algebra A must satisfy the following prop-
+erties:
+(1) The product map m must be associative.
+(2) The scalar multiplication through the η map must have a unit.
+The concept of a K-algebra morphism is defined next.
+Definition 3.2. Let (A , m, η), (A ′, m′, η′) be K-algebras. A map f : A −→ A ′ is called
+a K-algebra morphism provided the following diagrams commute.
+A ⊗ A
+m
+�
+f⊗f
+�
+A
+f
+�
+A ′ ⊗ A ′
+m′
+� A ′
+K
+η
+�
+η′
+�❋
+❋
+❋
+❋
+❋
+❋
+❋
+❋
+❋
+❋
+❋
+❋
+❋
+❋
+❋
+❋
+A
+f
+�①①①①①①①①①①①①①①①①
+A ′
+Definition 3.3. Let P and Q be modules over K. The twisting morphism τ of K-modules
+is τ : P ⊗ Q −→ Q ⊗ P with
+τ(p ⊗ q) = q ⊗ p
+∀ q ∈ Q, p ∈ P.
+A K-algebra A is commutative if and only if the following diagram commutes.
+A ⊗ A
+τ
+�
+m
+�■
+■
+■
+■
+■
+■
+■
+■
+■
+■
+■
+■
+■
+■
+■
+■
+■
+■
+A ⊗ A
+m
+� A
+A K-algebra A is a graded algebra if the underlying K-module structure is graded
+viz. A = �
+n∈N0 An, where An is a K-module for all n ∈ N0 such that m (Am ⊗ An) ⊆
+Am+n, for all m, n ∈ N0. The graded K-algebra is connected if η : K −→ A0 is a K-algebra
+isomorphism.
+3.2. Coalgebra. The notion of a K-coalgebra is a categorical structure dual to that of a
+K-algebra.
+Definition 3.4. A K-coalgebra C is a K-module with the K-module morphisms ∆ : C −→
+C ⊗ C , called the comultiplication or coproduct map, and ǫ : C −→ K, called the counit
+map, such that the following diagrams commute.
+(2)
+C
+∆
+�
+∆
+�
+C ⊗ C
+∆⊗idC
+�
+C ⊗ C
+idC ⊗∆
+� C ⊗ C ⊗ C
+C ⊗ C
+ǫ⊗idC
+� K ⊗ C
+∼
+=
+�
+C
+∆
+�❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑
+∆
+�sssssssssssssssssss
+C ⊗ C
+idC ⊗ǫ
+� C ⊗ K
+∼
+=
+�
+
+6
+VENKATESH G. S. AND KURUSCH EBRAHIMI-FARD
+The tuple (C , ∆, ǫ) is called a K-coalgebra.
+The commutative diagrams (2) imply that a K-coalgebra C must satisfy the following
+properties:
+(1) The coproduct map ∆ must be coassociative.
+(2) The counit map ǫ is the categorical dual to the unit map η for a K-algebra.
+The coalgebra C is called cocommutative if the following diagram commutes,
+C
+∆
+�
+∆
+�❑
+❑
+❑
+❑
+❑
+❑
+❑
+❑
+❑
+❑
+❑
+❑
+❑
+❑
+❑
+❑
+❑
+❑
+❑
+C ⊗ C
+τ
+� C ⊗ C
+where τ is the twisting morphism given in Definition 3.3. Sweedler’s notation is very useful
+in representing the coproduct map and is adopted in Sections 4 and 8.
+Definition 3.5. [Sweedler(1969)]. Given the K-coalgebra tuple (C , ∇, ǫ) and an element
+c ∈ C , then the Sweedler notation for the coproduct
+∆(c) =
+�
+(c)
+c(1) ⊗ c(2),
+where c(1), c(2) ∈ C are the components of the tensors resulting from the coproduct of c.
+Next, the definition of a K-coalgebra morphism is given.
+Definition 3.6. Let (C , ∆, ǫ), (C ′, ∆′, ǫ′) be K-coalgebras. A map f : C −→ C ′ is called a
+K-coalgebra morphism provided the following diagrams commute.
+C
+∆
+�
+f
+�
+C ⊗ C
+f⊗f
+�
+C ′
+∆′
+� C ′ ⊗ C ′
+C
+ǫ
+�
+f
+�❊
+❊
+❊
+❊
+❊
+❊
+❊
+❊
+❊
+❊
+❊
+❊
+❊
+❊
+❊
+❊
+K
+C ′
+ǫ′
+�②
+②
+②
+②
+②
+②
+②
+②
+②
+②
+②
+②
+��
+②
+②
+②
+3.3. Bialgebra. The bialgebra structure over a commutative ring is fundamental for defining
+a Hopf algebra. A bialgebra is an amalgamation of the algebra and coalgebra structures such
+that both are compatible with each other.
+Definition 3.7. A bialgebra H over K is a tuple (H, m, η, ∆, ǫ) such that
+(1) H is a K-module.
+(2) (H, m, η) is a K-algebra, where m and η are the product and unit maps, respectively.
+(3) (H, ∆, ǫ) is a K-coalgebra, where ∆ and ǫ are the coproduct and counit maps, respec-
+tively.
+such that the following diagrams commute.
+(3)
+H ⊗ H
+m
+�
+∆⊗∆
+�
+H
+∆
+� H ⊗ H
+H ⊗ H ⊗ H ⊗ H
+idH⊗τ⊗idH
+� H ⊗ H ⊗ H ⊗ H
+m⊗m
+�
+
+FORMAL SERIES APPROACH TO MULTIPLICATIVE DYNAMIC FEEDBACK CONNECTION
+7
+(4)
+H ⊗ H
+m
+�
+ǫ⊗ǫ
+�▼
+▼
+▼
+▼
+▼
+▼
+▼
+▼
+▼
+▼
+▼
+▼
+▼
+▼
+▼
+▼
+▼
+▼
+▼
+▼
+▼
+H
+ǫ
+�
+K ∼= K ⊗ K
+η⊗η
+�qqqqqqqqqqqqqqqqqqqqq
+η
+�
+H ⊗ H
+H
+∆
+�
+(5)
+H
+ǫ
+�❊
+❊
+❊
+❊
+❊
+❊
+❊
+❊
+❊
+❊
+❊
+❊
+❊
+❊
+❊
+❊
+K
+η
+�②
+②
+②
+②
+②
+②
+②
+②
+②
+②
+②
+②
+②
+②
+②
+②
+idK
+� K
+The diagrams (3) and (4) state that the product map m and the unit map η are K-
+coalgebra morphisms, while the coproduct map ∆ and the counit map ǫ are K-algebra
+morphisms. Diagram (5) describes that the unit map η is a section of the counit map ǫ in
+the category of K-modules.
+3.4. Hopf Algebra. Hopf algebras are an important class of bialgebras. A Hopf algebra is
+a bialgebra equipped with a particular K-linear map called antipode.
+Definition 3.8. A Hopf algebra H over K is a tuple (H, m, η, ∆, ǫ, S) such that the following
+conditions are satisfied:
+(1) (H, m, η, ∆, ǫ) is a K-bialgebra.
+(2) S : H −→ H is a K-linear map such that the following diagram commutes.
+(6)
+H ⊗ H
+idH⊗S
+� H ⊗ H
+m
+�❍
+❍
+❍
+❍
+❍
+❍
+❍
+❍
+❍
+❍
+❍
+❍
+❍
+❍
+❍
+❍
+❍
+❍
+H
+ǫ
+�
+∆
+�✈
+✈
+✈
+✈
+✈
+✈
+✈
+✈
+✈
+✈
+✈
+✈
+✈
+✈
+✈
+✈
+✈
+✈
+∆
+�❍
+❍
+❍
+❍
+❍
+❍
+❍
+❍
+❍
+❍
+❍
+❍
+❍
+❍
+❍
+❍
+❍
+❍
+K
+η
+� H
+H ⊗ H
+S⊗idH
+� H ⊗ H
+m
+�✈
+✈
+✈
+✈
+✈
+✈
+✈
+✈
+✈
+✈
+✈
+✈
+✈
+✈
+✈
+✈
+✈
+✈
+An element a ∈ H is called group-like if ∆(a) = a ⊗ a and thus a̸∈ker(ǫ), where ker(.)
+represents the kernel of a K-module map. A graded Hopf algebra H = �
+n∈N0 Hn is connected
+if and only if H0 ∼= Kη(1K) as K-modules.
+Equivalently, a graded Hopf algebra H is
+connected if and only if H+ := �
+k≥1 Hk is isomorphic to ker(ǫ) as K-modules viz. η◦ǫ = idH0
+and zero otherwise. For simplicity denote m (a, b) := ab, for all a, b, ∈ H. Using Sweedler’s
+
+8
+VENKATESH G. S. AND KURUSCH EBRAHIMI-FARD
+notation, diagram (6) implies that for all c ∈ H,
+�
+(c)
+S
+�
+c(1)
+�
+c(2) =
+�
+(c)
+c(1)S
+�
+c(2)
+�
+= ǫ (c) 1H ,
+where 1H is the multiplicative unit of the Hopf algebra H. The computation of the antipode
+of an element c becomes easier when the algebra structure of H is graded and connected.
+Theorem 3.1. If the Hopf algebra H is graded and connected, then the antipode can be
+computed for any a ∈ H+ := �
+k≥1 Hk as
+S(a) = −a −
+�
+a′
+(1)S(a′
+(2)),
+where the summation is taken over all components of the reduced coproduct ∆′ defined as:
+∆′ (a) := ∆ (a) − a ⊗ η (1K) − η (1K) ⊗ a.
+4. Unshuffle Hopf algebra and its Coaction
+The goal of this section is to explain and illustrate the computational framework to
+compute the shuffle product of two series and the shuffle inverse using the coordinate
+maps of the series. The framework is well-developed in the literature [Foissy(2015)] and
+was utilized in study of interconnections of Chen–Fliess series [Venkatesh & Gray(2022),
+Venkatesh & Gray(2021), Venkatesh & Gray (2020), Gray, et al.(2014b), Gray, et al.(2014a)].
+4.1. Unshuffle Hopf Algebra. We construct a dual Hopf algebra reflecting the group
+structure of M
+as defined in Section 2. The antipode constructed in the Hopf algebra
+provides a framework for computing the shuffle inverse of purely improper series c.
+Let the set Wb ⊂ Rm⟨⟨X⟩⟩∗ (dual module of Rm⟨⟨X⟩⟩) be defined as the collection of
+coordinate maps:
+Wb = {aη : aη(c) = (c, η), η ∈ X∗, c ∈ Rm⟨⟨X⟩⟩}.
+Define W to be the free Rm-module spanned by the set Wb. Let H
+denote the reduced
+symmetric algebra generated by the module W. The Rm-algebra H
+can equivalently be
+seen as the polynomial algebra of coordinate maps (corresponding to non-empty words) of
+Rm⟨⟨X⟩⟩. The unit map ξ : Rm −→ H
+is defined by ξ( ll) = a∅. Observe that a∅ : c �→ ll,
+for all c ∈ M
+. By construction, H
+is an Rm-associative, commutative and unital algebra
+with addition and scalar multiplication defined, respectively, as
+(aη + aζ)(c) = aη(c) + aζ(c)
+(kaη)(c) = k(aη(c)),
+where c ∈ Rm⟨⟨X⟩⟩ and k ∈ Rm, and product
+m(aη, aζ)(c) = aη(c).aζ(c),
+for c ∈ M
+. Then H
+is equipped with a coproduct ˆ∆
+: H
+−→ H
+� H
+such that
+ˆ∆
+aη(c, d) = (c
+d, η), for all c, d ∈ M
+and η ∈ X∗. The counit map ǫ : H
+−→ Rm is
+defined as
+ǫ(h) =
+� ll : h = a∅
+0 : otherwise.
+Since the shuffle product is associative and commutative, thus dually the coproduct ˆ∆
+is
+coassociative and cocommutative. Therefore, (H
+, m, ξ, ˆ∆
+, ǫ) forms a Rm-bialgebra. The
+
+FORMAL SERIES APPROACH TO MULTIPLICATIVE DYNAMIC FEEDBACK CONNECTION
+9
+following lemma is vital in the framework for computing both shuffle product and dynamic
+feedback group product. Define a collection of linear endomorphisms {θi}m
+i=0 on W
+θi : W −→ W
+aη �−→ axiη,
+for all xi ∈ X, η ∈ X∗. Thus θi (aη) (c) = aη
+�
+x−1
+i
+(c)
+�
+.
+The coproduct ˆ∆
+can be recursively constructed as defined in the following proposition.
+Proposition 4.1. [Foissy(2015)] On the module W
+ˆ∆
+◦ θk = (θk ⊗ id + id ⊗ θk) ◦ ˆ∆
+,
+for all i = 1, 2, . . . , m and k = 0, 1, . . . , m with base case being ˆ∆
+a∅ = a∅ ⊗ a∅.
+Proposition 4.1 infers that the maps θi, for i = 1, 2, . . . , m, are coderivations on the
+underlying coalgebra of H
+.
+We note that the unshuffle coproduct ˆ∆
+was utilized in the design of an algorithmic
+framework for computation of Wiener-Fliess composition product and subsequently additive
+static feedback product [Venkatesh & Gray(2021), Venkatesh & Gray(2022), Venkatesh(2021)]
+and also in the computation of shuffle-rational series from its representation [Venkatesh & Gray (2020),
+Venkatesh(2021)]. Moreover, the unshuffle coproduct was also crucial in the computational
+framework for the multivariate additive output feedback [Gray, et al.(2014a), Gray, et al.(2014b)]
+and for SISO affine output feedback [Gray & Ebrahimi-Fard(2017)].
+Let {πi}m
+i=1 be the collection of co-ordinate projection maps on the module W defined as
+ai
+η(c) := πi(aη)(c) = (c, η)i = (ci, η),
+for all η ∈ X∗. Thus, define the following notation
+ˆ∆j ai
+η := (πi ⊗ πj) ◦ ˆ∆
+aη.
+Note that the projection maps {πi}m
+i=1 commute with the maps {θj}m
+j=0 viz. θi
+�
+aj
+η
+�
+= aj
+xiη.
+The significance of these notations are well-reflected in the computational framework in
+Section 8. The following example is to demonstrate the result of Proposition 4.1 for a few
+words.
+Example 4.1. A few examples of the computation of deshuffle coproduct ˆ∆
+on W (akin
+to Example 4.3) using Proposition 4.1 are given as follows(indices i = 1, 2, . . . , m and k, s =
+0, 1, . . . , m):
+ˆ∆j ai
+xk = ai
+xk ⊗ aj
+∅ + ai
+∅ ⊗ aj
+xk.
+ˆ∆j ai
+xkxk = ai
+xkxk ⊗ aj
+∅ + 2ai
+xk ⊗ aj
+xk + ai
+∅ ⊗ aj
+xkxk.
+ˆ∆j ai
+xkxs = ai
+xkxs ⊗ aj
+∅ + ai
+xk ⊗ aj
+xs + ai
+xs ⊗ aj
+xk + ai
+∅ ⊗ aj
+xkxs.
+The connected Rm-bialgebra H
+is endowed with an antipode map S
+given as:
+S
+: H
+−→ H
+aη �→ S
+aη
+such that S
+aη (c) = (c
+−1, η), for η ∈ X∗, c ∈ M
+.
+
+10
+VENKATESH G. S. AND KURUSCH EBRAHIMI-FARD
+4.2. Gradation of Bialgebra H
+. The Hopf algebra H
+can be equipped with a grading
+such that it is connected and all its homogeneous components are finite-dimensional.
+Definition 4.1. Given η ∈ X+, define the degree of aη as deg (aη) = |η|.
+(1) Define gradation on the Rm-module W viz.
+W =
+�
+k≥1
+Wk,
+where Wk is the free Rm-module spanned by the aη of deg (aη) = k.
+(2) The gradation on the module W induces a graded structure on the algebra H
+as
+H
+=
+�
+n∈N0
+ˆHn,
+with ˆH0 ∼= Rm in the category of Rm-modules.
+The following proposition asserts that the above gradation is connected and all its homo-
+geneous components are finite-dimensional.
+Proposition 4.2. Given the gradation for the Hopf algebra H
+,
+(1) H
+is a graded and connected Hopf algebra viz.
+ˆ∆
+�
+ˆHn
+�
+⊆
+�
+i+j=n
+i,j≥0
+ˆHi ⊗ ˆHj.
+(2) For all k: define wk = dim (Wk) and FW = �
+k≥1 wkZk is the geometric series given
+by
+FW =
+1
+1 − mZ ,
+where m = |X| and for all k ≥ 1:
+wk = dim (Wk) = mk.
+(3) Define F ˆ
+H = �
+n≥1 hnZn where hn = dim( ˆHn) then
+F ˆ
+H =
+∞
+�
+k=1
+1
+(1 − Zk)wk .
+Proof:
+(1) The Hopf algebra H
+follows from the fact that if γ(̸= η, ζ) ∈ supp(η
+ζ) then
+deg (γ) = |γ| = |η| + |ζ| = deg (η) + deg (ζ) ,
+for all η, ζ, γ ∈ X∗.
+(2) Define the formal power series
+F(Z0, Z1, . . . , Zm) =
+�
+k≥1
+�
+i0,i1,...,im≥0
+i0+i1+···+im=k
+#{η : |η|xj = ij ∀ j = 0, 1, 2, . . . , m}Zi0
+0 Zi1
+1 · · · Zim
+m
+=
+(Z0 + Z1 + · · · + Zm)
+1 − (Z0 + Z1 + · · · + Zm).
+
+FORMAL SERIES APPROACH TO MULTIPLICATIVE DYNAMIC FEEDBACK CONNECTION
+11
+Since each letter contributes equally to the degree (viz. length), thus
+FW = F(Z, Z, . . ., Z) =
+mZ
+1 − mZ .
+(3) The proposition follows from the item 2 as ˆH is the symmetric algebra generated by
+the Rm-module W.
+Table 1. Dimensions of the homogeneous components of module W and H
+(when m = 2)
+k
+0
+1
+2
+3
+4
+5
+6
+7
+8
+9
+10
+dim (Wk)
+1
+2
+4
+8
+16
+32
+64
+128
+256
+512
+1024
+dim( ˆHk)
+1
+2
+7
+20
+59
+162
+449
+1200
+3194
+8348
+21646
+. . .
+Example 4.2. The dimensions of the homogeneous components of the graded module W
+(up to k = 10) and the graded algebra H
+for m = 2 viz when X = {x0, x1} is tabulated in
+Table 1.
+The sequence {dim( ˆHk)}k∈N0 is the sequence A034899 in [OEIS(2022)] which corresponds
+to the number of multisets of binary words of total length n.
+4.3. Coaction of H
+. The subsection explains the coaction of the Hopf algebra H
+(4.1)
+on the algebra of coordinate functions. It is utilized subsequently to develop an algorithm to
+compute the multiplicative mixed composition product explained in Section 5.2 and dynamic
+feedback product as defined in Theorem 6.2. Let W to be the Rm-module as described in
+Section 4.1. Let S+ (W) denote the reduced symmetric algebra generated by the module W.
+The non-unital Rm-algebra S+(W) are equivalently the polynomials without constant term
+of coordinate maps of Rm⟨⟨X⟩⟩. By construction S+(W) has a non-unital Rm-associative,
+commutative algebra structure with addition, scalar multiplication and product defined,
+respectively, as
+(aη + aζ)(c) = aη(c) + aζ(c)
+(kaη)(c) = k(aη(c))
+where c ∈ Rm⟨⟨X⟩⟩, and
+m(aη, aζ)(c) = aη(c).aζ(c),
+where c ∈ M
+. The Rm-algebra S+(W) is isomorphic to the algebra structure of H
+with
+forgetting of the unit map ξ. The right coaction map ρ
+: S+ (W) −→ S+ (W) ⊗ H
+is
+recursively defined on the module V as given by the following proposition.
+Proposition 4.3. For all i = 0, 1, 2, . . . , m :
+ρ
+◦ θi = (θi ⊗ id + id ⊗ θi) ◦ ρ
+,
+with base case being ρ
+a∅ = a∅ ⊗ a∅.
+Proposition 4.3 might appear as repetition of Proposition 4.1. It is vital to note that
+Proposition 4.1 is for defining the coproduct of Hopf algebra H
+, where a∅ is the unit
+element. Observe that,
+ρ
+ai
+η(c, d) = ai
+η(c
+d),
+
+12
+VENKATESH G. S. AND KURUSCH EBRAHIMI-FARD
+where c ∈ R⟨⟨X⟩⟩ (not necessarily in M
+) and d ∈ M
+.
+The coaction ρ
+thus is a
+corepresentation of the Hopf algebra H
+on the algebra S+ (W) or equivalently, ρ
+makes
+S+ (W), a H
+-algebra. Let {πi}m
+i=1 be the collection of co-ordinate projection maps on the
+module W defined as
+ai
+η(c) := πi(aη)(c) = (c, η)i = (ci, η),
+for all η ∈ X∗ and thus the following notation is well-defined,
+ρj ai
+η := (πi ⊗ πj) ◦ ρ
+aη.
+These notations are very much utilized in developing computational framework for the
+multiplicative mixed composition product as discussed in Section 8.
+Corollary 4.1. If n ∈ N0, then for all i = 0, 1, 2, . . ., m and j, k = 1, 2, . . . , m (defining
+x0
+j := ∅):
+ρj ak
+xin =
+n
+�
+r=0
+�n
+r
+�
+ak
+xir ⊗ aj
+xin−r .
+Proof: The statement is proved by induction on n ∈ N0. The base case (n = 0) follows from
+Proposition 4.3. Assume the statement is true for n = p − 1, then
+ρj ak
+xip = ρj ◦ θiak
+xip−1
+= (θi ⊗ id + id ⊗ θi) ◦ ∆j ak
+xip−1.
+Using the induction hypothesis,
+ρj ak
+xip = (θi ⊗ id + id ⊗ θi)
+�p−1
+�
+r=0
+�p − 1
+r
+�
+ak
+xir ⊗ aj
+xip−1−r
+�
+=
+p
+�
+r=1
+�p − 1
+r − 1
+�
+ak
+xir ⊗ aj
+xip−r +
+p−1
+�
+r=0
+�p − 1
+r
+�
+ak
+xir ⊗ aj
+xip−r.
+=
+p
+�
+r=0
+�n
+r
+�
+ak
+xir ⊗ aj
+xip−r.
+Since the S+ (W) and H
+are isomorphic as Rm-modules, the following lemma states the
+coaction of H
+on S+ (W) and the unshuffle coproduct coincide when the evaluation of
+coordinate maps are restricted to the group M
+.
+Lemma 4.1. Given c, d ∈ M
+, η ∈ X∗ and i = 1, 2, . . . , m,
+ˆ∆
+aη (c, d) = (c
+d, η) = ρ
+aη (c, d) ,
+where c, d ∈ M
+and ˆ∆i
+is the coproduct from the bialgebra H
+constructed in Section 4.3.
+Example 4.3. A few examples of the computation of the coaction map ρ
+on W using
+Proposition 4.3 are given as follows(indices i, j = 1, 2, . . . , m and k, s = 0, 1, . . . , m):
+∆j ai
+∅ = ai
+∅ ⊗ aj
+∅.
+∆j ai
+xk = ai
+xi ⊗ aj
+∅ + ai
+∅ ⊗ aj
+xi.
+∆j ai
+xkxk = ai
+xkxk ⊗ aj
+∅ + 2ai
+xk ⊗ aj
+xk + ai
+∅ ⊗ aj
+xkxk.
+
+FORMAL SERIES APPROACH TO MULTIPLICATIVE DYNAMIC FEEDBACK CONNECTION
+13
+∆j ai
+xkxs = ai
+xkxs ⊗ aj
+∅ + ai
+xk ⊗ aj
+xs + ai
+xs ⊗ aj
+xk + ai
+∅ ⊗ aj
+xkxs.
+The following example illustrates the application of the deshuffle coproduct ∆
+in the
+computation of the shuffle product of two series.
+Example 4.4. Let X = {x0, x1} and c, d ∈ R2⟨⟨X⟩⟩ described as
+c =
+�
+1 + x1 + x2
+1 + x3
+1 + · · ·
+x0 + x0x1 + x100
+1
+�
+&
+d =
+�
+1 + x2
+0 + exp (x1)
+1 + x2
+0x1
+�
+,
+where exp(.) is the standard exponential function expressed in its Taylor series. Note that
+c ̸∈ M
+but d ∈ M
+. The coefficient of x0x2
+1 in series c2
+d1 can be computed as:
+�
+c2
+d1, x0x2
+1
+�
+= ∆1 a2
+x0x2
+1 (c, d) = (π2 ⊗ π1) ◦ ∆
+ax0x2
+1 (c, d)
+= ∆1 ◦ θ0ax2
+1 (c, d) .
+Using Proposition 4.3,
+�
+c2
+d1, x0x2
+1
+�
+= (θ0 ⊗ id + id ⊗ θ0) ◦ ∆1 a2
+x2
+1 (c, d) .
+Using Corollary 4.1,
+�
+c2
+d1, x0x2
+1
+�
+= (θ0 ⊗ id + id ⊗ θ0) ◦
+�
+a2
+x12 ⊗ a1
+∅ + 2a2
+x1 ⊗ a1
+x1 + a2
+∅ ⊗ a1
+x12
+�
+(c, d)
+=
+�
+a2
+x0x12 ⊗ a1
+∅ + 2a2
+x0x1 ⊗ a1
+x1 + a2
+x0 ⊗ a1
+x12 + a2
+x12 ⊗ a1
+x0+
+2a2
+x1 ⊗ a1
+x0x1 + a2
+∅ ⊗ a1
+x0x12
+�
+(c, d)
+= (0)(1) + 2(1)(1) + (1)(0.5) + (0)(0) + 2(0)(0) + (0)(0) = 2.5.
+Therefore (c2
+d1, x0x2
+1) = 2.5.
+5. Chen–Fliess Series and its Interconnections
+The objective of the section is to describe Chen–Fliess series and the necessary non-
+recursive interconnections of Chen–Fliess series to understand the results about the multi-
+plicative dynamic feedback product in Section 6.
+5.1. Chen–Fliess Series. Let p ≥ 1 and t0 < t1 be given. For a Lebesgue measurable
+function u : [t0, t1] → Rm, define ∥u∥p = max{∥ui∥p :
+1 ≤ i ≤ m}, where ∥ui∥p is the
+usual Lp-norm for a measurable real-valued function, ui, defined on [t0, t1]. Let Lm
+p [t0, t1]
+denote the set of all measurable functions defined on [t0, t1] having a finite ∥ · ∥p norm
+and Bm
+p (R)[t0, t1] := {u ∈ Lm
+p [t0, t1] : ∥u∥p ≤ R}.
+Given any series c ∈ Rℓ⟨⟨X⟩⟩, the
+corresponding Chen–Fliess series is
+(7)
+Fc[u](t) =
+�
+η∈X∗
+(c, η) Fη[u](t, t0),
+where E∅[u] = 1 and
+Fxi¯η[u](t, t0) =
+� t
+t0
+ui(τ)F¯η[u](τ, t0) dτ
+with xi ∈ X, ¯η ∈ X∗, and u0 = 1 [Fliess(1981)]. If there exist constants K, M > 0 such that
+|(ci, η)| ≤ KM|η||η|!, ∀η ∈ X∗, ∀i = 1, . . . , ℓ ,
+(8)
+
+14
+VENKATESH G. S. AND KURUSCH EBRAHIMI-FARD
+then Fc constitutes a well-defined mapping from Bm
+p (R)[t0, t0 + T] into Bℓ
+q(S)[t0, t0 + T]
+for sufficiently small R, T > 0, where the numbers p, q ∈ [1, ∞] are conjugate exponents,
+i.e., 1/p + 1/q = 1 [Gray & Wang(2002)].
+This map is referred to as a Fliess operator.
+A series c ∈ Rℓ⟨⟨X⟩⟩ obeying the growth condition in (8) is called a locally convergent
+generating series. The set of all locally convergent generating series is denoted by Rℓ
+LC⟨⟨X⟩⟩.
+The supremum of the set of all max{R, T} for which a Fliess operator Fc is a well-defined
+mapping from Bm
+p (R)[t0, t0 + T] into Bℓ
+q(S)[t0, t0 + T] is called the radius of convergence
+of the Fliess operator Fc and is denoted by ρ (Fc). A Fliess operator Fc is called locally
+convergent if ρ (Fc) > 0. If there exist constants K, M > 0 and γ ∈ [0, 1[ such that
+|(ci, η)| ≤ KM|η| (|η|!)γ , ∀η ∈ X∗, ∀i = 1, . . . , ℓ ,
+(9)
+then Fc constitutes a well defined mapping from Bm
+p (R)[t0, t0 + T] into Bℓ
+q(S)[t0, t0 + T]
+for all R, T > 0 [Winter-Arboleda(2019), Winter-Arboleda, et al.(2015)]. The infimum of all
+the γ ∈ [0, 1[ such that (9) is satisfied for a series c ∈ Rℓ⟨⟨X⟩⟩ is called the Gevrey order of
+the series c.
+A series c ∈ Rℓ⟨⟨X⟩⟩ obeying the growth condition in (9) is called a globally convergent
+series. The set of all globally convergent series in Rℓ⟨⟨X⟩⟩ is denoted as Rℓ
+GC⟨⟨X⟩⟩. A Fliess
+operator Fc is globally convergent if and only if there exists no real number M > 0 such
+that ρ (Fc) < M. Observe that a noncommutative polynomial R⟨X⟩ is a globally convergent
+series with Gevrey degree 0. As described above, a series c ∈ Rℓ
+GC⟨⟨X⟩⟩ is only a sufficient
+condition for the corresponding Fliess operator Fc to be globally convergent.
+Necessary
+conditions are well-detailed in the literature [Winter-Arboleda(2019), Venkatesh(2021)]. In
+the absence of any convergence criterion, (7) only defines an operator in a formal sense.
+5.2. Interconnections of Chen–Fliess Series: Parallel and Cascade Connections.
+Given Chen–Fliess series Fc and Fd, where c, d ∈ Rℓ⟨⟨X⟩⟩, the parallel and product connec-
+tions satisfy Fc + Fd = Fc+d and FcFd = Fc
+d, respectively [Ree(1958), Fliess(1981)]. The
+parallel and product connections preserve local convergence and hence the interconnected
+systems has a Fliess operator representation [Thitsa & Gray(2012), Venkatesh(2021)]. When
+Chen–Fliess series Fc and Fd with c ∈ Rk⟨⟨X′⟩⟩ and d ∈ Rℓ⟨⟨X⟩⟩ are interconnected in a
+cascade fashion, where |X′| = ℓ + 1, the composite system Fc ◦ Fd has a Chen–Fliess series
+representation Fc◦d, where the composition product of c and d is given by
+(10)
+c ◦ d =
+�
+η∈X′∗
+(c, η) ψd(η)(1)
+[Ferfera(1979), Ferfera(1980)]. Here 1 denotes the monomial 1∅, and ψd is the continuous
+(in the ultrametric sense) algebra homomorphism from R⟨⟨X′⟩⟩ to the set of vector space
+endomorphisms on R⟨⟨X⟩⟩, End (R⟨⟨X⟩⟩), uniquely specified by
+ψd(x′
+iη) = ψd(x′
+i) ◦ ψd(η)
+with ψd(x′
+i)(e) = x0(di
+e), i = 0, 1, . . . , m for any e ∈ R⟨⟨X⟩⟩, and where di is the i-th
+component series of d (d0 := 1). By definition, ψd(∅) is the identity map on R⟨⟨X⟩⟩. The
+cascade interconnection preserves local convergence and thus the composite has a Fliess
+operator representation [Thitsa & Gray(2012)]. The linearity of the composition product in
+the left argument is evident form the definition. However, the following theorem states that
+the composition product distributes over the shuffle product from the right.
+Theorem 5.1. [Gray & Li(2005)] Let c, d ∈ Rk⟨⟨X′⟩⟩ and e ∈ Rℓ⟨⟨X⟩⟩, such that |X′| =
+ℓ + 1, then (c
+d) ◦ e = (c ◦ e)
+(d ◦ e).
+
+FORMAL SERIES APPROACH TO MULTIPLICATIVE DYNAMIC FEEDBACK CONNECTION
+15
+Given a series e ∈ Rℓ⟨⟨X⟩⟩, define a map Υe : Rk⟨⟨X′⟩⟩ −→ Rk⟨⟨X⟩⟩ defined as c �→
+c ◦ e. Theorem 5.1 infers that Υe is an R-algebra homomorphism from the shuffle algebra of
+Rk⟨⟨X′⟩⟩ to the shuffle algebra of Rℓ⟨⟨X⟩⟩. The composition product preserves the purely
+improper property of the left argument which is stated in the following theorem.
+Theorem 5.2. If c ∈ Rk⟨⟨X′⟩⟩ and d ∈ Rℓ⟨⟨X⟩⟩ such that |X′| = ℓ + 1, then (c ◦ d, ∅) =
+(c, ∅). Hence, if c ∈ Rk
+pi ⟨⟨X′⟩⟩ then c ◦ d ∈ Rk
+pi ⟨⟨X⟩⟩ and vice-versa. Similarly if c is a
+proper series then c ◦ d is also a proper series and vice-versa.
+Proof: The proof follows immediately from (10).
+The composition product is a strong contraction map with respect to its right argument
+in the ultrametric topology and is stated in the following theorem.
+Theorem 5.3. [Gray & Li(2005)] Let c ∈ Rk⟨⟨X′⟩⟩ and d, e ∈ Rℓ⟨⟨X⟩⟩, such that |X′| =
+ℓ + 1, then κ (c ◦ d, c ◦ e) ≤ σκ (d, e) where σ ∈ [0, 1[.
+5.3. Cascading of Chen–Fliess with Multiplicative Feedforward of Input. The cas-
+cade interconnection of a Chen–Fliess series Fc and Fd along with the multiplicative feed-
+forward of the input, as shown in Figure 1, arises primarily in the analysis of multiplicative
+feedback interconnection discussed in Section 6. A semblance of such an interconnection
+has appeared in Definition 3.1 of [Gray & Ebrahimi-Fard(2017)], without being explicit and
+limited to the SISO case. With respect to Figure 1, the map u �→ y viz. y = Fc[u.Fd[u]] has
+Chen–Fliess series representation denoted by Fc↶d, where c ↶ d denotes the multiplicative
+mixed composition product of c ∈ Rp⟨⟨X⟩⟩ and d ∈ Rm⟨⟨X⟩⟩ defined as
+c ↶ d =
+�
+η∈X∗
+(c, η) η ↶ d :=
+�
+η∈X∗
+(c, η) ¯φd (η) (1) .
+(11)
+Here, ¯φd : R⟨⟨X⟩⟩ −→ End (R⟨⟨X⟩⟩) is an R-algebra homomorphism such that
+¯φd(x0)(e) = x0e
+and
+¯φd(xi)(e) = xi(di
+e).
+Recall that R⟨⟨X⟩⟩ is an R-algebra under Cauchy product and End (R⟨⟨X⟩⟩). The multi-
+plicative mixed composition defined in (11) asserts that, for all η ∈ X∗ and d ∈ Rm⟨⟨X⟩⟩,
+∅ ↶ d = ∅
+x0η ↶ d = x0 (η ↶ d)
+xiη ↶ d = xi (di
+(η ↶ d))
+∀ i = 1, 2, . . . , m.
+For later reference, we summarise the properties of (11) in the following
+Theorem 5.4. The multiplicative mixed composition product (11) is linear in its left argu-
+ment and (c ↶ d, ∅) = (c, ∅), for all c ∈ Rp⟨⟨X⟩⟩ and d ∈ Rm⟨⟨X⟩⟩.
+The following results are already known in the single-input single-output (SISO) setting.
+However, their multi-input multi-output (MIMO) extensions are straightforward and to avoid
+reiteration of the proofs, only the statements are provided in this document. The foremost
+of the theorems asserts that the multiplicative mixed composition product distributes over
+shuffle product from the right.
+Theorem 5.5. [Gray & Ebrahimi-Fard(2017)] Let c, d ∈ Rp⟨⟨X⟩⟩ and e ∈ Rm⟨⟨X⟩⟩, then
+(c
+d) ↶ e = (c ↶ e)
+(d ↶ e).
+
+16
+VENKATESH G. S. AND KURUSCH EBRAHIMI-FARD
+Fd
+Fc
+u
+y
+Figure 1. Cascade connection of Chen–Fliess Fd with Fc along with multi-
+plicative feedforward of input
+The inference of Theorem 5.5 is that for any e ∈ Rm⟨⟨X⟩⟩, the map Γe : Rp⟨⟨X⟩⟩ −→
+Rp⟨⟨X⟩⟩ given by d �→ d ↶ e is an R-algebra endomorphism on the shuffle algebra Rp⟨⟨X⟩⟩.
+The next lemma is essential in proving that multiplicative mixed composition product is a
+strong contraction map in its right argument in the ultrametric topology.
+Lemma 5.1. [Gray & Ebrahimi-Fard(2017)] Let η ∈ X∗ and d, e ∈ Rm⟨⟨X⟩⟩, then
+κ (η ↶ d, η ↶ e) ≤ σ|η|κ (d, e) where σ ∈ [0, 1[.
+The following theorem states the strong contraction property of the multiplicative mixed
+composition product which is an essential result in Section 6.
+Theorem 5.6. [Gray & Ebrahimi-Fard(2017)] Let d, e ∈ Rm⟨⟨X⟩⟩ and c ∈ Rp⟨⟨X⟩⟩, then
+κ (c ↶ d, c ↶ e) ≤ σord(c′)κ (d, e), where c′ = c − (c, ∅), the proper part of c.
+Since ord (c′) ≥ 1 and σ ∈]0, 1[, then from Theorem 5.6, the map ¯Γc : e �→ c ↶ e is a strong
+contraction map in the ultrametric topology. The following lemma is essential in proving
+the mixed associativity of the composition and multiplicative mixed composition product.
+The result, along with Theorem 5.7 can be inferred in the SISO setting from Lemma 3.6 in
+[Gray & Ebrahimi-Fard(2017)], and its extension to the MIMO case is straightforward.
+Lemma 5.2. [Gray & Ebrahimi-Fard(2017)] Let X′ = {x′
+0, . . . , x′
+p} and η ∈ X′∗. Let d ∈
+Rp⟨⟨X⟩⟩ and e ∈ Rm⟨⟨X⟩⟩, then η ◦ (d ↶ e) = (η ◦ d) ↶ e.
+The following theorem states that the composition product and multiplicative mixed com-
+position product are associative in combination.
+Theorem 5.7. [Gray & Ebrahimi-Fard(2017)] Let X′ = {x′
+0, . . . , x′
+p} and c ∈ Rq⟨⟨X′⟩⟩. Let
+d ∈ Rp⟨⟨X⟩⟩ and e ∈ Rm⟨⟨X⟩⟩, then c ◦ (d ↶ e) = (c ◦ d) ↶ e.
+5.4. Multiplicative Dynamic Output Feedback Group. The dynamic multiplicative
+feedback group plays a vital role in computation of the multiplicative dynamic feedback
+formula, as well as in assessing the feedback as a group action in Section 6. Indeed, consider
+the cascade interconnection of two Chen–Fliess series Fc and Fd along with their multiplica-
+tive feedforward of inputs displayed in Figure 2, where c, d ∈ Rm⟨⟨X⟩⟩. The input-output
+relation of the composite system, u �→ y is u.Fd[u]Fc[u.Fd[u]] and can be represented by
+Chen–Fliess series as follows. Consider
+u.Fc⋆d[u] := u.Fd[u]Fc[u.Fd[u]],
+where the multiplicative composition product of c and d is defined as
+c ⋆ d = d
+(c ↶ d) .
+(12)
+The following theorems appeared in [Gray & Ebrahimi-Fard(2017)] in the SISO setting.
+We underline that the latter restriction is not essential, that is, the statements along with
+the proofs naturally extend to the MIMO setting.
+
+FORMAL SERIES APPROACH TO MULTIPLICATIVE DYNAMIC FEEDBACK CONNECTION
+17
+Figure 2. Cascade connection of Chen–Fliess Fd with Fc along with multi-
+plicative feedforward of their inputs.
+Theorem 5.8. [Gray & Ebrahimi-Fard(2017)] Let c, d, e ∈ Rm⟨⟨X⟩⟩, then, (c ⋆ d) ⋆ e =
+c ⋆ (d ⋆ e).
+Observe that (12) and Theorem 5.8 infer that Rm⟨⟨X⟩⟩ forms a non-commutative monoid
+under multiplicative composition product, with the identity element ll. The following theo-
+rem states that the multiplicative mixed composition product is a right action on Rq⟨⟨X⟩⟩
+by the monoid (Rm⟨⟨X⟩⟩, ⋆, ll).
+Theorem 5.9. [Gray & Ebrahimi-Fard(2017)] Let c ∈ Rq⟨⟨X⟩⟩ and d, e ∈ Rm⟨⟨X⟩⟩, then
+(c ↶ d) ↶ e = c ↶ (d ⋆ e).
+The prominent question is to find the invertible elements of the monoid (Rm⟨⟨X⟩⟩, ⋆) and
+the motivation to find the unit elements of the monoid shall be evident in Section 6. Let
+d, e ∈ Rm
+pi ⟨⟨X⟩⟩ and suppose
+d ⋆ e = ll.
+Observe that d ∈ Rm
+pi ⟨⟨X⟩⟩ implies (d ↶ e) ∈ Rm
+pi ⟨⟨X⟩⟩ and using Theorem 5.5,
+e = (d ↶ e)
+−1 = d
+−1 ↶ e.
+Hence, for e to be right inverse of d, the purely improper series e has to satisfy the fixed
+point equation
+e = d
+−1 ↶ e
+(13)
+Observe from Theorem 5.6 that the map e �→ d
+−1 ↶ e is a strong contraction in the
+ultrametric space inferring that (13) has a unique fixed point. Suppose e is the left inverse
+of d viz. e ⋆ d, then a similar procedure shows that e has to satisfy the equation
+d = e
+−1 ↶ d
+(14)
+Note that if e is a solution of (13), then e satisfies (14) and also the converse holds true.
+Hence, e is the unique inverse of d and is given the notation d⋆−1 for d ∈ Rm
+pi ⟨⟨X⟩⟩. Thus,
+Rm
+pi ⟨⟨X⟩⟩ forms a group under multiplicative composition product, ⋆, and is termed as the
+multiplicative dynamic output feedback group and is formally stated in the following theorem.
+Theorem 5.10.
+�
+Rm
+pi ⟨⟨X⟩⟩, ⋆
+�
+forms a group with the identity element ll.
+It is worth noting that [Gray & Ebrahimi-Fard(2017)] proved Theorem 5.10 for one-
+dimensional case viz. m = 1. In light of Theorem 5.10, Theorem 5.5 and (12) one obtains
+the following relations for c ∈ Rm
+pi ⟨⟨X⟩⟩:
+c⋆−1 = c
+−1 ↶ c⋆−1
+(15)
+�
+c⋆−1�
+−1 = c ↶ c⋆−1.
+The following lemma is essential in defining a subgroup of the multiplicative dynamic out-
+put feedback group upon which the computational framework for the multiplicative feedback
+products is discussed in Section 8.
+
+F
+F
+n18
+VENKATESH G. S. AND KURUSCH EBRAHIMI-FARD
+Lemma 5.3. Let c, d ∈ Rm
+pi ⟨⟨X⟩⟩, then (c ⋆ d, ∅) = (c, ∅) (d, ∅).
+Proof: Observe from (12) that,
+(c ⋆ d, ∅) = (d
+(c ↶ d) , ∅)
+= (c ↶ d, ∅) (d, ∅)
+Since (c ↶ d, ∅) = (c, ∅),
+(c ⋆ d, ∅) = (c, ∅) (d, ∅) .
+Lemma 5.3 thus proves that the set of all series which are of the form ll + c, where c is
+a proper series, forms a subgroup of the multiplicative dynamic feedback group, which is
+stated in the following theorem.
+Theorem 5.11. Let M = { ll + c : c ∈ Rm
+p ⟨⟨X⟩⟩}, then (M, ⋆, ll) forms a subgroup of the
+multiplicative dynamic feedback group.
+The algorithmic framework for the computation of multiplicative feedback products is
+fundamentally based on the subgroup M as asserted in Theorem 5.11. The group M is
+isomorphic to the character group of the Hopf algebra H which is used for computation of
+feedback and the framework is explained in detail in Section 8.
+6. Chen–Fliess Series Under Multiplicative Dynamic Output Feedback
+Let Fc be a Chen–Fliess series with a generating series c ∈ Rq⟨⟨X⟩⟩. Assume it is intercon-
+nected with a Chen–Fliess series Fd with a purely improper generating series d ∈ Rm
+pi ⟨⟨X′⟩⟩,
+as shown in Figure 3. Note that, |X| = m + 1 and |X′| = q + 1. The primary goal of this
+section is to show that the closed-loop system has a Chen–Fliess series representation, say
+y = Fe[v], where e ∈ Rq⟨⟨X⟩⟩. If this is the case, then necessarily
+y = Fe[v] = Fc[u] = Fc[vFd[y]]
+= Fc[vFd[Fe[v]]] = Fc[vFd◦e[v]]
+= Fc↶(d◦e)[v]
+for any admissible input v. Therefore, the series e has to satisfy the fixed point equation
+e = c ↶ (d ◦ e) .
+(16)
+Observe that, in light of Theorem 5.3 and Theorem 5.6 the map e �→ c ↶ (d ◦ e) is a
+strong contraction map in the ultrametric space and thus (16) has a unique fixed point. The
+following thoerem establishes the first main result of this section, which follows immediately.
+Theorem 6.1. The series c ↶ (d
+−1 ◦ c)⋆−1 ∈ Rq⟨⟨X⟩⟩ is the unique fixed point of the map
+e �→ c ↶ (d ◦ e).
+Proof: If e := c ↶ (d
+−1 ◦ c)⋆−1, then
+c ↶ (d ◦ e) = c ↶
+�
+d ◦
+�
+c ↶
+�
+d
+−1 ◦ c
+�⋆−1��
+Using Theorem 5.7 and then Theorem 5.5,
+c ↶ (d ◦ e) = c ↶
+�
+(d ◦ c) ↶
+�
+d
+−1 ◦ c
+�⋆−1�
+
+FORMAL SERIES APPROACH TO MULTIPLICATIVE DYNAMIC FEEDBACK CONNECTION
+19
+Fc
+v
+Fd
+y
+u
+Figure 3. Chen–Fliess series Fc in multiplicative output feedback with Chen-
+Flies series Fd
+= c ↶
+�
+(d ◦ c)
+−1 ↶
+�
+d
+−1 ◦ c
+�⋆−1�
+−1
+.
+Using Theorem 5.1,
+c ↶ (d ◦ e) = c ↶
+��
+d
+−1 ◦ c
+�
+↶
+�
+d
+−1 ◦ c
+�⋆−1�
+−1
+.
+Using the relations (15),
+c ↶ (d ◦ e) = c ↶
+���
+d
+−1 ◦ c
+�⋆−1�
+−1�
+−1
+= c ↶
+�
+d
+−1 ◦ c
+�⋆−1 = e.
+Theorem 6.2. Given a series c ∈ Rq⟨⟨X⟩⟩ and a purely improper series d ∈ Rm
+pi ⟨⟨X′⟩⟩ (such
+that |X| = m + 1 and |X′| = q + 1), then the generating series for the closed-loop system in
+Figure 3 is given by the multiplicative dynamic feedback product cˇ@d := c ↶ (d
+−1 ◦ c)⋆−1.
+The notion that feedback can described mathematically as a transformation group acting
+on the plant is well established in control theory [Brockett(1978)]. The following theorem
+describes the situation in the present context.
+Theorem 6.3. The multiplicative dynamic feedback product is a right group action by the
+multiplicative group
+�
+Rm
+pi ⟨⟨X′⟩⟩,
+, ll
+�
+on the set Rq⟨⟨X⟩⟩, where |X| = m + 1 and |X′| =
+q + 1.
+Proof: Let c ∈ Rq⟨⟨X⟩⟩. Observe that from Theorem 6.2,
+cˇ@ ll = c ↶
+�
+ll
+−1 ◦ c
+�⋆−1
+= c ↶ ll = c.
+Let d1, d2 ∈ Rm
+pi ⟨⟨X′⟩⟩. It needs to be proven that
+�
+cˇ@d1
+� ˇ@d2 = cˇ@ (d1
+d2). From Theo-
+rem 6.2, observe that
+�
+cˇ@d1
+� ˇ@d2 =
+�
+cˇ@d1
+�
+↶
+�
+d
+−1
+2
+◦
+�
+cˇ@d1
+��⋆−1
+=
+�
+c ↶
+�
+d
+−1
+1
+◦ c
+�⋆−1�
+↶
+�
+d
+−1
+2
+◦
+�
+c ↶
+�
+d
+−1
+1
+◦ c
+�⋆−1��⋆−1
+.
+Applying Theorem 5.7,
+�
+cˇ@d1
+� ˇ@d2 =
+�
+c ↶
+�
+d
+−1
+1
+◦ c
+�⋆−1�
+↶
+��
+d
+−1
+2
+◦ c
+�
+↶
+�
+d
+−1
+1
+◦ c
+�⋆−1�⋆−1
+.
+
+20
+VENKATESH G. S. AND KURUSCH EBRAHIMI-FARD
+Applying Theorem 5.9 and fact that the group inverse is anti-homomorphism with respect
+to the group product,
+�
+cˇ@d1
+� ˇ@d2 = c ↶
+� �
+d
+−1
+1
+◦ c
+�⋆−1 ⋆
+��
+d
+−1
+2
+◦ c
+�
+↶
+�
+d
+−1
+1
+◦ c
+�⋆−1�⋆−1 �
+= c ↶
+� ��
+d
+−1
+2
+◦ c
+�
+↶
+�
+d
+−1
+1
+◦ c
+�⋆−1�
+⋆
+�
+d
+−1
+1
+◦ c
+� �⋆−1
+.
+Applying (12),
+�
+cˇ@d1
+� ˇ@d2 = c ↶
+�
+�
+d
+−1
+1
+◦ c
+�
+�� �
+d
+−1
+2
+◦ c
+�
+↶
+�
+d
+−1
+1
+◦ c
+�⋆−1 �
+↶
+�
+d
+−1
+1
+◦ c
+�
+��⋆−1
+.
+Using Theorem 5.9,
+�
+cˇ@d1
+� ˇ@d2 = c ↶
+�
+�
+d
+−1
+1
+◦ c
+�
+�
+�
+d
+−1
+2
+◦ c
+�
+↶
+��
+d
+−1
+1
+◦ c
+�⋆−1 ⋆
+�
+d
+−1
+1
+◦ c
+�� ��⋆−1
+= c ↶
+��
+d
+−1
+1
+◦ c
+�
+��
+d
+−1
+2
+◦ c
+�
+↶ ll
+��⋆−1
+= c ↶
+��
+d
+−1
+1
+◦ c
+�
+�
+d
+−1
+2
+◦ c
+��⋆−1 .
+In light of Theorem 5.1,
+�
+cˇ@d1
+� ˇ@d2 = c ↶
+��
+d
+−1
+1
+d
+−1
+2
+�
+◦ c
+�⋆−1
+= c ↶
+�
+(d1
+d2)
+−1 ◦ c
+�⋆−1 .
+Therefore,
+�
+cˇ@d1
+� ˇ@d2 = cˇ@ (d1
+d2) .
+It is worth noting that for the additive dynamic feedback product the transformation group
+is the additive group (Rm⟨⟨X′⟩⟩, +, 0) while here (Rm
+pi ⟨⟨X′⟩⟩,
+, ll) plays the role.
+7. Invariance of Class and Relative Degree under multiplicative dynamic
+feedback connection
+The notion of relative degree of a plant is very essential and prime in the studies of
+feedback linearization [Isidori(1995)], flatness and system inversion etc. The existence and
+quantification of relative degree of a interconnection of systems is vital in systems theory.
+The notion of class and relative degree of a SISO Chen–Fliess series is equivalently char-
+acterized by the notion of relative degree of its generating series and the definition was
+furnished in [Gray, et al.(2014b), Gray & Venkatesh(2019)] and the existence and quantifi-
+cation of relative degree of interconnected system of Chen–Fliess series was described in
+[Gray & Venkatesh(2019), Venkatesh(2021)]. In addition, this definition of relative degree is
+consistent with the classical definition whenever y = Fc[u] has an input-affine analytic state
+space realization [Gray, et al.(2014b), Gray & Ebrahimi-Fard(2017)]. Let X = {x0, x1} and
+the following definition explains the concept of a class, a weaker notion than the relative
+degree of a series in R⟨⟨X⟩⟩.
+Definition 7.1.
+[Gray & Venkatesh(2019)] A series c ∈ R⟨⟨X⟩⟩ is said to be of r-class,
+denoted by C (c) = r, if supp(cF) ⊆ xr−1
+0
+X+ and supp(cF) ⊈ xr
+0X+.
+By definition, let
+C (c) = ∞ if cF = 0.
+The notion of class is universal and is versed in the following theorem.
+
+FORMAL SERIES APPROACH TO MULTIPLICATIVE DYNAMIC FEEDBACK CONNECTION
+21
+Lemma 7.1. [Gray & Venkatesh(2019)] Every series c ∈ R⟨⟨X⟩⟩ has a class.
+Definition 7.1 of class is illustrated in the following example.
+Example 7.1. Let c = 1 + x0x2
+1 + x2
+0x1, so that cF = x0x2
+1 + x2
+0x1. Observe that supp(cF) ⊆
+x0X+ but supp(cF) ⊈ x2
+0X+. Thus, C (c) = 2.
+The following lemma is essential in the proof of quantification of class for the multiplicative
+mixed composition product.
+Lemma 7.2. Let c, c′, d ∈ Rm⟨⟨X⟩⟩ such that supp (c′) ̸⊆ x0X∗. Then the following state-
+ments are true:
+(1) xk
+0 ↶ d = xk
+0 ∀k ∈ N0.
+(2) cN ↶ d = cN where cN is the natural part of the series c.
+(3) supp (c′ ↶ d) ̸⊆ x0X∗.
+Proof:
+(1) The proof is by induction on k ∈ N0. The base case being k = 0 is true viz ∅ ↶ d = ∅
+from (11). Assume the proposition is true for k = n − 1, then using (11)
+xn
+0 ↶ d = x0
+�
+xn−1
+0
+↶ d
+�
+= x0
+�
+xn−1
+0
+�
+= xn
+0.
+Hence proved by induction on N0.
+(2) Observe that from Definition 2.1, supp (cN) ⊆ {xk
+0 : k ∈ N0}. Thus, using the previ-
+ous statement (1) and Theorem 5.4 it follows that cN ↶ d = cN.
+(3) Since supp (c′) ̸⊆ x0X∗, there exists a word xiη ∈ supp (c′) where xi ̸= x0 and η ∈ X∗.
+Using (11),
+xiη ↶ d = xi (di
+(η ↶ d)) .
+Thus, supp (xiη ↶ d) ⊆ xiX∗, where xi ̸= x0. Therefore, supp (c′ ↶ d) ̸⊆ x0X∗.
+The following theorem quantifies that class is invariant under the multiplicative mixed
+composition product
+Theorem 7.1. Let c, d ∈ R⟨⟨X⟩⟩, then C (c ↶ d) = C (c).
+Proof: Suppose the series c ∈ R⟨⟨X⟩⟩ is of r-class, then the series c can be written as:
+c = cN + xr−1
+0
+c′,
+where c′ is a proper series such that supp (c′) ̸⊆ x0X∗. Hence by Theorem 5.4,
+c ↶ d = (cN ↶ d) +
+�
+xr−1
+0
+c′ ↶ d
+�
+.
+Using (11),
+c ↶ d = (cN ↶ d) + xr−1
+0
+(c′ ↶ d) .
+Since supp (c′) ̸⊆ x0X∗, then by applying Lemma 7.2,
+c ↶ d = cN + xr−1
+0
+(c′ ↶ d) ,
+
+22
+VENKATESH G. S. AND KURUSCH EBRAHIMI-FARD
+with supp (c′ ↶ d) ̸⊆ x0X∗. Given that c′ ∈ Rp ⟨⟨X⟩⟩, whence supp (c ↶ d)F ⊆ xr−1
+0
+X+ and
+supp (c ↶ d)F ̸⊆ xr
+0X+. Therefore, C (c ↶ d) = r = C (c).
+Example 7.2. Consider the series c in Example 7.1, given by c = 1 + x2
+0x1 + x0x2
+1 and
+d = 1 + x1 ∈ R⟨⟨X⟩⟩. Using (11), the multiplicative mixed composition product of c and d
+is computed as:
+c ↶ d = 1 + x0x2
+1 + 3x0x3
+1 + 3x0x4
+1 + x2
+0x1 + x2
+0x2
+1.
+Observe that C (c ↶ d) = 2 = C (c), as in Example 7.1.
+The following theorem asserts that class of a series is preserved under the multiplicative
+dynamic feedback product which is one of the prime goal of this subsection.
+Theorem 7.2. If c ∈ R⟨⟨X⟩⟩ with C (c) = r, and d ∈ Rpi ⟨⟨X⟩⟩, then C
+�
+cˇ@d
+�
+= r = C (c).
+Proof: From Theorem 6.2,
+cˇ@d = c ↶
+�
+d
+−1 ◦ c
+�⋆−1 .
+Since C (c) = r, whence applying Theorem 7.1,
+C
+�
+cˇ@d
+�
+= C
+�
+c ↶
+�
+d
+−1 ◦ c
+�⋆−1�
+= r = C (c) .
+The preservation of class under the multiplicative dynamic feedback connections as as-
+serted in Theorem 7.2 is further illustrated in the following example.
+Example 7.3. Let c, d ∈ R⟨⟨X⟩⟩ c = x1 and d = 1 + �
+k∈N k!xk
+1. Note that the class of
+series C (c) = 1. Using Theorem 6.2 the multiplicative feedback product is computed as:
+cˇ@d = x1 + x1x0x1 + 3x1x0x1x0x1 + 4x1x2
+0x2
+1 + · · · .
+Infer from Definition 7.1 that C
+�
+cˇ@d
+�
+= C (c) = 1.
+Finally, the main definition of the section details the concept of relative degree in the
+context of Chen–Fliess series which is characterized on its generating series.
+Definition 7.2.
+[Gray & Venkatesh(2019)] A series c ∈ R⟨⟨X⟩⟩ has relative degree r if
+C (c) = r and the word xr−1
+0
+x1 ∈ supp(cF). Otherwise, c does not have relative degree.
+The following theorem asserts the quantification of relative degree under multiplicative
+mixed composition product.
+Theorem 7.3. If c ∈ R⟨⟨X⟩⟩ with relative degree rc and d ∈ R⟨⟨X⟩⟩ be non-proper, then
+c ↶ d has relative degree rc.
+Proof: From Theorem 7.1, C (c ↶ d) = rc. It remains to prove that xrc−1
+0
+x1 ∈ supp (c ↶ d).
+Given that c ∈ R⟨⟨X⟩⟩ has relative degree rc, then c can be decomposed as:
+c = cN + λxrc−1
+0
+x1 + xrc−1
+0
+c′,
+where λ ̸= 0 and c′ is a proper series such that x1 ̸∈ supp (c′). Then,
+c ↶ d =
+�
+cN + λxrc−1
+0
+x1 + xrc−1
+0
+c′�
+↶ d.
+Applying Theorem 5.4,
+c ↶ d = (cN ↶ d) + λ
+�
+xrc−1
+0
+x1 ↶ d
+�
++
+�
+xrc−1
+0
+c′ ↶ d
+�
+.
+
+FORMAL SERIES APPROACH TO MULTIPLICATIVE DYNAMIC FEEDBACK CONNECTION
+23
+Using (11) and Lemma 7.2,
+c ↶ d = cN + λxrc−1
+0
+x1d + xrc−1
+0
+(c′ ↶ d) .
+Since d ∈ Rpi ⟨⟨X⟩⟩ −→ d = α + d′, where α ̸= 0 and d′ is a proper series. Hence,
+c ↶ d = cN + λαxrc−1
+0
+x1 + xrc−1
+0
+x1d′ + xrc−1
+0
+(c′ ↶ d) .
+Observe from (11), x1 ̸∈ supp (c′) =⇒ x1 ̸∈ supp (c′ ↶ d) and also that αλ ̸= 0.
+Therefore xrc−1
+0
+x1 ∈ supp (c ↶ d), whence the relative degree of c ↶ d is rc, when d is a
+non-proper series.
+The following example illustrates the statement from Theorem 7.3.
+Example 7.4. Let c = 1 + x2
+0 + x0x1 + x2
+0x1 and d = 1 + x1. Observe that by Definition 7.2,
+the relative degree of c is rc = 2 and also that d is non-proper. The multiplicative mixed
+composition product of c and d to computed as:
+c ↶ d = 1 + x2
+0 + x0x1 + x0x2
+1 + x2
+0x1 + x2
+0x2
+1.
+Using Definition 7.2, note that the relative degree of c ↶ d is 2 = rc.
+The following theorem is the prime objective of this section stating that the relative degree
+of a series remains invariant under multiplicative dynamic feedback product.
+Theorem 7.4. If c ∈ R⟨⟨X⟩⟩ with relative degree rc and d ∈ Rpi ⟨⟨X⟩⟩, then the relative
+degree of
+�
+cˇ@d
+�
+is rc.
+Proof: Since c ∈ R⟨⟨X⟩⟩ and d ∈ Rpi ⟨⟨X⟩⟩, then by Theorem 6.2,
+cˇ@d = c ↶
+�
+d
+−1 ◦ c
+�⋆−1 .
+Observe that d ∈ Rpi ⟨⟨X⟩⟩ ⇔ d
+−1 ∈ Rpi ⟨⟨X⟩⟩.
+Then by Theorem 5.2 (d
+−1 ◦ c) ∈
+Rpi ⟨⟨X⟩⟩. As per Theorem 5.10, the group inverse
+�
+d
+−1 ◦ c
+�⋆−1 ∈ Rpi ⟨⟨X⟩⟩.
+Hence by Theorem 7.3,
+cˇ@d = c ↶
+�
+d
+−1 ◦ c
+�⋆−1 .
+has relative degree rc.
+The invariance of the relative degree of a Chen–Fliess series under multiplicative dynamic
+feedback connections as stated in Theorem 7.4 is illustrated through the following example.
+Example 7.5. Consider the Example 7.3 again where c = x1 and d = 1 + �
+k∈N k!xk
+1.
+Observe that by Definition 7.2, the relative degree of c is rc = 1. The multiplicative feedback
+product is computed as:
+cˇ@d = x1 + x1x0x1 + 3x1x0x1x0x1 + 4x1x2
+0x2
+1 + · · ·
+Infer that the relative degree of cˇ@d = 1 = rc as stated in Theorem 7.4.
+
+24
+VENKATESH G. S. AND KURUSCH EBRAHIMI-FARD
+8. Computational Framework for Multiplicative Mixed Composition &
+Dynamic Feedback Product
+The goal of this section is to describe the computational framework for multiplicative
+dynamic feedback product as explained in Section 6.
+The section further illustrates the
+framework with examples but prior to that it is imperative to understand the dual bialgebra
+and Hopf algebra constructions corresponding to the multiplicative dynamic output feedback
+group.
+8.1. Hopf Algebra Corresponding to the Multiplicative Dynamic Feedback Sub-
+group. The goal of the subsection is to construct a dual Hopf algebra reflecting the group
+structure of the multiplicative dynamic feedback subgroup M as asserted in Theorem 5.11.
+The group inverse is computed the antipode of the constructed Hopf algebra and thus pro-
+vides a computational framework to compute the multiplicative dynamic feedback group
+inverse. As a recall, the group M is defined as
+M = { ll + d : d ∈ Rm
+p ⟨⟨X⟩⟩},
+where ll = [1 · · ·1 1]T ∈ Rm. In light of Theorem 5.11, (M, ⋆) forms a subgroup of the
+multiplicative dynamic feedback group. The algebra structure is same as the algebra of H
+in Section 4.1. Let the set Wb ⊂ Rm⟨⟨X⟩⟩∗ (dual module of Rm⟨⟨X⟩⟩) be defined as the
+collection of coordinate maps defined as:
+Wb = {aη : aη(c) = (c, η) : η ∈ X∗},
+where c ∈ Rm⟨⟨X⟩⟩.
+Define W to be the free Rm-module spanned by the set Wb.
+Let
+H denote the reduced symmetric algebra generated by the module W. The unit map ξ :
+Rm −→ W is defined by ξ( ll) = a∅. Note that a∅ (c) = ll ∀c ∈ M. By construction H is
+an Rm-associative, commutative and unital algebra with addition, scalar multiplication and
+product defined, respectively, as
+(aη + aζ)(c) = aη(c) + aζ(c)
+(kaη)(c) = k(ai
+η(c))
+m(aη, aζ)(c) = aη(c)aζ(c),
+where c ∈ Rm⟨⟨X⟩⟩. Then H is given a coproduct ∆H : H −→ H � H such that for all
+c, d ∈ M: ∆Hai
+η(c, d) = ai
+η(c ⋆ d) = ((c ⋆ d)i , η) ∀η ∈ X+. The counit map ǫ : H −→ R is
+defined as
+ǫ(h) =
+� ll : h = a∅
+0 : otherwise.
+Since ◦ is associative (from Theorem 5.8), thus by the dual the coproduct ∆H is coasso-
+ciative. Therefore, (H, m, ξ, ∆H, ǫ) forms a Rm-bialgebra. Owing to the group structure of
+(M, ◦), the bialgebra H is equipped with antipode S defined as:
+Saη (c) = aη
+�
+c⋆−1�
+=
+�
+c⋆−1, η
+�
+,
+for all i = 1, 2, . . . , m and η ∈ X+. Hence H is a Rm-Hopf algebra. The computation of
+coproduct ∆H is well-understood through the right coaction of Hopf algebra H on the Hopf
+algebra H
+. Prior to that, it is imperative to understand the right coaction of Hopf algebra
+H on the non-unital algebra of coordinate functions.
+
+FORMAL SERIES APPROACH TO MULTIPLICATIVE DYNAMIC FEEDBACK CONNECTION
+25
+8.2. Coaction of Hopf algebra H on Algebra of Coordinate Map. The subsection
+explains the coaction of the Hopf algebra H defined in Section 8.1 on the algebra of coordinate
+functions. The results in this subsection are utilized subsequently to explain the coaction of
+H on the bialgebra H
+, particularly in proofs in Section 8.3. The right coaction of the Hopf
+algebra H is on Rm-algebra of coordinate maps S+ (W) constructed in Section 4.3.
+The right coaction map ˜∆ : S+ (W) −→ S+ (W) � H is defined such that for all c ∈
+Rm⟨⟨X⟩⟩, d ∈ M and η ∈ X∗,
+˜∆aη (c, d) = (c ↶ d, η) .
+(17)
+The map ˜∆ being a right coaction map is a reflection of Theorem 5.9. It remains to
+show how the coaction map ˜∆ is computed on S+(W), for which it is sufficient to define its
+computation on the module W. Observe that for all aη ∈ W,
+˜∆aη =
+�
+˜∆ ◦ π1 ˜∆ ◦ π2 · · · ˜∆ ◦ πm�t
+aη.
+On the dual side, the above statement infers that for all c ∈ Rm⟨⟨X⟩⟩, d ∈ M and η ∈ X∗,
+(c ↶ d, η) = [((c ↶ d)1 , η) · · · ((c ↶ d)m , η)]t .
+Hence, the notation ˜∆ai
+η := ˜∆ ◦ πiaη for all η ∈ X∗ and i = 1, 2, . . . , m. The following
+proposition provides a recursive definition to compute ˜∆ on the module V viz to compute
+the ˜∆
+�
+aj
+η
+�
+∀η ∈ X∗ and j = 1, 2, . . . , m.
+Proposition 8.1. For all i = 1, . . . , m:
+(1) ˜∆ai
+∅ = ai
+∅ ⊗ ai
+∅.
+(2) ˜∆ ◦ θ0 = (θ0 ⊗ id) ◦ ˜∆.
+(3) ˜∆ ◦ θi = (θi ⊗ m) ◦
+�
+˜∆ ⊗ id
+�
+◦ ρi ,
+where ρ
+is the coaction map of Hopf algebra H
+on S+ (W) as defined in Section 4.3.
+Proof: Observe that ∀c ∈ Rm⟨⟨X⟩⟩ and d ∈ M,
+c = (c, ∅) +
+m
+�
+j=0
+xj
+�
+x−1
+j
+(c)
+�
+.
+Hence by Theorem 5.4,
+c ↶ d = (c, ∅) + x0
+�
+x−1
+0 (c) ↶ d
+�
++
+m
+�
+j=1
+xj
+�
+dj
+�
+x−1
+j
+(c) ↶ d
+��
+.
+(18)
+The proof for each of the statement as follows:
+(1) Let c, d ∈ Rm⟨⟨X⟩⟩. From (17) and (18),
+˜∆ai
+∅ (c, d) = ((c ↶ d)i , ∅)
+= (ci ↶ d, ∅) = (ci, ∅) .1 = ai
+∅ ⊗ ai
+∅ (c, d) .
+Therefore, ˜∆ai
+∅ = ai
+∅ ⊗ ai
+∅.
+
+26
+VENKATESH G. S. AND KURUSCH EBRAHIMI-FARD
+(2) Let c, d ∈ Rm⟨⟨X⟩⟩, η ∈ X∗ and ∀ j = 1, 2, . . . , m. Then,
+�
+˜∆ ◦ θ0
+�
+aj
+η (c, d) =
+�
+(c ↶ d)j , x0η
+�
+=
+�
+x−1
+0 (c ↶ d)j , η
+�
+From (18),
+�
+˜∆ ◦ θ0
+�
+aj
+η (c, d) =
+�
+x−1
+0 (cj) ↶ d, η
+�
+= ˜∆aj
+η
+�
+x−1
+0 (c) , d
+�
+= (θ0 ⊗ id) ◦ ˜∆aη (c, d) .
+Therefore, ˜∆ ◦ θ0 = (θ0 ⊗ id) ◦ ˜∆.
+(3) Let c, d ∈ Rm⟨⟨X⟩⟩ and η ∈ X∗. Then ∀ i, j = 1, 2, . . . , m,
+�
+˜∆ ◦ θi
+�
+aj
+η (c, d) =
+�
+(c ↶ d)j , xiη
+�
+=
+�
+x−1
+i
+(c ↶ d)j , η
+�
+From (18),
+�
+˜∆ ◦ θi
+�
+aj
+η (c, d) =
+�
+di
+x−1
+i
+(cj) ↶ d, η
+�
+= ρi aj
+η
+�
+x−1
+i
+(c) ↶ d, d
+�
+= ρi aj
+η
+�
+x−1
+i
+(c) ↶ d
+�
+= ρi aj
+η
+�
+x−1
+i
+(c) ↶ d, d
+�
+=
+�
+˜∆ ⊗ id
+�
+◦ ρi aj
+η
+�
+x−1
+i
+(c) , d, d
+�
+= (θi ⊗ m) ◦
+�
+˜∆ ⊗ id
+�
+◦ ρi aj
+η (c, d) .
+Therefore, ˜∆ ◦ θi = (θi ⊗ m) ◦
+�
+˜∆ ⊗ id
+�
+◦ ρi
+∀i = 1, 2, . . . , m.
+Example 8.1. A few examples of the computation of ˜∆ on V using Proposition 8.1 are
+given as follows(indices i, j, k = 1, 2, . . . , m.):
+˜∆ai
+∅ = ai
+∅ ⊗ ai
+∅.
+˜∆ai
+x0 = ai
+x0 ⊗ ai
+∅.
+˜∆aj
+xi = aj
+xi ⊗ ai
+∅.
+˜∆ai
+x2
+0 = ai
+x2
+0 ⊗ ai
+∅.
+˜∆aj
+x0xi = aj
+x0xi ⊗ ai
+∅.
+˜∆aj
+xix0 =
+�
+aj
+xix0 ⊗ ai
+∅
+�
++
+�
+aj
+xi ⊗ ai
+x0
+�
+.
+
+FORMAL SERIES APPROACH TO MULTIPLICATIVE DYNAMIC FEEDBACK CONNECTION
+27
+˜∆ak
+xixj =
+�
+ak
+xixj ⊗ aj
+∅ai
+∅
+�
++
+�
+ak
+xi ⊗ ai
+xj
+�
+.
+The coaction map ˜∆ thus provides a framework to compute the multiplicative mixed com-
+position product and multiplicative dynamic feedback group product whenever c ∈ Rm⟨⟨X⟩⟩
+and d ∈ M ⊊ Rm⟨⟨X⟩⟩. For computing the multiplicative mixed composition product for
+c ∈ Rp⟨⟨X⟩⟩ and d ∈ M ⊊ Rm⟨⟨X⟩⟩ where p = m,
+(1) If p < m, then define ˇc ∈ Rm⟨⟨X⟩⟩ such that ˇci = ci ∀ i = 1, 2, . . . , p and ˇci = 0 ∀ i =
+p + 1, p + 2, . . . , m. Then for all η ∈ X∗,
+((c ↶ d)i , η) = ˜∆ai
+η (ˇc, d)
+∀i = 1, 2, . . . , p.
+Note that (ˇc ↶ d)j = 0 ∀j = p + 1, p + 2, . . . , m.
+(2) If p > m, then this can be reduced to Case 1 by performing computations component
+wise viz computing ci ↶ d for all i = 1, 2, . . . , p.
+Thus the computational framework to compute the multiplicative mixed composition prod-
+uct of c ∈ Rp⟨⟨X⟩⟩ and d ∈ M, denoted by c ↶ d for arbitrary p and m is well-defined via
+the coaction map ˜∆. The computations of the coproduct ∆H and antipode S (defined in
+Section 8.1) are well-understood once the right coaction of Hopf algebra H on Hopf algebra
+H
+.
+8.3. Coaction of Hopf algbera H on the Hopf algebra H
+. The objective of the
+subsection is to define the right coaction map of Hopf algebra H on the unshuffle Hopf algebra
+H
+defined in Section 4.1. The right coaction is pivotal in computation of the coproduct
+and antipode of Hopf algebra H which in turn are essential to compute the multiplicative
+dynamic feedback product.
+The right coaction map of H on H
+is defined to be ˜∆H : H
+−→ H
+� H such that
+for all c, d ∈ M (the underlying sets of M and M
+are identical) and η ∈ X∗,
+˜∆Haη (c, d) = (c ↶ d, η) .
+(19)
+Observe that the algebra of coordinate functions S+(W) and H
+are isomorphic as Rm-
+modules. Thus it is vital to understand the relationship between the operator ˜∆ operating
+on the module S+(W) and operator ˜∆H operating on H
+, which is stated in the following
+lemma.
+Lemma 8.1. If c, d ∈ M, then for all η ∈ X∗
+˜∆Haη (c, d) = ˜∆aη (c, d) .
+Proof: If c, d ∈ M and η ∈ X+,
+˜∆Haη (c, d) = (c ↶ d, η) = ˜∆aη (c, d) .
+Despite the statement of Lemma 8.1, it is vital to understand the difference between the
+coaction maps ˜∆ and ˜∆H.
+The coaction map ˜∆H is compatible with the Hopf algebra
+structure of H
+viz.
+m1,3,24 ◦
+�
+˜∆H ⊗ ˜∆H
+�
+◦ ∆
+= (∆
+⊗ id) ◦ ˜∆H,
+˜∆H ◦ S = (S
+⊗ id) ◦ ˜∆H,
+where m1,3,24 = (m ⊗ m) ◦ (id ⊗ τ ⊗ id).
+
+28
+VENKATESH G. S. AND KURUSCH EBRAHIMI-FARD
+Thus the coaction map ˜∆H makes H
+a comodule-Hopf algebra over H. Equivalently,
+the coaction map ˜∆H is a corepresentation of Hopf algebra H over unshuffle Hopf algebra
+H
+. Similar to Section 8.2, for all aη ∈ W,
+˜∆Haη =
+�
+˜∆H ◦ π1 ˜∆H ◦ π2 · · · ˜∆H ◦ πm�
+aη.
+The map to compute the ˜∆H
+�
+aj
+η
+�
+∀η ∈ X∗ and j = 1, 2, . . . , m is g module W is stated
+in the following proposition.
+Proposition 8.2. For all i, j = 1, 2 . . . , m and η ∈ X∗:
+(1) ˜∆Hai
+∅ = ai
+∅ ⊗ ai
+∅.
+(2) ˜∆H ◦ θ0aj
+η = (θ0 ⊗ id) ◦ ˜∆Haj
+η.
+(3)
+�
+˜∆H ◦ θi
+�
+aj
+η = (θi ⊗ m) ◦
+�
+˜∆H ⊗ id
+�
+◦ ∆i aj
+η,
+where ∆
+is the unshuffle coproduct defined in Section 4.1.
+Proof: Observe that ∀c ∈ M,
+c = ll +
+m
+�
+j=0
+xj
+�
+x−1
+j
+(c)
+�
+.
+Hence by Theorem 5.4,
+c ↶ d = ll + x0
+�
+x−1
+0 (c) ↶ d
+�
++
+m
+�
+j=1
+xj
+�
+dj
+�
+x−1
+j
+(c) ↶ d
+��
+.
+(20)
+The proof for each of the statement as follows:
+(1) Let c, d ∈ M. From (19) and (20),
+˜∆Hai
+∅ (c, d) = ((c ↶ d)i , ∅)
+= (ci ↶ d, ∅) = 1 = (ci, ∅)(di, ∅)
+= ai
+∅ ⊗ ai
+∅(c, d).
+Therefore, ˜∆Hai
+∅ = ai
+∅ ⊗ ai
+∅.
+(2) Let c, d ∈ M, η ∈ X∗ and ∀ j = 1, 2, . . . , m. Then,
+�
+˜∆H ◦ θ0
+�
+aj
+η (c, d) =
+�
+(c ↶ d)j , x0η
+�
+=
+�
+x−1
+0 (c ↶ d)j , η
+�
+Observe that x−1
+0 (c) may not belong to M and from (20),
+�
+˜∆H ◦ θ0
+�
+aj
+η (c, d) =
+�
+x−1
+0 (cj) ↶ d, η
+�
+= ˜∆aj
+η
+�
+x−1
+0 (c) , d
+�
+= (θ0 ⊗ id) ◦ ˜∆aη (c, d) .
+
+FORMAL SERIES APPROACH TO MULTIPLICATIVE DYNAMIC FEEDBACK CONNECTION
+29
+Since η ∈ X+ and c, d ∈ M, then by Lemma 8.1
+�
+˜∆H ◦ θ0
+�
+aj
+η (c, d) = (θ0 ⊗ id) ◦ ˜∆Haη (c, d) .
+Therefore, ˜∆H ◦ θ0 = (θ0 ⊗ id) ◦ ˜∆H.
+(3) Let c, d ∈ M and η ∈ X∗. Then ∀ i, j = 1, 2, . . . , m,
+�
+˜∆H ◦ θi
+�
+aj
+η (c, d) =
+�
+(c ↶ d)j , xiη
+�
+=
+�
+x−1
+i
+(c ↶ d)j , η
+�
+From (20),
+�
+˜∆H ◦ θi
+�
+aj
+η (c, d) =
+�
+di
+x−1
+i
+(cj) ↶ d, η
+�
+.
+Since x−1
+i
+(c) may not belong to group M (also M
+),
+= ρi aj
+η
+�
+x−1
+i
+(c) ↶ d, d
+�
+= ρi aj
+η
+�
+x−1
+i
+(c) ↶ d
+�
+= ρi aj
+η
+�
+x−1
+i
+(c) ↶ d, d
+�
+=
+�
+˜∆ ⊗ id
+�
+◦ ρi aj
+η
+�
+x−1
+i
+(c) , d, d
+�
+= (θi ⊗ m) ◦
+�
+˜∆ ⊗ id
+�
+◦ ρi aj
+η (c, d) .
+Since η ∈ X+ and c, d ∈ M, then by Lemma 8.1 and Lemma 4.1,
+�
+˜∆H ◦ θi
+�
+aj
+η (c, d) = (θi ⊗ m) ◦
+�
+˜∆H ⊗ id
+�
+◦ ∆i aj
+η.
+Therefore,
+�
+˜∆H ◦ θi
+�
+= (θi ⊗ m) ◦
+�
+˜∆H ⊗ id
+�
+◦ ∆i
+for all i = 1, 2, . . . , m.
+8.4. Coproduct, Antipode Computations and Grading of Hopf algebra H. The
+objective of this subsection is to define and illustrate the computation of coproduct ∆H of
+the bialgebra H. Further, a graded and connected structure is endowed with the bialgebra
+owing to which the antipode computation is possible owing to Theorem 3.1. The following
+proposition asserts the essential reason behind the definition of ˜∆H.
+Proposition 8.3. For all η ∈ X∗ and i = 1, 2, . . . , m,
+∆Hai
+η = (id ⊗ m) ◦
+�
+˜∆H ⊗ id
+�
+◦ ˆ∆i ai
+η.
+Proof: Proof: Observe that for all c, d ∈ M and η ∈ X∗,
+∆ai
+η (c, d) = ((c ⋆ d)i , η)
+∀ i = 1, 2, . . . , m.
+Using (12),
+∆Hai
+η (c, d) = (di
+ci ↶ d, η)
+
+30
+VENKATESH G. S. AND KURUSCH EBRAHIMI-FARD
+= ˆ∆i ai
+η(c ↶ d, d)
+=
+�
+˜∆H ⊗ id
+�
+◦ ˆ∆i ai
+η (c, d, d)
+= (id ⊗ m) ◦
+�
+˜∆H ⊗ id
+�
+◦ ˆ∆i ai
+η (c, d) .
+Proposition 8.3 asserts that the computation of coproduct ∆H on the module W (sub-
+sequently on the algebra H) can be carried out post the computation of the operator ˜∆H
+on W. The computation of the coproduct ∆H for the some coordinate maps are given as
+follows:
+∆Hai
+∅ = ai
+∅ ⊗ ai
+∅.
+∆Hai
+x0 =
+�
+ai
+x0 ⊗ ai
+∅
+�
++
+�
+ai
+∅ ⊗ ai
+x0
+�
+.
+∆Haj
+xi =
+�
+aj
+xi ⊗ ai
+∅aj
+∅
+�
++
+�
+aj
+∅ ⊗ aj
+xi
+�
+.
+∆Hai
+x2
+0 =
+�
+ai
+x2
+0 ⊗ ai
+∅
+�
++ 2
+�
+ai
+x0 ⊗ ai
+x0
+�
++
+�
+ai
+∅ ⊗ ai
+x2
+0
+�
+.
+∆Haj
+x0xi =
+�
+aj
+x0xi ⊗ aj
+∅
+�
++
+�
+aj
+x0 ⊗ aj
+xi
+�
++
+�
+aj
+xi ⊗ ai
+∅aj
+x0
+�
++
+�
+aj
+∅ ⊗ aj
+x0xi
+�
+.
+∆Haj
+xix0 =
+�
+aj
+xix0 ⊗ ai
+∅aj
+∅
+�
++
+�
+aj
+xi ⊗ ai
+x0aj
+∅
+�
++
+�
+aj
+xi ⊗ ai
+∅aj
+x0
+�
++
+�
+aj
+x0 ⊗ aj
+xi
+�
++
+�
+aj
+∅ ⊗ aj
+xix0
+�
+.
+∆Hak
+xixj =
+�
+ak
+xixj ⊗ aj
+∅ai
+∅ak
+∅
+�
++
+�
+ak
+xi ⊗ ai
+xjak
+∅
+�
++
+�
+ak
+xi ⊗ ai
+∅ak
+xj
+�
++
+�
+ak
+xj ⊗ aj
+∅ak
+xi
+�
++
+�
+ak
+∅ ⊗ ak
+xixj
+�
+.
+If m = 2 (two input-two output MIMO case) viz. X = {x0, x1, x2}, then from above
+computations
+∆Hax1x2 =
+
+
+�
+a1
+x1x2 ⊗
+�
+a1
+∅
+�2 a2
+∅
+�
++
+�
+a1
+x1 ⊗ a1
+x2a1
+∅
+�
++
+�
+a1
+x1 ⊗ a1
+∅a1
+x2
+�
++
+�
+a1
+x2 ⊗ a2
+∅a1
+x1
+�
++
+�
+a1
+∅ ⊗ a1
+x1x2
+�
+�
+a2
+x1x2 ⊗ a1
+∅
+�
+a2
+∅
+�2�
++
+�
+a2
+x1 ⊗ a1
+x2a2
+∅
+�
++
+�
+a2
+x1 ⊗ a1
+∅a2
+x2
+�
++
+�
+a2
+x2 ⊗ a2
+∅a2
+x1
+�
++
+�
+a2
+∅ ⊗ a2
+x1x2
+�
+
+
+which can be rewritten as
+∆Hax1x2 =
+�
+ax1x2 ⊗ (a1
+∅a2
+∅ ll)a∅
+�
++
+�
+ax1 ⊗ (a1
+x2 ll)a∅
+�
++
+�
+ax1 ⊗ (a1
+∅ ll)ax2
+�
++
+�
+ax2 ⊗ (a2
+∅ ll)ax1
+�
++ (a∅ ⊗ ax1x2) ,
+where ll = [1 1]t. It is vital to observe that the term
+�
+ax1x2 ⊗ (a1
+∅a2
+∅ ll)a∅
+�
+is a primitive
+term of the coproduct as a1
+∅a2
+∅ ll ∼= ll since a∅ is the unit of H.
+The following corollary is resultant of the Proposition 8.2 to the words of the form xn
+0 for
+all n ≥ 0.
+Corollary 8.1. If n ∈ N0, then for all i = 1, 2, . . . , m (defining x0
+0 := ∅):
+˜∆Hai
+xn
+0 = ai
+xn
+0 ⊗ ai
+∅.
+∆Hai
+xn
+0 =
+n
+�
+k=0
+�n
+k
+�
+ai
+xk
+0 ⊗ ai
+∅ai
+xn−k
+0
+.
+Proof: The proof is by induction on n ∈ N0. The base case (n = 0) :
+˜∆Hai
+∅ = ai
+∅ ⊗ ai
+∅,
+
+FORMAL SERIES APPROACH TO MULTIPLICATIVE DYNAMIC FEEDBACK CONNECTION
+31
+is proved in Proposition 8.1. Assume the statement is true for n = k, then
+˜∆ai
+xk+1
+0
+=
+�
+˜∆ ◦ θ0
+�
+ai
+xk
+0.
+Using Proposition 8.1,
+˜∆ai
+xk+1
+0
+= (θ0 ⊗ id) ◦ ˜∆ai
+xk
+0
+= (θ0 ⊗ id) {ai
+xk
+0 ⊗ ai
+∅}
+= ai
+xk+1
+0
+⊗ ai
+∅.
+Hence proved by induction on n ∈ N0 that: ˜∆ai
+xn
+0 = ai
+xn
+0 ⊗1. Observe that from Proposition ??
+∆ai
+xn
+0 = (id ⊗ m) ◦
+�
+˜∆ ⊗ id
+�
+◦ ∆i ai
+xn
+0 .
+Using Corollary 4.1,
+∆ai
+xn
+0 = (id ⊗ m) ◦
+�
+˜∆ ⊗ id
+� � n
+�
+k=0
+�n
+k
+�
+ai
+xk
+0 ⊗ ai
+xn−k
+0
+�
+= (id ⊗ m)
+� n
+�
+k=0
+�n
+k
+�
+˜∆ai
+xk
+0 ⊗ ai
+xn−k
+0
+�
+= (id ⊗ m)
+� n
+�
+k=0
+�n
+k
+�
+ai
+xk
+0 ⊗ ai
+∅ ⊗ ai
+xn−k
+0
+�
+=
+n
+�
+k=0
+�n
+k
+�
+ai
+xk
+0 ⊗ ai
+∅ai
+xn−k
+0
+.
+Proposition 8.3 asserted that the calculation of coproduct ∆H is carried out post the
+computation of ˜∆H. However the converse is also true viz. the computation of ˜∆H can be
+carried out if the evaluation fo the coproduct ∆H is known a priori which is well-asserted in
+the following proposition.
+Proposition 8.4. For all η ∈ X+ and for all i = 1, 2, . . . , m,
+˜∆Hai
+η (c, d) = (id ⊗ m) ◦ (∆H ⊗ S
+) ◦ ˆ∆i ai
+η.
+Proof: Given c, d ∈ M, by Theorem (12)
+(c ⋆ d) = (d
+(c ↶ d)) .
+Observe that d ∈ M implies that d is shuffle invertible. Thus for any η ∈ X+,
+((c ↶ d)i , η) =
+�
+d
+−1
+i
+(c ⋆ d)i , η
+�
+,
+for all i = 1, 2, . . . , m. Hence,
+((c ↶ d)i , η) = ˜∆Hai
+η (c, d)
+= ˆ∆i ai
+η
+�
+c ⋆ d, d
+−1�
+.
+= (∆H ⊗ S
+) ◦ ˆ∆i ai
+η (c, d, d) .
+= (id ⊗ m) ◦ (∆H ⊗ S
+) ◦ ˆ∆i ai
+η (c, d) .
+
+32
+VENKATESH G. S. AND KURUSCH EBRAHIMI-FARD
+The key point of the Proposition 8.4 is the shuffle-invertibility of a series c ∈ M
+The goal of this subsection is to provide a graded structure on the R-module W and
+consequently on the underlying R-module structure of the Hopf algebra H such that H is
+connected and the homogeneous components of H are finite-dimensional.
+Definition 8.1. Given a word η ∈ X+, denote the degree of the word as deg (η) and define
+deg (η) = |η| and for all k ≥ 1:
+Xk := {aη : deg (η) = k}.
+(1) Define gradation on the R-module W viz.
+W =
+�
+k≥1
+Wk,
+where Wk is the free R-module spanned by Xk.
+(2) The gradation on the module W induces a graded structure on the algebra H as
+H =
+�
+n∈N0
+Hn,
+with H0 ∼= R in the category of R-modules.
+The following lemma aids in proving that the gradation in Definition 8.1 makes the Hopf
+algebra H is well-defined.
+Lemma 8.2. If η ∈ X∗ such that deg (η) = n then
+˜∆H (aη) ∈
+�
+i+j=n
+Wi ⊗ Hj,
+for all k = 1, 2, . . . , m.
+Proof: The following observations will help in proving the lemma.
+(1) The map {θi}m
+i=0 is a homogeneous operator of degree 1 on the module W. If deg (η) =
+|η| = n for some η ∈ X∗, then |xiη| = n + 1 for all i = 0, 1, . . . , m. Hence,
+θi : Wn −→ Wn+1
+for all i = 0, 1, . . . , m and n ≥ 1.
+(2) Observe that if η, ζ, γ ∈ X∗ such that |γ| = n and γ ∈ supp (η
+ζ) then |γ| = n =
+|ζ|+|η|. Thus, the reduced coproduct ˆ∆
+: W −→ W ⊗W is homogeneous operator
+of degree 0 viz.
+ˆ∆
+: Wn −→ (W ⊗ W)n .
+Let us prove the statement ot the lemma by induction on degree (equivalently length) n
+of the word η ∈ X∗. The base case is n = 0 ⇔ η = ∅. From Proposition 8.2,
+˜∆Ha∅ = a∅ ⊗ a∅
+∈ W0 ⊗ H0,
+Thus the statement holds true for the base case. Assume that the statement of theorem
+holds true for all η ∈ X∗ such that deg (η) ≤ k. Let η′ such that deg (η′) = k + 1. Then two
+cases can occur.
+(1) Let η′ = x0η where |η| = k. Then
+˜∆Haη′ = ˜∆H ◦ θ0aj
+η.
+
+FORMAL SERIES APPROACH TO MULTIPLICATIVE DYNAMIC FEEDBACK CONNECTION
+33
+By Proposition 8.2,
+˜∆Haη′ = (θ0 ⊗ id) ◦ ˜∆Haη.
+Since aη ∈ Wk, then by the induction hypothesis ˜∆H (aη) ⊆ �
+i+j=k Wi ⊗ Hj. Then,
+(θ0 ⊗ id)
+� �
+i+j=k
+Wi ⊗ Hj
+�
+⊆
+�
+i+j=k
+Wi+1 ⊗ Hj
+⊆
+�
+i+j=k+1
+Wi ⊗ Hj.
+Thus, ˜∆Haη′ ∈ �
+i+j=k+1 Wi ⊗ Hj where |η′| = k + 1.
+(2) Let η′ = xiη where |η| = k and xi ̸= x0. Then from Proposition 8.2,
+�
+˜∆H ◦ πj
+�
+aη′ = (θi ⊗ m) ◦
+�
+˜∆H ⊗ id
+�
+◦ (πj ⊗ πi) ◦ ˜∆
+aη.
+= (θi ⊗ m) ◦
+�
+( ˜∆H ◦ πj) ⊗ πi
+�
+◦ ˜∆
+aη
+Thus,
+˜∆Haη′ = (θi ⊗ m) ◦
+�
+˜∆H ⊗ ll.πi
+�
+˜∆
+aη,
+where ll.πi = [πi πi · · · πi]t. Since deg(η) = k,
+˜∆
+aη ⊆ (W ⊗ W)k.
+Note that ll.πi(aη)(c) = [ai
+η ai
+η · · · ai
+η](c) = aη[ci ci · · · ci]. Thus ll.πiaη ∈ W and then
+applying the induction hypothesis ˜∆HWn ⊆ (W ⊗ H)n for n ≤ k,
+�
+˜∆H ⊗ ll.πi
+�
+(W ⊗ W)k ⊆ (W ⊗ H ⊗ W)k .
+Finally,
+(θi ⊗ m) (W ⊗ H ⊗ W)k ⊆ (W ⊗ H)k+1,
+as θi is homogeneous operator of degree 1. Thus, ˜∆Haη′ ∈ �
+i+j=k+1 Wi ⊗ Hj where
+|η′| = k + 1.
+Hence proved by induction that for all n ≥ 0: ˜∆H (aη) ∈ �
+i+j=n Wi ⊗ Hj where |η| = n.
+The following proposition asserts that the grading on H in Definition 8.1 is compatible
+with bialgebraic structure of H.
+Proposition 8.5. With the grading on the Hopf algebra H as in Definition 8.1,
+∆H (Hn) ⊆
+�
+i+j=n
+Hi ⊗ Hj
+for all n ≥ 0.
+Proof: Observe that the statement is true for n = 0. Prior to the proving the statement for
+n ≥ 1, the following statement needs to be proved:
+∆H (Wn) ⊆
+�
+i+j=n
+Wi ⊗ Hj
+∀ n ≥ 0.
+
+34
+VENKATESH G. S. AND KURUSCH EBRAHIMI-FARD
+Observe that from Proposition 8.2,
+∆H ◦ πi = (id ⊗ m) ◦
+�
+˜∆H ⊗ id
+�
+◦ (πi ⊗ πi) ◦ ˜∆
+.
+Thus (by grouping them along the coordinate i),
+∆H = (id ⊗ m) ◦
+�
+˜∆ ◦ id
+�
+◦ ˜∆
+.
+Hence,
+∆H(Wn) = (id ⊗ m) ◦
+�
+˜∆ ◦ id
+�
+◦ ˜∆
+(Wn)
+⊆ (id ⊗ m) ◦
+�
+˜∆ ◦ id
+�
+(W ⊗ W)n.
+Using Proposition 8.2,
+∆H(Wn) ⊆ (id ⊗ m) (W ⊗ H ⊗ W)n
+⊆ (W ⊗ H)n.
+Therefore, the intermediate statement holds true viz.
+∆H (Wn) ⊆
+�
+i+j=n
+Wi ⊗ Hj
+∀ n ≥ 0.
+The statement of the theorem then holds true as ∆ is an Rn-algebra morphism from H to
+H ⊗ H.
+Thus Proposition 8.5 asserts that the grading defined on the Hopf algebra H in Defini-
+tion 8.1 is well-defined and connected. The homogeneous components are finite-dimensional
+and dimensions respect the Proposition 4.2 since the bialgebras H and H
+are isomorphic
+with respect to the underlying graded algebraic structures.
+The following example is rework of the Example 4.10 in [Gray & Ebrahimi-Fard(2017)]
+acting as a check for the computation of feedback group inverse in one-dimensional case.
+Example 8.2. Let c = 1−x1 ∈ R⟨⟨X⟩⟩. The series c◦−1 = 1+· · ·+· · · . Using the recursive
+computation formula for antipode as in Theorem 3.1
+ax1(c◦−1) = Sax1 (c) = −ax1(c) = 1.
+Observe that
+∆′
+Hax2
+1 = 3ax1 ⊗ ax1.
+Thus,
+ax2
+1
+�
+c◦−1�
+= Sax2
+1 (c)
+= −ax2
+1 − 3ax1.Sax1 = −ax2
+1 + 3a2
+x1.
+Therefore, ax2
+1 (c◦−1) = 0 + 3(1)2 = 3. In similar fashion the reduced coproduct of a3
+x1 is
+∆′
+Hax3
+1 = 4ax1 ⊗ ax2
+1 + 6ax2
+1 ⊗ ax1 + 3ax1 ⊗ a2
+x1.
+Thus,
+ax3
+1
+�
+c◦−1�
+=
+�
+−ax3
+1 − 4ax1.Sax2
+1 − 6ax2
+1.Sax1 − 3ax1. (Sax1)2�
+(c)
+= 0 − 4(−1)(3) − 6(0)(−1) − 3(−1)(1)2 = 15.
+Therefore c◦−1 = 1 + x1 + 3x2
+1 + 15x3
+1 + 105x4
+1 + · · · .
+The result matches exactly with that of Example 4.10 in [Gray & Ebrahimi-Fard(2017)].
+
+FORMAL SERIES APPROACH TO MULTIPLICATIVE DYNAMIC FEEDBACK CONNECTION
+35
+9. Conclusions and Future work
+It was shown that the closed-loop system of a plant in Chen–Fliess series description
+in multiplicative output feedback with another system, given by Chen–Fliess series, has a
+Chen–Fliess series representation. An explicit expression of the closed-loop generating series
+was derived and the multiplicative dynamic feedback connection has a natural interpretation
+as a transformation group acting on the plant. A computational framework has been devised
+utilizing the dual Hopf algebras corresponding to the shuffle group and multiplicative output
+dynamic feedback group. Future work will be to address the solemn problem regarding the
+local convergence of the both multiplicative dynamic and static output feedback connections
+and to identify both the multiplicative dynamic and static feedback invariants.
+References
+[Abe(2004)] Abe, E., Hopf Algebras, Cambridge University Press, Cambridge, UK, 2004.
+Abramowitz, M. and Stegun, I. A, Handbook of Mathematical Functions with Formulas, Graphs, and
+Mathematical Tables, Dover Publications, New York, 1970.
+[Berstel & Reutenauer(1988)] Berstel, J. and Reutenauer, C., Rational Series and Their Languages,
+Springer-Verlag, Berlin, 1988.
+[Brockett(1978)] Brockett, R. W., Feedback Invariants for Nonlinear Systems, IFAC Proceedings Volumes,
+11 (1978) 1115–1120.
+[Duffaut Espinosa, et al.(2016)] Duffaut Espinosa, L. A., Ebrahimi-Fard, K. and Gray, W. S., A Combina-
+torial Hopf Algebra for Nonlinear Output Feedback Control Systems, Journal of Algebra, 453 (2016)
+609–643.
+[Duffaut Espinosa & Gray(2017)] Duffaut Espinosa, L. A. and Gray, W. S., Integration of Output Tracking
+and Trajectory Generation via Analytic Left Inversion, Proc. 21st Int. Conf. on System Theory, Control
+and Computing, Sinaia, Romania, 2017, pp. 802–807.
+[Ferfera(1979)] Ferfera, A., Combinatoire du Mono¨ıde Libre Appliqu´ee `a la Composition et aux Variations de
+Certaines Fonctionnelles Issues de la Th´eorie des Syst`emes, Ph.D. Dissertation, University of Bordeaux
+I, 1979.
+[Ferfera(1980)] Ferfera, A., Combinatoire du Mono¨ıde Libre et Composition de Certains Syst`emes Non
+Lin´eaires, Ast´erisque, 75-76 (1980) 87–93.
+[Fliess(1981)] Fliess, M., Fonctionnelles Causales Non Lin´eaires et Ind´etermin´ees Non Commutatives, Bul-
+letin de la Soci´et´e Math´ematique de France, 109 (1981) 3–40.
+[Fliess(1983)] Fliess, M., R´ealisation Locale des Syst`emes Non Lin´eaires, Alg`ebres de Lie Filtr´ees Transitives
+et S´eries G´en´eratrices Non Commutatives, Inventiones Mathematicae, 71 (1983) 521–537.
+[Foissy(2015)] Foissy, L., The Hopf Algebra of Fliess Operators and Its Dual Pre-Lie Algebra, Communica-
+tions in Algebra, 43 (2015) 4528–4552.
+[Gray, et al.(2014a)] Gray, W. S., Duffaut Espinosa, L. A., and Ebrahimi-Fard, K., Fa`a di Bruno Hopf
+Algebra of the Output Feedback Group for Multivariable Fliess Operators, Systems & Control Letters,
+74 (2014) 64–73.
+[Gray, et al.(2014b)] Gray, W. S., Duffaut Espinosa, L. A., and Thitsa, M., Left Inversion of Analytic Non-
+linear SISO Systems via Formal Power Series Methods, Automatica, 50 (2014) 2381–2388.
+[Gray & Ebrahimi-Fard(2017)] Gray, W. S. and Ebrahimi-Fard, K., SISO Output Affine Feedback Trans-
+formation Group and Its Fa`a di Bruno Hopf Algebra, SIAM Journal on Control and Optimization, 55
+(2017) 885–912.
+[Gray & Li(2005)] Gray, W. S. and Li, Y., Generating Series for Interconnected Analytic Nonlinear Systems,
+SIAM Journal on Control and Optimization, 44 (2005) 646–672.
+[Gray & Venkatesh(2019)] Gray, W. S. and Venkatesh, G. S., Relative Degree of Interconnected SISO Non-
+linear Control Systems, Systems & Control Letters, 124 (2019) 99–105.
+[Gray & Wang(2002)] Gray, W. S. and Wang, Y., Fliess Operators on Lp spaces: Convergence and Conti-
+nuity, Systems & Control Letters, 46 (2002) 67–74.
+[Isidori(1995)] Isidori, A., Nonlinear Control Systems, 3rd Ed., Springer-Verlag, London, 1995.
+[OEIS(2022)] OEIS Foundation Inc., The On-Line Encyclopedia of Integer Sequences, published electroni-
+cally at http://oeis.org, 2022.
+
+36
+VENKATESH G. S. AND KURUSCH EBRAHIMI-FARD
+[Ree(1958)] Ree, R., Lie Elements and an Algebra Associated with Shuffles, Annals of Mathematics (2), 68
+(1958) 210–220.
+[Sweedler(1969)] Sweedler, M. E., Hopf Algebras, Benjamin Inc., New York, 1969.
+[Thitsa & Gray(2012)] Thitsa, M. and Gray, W. S., On the Radius of Convergence of Interconnected Analytic
+Nonlinear Input-Output Systems, SIAM Journal on Control and Optimization, 50 (2012) 2786–2813.
+[Venkatesh(2021)] Venkatesh, G. S., Wiener-Fliess Composition of Formal Power Series: Additive Static
+Feedback and Shuffle Rational Series, Ph.D. Dissertation, Old Dominion University, 2021.
+[Venkatesh & Gray(2022)] Venkatesh
+G.
+S.,
+Gray,
+W.
+S.,
+Formal
+Power
+Series
+Approach
+to
+Nonlinear
+Systems
+with
+Additive
+Static
+Feedback,
+International
+Journal
+of
+Control,
+https://doi.org/10.1080/00207179.2022.2059013 (appeared online).
+[Venkatesh & Gray(2021)] Venkatesh G. S., Gray, W. S., Formal Power Series Approach to Nonlinear Sys-
+tems with Static Output Feedback, Proc. 24th Int. Symp. on Mathematical Theory of Networks and
+Systems, Cambridge, UK, 2021, pp. 192–198.
+[Venkatesh & Gray (2020)] Venkatesh G. S. and Gray, W. S., Shuffle-Rational Series: Recognizability and
+Realizations, Proc. 24th Int. Conf. on System Theory, Control and Computing, Sinaia, Romania, 2020,
+pp. 404–411.
+[Winter-Arboleda(2019)] Winter-Arboleda, I. M., On Analytic Nonlinear Input-output Systems: Expanded
+Global Convergence and System Interconnections, Ph.D. Dissertation, Old Dominion University, 2019.
+[Winter-Arboleda, et al.(2015)] Winter-Arboleda, I. M., Gray, W. S. and Duffaut Espinosa, L. A., Frac-
+tional Fliess Operators: Two Approaches, Proc. 49th Conference on Information Sciences and Systems,
+Baltimore, MD, 2015, pp. 1–6
+Department of Mathematical Sciences, Norwegian University of Science and Technology
+(NTNU), 7491 Trondheim, Norway
+Email address: subbarao.v.guggilam@ntnu.no
+Department of Mathematical Sciences, Norwegian University of Science and Technology
+(NTNU), 7491 Trondheim, Norway
+Email address: kurusch.ebrahimi-fard@ntnu.no
+URL: https://folk.ntnu.no/kurusche/
+

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@@ -0,0 +1,2493 @@
+arXiv:2301.00164v1  [eess.SP]  31 Dec 2022
+1
+Design of a Multi-User Wireless Powered
+Communication System Employing Either
+Active IRS or AF Relay
+Omid Rezaei, Maryam Masjedi, Ali Kanaani, Mohammad Mahdi Naghsh∗,
+Saeed Gazor, and Mohammad Mahdi Nayebi
+Abstract
+In this paper, we optimize a Wireless Powered Communication (WPC) system including multiple
+pair of users, where transmitters employ single-antenna to transmit their information and power to their
+receivers with the help of one multiple-antennas Amplify-and-Forward (AF) relay or an active Intelligent
+Reflecting Surface (IRS). We propose a joint Time Switching (TS) scheme in which transmitters,
+receivers, and the relay/IRS are either in their energy or information transmission/reception modes.
+The transmitted multi-carrier unmodulated and modulated waveforms are used for Energy Harvesting
+(EH) and Information Decoding (ID) modes, respectively. In order to design an optimal fair system, we
+maximize the minimum rate of all pairs for both relay and IRS systems through a unified framework.
+This framework allows us to simultaneously design energy waveforms, find optimal relay/IRS amplifi-
+cation/reflection matrices, allocate powers for information waveforms, and allocate time durations for
+various phases. In addition, we take into account the non-linearity of the EH circuits in our problem. This
+problem turns out to be non-convex. Thus, we propose an iterative algorithm by using the Minorization-
+Maximization (MM) technique, which quickly converges to the optimal solution. Numerical examples
+show that the proposed method improves the performance of the multi-pair WPC relay/IRS system
+under various setups.
+O. Rezaei and M. M. Nayebi are with the Department of Electrical Engineering, Sharif University of Technology, Tehran,
+11155-4363, Iran. M. Masjedi, A. Kanaani, and M. M. Naghsh are with the Department of Electrical and Computer Engineering,
+Isfahan University of Technology, Isfahan, 84156-83111, Iran. S. Gazor is with the Department of Electrical and Computer
+Engineering, Queen’s University, Kingston, Ontario, K7L 3N6, Canada.
+∗Please address all the correspondence to M. M. Naghsh, Phone: (+98) 31-33912450; Fax: (+98) 31-33912451; Email:
+mm naghsh@cc.iut.ac.ir
+January 3, 2023
+DRAFT
+
+2
+Index Terms
+Fair Throughput Maximization, Intelligent Reflecting Surface (IRS), Minorization-Maximization
+(MM), Relay Networks, Wireless Powered Communication (WPC).
+I. INTRODUCTION
+Wireless Power Transfer (WPT) technology is introduced to extend the lifetime of devices
+in wireless networks in which the energy is emitted from the dedicated power sources to the
+devices [1]. Interestingly, WPT enables Simultaneous Wireless Information and Power Transfer
+(SWIPT) [2], where devices not only receive and decode information, but also harvest the
+energy from Radio Frequency (RF) signals. The Time Switching (TS) and Power Splitting
+(PS) schemes are two well-known implementing protocols of SWIPT [3]. Recently, SWIPT
+models are designed to employ relays to further enhance the coverage and spectral efficiency
+of wireless networks [4]–[6]. In [4], a Multiple-Input Multiple-Output (MIMO) two-way relay
+system is introduced in which two transceivers exchange their information through a relay. The
+authors in [5] designed the relay and source precoders by minimizing the bit error rate at the
+destination for a full-duplex MIMO relay system and SWIPT-enabled users. A similar system
+with a half-duplex two-way relay is designed in [6] by minimizing the mean square error at the
+destination.
+Another impactful technology that is currently emerging is to use of the Intelligent Reflecting
+Surface (IRS) in wireless communication systems. This promising solution not only is capable
+of improving energy delivery but also can enhance the spectral efficiency of future wireless
+communication networks. [7]. An IRS is an array of large number of Reflecting Elements (REs)
+designed to have controllable electromagnetic properties. Each RE introduces a phase shift on
+the impinging signal, allowing beamforming/manipulation of the reflection waveforms. Precisely,
+the IRS matrix allows controlling the reflected signal (amplify, attenuated, steer in the desired
+direction, and so on) toward optimal desired directions by purposefully designing the phase
+shift matrix. The IRS is exploited recently in SWIPT systems in [8]–[11]. The weighted sum
+harvested power maximization problem was studied in [8] for an IRS-aided SWIPT model in
+which a multiple-antennas access point serves multiple single-antenna users. In [9], the model
+in [8] is extended for a more practical multi-objective optimization problem by taking into
+account the trade-off between sum rate and sum harvested power maximization. In [10], the
+total transmission power is minimized for a Multiple-Input Single-Output (MISO) SWIPT system
+January 3, 2023
+DRAFT
+
+3
+employing multiple IRSs. In [11], the MISO SWIPT in [8]–[10] is extended to the MIMO case,
+and the weighted sum rate is maximized in an IRS-assisted system.
+The design of the energy waveform remarkably affects the performance of WPT-based systems.
+Indeed, an efficient waveform leads to significant improvement in the efficacy of power delivery.
+Experiments reveal that signals with high Peak to Average Power Ratio (PAPR) such as multi-
+sine signals provide more DC power at Energy Harvesting (EH) circuits than constant envelope
+signals with the same average RF power [12]. Based on this interesting observation, a multi-sine
+waveform design for WPT has been examined in several works [13]–[21]. Waveform design
+with a non-linear EH model was considered in [13] and [14] for MISO and Single-Input Single-
+Output (SISO) systems, respectively. The authors of [15] proposed a low-complexity method for
+a waveform design in a SISO WPT system. In [16], the transmit waveform was designed based
+on limited Channel State Information (CSI) feedback WPT system. Then, the authors in [17]
+studied waveform design for an IRS-aided SWIPT MISO system. The aforementioned methods
+for design of single-user systems were extended to the multi-user case in [18]. Also, a waveform
+design was performed in [19] for wireless powered backscatter communication networks, and this
+work was extended to multi-user backscatter systems in [20]. The authors of [21] investigated
+the waveform and transceiver design problem in a MISO SWIPT system and determined the
+multi-sine waveforms for Information Decoding (ID) and EH phases.
+In this paper, we optimize a multi-user wireless powered relay/IRS system using a multi-sine
+waveform with the following main contributions:
+• Relay model: To the best of our knowledge, a multi-sine waveform design for multiple
+user pairs in a wireless powered relay system has not been addressed in the literature. In
+this paper, we consider a multi-carrier Wireless Powered Communication (WPC) system
+for multi-user relay channels. Precisely, in our proposed model, an amplify-and-forward
+(AF) relay provides energy/information transmission from K transmitters to their receivers
+by adopting the TS scheme in all nodes1. Herein, the aim is to design the multi-carrier
+unmodulated energy waveforms and the allocated power for information waveforms at the
+transmitters, the amplification matrices in a relay, and the time durations for the EH and ID
+modes in order to maximize the minimum rate of the user pairs. In addition, we consider
+1Note that the system in [22], where a TS scheme is only applied for a receiver, is considered as a special case of the proposed
+joint TS scheme.
+January 3, 2023
+DRAFT
+
+4
+the effect of the non-linearity of EH circuits in the design problem.
+• IRS model: In the case of IRS-assisted communication, multi-pair WPC has not been
+considered in the literature, and therefore, herein, we consider this type of IRS-assisted
+systems. Precisely, in this case, an active IRS2 replaced with the AF relay in the proposed
+system model mentioned in the above paragraph. Also, some comparisons are made between
+relay and IRS models in terms of architecture and performance (see Remark 1-2).
+• Unified consideration of relay and IRS: Both proposed AF relay and active IRS-aided
+systems are modeled under a unified formulation, and we handle the resulting optimization
+problems under a unified mathematical umbrella. We show that the problem is non-convex
+and consequently, is hard to solve. To deal with this problem, we devise a method based on
+the Minorization-Maximization (MM) technique. Interestingly, the proposed algorithm can
+deal with relay and IRS systems by switching between Kronecker and Hadamard products
+for parameters used in the algorithm (see Lemma 2).
+• Sub-optimal methods: Some sub-optimal methods with lower signaling overhead and com-
+putational complexity are proposed and then, their performance are compared.
+• Numerical result: Simulation results are reported to illustrate the effectiveness of the pro-
+posed method; particularly, the impact of the relay/IRS matrix and energy waveform design.
+Also, numerical examples show that the minimum rate of users increases linearly/super-
+linearly with the number of antennas/REs in relay/IRS systems.
+The rest of this paper is organized as follows: The signal and system models are explained
+in Section II. In Section III, the minimum rate maximization problem is formulated, and a
+unified optimization framework is proposed for both relay and IRS models. Section IV presents
+numerical examples to illustrate the effectiveness of the proposed method. Finally, conclusions
+are drawn in Section V.
+Notation: Bold lowercase (uppercase) letters are used for vectors (matrices). The notations
+arg(·), E[·], ℜ{·}, ∥·∥2, (·)T, (·)H, (·)∗, tr{·}, λmax(·), vec(·), Diag(·), ∇xf(·) and ∇2
+xf(·) indicate
+the phase of a complex number, statistical expectation, real-part, l2-norm of a vector, transpose,
+Hermitian, complex conjugate, trace of a matrix, the principal eigenvalue of a matrix, stacking
+of the column of a matrix, a diagonal matrix formed by the entries, the gradient of a function
+with respect to (w.r.t.) x and the Hessian of a function w.r.t. x, respectively. The symbols ⊗ and
+2Note that in an active IRS, REs can amplify the reflected signals using their reflection-type amplifiers [23].
+January 3, 2023
+DRAFT
+
+5
+T1 ...
+τ
+T − τ
+Information
+Transmitter
+Energy
+Transmitter
+Tk ...
+TK
+AF Relay or Active IRS
+R1
+...
+Rk
+...
+τ
+T − τ
+Information
+Decoder
+Energy
+Harvester
+RK
+Blockage
+Fig. 1. Multi-user wireless powered relay/IRS system based on TS scheme with blocked direct path.
+⊙ stand for the Kronecker and Hadamard products of two matrices. We denote CN (ω, Σ) as
+a circularly symmetric complex Gaussian (CSCG) distribution with mean ω and covariance Σ.
+The set R+ represents non-negative real numbers and CN×N and DN×N are the set of N × N
+complex and complex diagonal matrices, respectively. The set of N × N positive (semi-)definite
+and identity matrices are denoted by SN
+++ ⊂ CN×N (SN
++ ⊂ CN×N) and IN, respectively. The
+notation A ≻ B (A ⪰ B) means that A − B is positive (semi-)definite.
+II. SYSTEM MODEL
+We consider a multi-carrier wireless powered relay/IRS system with K user pairs {(Tk, Rk)}K
+k=1
+as shown in Fig. 1, where the direct links between the transmitters and receivers are likely blocked
+(see [24] and [25], [26] for similar models of multiple user pairs with blocked direct path for
+relay and IRS systems, respectively). The single-antenna transmitter Tk communicates with its
+receiver Rk through either an AF relay with MR antennas or an active IRS with MIRS REs.
+We assume that Rk harvests a part of its required power, whereas Tk and the relay/IRS have
+no energy concern [4], [5]. In each time duration T, the relay/IRS helps Rk not only harvest
+energy from the signal of all {Tk}K
+k=1, but also decode the information from its corresponding
+transmitter Tk by using a joint TS scheme. Precisely, Tk, relay/IRS, and Rk switch simultaneously
+at time t = τ from their energy delivery modes to their communication modes. We assume that all
+nodes are perfectly synchronized as shown in Fig. 2 for this switching [22]. We consider a multi-
+carrier system with a total bandwidth of Bt equally divided into N orthogonal subbands. We
+also model all channels to have a frequency-selective block fading, i.e., the channel coefficients
+January 3, 2023
+DRAFT
+
+6
+Relay:
+IRS:
+{Tk}K
+k=1 → IRS → {Rk}K
+k=1
+{Tk}K
+k=1 → relay relay → {Rk}K
+k=1
+{Tk}K
+k=1 → IRS → {Rk}K
+k=1
+{Tk}K
+k=1 → relay
+relay → {Rk}K
+k=1
+τ
+2
+τ (energy waveform)
+τ
+2
+T −τ
+2
+T − τ (information waveform)
+T −τ
+2
+Fig. 2. The transmission, amplification/reflection, and reception timeline for the proposed relay/IRS model.
+remain constant for at least T seconds. Let the complex random matrices HR
+n ∈ CMR×K and
+GR
+n ∈ CK×MR denote the channels from transmitters to the relay and the channels from the
+relay to the receivers for nth subband, respectively. The elements of HR
+n and GR
+n are zero mean
+CSCG random variables in the case of Rayleigh fading. Similarly, we denote the channels from
+transmitters to the IRS and the channels from the IRS to the receivers by HIRS
+n
+∈ CMIRS×K
+and GIRS
+n
+∈ CK×MIRS, respectively. In the sequel, Hn and Gn refer to either HR
+n and GR
+n or
+HIRS
+n
+and GIRS
+n , depending on the case under discussion. In addition, we assume that we can
+control the relay and IRS by collecting and using the CSI of all links [25]–[27]. For example,
+a relay itself can act as a controller. The CSI may be estimated in various ways, e.g., by using
+orthogonal pilot sequences ( see [28], [29] for more details). The CSI estimation is out of the
+scope of this paper. Also, we propose two low-complexity implementation methods mentioned
+in Remark 3 to reduce the signaling overhead in the controller node.
+Each Tk transmits a multi-sine energy waveform xE,k(t) and a multi-carrier modulated infor-
+mation waveform xI,k(t) to the relay/IRS during the first-hop transmission at the EH and ID
+time slots, respectively, as follows
+xE,k(t) =
+N
+�
+n=1
+aE,k,ncos(2πfnt + φE,k,n), = ℜ
+� N
+�
+n=1
+sE,k,nej2πfnt
+�
+,
+(1)
+xI,k(t) =
+N
+�
+n=1
+aI,k,n(τ)cos(2πfnt + φI,k,n) = ℜ
+� N
+�
+n=1
+sI,k,nej2πfnt
+�
+,
+(2)
+where sE,k,n = aE,k,nejφE,k,n and sI,k,n = aI,k,nejφI,k,n are the baseband complex signal represen-
+tations for the energy and information waveforms, respectively. We assume that the baseband
+information signals are i.i.d. CSCG random variable variables, i.e., sI,k,n ∼ CN (0, pI,k,n). The
+transmitted energy by Tk is constrained by
+τ
+2ρ|sE,k,n|2 + T − τ
+2ρ
+pI,k,n ≤ Tprf
+k,n, ∀k, n,
+(3)
+January 3, 2023
+DRAFT
+
+7
+where prf
+k,n is the maximum power budget at Tk for the nth subband and ρ addresses both ρR = 2
+for relay and ρIRS = 1 for IRS system according to the proposed timeline in Fig. 2 (see also
+Remark 1). By defining sE,n = [sE,1,n, · · · , sE,K,n]T and sI,n = [sI,1,n, · · · , sI,K,n]T, the received
+signal at the relay/IRS is expressed as
+r(t) =
+
+
+
+
+
+
+
+N�
+n=1
+{HnsE,n + zn}, t ∈ TEH, for EH,
+N�
+n=1
+{HnsI,n + zn}, t ∈ TID, for ID,
+(4)
+where
+TEH =
+
+
+
+
+
+0 ≤ t ≤ τ
+2, for relay,
+0 ≤ t ≤ τ, for IRS,
+TID =
+
+
+
+
+
+τ ≤ t ≤ τ + T−τ
+2 , for relay,
+τ ≤ t ≤ T, for IRS,
+(5)
+and r denotes either rR or rIRS. Furthermore, the AWGN zn denotes either zR
+n ∼ CN (0, σ2
+R,nIMR)
+or zIRS
+n
+∼ CN (0, σ2
+IRS,nIMIRS) for relaying or reflecting modes. In contrast to the passive IRS,
+an active IRS adds non-negligible noise (which is introduced by the active elements [23], [30]);
+however, the added noise of an active IRS has considerably less impact compared to the relay
+noise (which is introduced by RF chains), i.e., σ2
+IRS,n ≤ σ2
+R,n [31].
+In the second-hop transmission, the relay/IRS amplifies the energy and information signals of
+Tk by amplification/reflection matrices and then forwards them to Rk. For AF relay system, the
+amplification matrices is introduced as UR
+E,n and UR
+I,n ∈ CMR×MR, ∀n for energy and information
+phases, respectively. In the case of IRS-aided system, the reflection matrices is defined as UIRS
+E
+=
+Diag(θE) and UIRS
+I
+= Diag(θI) for energy and information time slots, respectively, where θE =
+[ηE,1ejθE,1, ηE,2ejθE,2, · · · , ηE,MIRSejθE,MIRS]T and θI = [ηI,1ejθI,1, ηI,2ejθI,2, · · · , ηI,MIRSejθI,MIRS]T
+with ηE,m, ηI,m ≥ 1 and θE,m, θI,m ∈ [0, 2π] respectively denote the reflection amplitude and
+the phase shift at the mth RE3.
+Remark 1. An active IRS amplifies the signal without any significant delay. However, in an
+AF relay, the signal reception, amplification, and transmission at the RF chain cause a long
+delay. Therefore, in practice, the AF relay requires twice time compared to the active IRS for
+transmission one information symbol [23].
+3Note that passive and passive lossless IRS require ηE,m, ηI,m ∈ [0, 1] and ηE,m = ηI,m = 1, respectively.
+January 3, 2023
+DRAFT
+
+8
+We define UE,n and UI,n to address both UR
+E,n, UIRS
+E
+and UR
+I,n, UIRS
+I
+, respectively. The
+forwarded signal by the relay/IRS is given by
+�r(t)=
+
+
+
+
+
+�N
+n=1 UE,n (HnsE,n + zn) , t ∈ �TEH, for EH,
+�N
+n=1 UI,n (HnsI,n + zn) , t ∈ �TID, for ID,
+where
+�TEH =
+
+
+
+
+
+τ
+2 ≤ t ≤ τ, for relay,
+0 ≤ t ≤ τ, for IRS,
+�TID =
+
+
+
+
+
+τ + T−τ
+2
+≤ t ≤ T, for relay,
+τ ≤ t ≤ T, for IRS,
+(6)
+and �r denotes either �rR or �rIRS for relay or IRS system, with a slight abuse of notation.
+Then, the power of �r(t) from the relay/IRS is written as
+E
+�
+∥�r(t)∥2
+2
+�
+=
+
+
+
+
+
+
+
+1
+2
+N�
+n=1
+�
+sH
+E,nVE,nsE,n + σ2
+ntr
+�
+UE,nUH
+E,n
+��
+, for EH,
+1
+2
+N�
+n=1
+�
+tr {QI,nVI,n} + σ2
+ntr
+�
+UI,nUH
+I,n
+��
+, for ID,
+(7)
+where σ2
+n addresses both σ2
+R,n and σ2
+IRS,n, QI,n = Diag(pI,1,n, pI,2,n, · · · , pI,K,n) and
+VE,n = HH
+n UH
+E,nUE,nHn, ∀n,
+VI,n = HH
+n UH
+I,nUI,nHn, ∀n.
+(8)
+Using (7), the total consumed energy is bounded at the relay/IRS in t ∈ [0, T] as
+τ
+2ρ
+�
+sH
+E,nVE,nsE,n + σ2
+ntr{UE,nUH
+E,n}
+�
++ T − τ
+2ρ
+�
+tr{QI,nVI,n} + σ2
+ntr{UI,nUH
+I,n}
+�
+≤ Tprf
+n , ∀n, (9)
+where prf
+n denotes either the maximum power budget at the relay prf
+R,n or IRS prf
+IRS. We can write
+received signal at Rk as
+yk(t) =
+
+
+
+
+
+
+
+N�
+n=1
+�
+gT
+k,nUE,n (HnsE,n + zn) + zk,n
+�
+, t ∈ �TEH, ∀k, for EH,
+N�
+n=1
+�
+gT
+k,nUI,n (HnsI,n + zn) + zk,n + �zk,n
+�
+, t ∈ �TID, ∀k, for ID,
+where gk,n is the kth column vector of GT
+n, and zk,n as well as �zk,n are the AWGN from
+the antenna and baseband processing noises at Rk, respectively, with zk,n ∼ CN(0, σ2
+k,n) and
+�zk,n ∼ CN (0, δ2
+k,n). The information signals at Rk corresponding to the nth subband can be
+expanded as
+yk,n(t) =gT
+k,nUI,nhk,nsI,k,n + gT
+k,nUI,n
+K
+�
+j=1,j̸=k
+hj,nsI,j,n + gT
+k,nUI,nzn + zk,n + �zk,n, ∀k, n,
+(10)
+January 3, 2023
+DRAFT
+
+9
+where hk,n and gk,n are the kth column vector of Hn and GT
+n, respectively. By defining pI,n =
+[pI,1,n, pI,2,n, · · · , pI,K,n]T, the SINR at the ID part for the nth subband is given by
+γk,n(pI,n, UI,n) =
+pI,k,nψk,k,n
+�K
+j=1,j̸=k pI,j,nψk,j,n + σ2n �ψk,n + δ2
+k,n + σ2
+k,n
+, ∀k, n,
+(11)
+where ψk,j,n = gT
+k,nUI,nhj,nhH
+j,nUH
+I,ng∗
+k,n and �ψk,n = gT
+k,nUI,nUH
+I,ng∗
+k,n. From Remark 1, we obtain
+the achievable rate at the kth pair as follows
+Rk
+�
+{pI,n}N
+n=1, {UI,n}N
+n=1, τ
+�
+= T − τ
+ρT
+N
+�
+n=1
+log2
+�
+1 + γk,n(pI,n, UI,n)
+�
+.
+(12)
+For the EH stream, we assume the noise power is negligible compared to the received signal
+power. We take into account the rectifier non-linearity by employing the results from [32] where
+the harvested energy at Rk is approximated by
+Ek
+�
+{sE,n}N
+n=1, {UE,n}N
+n=1, τ
+�
+= τ
+ρexp
+�
+�alog2pE,k
+�
+p
+�b
+E,kexp�c, ∀k,
+(13)
+where �a, �b, and �c are the curve fitting constants and pE,k is the average input power to Rk’s
+harvester as
+pE,k
+�
+{sE,n}N
+n=1, {UE,n}N
+n=1
+�
+= 1
+2
+N
+�
+n=1
+sH
+E,nΞk,nsE,n, ∀k,
+(14)
+with
+Ξk,n = HH
+n UH
+E,ng∗
+k,ngT
+k,nUE,nHn, ∀k, n.
+(15)
+Remark 2. Note that the reflection matrix cannot be designed separately for each subband in the
+IRS system, while, thanks to the RF chain circuits in a relay, the amplification matrix design is
+considered for each subband. We note that an active IRS is considerably less expensive than an
+AF relay. This is because an AF relay requires massive integrated circuits (including analog-to-
+digital/digital-to-analog converter, self-interference cancellation circuits, etc). The delay caused
+by RF chain processing of an AF relay contributes to latency, leads to lower transmission time,
+and requires more power for energy and information signals (see (3) and (9)). Therefore, a
+relay-IRS trade-off exists in the system performance (see (12) and (13)).
+Remark 3. An approach with lower implementation complexity is considered in which only one
+amplification/reflection matrix needs to be designed for both energy and information time slots,
+called the t-static approach. Also, one can consider another approach with only one amplification
+matrix design in both time slots and all subbands, referred to as t-f-static in the relay system.
+These design methodologies lead to a lower signaling overhead and system performance.
+January 3, 2023
+DRAFT
+
+10
+III. THE PROPOSED MINIMUM RATE MAXIMIZATION METHOD
+In this section, the aim is to maximize the minimum rate of the multi-user relay/IRS WPC
+system w.r.t. multi-sine energy waveforms sE,n, allocated power pI,n, amplification/reflection
+matrices UE,n, UI,n, and the time allocation parameter τ. The unified minimum rate maximization
+problem for both relay and IRS systems is cast as
+max
+τ,{sE,n,pI,n,UE,n,UI,n}N
+n=1
+min
+1≤k≤K
+Rk
+(16)
+s.t.
+�
+τ, {sE,n, pI,n, UE,n, UI,n}N
+n=1
+�
+∈ Ω,
+where Ω = Ω0 ∩ Ωind with
+Ω0 =
+�
+C1 : 0 ≤ τ ≤ T, C2 : (3), pI,k,n ≥ 0, ∀k, n, C3 : (9), C4 : Ek ≥ Emin,k, ∀k
+�
+,
+(17)
+Ωind =
+
+
+
+
+
+CR : UE,n, UI,n ∈ CMR×MR, ∀n, for relay,
+CIRS : UE,n, UI,n ∈ DMIRS×MIRS, ∀n, |θE,m| ≥ 1, |θI,m| ≥ 1, ∀m, for IRS,
+(18)
+and Emin,k in C4 is the minimum required harvested energy for the kth user.
+The problem in (16) is non-convex due to the coupled design variables in the objective function
+and the constraints C2 − C4 and CIRS. To deal with this non-convex problem, we first solve
+the problem w.r.t. {UE,n, UI,n} for fixed {sE,n, pI,n, τ}, then optimize {sE,n, pI,n} for given
+{UE,n, UI,n, τ}, and finally, solve the problem w.r.t. τ via a closed-form solution. The procedure
+is repeated until convergence.
+A. Maximization over {UE,n, UI,n}
+Here, we first consider the relay problem, and then the IRS problem is investigated.
+1) Relay System: The problem in (16) for fixed {sE,n, pI,n} reduces to the following opti-
+mization
+max
+{UE,UI}N
+n=1
+min
+1≤k≤K
+N
+�
+n=1
+log2 (1 + γk,n(UI,n))
+(19)
+s.t.
+C3, C4,
+January 3, 2023
+DRAFT
+
+11
+which is still a non-convex problem. To start solving the problem, first we need to reformulate
+the obtained expressions for the relay power constraint (7), SINR (11), and the input power of
+harvesters (14) from Section II. We can rewrite (7) as (see Appendix A for the derivation)
+E
+�
+∥�r(t)∥2
+2
+�
+=
+
+
+
+
+
+
+
+1
+2
+N�
+n=1
+uH
+E,n �AR
+E,nuE,n, 0 ≤ t ≤ τ, for EH,
+1
+2
+N�
+n=1
+uH
+I,n �AR
+I,nuI,n, τ ≤ t ≤ T, for ID,
+(20)
+where uE,n = vec(UE,n), uI,n = vec(UI,n), and
+�AR
+E,n =
+�
+HnsE,nsH
+E,nHH
+n
+�T ⊗ IMR + σ2
+nIM2
+R,
+�AR
+I,n =
+�
+HnQI,nHH
+n
+�T ⊗ IMR + σ2
+nIM2
+R.
+(21)
+Therefore, we rewrite the relay power constraint in (9) by using (20) as
+τ
+2ρuH
+E,n �AR
+E,nuE,n + T − τ
+2ρ
+uH
+I,n �AR
+I,nuI,n ≤ Tprf
+R,n, ∀n.
+(22)
+Next, we rewrite the SINR and the input power at Rk’s harvester in (11) and (14) as
+γk,n(uI,n) =
+uH
+I,nAR
+k,nuI,n
+uH
+I,n �AR
+k,nuI,n + δ2
+k,n + σ2
+k,n
+, ∀k, n,
+(23)
+pE,k
+�
+{uE,n}N
+n=1
+�
+= 1
+2
+N
+�
+n=1
+uH
+E,n ¯AR
+k,nuE,n, ∀k,
+(24)
+where
+AR
+k,n = pI,k,n
+�
+hk,nhH
+k,n
+�T ⊗ g∗
+k,ngT
+k,n,
+(25)
+�AR
+k,n=
+K
+�
+j=1,j̸=k
+pI,j,n
+�
+hj,nhH
+j,n
+�T ⊗ g∗
+k,ngT
+k,n + σ2
+nIMR ⊗ g∗
+k,ngT
+k,n,
+(26)
+¯AR
+k,n =
+�
+HnsE,nsH
+E,nHH
+n
+�T ⊗ g∗
+k,ngT
+k,n.
+(27)
+By using (22), (23), and (24) with an auxiliary variable αa the optimization problem in (19) can
+be equivalently rewritten as
+max
+αa,{uE,uI}N
+n=1
+αa
+(28)
+s.t.
+C3 : (22), C4 : Ek
+�
+{uE,n}N
+n=1
+�
+≥ Emin,k, ∀k,
+C5 :
+N
+�
+n=1
+log2
+�
+1 +
+uH
+I,nAR
+k,nuI,n
+uH
+I,n �AR
+k,nuI,n + ζk,n,a
+�
+≥ αa, ∀k,
+where ζk,n,a = σ2
+k,n + δ2
+k,n. The constraint C5 can be equivalently rewritten as
+C5 :
+N
+�
+n=1
+�
+log2
+�
+uH
+I,nBk,nuI,n + ζk,n,a
+�
+− log2
+�
+uH
+I,n �AR
+k,nuI,n + ζk,n,a
+� �
+≥ αa,
+(29)
+January 3, 2023
+DRAFT
+
+12
+where Bk,n = �AR
+k,n + AR
+k,n. It is observed that this constraint is non-convex. Therefore, we
+employ the MM technique to tackle its non-convexity. Precisely, we minorize the denominator
+term − log2
+�
+uH
+I,n �AR
+k,nuI,n +ζk,n,b
+�
+by the using the following inequality
+log2(x) ≤ log2(x0) + log2 e
+x0
+(x − x0).
+(30)
+By setting x = uH
+I,n �AR
+k,nuI,n + ζk,n,a and x0 =
+�
+u0
+I,n
+�H �AR
+k,nu0
+I,n + ζk,n,a in (30) we obtain
+− log2
+�
+uH
+I,n �AR
+k,nuI,n + ζk,n,a
+�
+≥ − log2
+��
+u0
+I,n
+�H �AR
+k,nu0
+I,n + ζk,n,a
+�
+(31)
+−
+log2 e
+�
+uH
+I,n �AR
+k,nuI,n −
+�
+u0
+I,n
+�H �AR
+k,nu0
+I,n
+�
+�
+u0
+I,n
+�H �AR
+k,nu0
+I,n + ζk,n,a
+.
+Applying the above minorizer, the constraint C5 in (29) is rewritten at the ith iteration of the
+MM technique as
+N
+�
+n=1
+�
+log2
+�
+uH
+I,nBk,nuI,n + ζk,n,a
+�
+− log2
+��
+u(i−1)
+I,n
+�H �AR
+k,nu(i−1)
+I,n
++ ζk,n,a
+�
+(32)
+−
+log2 e
+�
+u(i−1)
+I,n
+�H �AR
+k,nu(i−1)
+I,n
++ ζk,n,a
+�
+uH
+I,n �AR
+k,nuI,n −
+�
+u(i−1)
+I,n
+�H �AR
+k,nu(i−1)
+I,n
+� �
+≥ αa.
+The following lemma lays the ground for dealing with the first non-concave logarithmic term
+in (32) in light of the MM technique.
+Lemma 1. Let s(x) = − log2
+�
+xHTx + ν
+�
+and xHQx ≤ P for any positive-definite matrices
+T, Q ∈ SN
+++ and P ∈ R+. Then, s(x) is bounded for all x and x0 as follows
+s(x) ≤ s(x0) + ℜ
+�
+bH(x − x0)
+�
++ (x − x0)HD(x − x0),
+where b =
+−2 log2 e
+xH
+0 Tx0+νTx0, D =
+4P
+wH
+1 Qw1IM2
+R, and w1 is the principal eigenvector of T and ǫ > 0.
+Proof. See Appendix B.
+Using Lemma 1 and noting that the term τ
+2uH
+E,n �AR
+E,nuE,n in (22) is positive, we obtain the
+following minorizer for the term log2(uH
+I,nBk,nuI,n + ζk,n,a) in (32) at any given u0
+I,n
+log2(uH
+I,nBk,nuI,n + ζk,n,a) ≥ log2
+��
+u0
+I,n
+�H Bk,nu0
+I,n + ζk,n,a
+�
+− ℜ
+�
+bH
+k,n(uI,n − u0
+I,n)
+�
+(33)
+−
+�
+uI,n − u0
+I,n
+�H Dk,n(uI,n − u0
+I,n),
+where
+bk,n =
+−2 log2 e
+�
+u0
+I,n
+�H Bk,nu0
+I,n + ζk,n,a
+Bk,nu0
+I,n,
+Dk,n =
+16T
+T−τ prf
+R,n
+�wH
+k,n �AR
+I,n �wk,n
+IM2
+R,
+January 3, 2023
+DRAFT
+
+13
+and �wk,n denotes the principal eigenvector of Bk,n. Applying (33), the constraint in (32) is
+restated as
+−
+N
+�
+n=1
+�
+log2 e uH
+I,n �AR
+k,nuI,n
+�
+u(i−1)
+I,n
+�H �AR
+k,nu(i−1)
+I,n
++ ζk,n,a
++ uH
+I,nDk,nuI,n + ℜ
+��
+bk,n − 2Dk,nu(i−1)
+I,n
+�H
+uI,n
+�
++ d(i)
+k,n
+�
+≥ αa, ∀k,
+(34)
+where
+d(i)
+k,n = log2
+�
+u(i−1)
+I,n
+�H �AR
+k,nu(i−1)
+I,n
++ ζk,n,a
+�
+u(i−1)
+I,n
+�H
+Bk,nu(i−1)
+I,n
++ ζk,n,a
+− ℜ
+�
+bH
+k,nu(i−1)
+I,n
+�
++
+�
+u(i−1)
+I,n
+�H
+Dk,nu(i−1)
+I,n
+(35)
+−
+log2 e
+�
+u(i−1)
+I,n
+�H �AR
+k,nu(i−1)
+I,n
+�
+u(i−1)
+I,n
+�H �AR
+k,nu(i−1)
+I,n
++ ζk,n,a
+.
+Then, we can simplify constraint in (34) as
+−
+N
+�
+n=1
+�
+uH
+I,nF(i)
+k,nuI,n + ℜ
+�
+(f(i)
+k,n)HuI,n
+�
++ d(i)
+k,n
+�
+≥ αa, ∀k,
+(36)
+where
+F(i)
+k,n =
+log2 e �AR
+k,n
+�
+u(i−1)
+I,n
+�H �AR
+k,nu(i−1)
+I,n
++ ζk,n,a
++ Dk,n,
+f(i)
+k,n = bk,n − 2Dk,nu(i−1)
+I,n
+.
+(37)
+Finally, we focus on the constraint C4. From (13) and (24), we see that in the left-hand side
+(LHS) of C4, Ek is neither convex nor concave w.r.t. uE,n. To apply the MM technique on LHS
+of C4, we first define a parameter4 βk,n,a such that ∇2
+uE,nEk
+�
+{uE,n}N
+n=1
+�
++βk,n,aIM2
+R ⪰ 0, ∀k, n,
+and write Ek as the sum of a convex and a concave function as
+Ek
+�
+{uE,n}N
+n=1
+�
+=Ek
+�
+{uE,n}N
+n=1
+�
++ 1
+2
+N
+�
+n=1
+βk,n,auH
+E,nuE,n − 1
+2
+N
+�
+n=1
+βk,n,auH
+E,nuE,n, ∀k.
+(38)
+We now apply the MM technique to C4 and obtain a convex constraint. To do so, we keep the
+concave part and minorize the convex part of (38) and rewrite C4 as
+Ek
+�
+{u(i−1)
+E,n }N
+n=1
+�
++ 1
+2
+N
+�
+n=1
+βk,n,a
+�
+u(i−1)
+E,n
+�H
+u(i−1)
+E,n
++
+N
+�
+n=1
+ℜ
+�
+ϑ(i)
+k,n,a
+�
+uE,n − u(i−1)
+E,n
+��
+(39)
+− 1
+2
+N
+�
+n=1
+βk,n,auH
+E,nuE,n ≥ Emin,k, ∀k,
+4See Appendix C for a selection of βk,n,a.
+January 3, 2023
+DRAFT
+
+14
+where
+ϑ(i)
+k,n,a =βk,n,a
+�
+u(i−1)
+E,n
+�H
++ τexp�c
+2
+exp
+�
+�alog2ω(i)
+k,a
+� �
+ω(i)
+k,a
+��b−1 �
+2�a log ω(i)
+k,a + �b
+� �
+u(i−1)
+E,n
+�H ¯AR
+k,n,
+with ω(i)
+k,a = 1
+2
+�N
+n=1
+�
+u(i−1)
+E,n
+�H ¯AR
+k,nu(i−1)
+E,n . Therefore, the ith MM iteration for (19) is the solution
+of the following convex problem
+max
+αa,{uE,n,uI,n}N
+n=1
+αa
+(40)
+s.t. C3 : (22), C4 : (39), C5 : (36),
+which can be solved efficiently.
+2) IRS System: By considering UE,n = Diag(θE), UI,n = Diag(θI) and adding the constraint
+CIRS in (18), the optimization problem in (19) is considered in this subsection. Since UE,n and
+UI,n are diagonal matrices, the expressions in (22)-(24) are modified as
+τ
+2θH
+E �AIRS
+E,nθE + T − τ
+2
+θH
+I �AIRS
+I,n θI ≤ Tprf
+IRS, ∀n,
+(41)
+γk,n(θI) =
+θH
+I AIRS
+k,n θI
+θH
+I �AIRS
+k,n θI + δ2
+k,n + σ2
+k,n
+, ∀k, n,
+(42)
+pE,k (θE) = 1
+2
+N
+�
+n=1
+θH
+E ¯AIRS
+k,n θE, ∀k,
+(43)
+where their parameters are defined in Lemma 2 below.
+Lemma 2. The parameters �AIRS
+E,n, �AIRS
+I,n , AIRS
+k,n , �AIRS
+k,n , and ¯AIRS
+k,n are expressed as follows:
+�AIRS
+E,n =
+�
+HnsE,nsH
+E,nHH
+n
+�T ⊙ IMIRS + σ2
+nIMIRS,
+(44)
+�AIRS
+I,n =
+�
+HnQI,nHH
+n
+�T ⊙ IMIRS + σ2
+nIMIRS,
+(45)
+AIRS
+k,n = pI,k,n
+�
+hk,nhH
+k,n
+�T ⊙ g∗
+k,ngT
+k,n,
+(46)
+�AIRS
+k,n =
+K
+�
+j=1,j̸=k
+pI,j,n
+�
+hj,nhH
+j,n
+�T⊙ g∗
+k,ngT
+k,n+ σ2
+nIMIRS⊙ g∗
+k,ngT
+k,n,
+(47)
+¯AIRS
+k,n =
+�
+HnsE,nsH
+E,nHH
+n
+�T ⊙ g∗
+k,ngT
+k,n.
+(48)
+It is worth pointing out that the only difference between the parameters above and their corre-
+sponding expressions in (21) and (25)-(27), is the symbol of multiplication, i.e., ⊗ and ⊙, in a
+January 3, 2023
+DRAFT
+
+15
+proper dimension. The proper dimension consideration means MR → MIRS for all of the above
+parameters and IM2
+R → IMIRS for the second terms of �AIRS
+E,n and �AIRS
+I,n .
+Proof. See Appendix D.
+Next, we focus on constraint CIRS. First, let us introduce the following minorizer [33]
+|x| ≥ ℜ
+�
+x∗ x0
+|x0|
+�
+.
+(49)
+Then, considering the above minorizer, the constraint CIRS is expressed as the ith iteration of
+MM as
+ℜ
+�
+θ∗
+E,m
+θ(i−1)
+E,m
+|θ(i−1)
+E,m |
+�
+≥ 1, ℜ
+�
+θ∗
+I,m
+θ(i−1)
+I,m
+|θ(i−1)
+I,m |
+�
+≥ 1,
+∀m.
+(50)
+Therefore, the optimization problem in (28) is modified as
+max
+αa,θE,θI
+αa
+(51)
+s.t.
+C3 : (41), C4 : Ek (θE) ≥ Emin,k, ∀k, CIRS : (50),
+C5 :
+N
+�
+n=1
+log2
+�
+1 +
+θH
+I AIRS
+k,n θI
+θH
+I �AIRS
+k,n θI + ζk,n,a
+�
+≥ αa, ∀k,
+where the steps for constraints C3-C5 in Subsection III-A1 are used exactly here.
+B. Maximization over {sE,n, pI,n}
+By introducing an auxiliary variable αb, the relay/IRS problem in (16) for fixed {UE,n, UI,n, τ}
+boils down to the following optimization:
+max
+αb,{sE,n,pI,n}N
+n=1
+αb
+(52)
+s.t.
+C2 : (3), pI,k,n ≥ 0, ∀k, n, C3 : (9), C4 : Ek
+�
+{sE}N
+n=1
+�
+≥ Emin,k, ∀k,
+C5 :
+N
+�
+n=1
+log2 (1 + γk,n(pI,n)) ≥ αb, ∀k.
+The constraints C4 and C5 of this sub-problem are non-convex. We first rewrite the SINR
+associated with the kth pair in (11) as
+γk,n(pI,n) =
+aT
+k,npI,n
+bT
+k,npI,n + σ2n �ψk,n + δ2
+k,n + σ2
+k,n
+,
+(53)
+where ak,n = ψk,k,nek, bk,n = [ψk,1,n, ψk,2,n, · · · , ψk,k−1,n, 0 , ψk,k+1,n, · · · , ψk,K,n]T, and ek is
+the kth unit vector. Therefore, the LHS of C5 in (52) is written as
+N
+�
+n=1
+�
+log2
+�
+qT
+k,npI,n + ζk,n,b
+�
+− log2
+�
+bT
+k,npI,n + ζk,n,b
+� �
+,
+(54)
+January 3, 2023
+DRAFT
+
+16
+where qk,n = ak,n +bk,n and ζk,n,b = σ2
+n �ψk,n +σ2
+k,n +δ2
+k,n. Then, similar to the procedure in Sub-
+section III-A for C5, we resort to the MM technique. Precisely, considering the inequality in (30),
+the second term in (54) is minorized by setting x = bT
+k,npI,n + ζk,n,b and x0 = bT
+k,np0
+I,n + ζk,n,b.
+By substituting the minorizer in (54), the constraint C5 at the ith iteration is obtained as
+C5 :
+N
+�
+n=1
+�
+log2
+�
+qT
+k,npI,n + ζk,n,b
+�
+− log2(bT
+k,np(i−1)
+I,n
++ ζk,n,b)
+(55)
+−
+log2 e
+bT
+k,np(i−1)
+I,n
++ ζk,n,b
+bT
+k,n
+�
+pI,n − p(i−1)
+I,n
+� �
+≥ αb.
+Next, we consider the non-convex constraint C4. It is observed that the term Ek
+�
+{sE,n}N
+n=1
+�
+in
+the LHS of the this constraint is neither convex nor concave w.r.t. sE,n. Therefore, similar to
+the procedure in Subsection III-A, we apply the MM by selecting βk,n,b (see Appendix C) and
+minorize C4 at the ith iteration by
+Ek
+��
+s(i−1)
+E,n
+�N
+n=1
+�
++ 1
+2
+N
+�
+n=1
+βk,n,b
+�
+s(i−1)
+E,n
+�H
+s(i−1)
+E,n
++
+N
+�
+n=1
+ℜ
+�
+ϑ(i)
+k,n,b
+�
+sE,n − s(i−1)
+E,n
+��
+(56)
+− 1
+2
+N
+�
+n=1
+βk,n,bsH
+E,nsE,n ≥ Emin,k, ∀k,
+where we define
+ϑ(i)
+k,n,b =βk,n,b
+�
+s(i−1)
+E,n
+�H
++ τ
+ρexp�cexp
+�
+�alog2ω(i)
+k,b
+� �
+ω(i)
+k,b
+��b−1 �
+2�a log ω(i)
+k,b + �b
+� �
+s(i−1)
+E,n
+�H
+Ξk,n,
+with ω(i)
+k,b =
+1
+2
+�N
+n=1
+�
+s(i−1)
+E,n
+�H
+Ξk,ns(i−1)
+E,n . Consequently, the ith iteration of the MM update
+for (52) is obtained easily as the interior point solution of the following convex problem
+max
+αb,{sE,n,pI,n}N
+n=1
+αb
+(57)
+s.t.
+C2, C3, C4 : (56), C5 : (55).
+C. Maximization over τ
+The optimization problem in (16) w.r.t. τ becomes
+min
+τ
+τ
+(58)
+s.t.
+C1 : 0 ≤ τ ≤ T,
+C2 : τvk ≤ �vk, ∀k,
+C3 : τ�v1 ≤ �v2,
+C4 : τ ≥ ¯vk, ∀k,
+where
+vk = 1
+2ρ
+N
+�
+n=1
+�
+|sE,k,n|2 − pI,k,n
+�
+, ∀k,
+�vk = T
+N
+�
+n=1
+�
+prf
+k,n − pI,k,n
+2ρ
+�
+, ∀k,
+(59)
+January 3, 2023
+DRAFT
+
+17
+Algorithm 1 The Proposed Method for Minimum Rate Maximization in Relay/IRS Systems
+1. Relay: Initialize U(l)
+E,n, U(l)
+I,n ∈ CMR×MR, τ (l) ∈ R+, l ← 0.
+1. IRS: Initialize θ(l)
+E , θ(l)
+I
+∈ CMIRS, τ (l) ∈ R+, l ← 0.
+repeat
+2. Relay: Initialize U(i)
+E,n and U(i)
+I,n and set i = 0.
+2. IRS: Initialize θ(i)
+E , θ(i)
+I
+and set i = 0.
+repeat
+3. Relay: Solve (40) to obtain {UE,n, UI,n, αa}.
+3. IRS: Solve (51) to obtain {θE, θI, αa}.
+4. Update i ← i + 1.
+until convergence
+5. Relay/IRS: Initialize s(i)
+E,n, p(i)
+I,n and set i = 0.
+repeat
+6. Relay/IRS: Solve the convex problem in (57) to obtain {sE,n, pI,n, αb}.
+7. Update i ← i + 1.
+until convergence
+8. Relay/IRS: Compute τ (l) via the closed-form solution in (63).
+9. Update l ← l + 1.
+until convergence
+�v1 = 1
+2ρ
+N
+�
+n=1
+�
+sH
+E,nVE,nsE,n + σ2
+ntr
+�
+UE,nUH
+E,n
+�
+− tr{QI,nVI,n} − σ2
+ntr{UI,nUH
+I,n}
+�
+,
+(60)
+�v2 = T
+N
+�
+n=1
+�
+prf
+n − 1
+2ρ
+�
+tr{QI,nVI,n} + σ2
+ntr
+�
+UI,nUH
+I,n
+���
+,
+(61)
+¯vk =
+ρEmin,k
+exp
+�
+�alog2pE,k
+�
+p�b
+E,kexp (�c)
+, ∀k.
+(62)
+Therefore, a closed-form solution (for a non-empty feasible set5) can be obtained as
+τopt = max{¯v1, ¯v2, ..., ¯vK}.
+(63)
+5The following conditions lead to a non-empty feasible set for the problem:
+1) ¯vk ≤ T, ∀k, 2) �vk ≥ 0, ∀k, 3) �v2 ≥ 0, 4)
+�vj
+vj |K
+j=1 ≥ ¯vk, ∀k (for vj ≥ 0, ∀j), 5) �v2
+�v1 ≥ ¯vk, ∀k (for v1 ≥ 0).
+January 3, 2023
+DRAFT
+
+18
+TABLE I
+THE COMPUTATIONAL COMPLEXITY ORDER (PER INNER ITERATIONS) FOR STEP 3 OF THE ALGORITHM 1.
+Relay
+O
+��
+2NM 2
+R(1 + 2N)(1 + K)
+�3.5�
+Relay (t-static)
+O
+��
+NM 2
+R(1 + N)(1 + 2K)
+�3.5�
+Relay (t-f-static)
+O
+��
+2M 2
+R(1 + 2K)
+�3.5�
+IRS
+O
+�
+(6MIRS(N + K + 1))3.5�
+IRS (t-static)
+O
+�
+(2MIRS(N + 2K + 1))3.5�
+Algorithm 1 summarizes the discussions in Section III and represents the steps of the proposed
+method for maximizing the minimum rate of all user pairs in relay/IRS WPC systems. Note that
+similar mathematical derivations are used to develop t-f-static algorithm for relay system as well
+as t-static algorithm for both relay and IRS systems.
+Remark 4 (convergence). It has been shown that under some mild conditions, the MM technique
+converges to the stationary points of the problem [34], [35].
+D. Complexity Analysis
+The main computational burdens in Algorithm 1 are associated with steps 3, 6, and 8. At
+each inner iteration in step 3, the convex problems in (40) and (51) are solved via interior
+point methods for relay and IRS system design, respectively, with a computational complexity
+of O
+�
+(2NM2
+R(1 + 2N)(1 + K))3.5�
+and O
+�
+(6MIRS(N + K + 1))3.5�
+[36]. Table I compares
+the computational complexity of step 3 for other versions of relay/IRS models. Similar to step
+3, the complexity (per inner iterations) for step 6 which solves (57) (e.g., by using the interior
+point methods) is O
+�
+(KN(1 + 2N)(5 + 2K))3.5�
+for all versions of relay/IRS models. In step
+8, the closed-form expression in (63) must be calculated leading to the complexity of6 O(N3).
+IV. NUMERICAL EXAMPLES
+Here, we evaluate the proposed relay/IRS method in different scenarios. The channels from
+transmitters to the relay and the channels from the relay to the receivers are modeled as Hn =
+0.1
+� �d1
+d0
+� −�γ
+2 �Hn and Gn = 0.1
+� �d2
+d0
+� −�γ
+2 �Gn, respectively, where d0 = 1 m is a reference distance,
+�d1 is the distance between Tk and the relay, �d2 is the distance between the relay and Rk, and
+6This can be decreased to O(N 2.3) via finding the best order of matrix multiplications (see [37] for details).
+January 3, 2023
+DRAFT
+
+19
+T1
+T2
+TK
+Relay or IRS
+rT
+d3
+d1
+d2
+rR
+R1
+...
+...
+R2
+RK
+Fig. 3. Simulation setup for relay/IRS WPC systems with K user pairs.
+TABLE II
+THE BASELINE BENCHMARK METHODS
+Baseline 1
+Baseline 2
+Information Power Allocation
+✓
+✓
+Energy Waveform Design
+✗
+✓
+Time Allocation
+✓
+✓
+Energy/Information Relay Beamforming
+✓
+✗
+�γ = 3 is the path-loss exponent. It is assumed that the elements of �Hn and �Gn are i.i.d. CSCG
+random variables with zero mean and unit variance. As shown in Fig. 3, the transmitters and
+receivers are distributed uniformly within a circle with radius rT and rR, respectively. We set
+the distance parameters as d1 = d2 = d3 = 10 m and rT = rR = 5 m. The maximum power
+budget for Tk, relay, and IRS are set to prf
+k,n = prf
+R,n = 28 dBm, prf
+IRS = 20 dBm, ∀k, n, and
+the noise power at the relay, IRS and receivers are supposed to be σ2
+R,n = σ2
+k,n = δ2
+k,n = −80
+dBm, σ2
+IRS,n = −100 dBm, ∀k, n. The total bandwidth is fixed to Bt = 1 MHz. We further
+assume the total operation time T = 1. The curve fitting parameters for non-linear EH circuits
+are equal to �a = −0.11, �b = −1.17, and �c = −12 [32]. Also, we set K = 5, N = 8, MR = 6,
+and Emin,k = Emin = 10 µW, ∀k, unless otherwise specified. We solve the convex optimization
+problems using CVX [38].
+A. Relay System
+Here, we compare the results of the proposed algorithms with partially optimized methods
+(referred to as baseline schemes in the sequel) listed in Table II. For the first baseline method,
+the energy signals are not optimized, and in the second baseline method, there is no optimization
+January 3, 2023
+DRAFT
+
+20
+1
+2
+3
+4
+5
+0
+1
+2
+3
+4
+5
+6
+Minimum Rate (bps/Hz)
+Number of Outer Iterations
+0.2
+0.8
+5.75
+5.8
+Different Initializations
+(a)
+1
+2
+3
+4
+5
+6
+7
+0
+1
+2
+3
+4
+Minimum Rate (bps/Hz)
+Number of Inner Iterations
+(b)
+1
+2
+3
+4
+5
+6
+7
+8
+9
+10
+11
+4
+4.2
+4.4
+4.6
+4.8
+5
+Number of Inner Iterations
+Minimum Rate (bps/Hz)
+ 
+ 
+(c)
+Fig. 4.
+Convergence behavior of the proposed method in Algorithm 1: (a) outer iterations for three random initial points,
+(b) inner iterations associated with the sub-problem III-A1 in the first outer iteration, (c) inner iterations associated with the
+sub-problem III-B in the first outer iteration.
+for the relay beamformer; more precisely, the relay amplification matrices are assumed to be
+identity matrices, i.e. UR
+E,n = �αE,nIMR, ∀n, and UR
+I,n = �αI,nIMR, ∀n, where the scalar parameters
+�αE,n and �αI,n are employed to satisfy the feasible set Ω in (16). The convergence of the proposed
+algorithm for inner and outer iterations (see Algorithm 1) are plotted in Fig. 4. This figure shows
+that the proposed algorithm converges within a few outer iterations. Also, in this example, the
+three different initializations lead to almost the same final value.
+In Fig. 5.a, we illustrate the rate-energy region of the proposed method in comparison with the
+first baseline method for different number of subbands. We can observe that the minimum rate
+increases as N grows. The optimal time allocation parameter τopt w.r.t. the EH target is depicted
+in Fig. 5.b. It is seen that the increased energy threshold Emin leads to a larger τ. As τ increases,
+the duration of the ID phase decreases. Therefore, as we observe in Fig. 5.a, the minimum rate
+reduces with increasing Emin. Also, the impact of the energy waveform design is evident in both
+figures. In Fig. 6.a and Fig. 6.b, we compare the minimum rate of the proposed optimal and sub-
+optimal approaches with baseline methods. As we can see in Fig. 6.a, increasing the number
+of pairs results in lower minimum rate for all methods with MR = 9. Furthermore, Fig. 6.b
+shows that a larger MR increases the minimum rate with an almost linear trend. The importance
+of the energy waveform and relay beamforming design is observed through both figures. We
+can see that the method with no relay beamforming has the worst performance compared to
+other methods since without a relay amplification matrix design, inter-pair interference cannot
+be managed.
+January 3, 2023
+DRAFT
+
+21
+10 20 30 40 50 60 70 80 90 100110120130
+0
+1
+2
+3
+4
+5
+6
+7
+EH Target, Emin (µW )
+Minimum Rate (bps/Hz)
+ 
+ 
+Proposed - N = 8
+Baseline 1 - N = 8
+Proposed - N = 9
+Baseline 1 - N = 9
+(a) the rate-energy region
+10 20 30 40 50 60 70 80 90 100110120130
+0
+0.2
+0.4
+0.6
+0.8
+1
+EH Target, Emin (µW )
+Time Allocation Parameter (s)
+ 
+ 
+Proposed - N = 8
+Baseline 1 - N = 8
+Proposed - N = 9
+Baseline 1 - N = 9
+(b) the time allocation parameter τopt
+Fig. 5. Comparison of the proposed and baseline 1 methods for different number of subbands N = 8, 9.
+3
+4
+5
+6
+7
+8
+3
+4
+5
+6
+7
+8
+9
+10
+11
+12
+Number of Pairs, K
+Minimum Rate (bps/Hz)
+ 
+ 
+Proposed 
+Baseline 1
+Proposed (t−static)
+Proposed (t−f−static)
+Baseline 2
+(a) minimum rate versus number of pairs K
+5
+6
+7
+8
+9
+10
+2
+3
+4
+5
+6
+7
+8
+9
+Number of Antennas, MR
+Minimum Rate (bps/Hz)
+ 
+ 
+Proposed
+Baseline 1
+Proposed (t−static)
+Proposed (t−f−static)
+Baseline 2
+(b) minimum rate versus number of antennas MR
+Fig. 6. Comparison of the proposed optimal and sub-optimal methods with baseline methods.
+B. IRS System
+In this subsection, the performance of the proposed IRS-assisted WPC system is evaluated.
+Since most of the studied scenarios for relay (i.e., Fig. 4, Fig. 5, and Fig. 6.a) have similar trends
+for IRS, we only consider the scenario of Fig. 6.b, for the sake of brevity. As we can see from
+Fig.7, in the case of IRS, the minimum rate has a super-linear ascent property versus increasing
+MIRS.
+V. CONCLUSION
+In this paper, the max-min rate maximization in a multi-carrier relay/IRS WPC system with a
+joint TS scheme was considered. A unified framework was proposed to maximize the minimum
+rate of the user pairs in both relay and IRS systems by jointly designing the energy waveforms,
+January 3, 2023
+DRAFT
+
+22
+7
+10
+13
+16
+19
+22
+3
+3.5
+4
+4.5
+5
+5.5
+Number of REs, MIRS
+Minimum Rate (bps/Hz)
+ 
+ 
+Proposed
+Proposed (t−static)
+Fig. 7. The effect of the number of REs MIRS for the proposed optimal and sub-optimal IRS methods.
+power of information waveforms, amplification matrices, and the time allocation parameter.
+The non-linearity in EH circuits was also considered in the design problem. The non-convex
+problem was handled via the MM technique. Numerical results demonstrated the effectiveness of
+the proposed algorithm in terms of the minimum rate. As a extended future work in this area, it
+might be interesting to develop a distributed algorithm for design of multi-user relay/IRS WPC
+systems.
+APPENDIX A
+THE DERIVATION OF THE EXPRESSIONS IN (21) AND (25)-(27)
+Using tr
+�
+XHY
+�
+= vec(X)Hvec(Y) and vec(XYZ) = (ZT ⊗ X)vec(Y), the power of the
+relay signal for ID mode in (7) can be obtained as
+E
+�
+∥�r(t)∥2
+2
+�
+= 1
+2
+N
+�
+n=1
+�
+uH
+I,nvec
+�
+UI,nHnQI,nHH
+n
+�
++ σ2
+nuH
+I,nuI,n
+�
+= 1
+2
+N
+�
+n=1
+�
+uH
+I,n
+��
+HnQI,nHH
+n
+�T ⊗ IMR
+�
+uI,n + σ2
+nuH
+I,nuI,n
+�
+= 1
+2
+N
+�
+n=1
+uH
+I,n �AR
+I,nuI,n.
+Similarly, we can derive the power of the relay signal for the EH mode in (20) and the expressions
+in (25)–(27).
+January 3, 2023
+DRAFT
+
+23
+APPENDIX B
+PROOF OF LEMMA 1
+By defining a positive semi-definite matrix D such that ∇2
+xs(x) ⪯ D, we can write the
+following majorizer for s(x) as [39]
+s(x) ≤ s(x0) + ℜ
+�
+(∇xs(x))H |x=x0(x − x0)
+�
++ (x − x0)HD(x − x0),
+(64)
+where the gradient and Hessian of s(x) are respectively expressed as
+∇xs(x) = −2 log2 e
+xHTx + ν Tx,
+(65)
+∇2
+xs(x) =
+�
+−2T
+xHTx + ν +
+4TxxHT
+(xHTx + ν)2
+�
+log2 e.
+Since T ⪰ 0, the term
+−2T
+xHTx+ν is negative semi-definite, and thus we obtain ξ > 0 such that for
+any ν ⩾ 0
+4TxxHT
+(xHTx + ν)2 log2 e ⩽ 4TxxHT
+(xHTx)2 ⩽ ξIM2
+R.
+Also, as TxxHT is a rank-one matrix, we can choose ξ as ξ ⩾ 4φ, where φ is given as
+φ = max
+x
+xHT2x
+(xHTx)2.
+(66)
+Then by choosing a = VHx, where V is a full-rank matrix such that T = VVH, the following
+optimization is equivalently obtained from (66) as
+φ = max
+a
+aHVHVa
+aHa
+1
+aHa.
+(67)
+Using xHQx ≤ P and applying a similar procedure in [39, Appendix B], we can write
+φ ≤
+Pλmax(T)
+vH
+1 V−1QV−Hv1
+,
+where v1 is the principal eigenvector of VHV. Finally, from (64), (65), and ξ = 4φ, we obtain
+b = ∇s(x)|x=x0 =
+−2 log2 e
+xH
+0 Tx0+νTx0 and D =
+4P
+wH
+1 Qw1IM2
+R, where w1 is the principal eigenvector of
+T.
+January 3, 2023
+DRAFT
+
+24
+APPENDIX C
+A SELECTION OF βk,n,a AND βk,n,b
+The value of βk,n,b should be selected such that ∇2
+sE,nEk
+�
+{sE,n}N
+n=1
+�
++βk,n,bIK ⪰ 0. The term
+∇2
+sE,nEk
+�
+{sE,n}N
+n=1
+�
+is straightforwardly calculated as
+∇2
+sE,nEk
+�
+{sE,n}N
+n=1
+�
+=̺k
+N
+�
+n=1
+Ξk,n + ηk
+N
+�
+n=1
+N
+�
+n′=1
+Ξk,nsE,nsH
+E,n′Ξk,n′,
+(68)
+where
+̺k = τ
+ρexp�c exp
+�
+�alog2pE,k
+�
+p
+�b−1
+E,k
+�
+2�a log pE,k + �b
+�
+,
+ηk =
+τexp�c exp
+�
+�alog2pE,k
+�
+p
+�b−2
+E,k
+ρ
+�
+4�a2log2pE,k +
+�
+4�a�b − 2�a
+�
+log pE,k + �b2 −�b + 2�a
+�
+.
+As �a < 0,�b < 0, Ξk,n ⪰ 0, and Ξk,nsE,nsH
+E,nΞk,n ⪰ 0, it suffices to choose βk,n,b such that
+βk,n,bIK ⪰ − �̺k
+N
+�
+n=1
+Ξk,n − �ηk
+N
+�
+n=1
+N
+�
+n′=1
+Ξk,nsE,nsH
+E,n′Ξk,n′,
+(69)
+where
+�̺k = τ
+ρ
+�b exp�c exp
+�
+�alog2pE,k
+�
+p
+�b−1
+E,k ,
+�ηk =τ
+ρexp�c exp
+�
+�alog2pE,k
+�
+p
+�b−2
+E,k
+�
+log pE,k
+�
+4�a�b − 2�a
+�
++ 2�a
+�
+.
+Thus from (3), we can write
+∥sE,n∥2
+2 ≤ 2ρT
+τ
+K
+�
+k=1
+prf
+k,n.
+(70)
+Finally, using (16), (69), (70) and knowing that sH
+E,nΞk,nsE,n ≤ ∥sE,n∥2
+2λmax (Ξk,n), we can select
+βk,n,b > βt
+k,n,b where
+βt
+k,n,b = − τ
+ρexp�c exp
+�
+2�alog2T
+N
+�
+n=1
+λmax (Ξk,n)
+K
+�
+k=1
+prf
+k,n
+�
+�f
+�b−2
+k
+� ��
+4�a�b − 2�a
+�
+log �fk + 2�a
+�
+×
+N
+�
+n=1
+N
+�
+n′=1
+λmax (Ξk,nΞk,n′)
+K
+�
+k=1
+�
+prf
+k,nprf
+k,n′ + �b �fk
+N
+�
+n=1
+λmax (Ξk,n)
+�
+,
+with �fk = exp
+�
+−�b−
+�
+�b2−4�a log
+ρEmin,k
+τexp�c
+2�a
+�
+. We can take similar steps for selecting βt
+k,n,a.
+January 3, 2023
+DRAFT
+
+25
+APPENDIX D
+PROOF OF LEMMA 2
+The ID part of the relay power constraint in (20) is uH
+I,n �AR
+I,nuI,n. Only (iMIRS + i+ 1)th, 0 ≤
+i ≤ MIRS − 1 entries of uI,n = vec(Diag(θI)) are non-zero for the IRS system. Thus, we can
+rewrite uH
+I,n �AR
+I,nuI,n for IRS system as θH
+I �AIRS
+I,n θI, where �AIRS
+I,n contains only the (kMIRS + k +
+1, lMIRS +l+1)th, 0 ≤ k, l ≤ MIRS −1 entries of �AI,n which is the same as �AR
+I,n with replacing
+MR by MIRS. Therefore, from (21) and by using some matrix manipulations, we obtain
+�AIRS
+I,n =
+�
+HnQI,nHH
+n
+�T ⊙ IMIRS + σ2
+nIMIRS.
+(71)
+Other expressions in (44)-(48) are similarly obtained.
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+

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+arXiv:2301.11680v1  [cs.IT]  27 Jan 2023
+Codes for Correcting Asymmetric Adjacent
+Transpositions and Deletions
+Shuche Wang∗, Van Khu Vu§, and Vincent Y. F. Tan†‡∗
+∗ Institute of Operations Research and Analytics, National University of Singapore, Singapore
+† Department of Mathematics, National University of Singapore, Singapore
+‡ Department of Electrical and Computer Engineering, National University of Singapore, Singapore
+§ Department of Industrial Systems Engineering and Management, National University of Singapore, Singapore
+Emails: shuche.wang@u.nus.edu, isevvk@nus.edu.sg, vtan@nus.edu.sg
+Abstract
+Owing to the vast applications in DNA-based data storage, Gabrys, Yaakobi, and Milenkovic recently proposed to study codes
+in the Damerau–Levenshtein metric, where both deletion and adjacent transposition errors occur. In particular, they designed a
+code correcting a single deletion and s adjacent transpositions with at most (1 + 2s) log n bits of redundancy. In this work, we
+consider a new setting where both asymmetric adjacent transpositions (also known as right-shifts or left-shifts) and deletions occur.
+We present several constructions of the codes correcting these errors in various cases. In particular, we design a code correcting
+a single deletion, s+ right-shift, and s− left-shift errors with at most (1 + s) log(n + s + 1) + 1 bits of redundancy where
+s = s+ + s−. In addition, we investigate codes correcting t 0-deletions and s adjacent transpositions with both unique decoding
+and list-decoding algorithms. Our main contribution here is a construction of a list-decodable code with list-size O(nmin{s+1,t})
+and has at most (max{t, s + 1}) log n + O(1) bits of redundancy. Finally, we provide both non-systematic and systematic codes
+for correcting t blocks of 0-deletions with ℓ-limited-magnitude and s adjacent transpositions.
+I. INTRODUCTION
+The Levenshtein (edit) distance of two different strings is the smallest number of operations (including deletions, insertions,
+and substitutions) required to transform one string into the other. This metric has a long history and has attracted a lot of research
+in computer science in the past as well as recently [2]–[4]. Codes in the Levenshtein metric have been investigated extensively
+recently due to theoretical interests and their numerous applications, including racetrack memory [5]–[7] and DNA-based data
+storage [8]–[10].
+This paper was presented in part at the 2022 IEEE Information Theory Workshop (ITW) [1].
+
+1
+In some channels, such as DNA-based data storage ones, we observe that, besides deletions, insertions, and substitutions,
+there are also adjacent transpositions. Hence, there exists some recent work concerning the Damerau–Levenshtein distance
+which is motivated by applications to DNA-based data storage. The distance is a generalization of the well-known Levenshtein
+distance taking into account adjacent transpositions. More precisely, the Damerau–Levenshtein metric is the smallest number
+of operations (including deletions, insertions, substitutions, and adjacent transpositions) required to transform one string into
+another. We note that it is possible to compute the exact Damerau–Levenshtein distance of two strings in polynomial time [11]
+but it is not known if we can compute the distance in linear time. Recently, Gabrys, Yaaboki, and Milenkovic [12] proposed
+to study codes in the Damerau–Levenshtein distance. They provided several constructions of codes correcting both deletions
+and adjacent transpositions. However, these codes are not optimal in general. For example, to correct a single deletion and
+at most s adjacent transpositions, the authors require (1 + 2s) log n bits of redundancy. Designing an optimal code correcting
+both deletions and multiple adjacent transpositions has turned out to be a formidable challenge for coding theorists in recent
+times.
+The problem of constructing codes for correcting synchronization errors, including deletions and insertions, was first
+investigated by Levenshtein [13] and Ullman [14], [15]. Sticky deletions/insertions and duplication deletions can be considered
+as asymmetric deletions/insertions via the Gray mapping [16]. Owing to various applications, such as in flash memories [17],
+[18], racetrack memories [6], and DNA data storage systems [19], [20], codes for correcting asymmetric deletions/insertions
+have garnered significant attention recently. Tallini et al. [16], [21]–[24] provided a series of theories and code designs for
+correcting these kinds of errors. Especially, Mahdavifar and Vardy [18] provided some efficient encoding/decoding algorithms
+for an optimal code correcting sticky-insertion and thus for an optimal code correcting 0-deletion.
+Codes correcting adjacent transposition errors have been investigated for a long time as codes for shift errors [25]–[27].
+Codes correcting asymmetric shift errors have also been studied recently [28]. In this work, we are interested in codes correcting
+a combination of both asymmetric adjacent transposition errors and deletion errors. We aim to obtain some optimal codes with
+simple efficient encoding/decoding algorithms.
+We note that codes correcting substitutions, deletions, and their combinations have attracted a lot of research recently [29],
+[30]. However, there are only a few code constructions that correct a combination of adjacent transposition and other kinds of
+errors. Klove [31] proposed a class of perfect constant-weight codes capable of correcting a single deletion, a single insertion or
+an adjacent transposition. Gabrys, Yaakobi, and Milenkovic [12] presented several codes correcting a combination of deletions
+and adjacent transpositions. If there is a single adjacent transposition or a single deletion, there exist codes correcting the error
+with at most log n + O(log log n) bits of redundancy [32]. The best-known codes correcting a single deletion and at most s
+
+2
+adjacent transpositions require (1 + 2s) log n bits of redundancy [12]. In this work, we design several new families of codes
+in numerous cases. We provide our main contributions as follows.
+Our first contribution in this work is Construction 1, which presents a construction of an optimal code correcting a single
+adjacent transposition or a single 0-deletion. Analyzing the size of our code, we obtain the following result.
+Theorem 1. There is a code correcting a single 0-deletion or a single adjacent transposition with at most log n + 2 bits of
+redundancy.
+Next, we construct a code correcting t 0-deletions and s adjacent transpositions with at most (t + 2s) log n + o((t +
+2s) log n) bits of redundancy. The constructed code is the best known that corrects multiple 0-deletions and multiple adjacent
+transpositions. See Theorem 7 for the detail.
+Theorem2. There is a code correcting t 0-deletions and s adjacent transpositions with at most (t+2s) log n+o((t+2s) log n)
+bits of redundancy.
+Further, we construct an optimal code for correcting a single deletion, s+ right-shift and s− left-shift errors. Throughout
+this paper, we denote the adjacent transposition as 01 → 10 or 10 → 01, right-shift of 0 as 01 → 10 and left-shift of 0 as
+10 → 01. See Construction 2 and Theorem 8 for the detail.
+Theorem 3. There is a code correcting a single deletion, s+ right-shift and s− left-shift errors with at most (1 + s) log(n +
+s + 1) + 1 bits of redundancy where s = s+ + s−.
+Compare the results in [12], where the code for correcting a single deletion and s adjacent transpositions needs at most
+(1 + 2s) log(n + 2s + 1) redundancy. If we know the direction of these s adjacent transpositions containing s+ right-shifts
+of 0 and s− left-shifts of 0, the redundancy of the code can be further reduced to at most (1 + s) log(n + s + 1) + 1 where
+s = s+ + s−.
+We also investigate list-decodable codes of small list-size and construct a list-decodable code for at most t 0-deletions and
+s adjacent transpositions. See the proof of Theorem 9 for the construction. Our results are the first known list-decodable codes
+for the asymmetric Damerau–Levenshtein distance.
+Theorem 4. There is a list-decodable code that can correct t 0-deletions and s adjacent transpositions with list size
+O(nmin(t,s+1)) and has max(t, s + 1) log n + O(1) bits of redundancy.
+Finally, we construct both non-systematic and systematic codes for correcting t blocks of 0-deletions with ℓ-limited-magnitude
+and s adjacent transpositions. See the proof of Theorem 10 for the construction.
+
+3
+Theorem5. There is a code capable of correcting t blocks of 0-deletions with ℓ-limited-magnitude and s adjacent transpositions
+with at most ⌈2(t + 2s)(1 − 1/p)⌉ log(n + 1) + O(1) bits of redundancy, where p is the smallest prime larger than tℓ + 2.
+The rest of this paper is organized as follows. Section II provides the notation and preliminaries. Section III presents three
+uniquely-decodable codes for correcting asymmetric deletions and adjacent transpositions. Section IV proposes list-decodable
+codes for correcting asymmetric deletions and adjacent transpositions with low redundancy. In Section V, we construct codes
+both non-systematic and systematic codes are capable of correcting t blocks of 0-deletions with ℓ-limited-magnitude and s
+adjacent transpositions. Finally, Section VI concludes this paper.
+II. NOTATION AND PRELIMINARIES
+We now describe the notations used throughout this paper. Σq denotes the finite alphabet of size q and Σn
+q represents the set
+of all sequences of length n over Σq. Without loss of generality, we assume Σq = {0, 1, . . ., q − 1}. For two integers i < j,
+let [i, j] denote the set {i, i + 1, i + 2, . . . , j}. The size of a binary code C ⊆ Σn
+2 is denoted |C| and its redundancy is defined
+as n − log |C|, where all logarithms without a base in this paper are to the base 2.
+We write sequences with bold letters, such as x and their elements with plain letters, e.g., x = x1 · · · xn for x ∈ Σn
+q . The
+length of the sequence x is denoted |x|. The weight wt(x) of a sequence x represents the number of non-zero symbols in it.
+A run is a maximal substring consisting of identical symbols and nr(x) denotes the number of runs of the sequence x. For
+functions, if the output is a sequence, we also write them with bold letters, such as φ(x). The ith position in φ(x) is denoted
+φ(x)i. In addition, for a sequence u ∈ Σn
+q , denote (u mod a) = (u1 mod a, u2 mod a, . . . , un mod a), where a < q.
+For a binary sequence x ∈ Σn
+2, we can uniquely write it as x = 0u110u210u3 . . . 10uw+1, where w = wt(x).
+Definition 1. Define function φ
+:
+Σn
+2
+→
+Σw+1 and φ(x)
+def=
+(u1, u2, u3, . . . , uw+1)
+∈
+Σw+1, where x
+=
+0u110u210u3 . . . 10uw+1 with w = wt(x).
+Example 1. Suppose x = (0, 1, 1, 1, 0, 1, 0, 1, 0, 0). Then, φ(x) = (1, 0, 0, 1, 1, 2).
+Definition 2. Define function ψ : Σn
+2 → Σn
+2 such that ψ(x) = (x1, x1 + x2, . . . , x1 + x2 + · · · + xn).
+Definition 3. The Lee weight of an element xi ∈ Σq is defined by
+wL(xi) =
+
+
+
+
+
+
+
+
+
+xi,
+if 0 ≤ xi ≤ q/2
+q − xi,
+otherwise
+For a sequence x ∈ Σn
+q , the Lee weight of x is
+wL(x) =
+n
+�
+i=1
+wL(xi).
+
+4
+Define the Lee distance of two sequences x, x′ ∈ Σn
+q as
+dL(x, x′) = wL(x − x′).
+Example 2. Suppose x ∈ Σ7
+6 = (1, 4, 0, 5, 2, 3, 4). Then, wL(x) = 1 + 2 + 0 + 1 + 2 + 3 + 2 = 11.
+Example 3. Suppose x ∈ Σ7
+6 = (1, 4, 0, 5, 2, 3, 4) and x′ ∈ Σ7
+6 = (0, 3, 0, 5, 3, 3, 3). Then, x − x′ = (1, 1, 0, 0, 5, 0, 1) and
+dL(x, x′) = wL(x − x′) = 4.
+For any x ∈ Σn
+2, denote Bt,s(x) as the error ball of x under t 0-deletions and s adjacent transpositions. The code Ct,s(n) is
+a unique-decodable code for correcting t 0-deletions and s adjacent transpositions, for which holds that Bt,s(c1)∩Bt,s(c2) = ∅
+for all c1, c2 ∈ Ct,s(n). The code CList
+t,s (n) is a list-decodable code for correcting t 0-deletions and s adjacent transpositions
+with list size L such that for any corrupted sequence x′ ∈ Σn−t
+2
+there exist at most L codewords in CList
+t,s (n) that can be
+obtained by t 0-deletions and s adjacent transpositions.
+Example 4. Suppose x = (0, 1, 1, 1, 0, 1, 0, 1, 0, 0), the first and last 0 bits are deleted and two pairs of ((4th, 5th) and (7th,
+8th)) adjacent bits are transposed in x = (❆0, 1, 1, 1, 0, 1, 0, 1, 0, ❆0). Then, x′ = (1, 1, 0, 1, 1, 1, 0, 0) ∈ B2,2(x).
+Proposition 1. Once a 0-deletion occurs in x and we receive x′, there is an index i such that φ(x)i − 1 = φ(x′)i.
+Proposition 2. Suppose an adjacent transposition occurs in x at the ith 1, the corresponding changes in φ(x) can be shown
+as follows:
+1) 10 → 01: (φ(x)′
+i, φ(x)′
+i+1) = (φ(x)i + 1, φ(x)i+1 − 1).
+2) 01 → 10: (φ(x)′
+i, φ(x)′
+i+1) = (φ(x)i − 1, φ(x)i+1 + 1).
+Example 5. Suppose x = (0, 1, 1, 1, 0, 1, 0, 1, 0, 0), φ(x) = (1, 0, 0, 1, 1, 2) and the adjacent transposition is occurred in
+the 4-th bit 1 and the following bit 0 in x. Then, x′ = (0, 1, 1, 1, 0, 0, 1, 1, 0, 0) and φ(x′) = (1, 0, 0, 2, 0, 2), where
+(φ(x′)4, φ(x′)5) = (φ(x)4 + 1, φ(x)5 − 1).
+The well-known Varshamov–Tenengol’ts (VT) code will be use of in this paper, and we will introduce the following lemma.
+For x ∈ Σn
+2, we define the syndrome of VT code as VT(x) = �n
+i=1 ixi.
+Lemma 1 (Varshamov-Tenengol’ts (VT) code [33]). For integers n and a ∈ [0, n],
+VTa(n) = {x ∈ Σn
+2 : VT(x) ≡ a mod (n + 1)}
+is capable of correcting a single deletion.
+Define Mt,s(n) as maximal size of binary codes for correcting t deletions and s adjacent transpositions.
+
+5
+Lemma 2 (cf. Levenstein [2]). For enough large n, Mt,s(n) ≤ (s + t)! 2n
+ns+t .
+Proof. t deletions and s adjacent transpositions in x can be considered as t deletions and s substitutions in ψ(x). An asymptotic
+bound for the size of any codes is capable of correcting up to t deletions, insertions and substitutions have been shown in [2],
+which is (t! · 2n)/nt. Since the function ψ is a one-to-one mapping function, an upper bound of binary codes for correcting t
+deletions and s adjacent transpositions can be derived.
+From Lemma 2, we can obtain a lower bound of the minimal redundancy of the code for correcting t 0-deletions and s
+adjacent transpositions.
+Corollary1. A lower bound of the minimal redundancy of binary codes for correcting t 0-deletions and s adjacent transpositions
+is (t + s) log n − O(1).1
+III. UNIQUELY-DECODABLE CODES FOR ASYMMETRIC DELETIONS AND ADJACENT TRANSPOSITIONS
+In this section, we will present three uniquely-decodable codes for correcting asymmetric deletions and adjacent transposi-
+tions, that is, once there are some errors, we can correct these errors to recover the original codeword uniquely.
+A. Codes for correcting a single 0-deletion or a single adjacent transposition
+In this subsection, we present the first construction of an optimal code correcting a single 0-deletion or a single adjacent
+transposition.
+Construction 1. The code C1(n, a; p) is defined as the set of all x ∈ Σn
+2 such that the syndrome
+S(x) =
+w+1
+�
+i=1
+i2φ(x)i ≡ a mod p
+where w = wt(x) and p is a prime such that p > 2n.
+Theorem 6. The code C1(n, a; p) in Construction 1 can correct a single 0-deletion or a single adjacent transposition.
+Proof. Let x = (x1, . . . , xn) ∈ Σn
+2 be the original vector and x′ be the received vector after a single 0-deletion or a single
+adjacent transposition.
+If x′ ∈ Σn−1
+2
+, that is the length of x′ is n−1, then there is a single 0 deletion. In this case, we compute the vector φ(x′) and
+a′ < p such that a′ = S(x′) mod p. We note that dL(φ(x), φ(x′)) = 1 and there is an index i such that φ(x)i − 1 = φ(x′)i.
+Hence, S(x) − S(x′) = i2. That is, a − a′ = i2 mod p. Since i2 − j2 ̸= 0 mod p for all i ̸= j, i, j < n < p/2, we can
+determine the unique index i such that a − a′ = i2 mod p. And thus, we locate the error and can correct it.
+1The difference between the lower bound of the redundancy for correcting general t deletions and t 0-deletions is only O(1). [17]
+
+6
+If x′ ∈ Σn
+2, that is the length of x′ is n, then there is no 0 deletion and at most a single adjacent transposition. Similar to
+the previous case, we also compute the vector φ(x′) and a′ < p such that a′ = S(x′) mod p. Once an adjacent transposition
+occurs, there are two types of errors: a symbol 0 moves to the left and a symbol 0 moves to the right. If a symbol 0 moves
+to the left, there exists 0 ≤ j ≤ n − 1 such that a − a′ = 2j + 1 mod p. Otherwise, if a symbol 0 moves to the right,
+there is 0 ≤ j ≤ n − 1 such that a − a′ = −2j − 1 mod p. Since p > 2n, for all i, j < n < p/2 and i ̸= j, these four
+values, {2i + 1, −2i − 1, 2j + 1, −2j − 1} are distinct. Hence, we can determine the type of error and the unique j such that
+a − a′ = 2j + 1 mod p or a − a′ = −2j − 1 mod p. And thus, we can correct the error.
+In conclusion, either a 0 deletion occurs or an adjacent transposition occurs, we always can correct the error and recover
+the original vector. The theorem is proven.
+From the well-known Bertrand–Chebyshev theorem, there exists a prime p such that 2n < p < 4n. Hence, by the pigeonhole
+principle, there exists a code C1(n, a; p) of size at least 2n/(4n). That is, it is possible to construct the code C1(n, a; p) at most
+log n + 2 redundancy. Therefore, we can conclude that we can correct a single 0-deletion or a single adjacent transposition
+with at most log n + 2 redundancy.
+B. Codes for correcting t 0-deletions and s adjacent transpositions
+In this subsection, we explore the general case in the asymmetric Damerau–Levenshtein distance scheme. We investigate a
+code correcting at most t 0-deletions and s adjacent transpositions, given constants t and s.
+We observe that the asymmetric Damerau–Levenshtein distance between two vectors x and y is closely related to Lee
+distance between φ(x) and φ(y). Indeed, once an adjacent transposition occurs in x, the Lee weight of x is changed by two
+based on Proposition 2 and once a 0-deletion occurs in x, the Lee weight of x is changed by one. Hence, if there are at most
+s adjacent transpositions and t 0-deletions, the Lee weight of x is changed by at most t + 2s. Now, we present a well-known
+BCH code in the Lee distance.
+Lemma 3. ( [18], [34]) The systematic BCH code CBCH(n, t + 1; p) : x ∈ Σm
+2 → E(x) ∈ Σn
+p with the lower bound of
+minimum Lee distance
+dL(CBCH(n, t + 1; p)) ≥
+
+
+
+
+
+
+
+
+
+2(t + 1),
+if t ≤ (p − 3)/2
+p,
+if (p − 1)/2 ≤ t ≤ p
+can correct errors up to t Lee weight with redundancy t log n + o(t log n), where p is a prime.
+Furthermore, Mahdavifar and Vardy [18] used the above code to construct a code C(n, r) of length n correcting r 0
+insertions with at most r log n + o(r log n) bits of redundancy. It is known that for any two words c1, c2 ∈ C(n, r), we
+
+7
+have dL(φ(c1), φ(c2)) ≥ 2(r + 1) by Lemma 3. Hence, we can use the code C(n, r) to correct t 0-deletions and s adjacent
+transpositions.
+Theorem 7. The code C(n, r) can correct at most t 0-deletions and s adjacent transpositions, given t + 2s = r.
+Proof. Let x = (x1, . . . , xn) ∈ Σn
+2 be the original vector and x′ ∈ Σn−t
+2
+be the received vector after t 0-deletions and s
+adjacent transpositions. Hence, we obtain the vector y′ = φ(x′). We consider two vectors φ(x) and φ(x′). We observe that
+once an adjacent transposition occurs in x, the Lee weight of x is changed by at most two based on Proposition 2 and once
+a 0-deletion occurs in x, the Lee weight of x is changed by one. Hence, if there are at most s adjacent transpositions and t
+0-deletions, the Lee weight of x is changed by at most t + 2s. That is, the Lee distance between two vectors φ(x) and φ(x′)
+is dL(φ(x), φ(x′)) ≤ t + 2s. Therefore, we set r = t + 2s and then the code C(n, r) can correct at most t 0-deletions and s
+adjacent transpositions with redundancy (t + 2s) log n + o((t + 2s) log n).
+C. Codes for correcting a single deletion and multiple right-shifts
+In previous two subsections, we focus on the error type of 0-deletions and arbitrary adjacent transposition (both 01 → 10
+and 10 → 01 can occur) in the asymmetric Damerau-Levenshtein distance. In this subsection, we propose an optimal code for
+correcting a single deletion and s right-shifts of 0. We denote the adjacent transposition as 01 → 10 or 10 → 01, right-shift
+of 0 as 01 → 10 and left-shift of 0 as 10 → 01 throughout this subsection.
+Construction 2. The code C(n, a, b) is defined as follows.
+C(n, a, b) = {x ∈ Σn
+2 : VT(x) ≡ a mod (n + s + 1),
+n
+�
+i=1
+xi ≡ b mod 2, ψ(x) ∈ CH(n, 2s + 1)},
+where CH(n, 2s + 1) is a linear binary code capable of correcting errors with 2s + 1 distance.
+Proposition 3. (cf. [12]) A single adjacent transposition (01 → 10 or 10 → 01) in x is equivalent to a single substitution in
+ψ(x).
+Proposition 4. Suppose there are s right-shifts of 0 occurs in x, we have VT(x) − VT(x′) = s.
+Proof. Suppose a right-shift of 0 (01 → 10) occurs at the i-th 1 in x. The index of this 1 in x′ will be i − 1. Thus, for
+a single right-shift of 0, the change of the VT syndrome will be 1. If there are s right-shifts of 0 occurs in x, we have
+VT(x) − VT(x′) = s.
+Lemma 4. The following statements are true:
+
+8
+• Suppose a 0 is deleted before p-th 1 in x, and insert a 0 before (p + v)-th 1 to get ˆx. x can be obtained from ˆx by v
+adjacent transpositions.
+• Suppose a 1 is deleted after p-th 0 in x, and insert a 1 after (p − v)-th 0 to get ˆx. x can be obtained from ˆx by v
+adjacent transpositions.
+Proof. Denote the indexes of p-th 1, (p + 1)-th 1, . . . , (p + v − 1)-th 1 in x as ip, ip+1, . . . , ip+v−1. Then, we can see that
+the indexes of these 1s in ˆx should be ip − 1, ip+1 − 1, . . . , ip+v−1 − 1. Since 0 is inserted before (p + v)-th 1, we can swap
+the (ip+v−1 − 1)-th and ip+v−1-th bits and hence ˆx[ip+v−1,ip+v] = x[ip+v−1,ip+v]. Continuing this process, we can see that x
+can be recovered from ˆx by v adjacent transpositions. The case of deleting 1 is the same deleting 0, hence we can have the
+above two statements.
+Theorem 8. For all a ∈ [0, n + s] and b ∈ [0, 1], the code C(n, a, b) can correct a single deletion and s right-shifts of 0 with
+redundancy at most (1 + s) log(n + s + 1) + 1.
+Proof. Denote the retrieved sequence as x′ ∈ Σ2 through a single deletion and at most s right-shifts of 0. We first use the VT
+syndrome to correct the deletion and then apply the CH(n, 2s + 1) on ψ(x) to correct the right-shifts of 0.
+Further, let ∆ = VT(x) − VT(x′), w be the weight of x′ and p be the index of deletion. Then, let L0 be the number of 0s
+on the left of the deleted bits in x′ and R0 on its left. Similarly, denote L1, R1. We have the following cases when recover x
+by x′:
+• If x′ = Σn
+2, it means no deletion occurs in x and there are at most s right-shifts of 0. Based on Proposition 3, there are
+at most s substitutions in ψ(x). Hence we can recover ψ(x) by ψ(x′) since ψ(x) ∈ CH(n, 2s + 1), and then recover x.
+• If x′ = Σn−1
+2
+and suppose a 0 is deleted. From Proposition 4, then ∆ = R1 + k, where k is the actual number of
+right-shifts of 0s. We can first recover ˆx by inserting 0 in the rightmost index of (∆ − s) 1s. Since ∆ = R1 + k and we
+insert 0 in the rightmost index of (R1 + k − s) 1s. Based on the Case 1 of Lemma 4, we can have that there are at least
+(s − k) adjacent transpositions between ˆx and x. In addition, there are also k right-shifts of 0s occur in x. Therefore, x
+can be obtained from ˆx by total s adjacent transpositions. Hence, we can recover ψ(x) by ψ(ˆx) and then x.
+• If x′ = Σn−1
+2
+and suppose a 1 is deleted. From Proposition 4, then ∆ = p + R1 + k = w + L0 + k + 1. We recover ˆx
+by inserting 1 in the leftmost index of (∆ − w − s − 1) 0s. Similar as Case 2, since ∆ = w + L0 + k + 1 and we insert
+1 in the leftmost index of (L0 + k − s) 0s. Based on the Case 2 of Lemma 4, we can have that there are at least (s − k)
+adjacent transpositions between ˆx and x. Similarly, x can be obtained from ˆx by total s adjacent transpositions. Hence,
+we can recover ψ(x) by ψ(ˆx) and then x.
+
+9
+It is worth noticing that Case 1 and Case 2, 3 can be distinguished by the length of the retrieved sequence x′. Case 2 and
+Case 3 can distinguished based on the constraint of �n
+i=1 xi ≡ b mod 2, from where we can know the deleted bit is 0 or 1.
+There are three constraints on the sequence x ∈ C(n, a, b) including a VT code, a parity check bit and a linear binary
+(n, 2s + 1)-code. It can be easily shown that the redundancy of the code C(n, a, b) is log(n + s + 1) + s log n + 1. Thus, the
+redundancy of the code C(n, a, b) is at most (1 + s) log(n + s + 1) + 1.
+The decoding algorithm of the code C(n, a, b) for correcting a single deletion and s right-shifts of 0 is summarized in
+Algorithm 1.
+Algorithm 1: Decoding procedure of C(n, a, b)
+Input: Corrupted Sequence x′
+Output: Original Sequence x ∈ C(n, a, b)
+∆ = VT(x) − VT(x′), b = �n
+i=1 xi − �|x′|
+i=1 x′
+i and w = wt(x′).
+if |x′| = n then
+Recover ψ(x) by ψ(x′) and then x.
+else
+if b = 0 then
+Insert a 0 in the rightmost index of (∆ − s) 1s to get ˆx. Recover ψ(x) by ψ(ˆx) and then x.
+else
+Insert a 1 in the leftmost index of (∆ − w − s − 1) 0s to get ˆx. Recover ψ(x) by ψ(ˆx) and then x.
+end
+end
+Further, Construction 2 and Theorem 8 can be naturally extended to construct codes for correcting a single deletion, s+
+right-shifts of 0 and s− left-shifts of 0 with s = s+ + s−.
+Corollary 2. For all a ∈ [0, n + s] and b ∈ [0, 1], the code C2(n, a, b) such that
+C2(n, a, b) = {x ∈ Σn
+2 : VT(x) ≡ a mod (n + s + 1),
+n
+�
+i=1
+xi ≡ b mod 2, ψ(x) ∈ CH(n, 2s + 1)}.
+can correct a single deletion, s+ right-shifts of 0 and s− left-shifts of 0 with redundancy at most (1 + s) log(n + s + 1) + 1,
+where s = s+ + s−.
+Proof. Similar as Proposition 4, suppose there are at most s− left-shifts of 0s, the change of VT syndrome is VT(x) −
+VT(x′) = −s−. Suppose a 0 is deleted, and the same as the proof of Theorem 8 with the same notations, we can also have
+∆ = R1 + k+ − k−, where k+ and k− are actual number of right-shifts and left-shifts of 0 occur. Also, we still insert a 0 in
+
+10
+the index of rightmost of (∆ − s+ + s−) 1s to obtain ˆx. Based on the Case 1 of Lemma 4, we can have that there are at least
+((s+ − s−) − (k+ − k−)) adjacent transpositions between ˆx and x and there are k+ + k− adjacent transpositions occur in x.
+Therefore, the total number of adjacent transpositions that x can be obtained from ˆx is at most
+(s+ − s−) − (k+ − k−) + (k+ + k−) = s+ − s− + 2k− ≤ s+ + s− = s
+Hence, we can recover ψ(x) by ψ(ˆx) since there are at most s substitutions and then x. Also, the analysis of redundancy is
+the same as the proof of Theorem 8.
+Compare the results in [12], where the code for correcting a single deletion and s adjacent transpositions needs at most
+(1 + 2s) log(n + 2s + 1) redundancy. If we know the direction of these s adjacent transpositions containing s+ right-shifts
+of 0 and s− left-shifts of 0, the redundancy of the code can be further reduced to at most (1 + s) log(n + s + 1) + 1 where
+s = s+ + s−.
+IV. LIST-DECODABLE CODES FOR CORRECTING ASYMMETRIC DELETIONS AND ADJACENT TRANSPOSITIONS
+In this section, we aim to construct List-Decodable codes with low redundancy. For correcting t 0-deletions without s
+adjacent transpositions, Dolecek and Anatharam [17] proposed a well-known construction with optimal redundancy t log n.
+Inspired by this, we have the following construction:
+Construction 3. The construction CList
+t,s (n, K, a; p) is defined as the set of all x ∈ Σn
+2 such that
+w+1
+�
+i=1
+imφ(x)i ≡ am mod p, ∀m ∈ {1, . . . , K}.
+where the prime p such that p > 2n and a = (a1, a2, . . . , aK).
+Let x = (x1, . . . , xn) ∈ Σn
+2 be the original vector and x′ ∈ Σn−t
+2
+be the received vector after t 0-deletions and s adjacent
+transpositions. Hence, we obtain the vector φ(x′) and the corresponding a′ at the receiver. Let a′
+m = �w+1
+i=1 imφ(x′)i and
+a′′
+m = am − a′
+m, ∀m ∈ {1, . . . , K}.
+Proposition 5. Suppose there is only a single adjacent transposition occurs in x at the position of j-th 1, the change of
+syndrome a′′
+m can be shown as follows:
+1) 10 → 01:
+a′′
+m = (j + 1)m − jm mod p =
+m−1
+�
+i=0
+�m
+i
+�
+ji mod p
+2) 01 → 10:
+a′′
+m = jm − (j + 1)m mod p = −
+m−1
+�
+i=0
+�m
+i
+�
+ji mod p
+
+11
+Then, suppose t 0-deletions occur in the 0-run before the (d1, d2, . . . , dt)-th 1, respectively, where d1 ≤ d2 ≤ · · · ≤ dt.
+Also, ℓ (10 → 01) adjacent transpositions occur in (j1, j2, . . . , jℓ)-th 1 and r (01 → 10) adjacent transpositions occur in
+(k1, k2, . . . , kr)-th 1, respectively.
+Based on Proposition 5, considering all t 0-deletions and s adjacent transpositions and set K = t + s, we have a set of
+equations showing the change of syndromes for all m ∈ {1, . . ., t + s} as follows:
+a′′
+m ≡
+t
+�
+u=1
+dm
+u +
+m−1
+�
+i=0
+��m
+i
+��
+ℓ
+�
+v=1
+ji
+v −
+r
+�
+w=1
+ki
+w
+��
+mod p.
+(1)
+If there are only t 0-deletions without s adjacent transpositions, Dolecek and Anantharam [17] showed that the following
+system of equations has the unique solution.
+Lemma 5 (Dolecek and Anatharam [17]). Without s adjacent transpositions, (1) can be rewritten as the following set of
+constraints with t equations such that
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+a′′
+1 ≡ d1 + d2 + . . . + dt mod p,
+a′′
+2 ≡ d2
+1 + d2
+2 + . . . + d2
+t mod p,
+...
+a′′
+t ≡ dt
+1 + dt
+2 + . . . + dt
+t mod p.
+(2)
+which can uniquely determine the solution set {d1, d2, . . . , dt}, where p is a prime such that p > 2n and d1 ≤ d2 ≤ · · · ≤ dt.
+Following the technique in [17], if we can determine uniquely the solution set {d1, . . . , dt, j1, . . . , jℓ, k1, . . . , kr} of (1), we
+also can correct t 0-deletions and s adjacent transpositions with at most (t + s) log n bits of redundancy. However, the result
+is not known to us and is still open for future work.
+In this section, we focus on List-Decodable code CList
+t,s (n, κ, a; p) for correcting t 0-deletions and s adjacent transpositions.
+Set K = κ in Construction 3, where κ = max(t, s + 1) and p is a prime such that p > 2n. For the following system of
+equations, we can determine the solution set uniquely.
+Lemma 6. A set of constraints with s equations such that
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+b′′
+1 ≡ �ℓ
+v=1 j1
+v − �r
+w=1 k1
+w mod p,
+b′′
+2 ≡ �ℓ
+v=1 j2
+v − �r
+w=1 k2
+w mod p,
+...
+b′′
+s ≡ �ℓ
+v=1 js
+v − �r
+w=1 ks
+w mod p.
+(3)
+
+12
+is capable of uniquely determining the solution set {j1, . . . , jℓ, k1, . . . , kr}, where p is a prime such that p > 2n. Also, ℓ+r ≤ s,
+j1 < j2 < · · · < jℓ, k1 < k2 < · · · < kr and jv ̸= kw, ∀v ∈ {1, . . . , ℓ}, w ∈ {1, . . . , r}.
+We note that Lemma 6 is similar to Lemma 5. The only difference is that the coefficients of all terms in Lemma 5 are
+positive while the coefficients of all terms in Lemma 6 can be either positive or negative. Hence, we can use the same technique
+in Lemma 5 to prove Lemma 6.
+Proof. Define the polynomials
+σ+(x) =
+ℓ
+�
+v=1
+(1 − jvx)
+and
+σ−(x) =
+r
+�
+w=1
+(1 − kwx).
+Let σ(x) = �s
+m=0 σmxm be defined by
+σ(x) = σ+(x)/σ−(x) mod xs
+Then, we define σ∗(x) = σ(x) mod p.
+We also define
+S∗(x) =
+∞
+�
+m=1
+�
+ℓ
+�
+v=1
+jm
+v −
+r
+�
+w=1
+km
+w
+�
+xm.
+and S∗
+m = �ℓ
+v=1 jm
+v − �r
+w=1 km
+w mod p.
+Then, we have Newton’s identities over GF(p) as follows
+σ∗(x)S∗(x) + x(σ∗(x))′ = 0
+u−1
+�
+m=0
+σ∗
+mS∗
+u−m + uσ∗
+u = 0, u ≥ 1.
+(4)
+where (σ∗(x))′ is derivative of σ∗(x). (see [35, Lemma 10.3] for details)
+Using the similar technique as the proof of Lemma 5, from (4), σ∗
+m can be recursively obtained by {S∗
+1, . . . , S∗
+m} and
+{σ∗
+1, . . . , σ∗
+m−1}, where {S∗
+1, . . . , S∗
+m} = {b′′
+1, . . . , b′′
+m}, which follows that all the coefficients of the polynomial σ∗(x) =
+�s
+m=0 σ∗
+mxm mod p are known. Further, we know that the polynomial σ∗(x) has at most s solutions by Lagrange Theorem.
+Denote I0 = {j1, . . . , jℓ, k1, . . . , kr} with the value of each element in I0 is less than p and let Im = {j1 + mp, . . . , jℓ +
+mp, k1+mp, . . . , kr+mp} be one of the incongruent solution sets of I0. We can have I0∩Im = ∅ due to p > 2n, which follows
+that all incongruent solutions are distinguishable. Therefore, we can conclude that the solution set {j1, . . . , jℓ, k1, . . . , kr} is
+unique.
+Theorem 9. The list-decodable code CList
+t,s (n, κ, a; p) has redundancy κ log n, where κ = max(t, s + 1) and prime p > 2n. If
+there are at most t 0-deletions and s adjacent transpositions, we can do list-decoding with list size O(nmin(t,s+1)).
+
+13
+Proof. Let x = (x1, . . . , xn) ∈ Σn
+2 be the original vector and x′ be the received vector after t 0-deletions and s single adjacent
+transpositions. Hence, we can compute φ(x′) and a′ from x′. Also, we can obtain a′′ = a′ − a, where a′′ = {a′′
+1, . . . , a′′
+κ}.
+Suppose t ≥ s + 1 and expand (1). We have the following set of equations with κ = t:
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+a′′
+1 ≡ �t
+u=1 du + (ℓ − r) mod p,
+a′′
+2 ≡ �t
+u=1 d2
+u + (ℓ − r) + 2(�ℓ
+v=1 j1
+v − �r
+w=1 k1
+w) mod p,
+...
+a′′
+t ≡ �t
+u=1 dt
+u + (ℓ − r) + t(�ℓ
+v=1 j1
+v − �r
+w=1 k1
+w)
++ · · · + t(�ℓ
+v=1 jt−1
+v
+− �r
+w=1 kt−1
+w
+) mod p.
+(5)
+Recall that we can decode uniquely if we can determine the unique solution set of (5). However, the method to solve (5)
+uniquely is not known to us. We know that, given e = {e1, . . . , es+1}, we can solve the following equations uniquely.
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+e1 ≡ ℓ − r mod p,
+e2 ≡ (ℓ − r) + 2(�ℓ
+v=1 j1
+v − �r
+w=1 k1
+w) mod p,
+...
+es+1 ≡ (ℓ − r) + (s + 1)(�ℓ
+v=1 j1
+v − �r
+w=1 k1
+w)
++ · · · + (s + 1)(�ℓ
+v=1 js
+v − �r
+w=1 ks
+w) mod p.
+(6)
+Indeed, denote e′ = {e′
+1, . . . , e′
+s+1} with me′
+m = em − �m−1
+i=1
+�� m
+i−1
+�
+e′
+i
+�
+for all m ∈ {2, . . ., s + 1} and e���
+1 = e1, we can
+rearrange (6) to be similar to Lemma 6 as follows.
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+e′
+1 ≡ ℓ − r mod p,
+e′
+2 ≡ �ℓ
+v=1 j1
+v − �r
+w=1 k1
+w mod p,
+...
+e′
+s+1 ≡ �ℓ
+v=1 js
+v − �r
+w=1 ks
+w mod p.
+(7)
+Therefore, based on Lemma 6, we can obtain the unique solution set {j1, . . . , jℓ, k1, . . . , kr} from (7).
+Once the solution set {j1, . . . , jℓ, k1, . . . , kr} is obtained, we can compute the following values {es+2, . . . , et}.
+em =
+m−1
+�
+i=0
+��m
+i
+��
+ℓ
+�
+v=1
+ji
+v −
+r
+�
+w=1
+ki
+w
+��
+mod p.
+(8)
+where m ∈ {s + 2, . . . , t}.
+
+14
+Denote a∗ = {a∗
+1, . . . , a∗
+t } with a∗
+m = a′′
+m − em, ∀m ∈ {1, . . ., t}. Substituting (6) and (8) into (5), we obtain the following
+set of equations.
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+a∗
+1 ≡ �t
+u=1 du mod p,
+a∗
+2 ≡ �t
+u=1 d2
+u mod p,
+...
+a∗
+t ≡ �t
+u=1 dt
+u mod p.
+(9)
+The set of equations (9) provides the unique solution set {d1, . . . , dt} by Lemma 5. Therefore, the unique solution of all
+positions of 0-deletions and adjacent transpositions {d1, . . . , dt, j1, . . . , jℓ, k1, . . . , kr} can be obtained. So, for each set of
+s+1 values {e1, . . . , es+1}, we can obtain the set {d1, . . . , dt, j1, . . . , jℓ, k1, . . . , kr}. There are ps+1 sets of these values. One
+of these sets corresponds to the true value of x and gives us the correct vector x. So, we can do list-decoding with the list
+size O(ns+1) since p = O(n). Moreover, the size of the list-decodable code CList
+t,s (n, κ, a; p) with κ = t is at least 2n/(4n)t,
+that is, we need at most κ log n bits of redundancy to construct the code CList
+t,s (n, κ, a; p).
+When t < s + 1, we can do similarly to the case t ≤ s + 1. In this case, we can do list-decoding with the list-size O(nt).
+The size of the code CList
+t,s (n, κ, a; p) is at least 2n/(4n)s+1.
+Then, we can conclude that the list-decodable code CList
+t,s (n, κ, a; p) can correct t 0-deletions and s adjacent transpositions
+with list size at most O(nmin(t,s+1)) and has redundancy κ log n+O(1), where both t, s are constant and κ = max(t, s+1).
+The decoding algorithm of the list-decodable code CList
+t,s (n, κ, a; p) for correcting t 0-deletions and s adjacent transpositions
+is summarized in Algorithm 2, where t > s + 1.
+Algorithm 2: List decoding procedure
+Input: Corrupted Sequence x′ ∈ Σn−t
+2
+Output: O(ns+1) possible sequences, including the original codeword x ∈ CList
+t,s (n, κ, a; p)
+Compute φ(x′) based on x′ and compute a′′ to obtain (5).
+for e = (e1, . . . , es+1) such that ei ∈ {0, 1, . . ., p − 1}, ∀i ∈ {1, . . . , s + 1} do
+Get the solution set {j1, . . . , jℓ, k1, . . . , kr} by (6) and (7).
+Compute em from the solution set {j1, . . . , jℓ, k1, . . . , kr} using (8) for each s + 2 ≤ m ≤ t. Compute
+a∗
+m = a′′
+m − em. Solve (9) to obtain the unique solution set {d1, . . . , dt}.
+end
+For each fixed e, we can recover φ(x) from φ(x′) by a set of error positions {d1, . . . , dt, j1, . . . , jℓ, k1, . . . , kr} and
+then output x.
+
+15
+Next, we will present the result for a special case t = 1.
+Corollary 3. The list-decodable code CList
+1,s (n, s + 1, a; p) can correct a single 0-deletion and s adjacent transpositions with
+list size at most 2s and has redundancy (s + 1) log n + O(1).
+Proof. When t = 1, It can be noticed that when the deletion position is determined, means d is known. Since l, r ∈ {1, . . . , s}
+and a′′
+1 ≡ d + (ℓ − r) mod p, hence there are 2s choice for d, which means that the list size of CList
+1,s (n, s + 1, a; p) is at most
+2s.
+The above code CList
+1,s (n, s + 1, a; p) is capable of correcting a single 0-deletion and s adjacent transpositions with constant
+list size at most 2s and has redundancy (s + 1) log n + O(1). The list size is constant 2s, which is less than the list size O(n)
+when we directly substitute t = 1 to Theorem 9.
+V. CODES FOR CORRECTING LIMITED-MAGNITUDE BLOCKS OF 0-DELETIONS AND ADJACENT TRANSPOSITIONS
+In this section, we focus on studying the error of t blocks of asymmetric deletions with ℓ-limited-magnitude and s adjacent
+transpositions. t blocks of asymmetric deletions with ℓ-limited-magnitude denotes that there are at most t blocks of 0s are
+deleted with the length of each block is at most ℓ. Therefore, at most tℓ 0s are deleted and these t blocks of 0-deletions may
+occur in at most t 0 runs.
+For the sake of convenience in the following paper, we append a bit 1 at the end of x and denote it as x1. Since the sequence
+x1 always ends with 1, x1 can be always written as x1 = 0u110u210u3 . . . 0uw1, where w = wt(x1). In addition, we revisit
+the definition of function φ : Σn
+2 → Σw and φ(x)
+def= (u1, u2, u3, . . . , uw) �� Σw. Then, combining with Proposition 2, we can
+have that the length of each 0 run increase by at most 1 and decrease by at most tℓ + 1 through t blocks of 0-deletions with
+ℓ-limited-magnitude and s adjacent transpositions. Then, the definition of t blocks of 0-deletions with ℓ-limited-magnitude and
+s adjacent transpositions is provided as follows.
+Definition 4. Define the error ball B(n, t, k+, k−) such that
+B(n, t, k+, k−) = {u ∈ Σn
+q : −k− ≤ ui ≤ k+, wt(u) ≤ t}.
+where at most t entries increase by at most k+ and decrease by at most k− for a sequence with length n.
+Definition 5. t blocks of asymmetric deletions with ℓ-limited-magnitude and s adjacent transpositions denote that given a
+sequence x ∈ Σn
+2 , the retrieved sequence x′ through this type of error can be written as φ(x′1) = φ(x1) + v, where
+v ∈ B(w, t + 2s, 1, tℓ + 1) and w = wt(x′1) = wt(x1)
+
+16
+Example 6. Suppose we have x = 0100101001 ∈ Σ10
+2
+with ℓ = 2, t = 3 and s = 1, then φ(x1) = 12120. If the retrieved
+sequence x′ = 0110110 ∈ Σ6
+2 and the corresponding φ(x′1) = 10101, by comparing φ(x1) and φ(x′1), we can see
+v = (0, −2, 0, −2, 1) ∈ B(5, 5, 1, 7).
+Denote Φ be the set of mapping Σn
+2 by the function φ and Σn
+2 is the set containing all binary sequences with length n.
+Lemma 7. The cardinality of Φ is:
+|Φ| =
+n+1
+�
+w=1
+�
+n
+w − 1
+�
+= 2n.
+(10)
+Proof. For a binary sequence x ∈ Σn
+2, the corresponding sequence φ(x1) is with length w = w(x1) and wt(φ(x1)) = n+1−w.
+Also, the cardinality of Φ can be considered the number of ways of arranging n + 1 − w indistinguishable objects in w
+distinguishable boxes. Thus, we can get the cardinality of Φ as shown in Lemma 7.
+On the other side, since the mapping function φ is a one-to-one mapping function, the cardinality of Φ should be the same
+as |Σn
+2| = 2n.
+Proposition 6. (cf. [36]) The code C(n, t, ℓ, s) for correcting t blocks of 0-deletions with ℓ-limited-magnitude and s adjacent
+transpositions is equivalent to a packing to Σw by the error ball B(w, t+2s, 1, tℓ+1), where w = wt(x) and x ∈ C(n, t, ℓ, s).
+A. Non-systematic Code Construction
+In this section, we will provide a non-systematic construction for the code capable of correcting t blocks of 0-deletions with
+ℓ-limited-magnitude and s adjacent transpositions. Then, we present the decoding algorithm of this code and a lower bound
+of the code size.
+Construction 4. The code C(n, t, ℓ, s) is defined as
+C(n, t, ℓ, s) = {x ∈ Σn
+2 : φ(x1) mod p ∈ Cp, wt(φ(x1)) = n + 1 − w},
+where w = wt(x1) and Cp is a code over Σp with p is the smallest prime larger than tℓ + 2.
+Lemma 8. C(n, t, ℓ, s) is capable of correcting t blocks of 0-deletions with ℓ-limited-magnitude and s adjacent transpositions
+for x ∈ C(n, t, ℓ, s) if Cp is capable of correcting t + 2s symmetric errors for φ(x1).
+Lemma 9. ( [37], Theorem 10 ) Let p be a prime such that the distance 2 ≤ d ≤ p⌈m/2⌉−1 and n = pm − 1. Then, there
+exists a narrow-sense [n, k, d]-BCH code Cp over Σp with
+n − k = ⌈(d − 1)(1 − 1/p)⌉m.
+
+17
+Theorem 10. Let p be the smallest prime such that p ≥ tℓ + 2, w = pm − 1, w = wt(x1) and Cp is a primitive narrow-sense
+[w, k, 2(t + 2s) + 1]-BCH code with w − k = ⌈2(t + 2s)(1 − 1/p)⌉m, the code C(n, t, ℓ, s) such that
+C(n, t, ℓ, s) = {x ∈ Σn
+2 : φ(x1) mod p ∈ Cp, wt(φ(x1)) = n + 1 − w}.
+is capable of correcting t blocks of 0-deletions with ℓ-limited-magnitude and s adjacent transpositions.
+Proof. Let x ∈ C(n, t, ℓ, s) be a codeword, and x′ be the output through the channel that has t blocks of 0-deletions with
+ℓ-limited-magnitude and s adjacent transpositions. Let z′ = φ(x′1) mod p, where p is the smallest prime larger than tℓ + 2.
+Run the decoding algorithm of Cp on z′ and output z∗. Thus, z∗ is also a linear code in Cp and it can be shown that
+z∗ = φ(x1) mod p. Denote ǫ′ = (z′ − z∗) mod p, we can have that
+(φ(x′1) − φ(x1)) mod p = (z′ − z∗) mod p = ǫ′.
+(11)
+and the error vector ǫ satisfies
+ǫi =
+
+
+
+
+
+
+
+
+
+ǫ′
+i,
+if 0 ≤ ǫ′
+i ≤ 1
+ǫ′
+i − p,
+otherwise
+.
+(12)
+Hence, the output is φ(x1) = φ(x′1) − ǫ and then recover x from φ(x1).
+The detailed decoding steps are shown in Algorithm 3.
+Algorithm 3: Decoding Algorithm of C(n, t, ℓ, s)
+Input: Retrieved sequence x′
+Output: Decoded sequence x ∈ C(n, t, ℓ, s).
+Initialization: Let p be the smallest prime larger than tℓ + 2. Also, append 1 at the end of x′ and get φ(x′1).
+Step 1: z′ = φ(x′1) mod p. Run the decoding algorithm of Cp on z′ to get the output z∗.
+Step 2: ǫ′ = (z′ − z∗) mod p and then ǫ. φ(x1) = φ(x′1) − ǫ.
+Step 3: Output x1 = φ−1(φ(x1)) and then x.
+Example 7. Suppose x = 0100101001 and x′ = 0110110 ∈ Σ6
+2 with ℓ = 2, t = 3 and s = 1. Since the retrieved sequence
+x′ = 0110110, then φ(x′1) = 10101 and z′ = φ(x′) mod 11 = 10101, where p = 11 is smallest prime such that p ≥ tℓ + 2.
+Run the decoding algorithm of Cp on z′ ∈ Cp, we have the output sequence z∗ = 12120. Hence ǫ′ = (z − z∗) mod 11 =
+(0, 9, 0, 9, 1) and ǫ = (0, −2, 0, −2, 1). Thus, the output of the decoding algorithm φ(x1) = φ(x′1) − ǫ = (1, 0, 1, 0, 1) −
+(0, −2, 0, −2, 1) = (1, 2, 1, 2, 0). Finally, x1 = 01001010011 and x = 0100101001.
+Next, we will present a lower bound of the size of C(n, t, ℓ, s).
+
+18
+Theorem 11. The size of the code C(n, t, ℓ, s) in Theorem 10 is bounded by
+|C(n, t, ℓ, s)| ≥
+2n
+p(n + 1)⌈2(t+2s)(1−1/p)⌉ .
+where p is the smallest prime larger than tℓ + 2.
+Proof. Denote z = φ(x1) mod p. φ(x1) can be written as φ(x1) → (z, a) such that φ(x1) = z + p · a, where a is a
+vector with the same length as φ(x1) and z. Further, since z ∈ Cp and Cp is a linear code, the code Cp with length w can be
+considered as a set which is obtained by Σw
+p partitioned into pw−k classes.
+Denote φ(x1)w as the φ(x1) with length w. Thus, for any fixed number of weight w, the cardinality of φ(x1)w such that
+φ(x1)w mod p ∈ Cp with length w is:
+|φ(x1)w| =
+�
+n
+w−1
+�
+pw−k .
+Then, the size of the code C(n, t, ℓ, s) in Theorem 10 can be shown as:
+|C(n, t, ℓ, s)| =
+n+1
+�
+w=1
+|φ(x1)w| =
+n+1
+�
+w=1
+�� n
+w−1
+�
+pw−k
+�
+≥
+�n+1
+w=1
+�
+n
+w−1
+�
+pn+1−k
+=
+2n
+pn+1−k .
+(13)
+From Lemma 9 and Theorem 10, let d = 2(t + 2s) + 1 and m = logp(n + 1).
+pn−k+1 = p⌈2(t+2s)(1−1/p)⌉·logp(n+1)+1 = p(n + 1)⌈2(t+2s)(1−1/p)⌉.
+(14)
+Therefore, from (13) and (14), the size of the code C(n, t, ℓ, s) in Theorem 10 is bounded by
+|C(n, t, ℓ, s)| ≥
+2n
+p(n + 1)⌈2(t+2s)(1−1/p)⌉ .
+where p is the smallest prime larger than tℓ + 2.
+B. Systematic Code Construction
+In the previous subsection, we propose a non-systematic code C(n, t, ℓ, s) for correcting t blocks of 0-deletions with ℓ-limited-
+magnitude and s adjacent transpositions. In this subsection, we will provide the efficient encoding and decoding function based
+on the code C(n, t, ℓ, s) presented in Theorem 10.
+1) Efficient Encoding: Before providing the efficient systematic encoding algorithm, we now introduce a useful lemma
+proposed in [38] for encoding balanced sequences efficiently. The balanced sequence denotes the binary sequence with an
+equal number of 0s and 1s, which will be used for distinguishing the boundary of redundancy.
+
+19
+Lemma 10. (cf. [38]) Given the input x ∈ Σk
+2, let the function s′ : Σk
+2 → Σn
+2 such that s′(x) ∈ Σn
+2 is a balanced sequence,
+where n = k + log k.
+Definition 6. Given the input x ∈ Σk
+2, let the function s : Σk
+2 → Σn′
+2 such that s(x) ∈ Σn′
+2 whose first bit is 1 and s(x)[2,n′]
+is balanced sequence with (n′ − 1)/2 0s and (n′ − 1)/2 1s, where n′ = k + log k + 1.
+An adjacent transposition can be considered as two substitutions, hence the maximum total number of deletions and
+substitutions in the t blocks of 0-deletions with ℓ-limited-magnitude and s adjacent transpositions is r = tℓ+2s. The following
+lemma is used for correcting deletions, insertions and substitutions up to r = tℓ + 2s in a binary sequence.
+Lemma 11. (cf. [39]) Let t, ℓ, s be constants with respect to k. There exist an integer a ≤ 22r log k+o(log k) and a labeling
+function fr : Σk
+2 → Σ2Rr(k), where Rr(k) = O(r4 log k) such that {(x, a, fr(x) mod a) : x ∈ Σk
+2} can correct deletions,
+insertions and substitutions up to r = tℓ + 2s. Let gr(x) = (a, fr(x) mod a) ∈ Σ4r log k+o(log k)
+2
+for given x ∈ Σk
+2.
+Next, we define the mapping function from non-binary to binary.
+Definition 7. Given the input x ∈ Σk
+2, define the function b : Σk
+p → Σn
+2 such that b(u)[i·⌈log p⌉+1,(i+1)·⌈log p⌉] is the binary
+form of ui, where n = k · ⌈log p⌉.
+Given the parameters t, ℓ and s, let p be the smallest prime larger than tℓ + 2 and Cp in Lemma 9 be the p-ary primitive
+narrow-sense [n, k, 2(t + 2s) + 1]-BCH codes.
+Definition 8. Define the labeling function as g : Σk
+p → Σn−k
+p
+such that (x, g(x)) is a p-ary primitive narrow-sense [n, k, 2(t +
+2s) + 1]-BCH codes, where n = k + ⌈2(t + 2s)(1 − 1/p)⌉m and n = pm − 1.
+Suppose the input sequence is c ∈ Σk
+2, and we have φ(c1) with length rc = wt(c1). Then, let c′ = φ(c1) mod p ∈ Σrc
+p ,
+where p is the smallest prime larger than tℓ + 2, and append 0k+1−rc at the end of c′. Hence, we denote ¯c ∈ Σk+1
+p
+=
+(c′, 0k+1−rc).
+Next, encode ¯c via the labeling function g of the p-ary primitive narrow-sense [n, k, 2(t + 2s) + 1]-BCH code and output
+the redundancy part g(¯c). We map the redundancy part g(¯c) into binary sequence b(g(¯c)) and make b(g(¯c)) to the balanced
+sequence s(b(g(¯c))). Then, we prepend two 1s as the protecting bits at the beginning of s(b(g(¯c))) and denote h1(¯c) =
+(1, 1, s(b(g(¯c)))).
+Further, we need to protect the redundancy part h1(¯c). The idea is to apply the code in Lemma 11 on h1(¯c) since the code
+in Lemma 11 is capable of correcting at most tℓ + 2s deletions and substitutions. Then, we output gr(h1(¯c)). In addition,
+make gr(h1(¯c)) to balanced sequence s(gr(h1(¯c))) and repeat its each bit 2tℓ + 3 times. Let h2(¯c) = Rep2tℓ+3s(gr(h1(¯c))),
+
+20
+where Repkx is the k-fold repetition of x.
+Finally, we have the output Enc(c) = (c, h(c)), where h(c) = (h1(¯c), h2(¯c)). The detailed encoding steps are summarized
+in the following Algorithm 4.
+Algorithm 4: Encoding Algorithm
+Input: c ∈ Σk
+2
+Output: Encoded sequence Enc(c) ∈ ΣN
+2
+Initialization: Let p be the smallest prime larger than tℓ + 2.
+Step 1: Append 1 at the end of c and get φ(c1) with length rc = wt(c1).
+Step 2: c′ = φ(c1) mod p ∈ Σrc
+p . Append 0k+1−rc at the end of c′, then ¯c = (c′, 0k+1−rc).
+Step 3: Encode ¯c via Cp and output g(¯c). Mapping g(¯c) to balanced binary sequence s(b(g(¯c))) and introduce
+protecting bits h1(¯c) = (1, 1, s(b(g(¯c)))).
+Step 4: Protect h1(¯c) via gr and obtain the total redundancy h(c) = (h1(¯c), h2(¯c)).
+Step 5: Output Enc(c) = (c, h(c)) ∈ ΣN
+2 .
+Lemma 12. Given a sequence c ∈ Σk
+2, Algorithm 4 outputs an encoded sequence Enc(c) ∈ ΣN
+2 capable of correcting t blocks
+of 0-deletions with ℓ-limited-magnitude and s adjacent transpositions.
+Therefore, the redundancy of the code h(c) = (h1(¯c), h2(¯c)) via this encoding process can be shown as follows.
+Theorem 12. The total redundancy of the code Enc(c) ∈ ΣN
+2 by given input c ∈ Σk
+2 is
+N − k = ⌈2(t + 2s)(1 − 1/p)⌉ · ⌈log p⌉
+log p
+log(N + 1) + O(log log N).
+where p is smallest prime such that p ≥ tℓ + 2.
+Proof. Let m = logp(N + 1), hence N = pm − 1. The lengths of the redundancy parts are as follows:
+• n′′
+1 is the length of g(¯c): n′′
+1 = ⌈2(t + 2s)(1 − 1/p)⌉m;
+• n′
+1 is the length of b(g(¯c)): n′
+1 = n′′
+1 · ⌈log p⌉;
+• n1 is the length of h1(¯c): n1 = n′
+1 + log n′
+1 + 3;
+• n′′
+2 is the length of gr(h1(¯c)): n′′
+2 = 4(tℓ + 2s) log n1 + log n1;
+• n′
+2 is the length of s(f0(h1(¯c))): n′
+2 = n′′
+2 + log n′′
+2 + 1;
+• n2 is the length of h2(¯c): n2 = (2tℓ + 3)n′
+2;
+
+21
+Based on the above statement, we can see that N − k = n1 + n2, where
+n′
+1 = (⌈2(t + 2s)(1 − 1/p)⌉m) · ⌈log p⌉
+with m = logp(N + 1). Hence, we have
+n′
+1 = ⌈2(t + 2s)(1 − 1/p)⌉ · ⌈log p⌉
+log p
+log(N + 1)
+Since both t, p and s are constants, then log n′
+1 = O(log log N) and n2 = O(log log N). Therefore, the total redundancy of
+the code Enc(c) ∈ ΣN
+2 given the input c ∈ Σk
+2 can be shown as the Theorem 12.
+2) Decoding Algorithm: Without loss of generality, suppose the encoded sequence Enc(c) ∈ ΣN
+2 is transmitted through the
+t blocks of 0-deletions with ℓ-limited-magnitude and s adjacent transpositions channel, and we have the retrieved sequence
+d ∈ ΣN−tℓ
+2
+. In this subsection, we will introduce the decoding algorithm for obtaining Dec(d) ∈ Σk
+2 by given d ∈ ΣN−tℓ
+2
+.
+First, we need to distinguish where the redundancy part begins. Since the error type is at most t blocks of 0-deletions with
+ℓ-limited-magnitude and s adjacent transpositions, the number of 1s in d is the same as that of in Enc(c). Thus, we can
+count the number of 1s from the end of d to find the beginning of the redundancy since the redundancy part is the balanced
+sequence.
+Hence, we find the (n2 + 2tℓ + 3)/2-th 1 and (n1/2 + n2/2 + tℓ + 3)-th 1 from the end of d and denote their entries as
+ir2 and ir1, respectively. For the subsequence d[ir2,N−tℓ], since there are at most tℓ 0s deletions and s adjacent transpositions
+occur in Enc(c)[N−n2+1,N], the (2tℓ + 3)-fold repetition code can help recover s(gr(h1(¯c))). Further, we can obtain parity
+bits gr(h1(¯c)).
+For the subsequence d[ir1,ir2−1], there are also at most tℓ 0-deletions and 2s substitutions occur in Enc(c)[N−n1−n2+1,N−n2].
+The recovered parity bits gr(h1(¯c)) can help recover h1(¯c). Further, we remove the two 1 bits at the beginning of h1(¯c) and
+get the g(¯c) from h1(¯c) = s(b(g(¯c))).
+Finally, denote z = (φ(d[1,ir1−1], 1), 0k+1−rc) and z′ = z mod p, where rc is the length of φ(d[1,ir1−1], 1) and k =
+N − n1 − n2. Then, the following decoding steps are the same as Algorithm 3 where z′ is the input of Step 1 of Algorithm 3.
+The only difference is we need to first remove 0k+1−rc at the end before the last step of φ−1. Therefore, the main steps for
+decoding d ∈ ΣN−tℓ
+2
+is summerized in Algorithm 5.
+3) Time Complexity: For the encoding algorithm, it can be easily shown that the time complexity is dominated by the p-ary
+narrow-sense BCH code and the code in Lemma 11, which is O(tn log n + (log n)2(tℓ+2s)+1).
+For the decoding algorithm, the time complexity is also dominated by the decoding of the p-ary narrow-sense BCH code
+and decoding for the code in Lemma 11. Therefore, the total time complexity of decoding is O(tn + (log n)tℓ+2s+1).
+
+22
+Algorithm 5: Decoding Algorithm
+Input: d ∈ ΣN−tℓ
+2
+Output: Decoded sequence Dec(d) ∈ Σk
+2
+Initialization: Let p be the smallest prime larger than tℓ + 2.
+Step 1: Find the (n2 + 2tℓ + 3)/2-th 1 and (n1/2 + n2/2 + tℓ + 3)-th 1 from the end of d and denote their entries as
+ir2 and ir1, respectively.
+Step 2: Recover s(gr(h1(¯c))) from d[ir2,N−tℓ] and then get gr(h1(¯c)).
+Step 3: Recover h1(¯c) via gr(h1(¯c)) and then obtain h1(¯c).
+Step 4: Denote z′ = (φ(d[1,ir1−1], 1), 0k+1−rc) mod p. Input z′ to Step 1 of Algorithm 3 and run the remaining steps
+of Algorithm 3.
+Step 5: Output Dec(d).
+VI. CONCLUSION
+In this paper, motivated by the errors in the DNA data storage and flash memories, we presented codes for correcting
+asymmetric deletions and adjacent transpositions. We first present three uniquely-decodable codes for different types of
+asymmetric deletions and adjacent transpositions. We then construct a list-decodable code for correcting asymmetric deletions
+and adjacent transpositions with low redundancy. At last, we present the code for correcting t blocks of 0-deletions with
+ℓ-limited-magnitude and s adjacent transpositions.
+However, there still remain some interesting problems.
+• Construct codes that are capable of correcting symmetric t deletions and s adjacent transpositions with low redundancy.
+• Construct codes that are capable of correcting t deletions/insertions + k substitutions + s adjacent transpositions.
+• Construct codes for Damerau-Levenshtein distance for larger number of errors, not only constant t and s.
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+

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+On the Inconsistencies of Conditionals Learned by Masked Language
+Models
+Tom Young
+Yang You
+School of Computing,
+National University of Singapore
+tomyoung@nus.edu.sg
+youy@comp.nus.edu.sg
+Abstract
+Learning to predict masked tokens in a se-
+quence has been shown to be a powerful
+pretraining objective for large-scale language
+models. After training, such masked language
+models can provide distributions of tokens con-
+ditioned on bidirectional context.
+In this short draft, we show that such bidirec-
+tional conditionals often demonstrate consider-
+able inconsistencies, i.e., they can not be de-
+rived from a coherent joint distribution when
+considered together. We empirically quantify
+such inconsistencies in the simple scenario of
+bigrams for two common styles of masked lan-
+guage models: T5-style and BERT-style 1. For
+example, we show that T5 models often con-
+fuse its own preference regarding two similar
+bigrams.
+Such inconsistencies may represent a theoreti-
+cal pitfall for the research work on sampling
+sequences based on the bidirectional condi-
+tionals learned by BERT-style MLMs. This
+phenomenon also means that T5-style MLMs
+capable of infilling will generate discrepant
+results depending on how much masking is
+given, which may represent a particular trust
+issue.
+1
+Introduction
+Pretraining objectives of large language models
+can be roughly divided into two categories. First,
+vanilla next token prediction (Brown et al., 2020)
+aims to learn the distribution of the next token in a
+sequence given the context to the left. Second, the
+masked language modeling (MLM) objective (De-
+vlin et al., 2018; Raffel et al., 2020), which masks
+out a portion of the tokens in a sequence and asks
+the model to predict them, aims to learn the distri-
+bution of one or more tokens given bidirectional
+context.
+1https://github.com/tomyoung903/MLM_
+inconsistencies
+While the major breakthrough, aka, GPT3
+(Brown et al., 2020) was demonstrated using
+vanilla next token prediction, recent work (Tay
+et al., 2022; Zeng et al., 2022; Bavarian et al., 2022)
+has hinted that incorporating the masked language
+modeling objective may be highly beneficial. In ad-
+dition, (Tay et al., 2022) has demonstrated that such
+bidirectional conditionals provide strong infilling
+capabilities.
+One may notice that, unlike the unidirectional
+conditional distributions that vanilla next token pre-
+diction learns, the bidirectional conditionals that
+MLMs learn are overly abundant in terms of rep-
+resenting a coherent joint distribution. Therefore,
+they are not guaranteed to be self-consistent (see
+Chapter 2).
+A very simple example for such inconsisten-
+cies is shown in Figure 1. In this example, we
+obtain the bidirectional conditional distributions
+that the T5 model learned using two input masked
+sequences. The two similar sequences are designed
+with a small difference, in order to examine if the
+resulting conditionals satisfy a basic law of prob-
+abilities (hold consistency). Results clearly show
+otherwise. We design experiments to quantify such
+inconsistencies in Chapter 3.
+One interesting line of research in the litera-
+ture focused on whether and how the bidirectional
+conditionals that BERT-style MLMs provide can
+be used to construct the joint probability of a se-
+quence in a principled manner (Goyal et al., 2021;
+Ghazvininejad et al., 2019; Wang et al., 2019), just
+like vanilla next token prediction models. But the
+numerous papers on this topic have overlooked
+the concern of inconsistencies. (Yamakoshi et al.,
+2022) stated that “any deviations (supposedly) tend
+to be negligible with large datasets”. The experi-
+ments shown in Chapter 4 demonstrate that this is
+not the case at all. We thus posit that addressing the
+consistency issue should be treated as the first step
+in modeling the joint distribution with BERT-style
+arXiv:2301.00068v1  [cs.CL]  30 Dec 2022
+
+MLMs.
+2
+Why inconsistencies can occur in
+MLMs
+For a set of conditional distributions to be self-
+consistent, they need to be able to be derived from
+a single coherent joint distribution.
+One essential reason for the inconsistencies to
+occur among the conditionals provided by a trained
+MLM is that the number of conditionals it can
+calculate far exceeds the number of degrees of free-
+dom of a joint distribution.
+Consider a sequence of length L and with vo-
+cabulary V , the joint distribution of the tokens in
+such a sequence is defined by |V |L probabilities
+that sum to 1. Therefore, the number of degrees of
+freedom (D) of such a joint distribution is given
+by:
+Djoint = |V |L − 1,
+(1)
+Vanilla next token prediction models or MLMs
+essentially learn conditionals that predict some
+tokens in the sequence given others. Such con-
+ditional probabilities and probabilities from the
+joint distribution can be linearly derived from each
+other. Therefore, each free conditional that the
+language model is capable of specifying provides
+an additional constraint on the joint distribution.
+One can easily verify that a vanilla next token pre-
+diction based language model provides |V |L − 1
+free conditionals 2 to just exactly determine the
+joint distribution.
+Therefore, a vanilla next to-
+ken prediction model (no matter how it is trained,
+or even untrained) would never suffer from self-
+inconsistencies.
+MLMs, which can provide distributions of
+masked tokens given bidirectional context, could
+specify far more free conditionals.
+Even for the simplest case, where the MLM pre-
+dicts the distribution of only 1 (masked) token
+given L − 1 other (unmasked) tokens in the se-
+quence, the total number of free conditionals (N)
+is
+Nmlm(1) = L × (|V |L − |V |L−1),
+(2)
+Just Nmlm(1) is already far larger than Djoint.
+We leave the discussions for Nmlm(k) for later
+2A single softmax operation over V essentially gives |V |−
+1 free conditionals. Here we call conditionals free when they
+can be assigned any values decided by an underlying neural
+network.
+work. This fact sets up room for there to be in-
+consistencies among the conditionals an MLM pro-
+vides.
+We explain our strategies and quantification
+methods for diagnosing T5-style and BERT-style
+MLMs in the next 2 sections.
+3
+Diagnosing T5-style MLMs
+T5-style MLMs are capable of modeling the dis-
+tribution of segments of variable length in a given
+bidirectional context. Here we use the simple bi-
+gram scenario to expose the inconsistencies that
+exist among such distributions. Consider two bi-
+grams x1x21 and x1x22 that share a same token x1
+in the first position, the conditional distributions
+concerning such two bigrams should satisfy
+p(x21|x1)
+p(x22|x1) = p(x1x21)
+p(x1x22)
+(3)
+The left hand side can be obtained by only mask-
+ing the second token, leaving x1 in the context.
+While the right hand side can be obtained by mask-
+ing the whole bigram. For the example in Figure 1,
+“chicken” corresponds to x1. “Salad” and “breast”
+correspond to x21 and x22.
+We automatically build such a dataset of bigram
+pairs in a given context by running BART (Lewis
+et al., 2019) on a portion of the C4 dataset (Raffel
+et al., 2020) to generate another plausible bigram
+alternative to an existing one. We then use the two
+sequences to test T5’s inconsistencies regarding
+Equation 3 3.
+We can use relative difference (dr) of the left
+and right hand side of Equation 3 to quantify the
+inconsistency.
+dr = |lhs(3) − rhs(3)|
+lhs(3)
+(4)
+dr is expected to be 0 for a self-consistent MLM.
+Table 1 shows that dr is typically very large for
+the T5 family, although scaling up the model has a
+markable effect on reducing it.
+Another way to quantify the inconsistency re-
+garding the two bigrams is to count how often a
+severe case happens where the MLM disagrees
+with itself on which bigram it prefers. I.e., some-
+times lhs(3) > 1 and rhs(3) < 1, or lhs(3) < 1
+3We focus on plausible bigrams in this draft because they
+are most relevant in practice but Equation 3 should hold for
+all bigrams in all sentences in all corpora in a self-consistent
+MLM.
+
+The
+is a common choice of food.
+<mask>
+option
+𝑝
+…
+…
+breast
+0.030
+…
+…
+salad
+0.024
+…
+…
+The
+is a common choice of food.
+<mask>
+option
+𝑝
+…
+…
+chicken salad
+0.00028
+…
+…
+chicken breast
+0.00017
+…
+…
+Basic law of probabilities
+𝑝 salad chicken)
+𝑝 breast chicken) = 𝑝(chicken salad)
+𝑝(chicken breast)
+𝑝 breast chicken) > 𝑝 salad chicken)
+𝑝(chicken breast) < 𝑝(chicken salad)
+chicken
+Figure 1: A simple bigram example that exposes the inconsistencies in the T5 model. The conditional probabilities
+that the model learned (quoted from t5-11b fed with the shown masked sequences) contradict each other greatly.
+Not only are the ratios unbalanced, the model confuses its own preference of the two bigrams.
+and rhs(3) > 1.
+Figure 1 shows such a case,
+where t5-11b prefers “chicken salad” over “chicken
+breast” when considering the conditionals provided
+in rhs(3), yet its preference flips when considering
+lhs(3). Table 1 shows that disagreement on com-
+parison happens with considerable frequency, but
+scaling up models helps reduce it.
+4
+Diagnosing BERT-style MLMs
+Ever since the success of BERT (Devlin et al.,
+2018), there has been research effort (Goyal et al.,
+2021; Wang et al., 2019; Yamakoshi et al., 2022)
+on sampling sequences from it by modeling its
+implicitly specified joint distribution one way or
+another. For example, (Goyal et al., 2021) views
+it as an energy-based model defined using the
+bidirectional conditionals of the masked tokens.
+Such research effort is based on the intuition that
+bidirectional conditionals could be more robust
+than auto-regressive (unidirectional) conditionals
+(Goyal, 2021).
+This line of research operates based on the as-
+sumption that the overly abundant bidirectional
+conditionals that the BERT-style MLMs provide
+are self-consistent. (Yamakoshi et al., 2022) based
+on (Heckerman et al., 2000; Neville and Jensen,
+2007) stated that “any deviations (supposedly) tend
+to be negligible”.
+We demonstrate in this section that this is not
+the case at all. There are considerable inconsis-
+tencies that exist among the bidirectional condi-
+tionals that a trained BERT-style model provides.
+Figure 2 demonstrates such an example. Again
+we use bigrams as the simplest example to expose
+the inconsistencies. Because BERT-style MLMs
+cannot directly model the distribution of multiple
+tokens together (local joint distribution), we con-
+sider 4 bigrams this time: x11x21, x11x22, x12x21
+and x12x22. x11 and x12 are two possible tokens
+that the first position can take. x21 and x22 the sec-
+ond. One can easily verify 4 that the 8 conditional
+distributions concerning such four bigrams should
+theoretically satisfy
+p(x21|x11)
+p(x22|x11) × p(x11|x22)
+p(x12|x22) =
+p(x11|x21)
+p(x12|x21) × p(x21|x12)
+p(x22|x12)
+(5)
+One way to test the inconsistencies among the 8
+conditionals is to try to solve one using the other
+7 and compare the solved conditional with the
+original (inferred by model) one. We show the
+solved conditionals in the example in Figure 2. It
+clearly demonstrates that the probabilities given by
+4Clue: converting each term to local joint distributions.
+
+I had
+eggs
+<mask>
+I had
+at lunch.
+<mask>
+𝑝
+Inferred
+Solved
+𝑝 duck eggs)
+1.7 × 10−4
+9.0 × 10−6
+𝑝 chicken eggs)
+1.1 × 10−3
+0.020
+𝑝 duck soup)
+3.1 × 10−4
+5.8 × 10−3
+𝑝 chicken soup)
+0.17
+9.2 × 10−3
+𝑝 eggs duck)
+5.7 × 10−3
+0.11
+𝑝 soup duck)
+0.23
+0.012
+𝑝 eggs chicken)
+6.8 × 10−4
+3.7 × 10−5
+𝑝 soup chicken)
+0.13
+2.31
+𝑝(eggs|duck)
+𝑝(soup|duck) ×
+𝑝(duck|soup)
+𝑝(chicken|soup) =
+𝑝(duck|eggs)
+𝑝(chicken|eggs) × 𝑝(eggs|chicken)
+𝑝(soup|chicken)
+at lunch.
+I had
+soup
+<mask>
+at lunch.
+duck
+I had
+at lunch.
+<mask>
+chicken
+Figure 2: An example of inconsistencies in the BERT-style MLM. Each “inferred” value refers to the probability
+given by the MLM (RoBERTa-large in this figure). Each “solved” value is obtained by passing the other 7 “inferred”
+values to the equation in the red square. We see that the difference between each inferred and solved value is
+significant. And the solved value may even be larger than 1.
+Metric
+T5-base
+T5-large
+T5-3b
+T5-11b
+Relative difference (dr, median, %)
+47.5
+45.8
+44.7
+42.0
+Disagreement on comparison (%)
+9.64
+8.85
+7.53
+6.54
+Table 1: Inconsistencies in the T5 model tested on 19399 pairs of bigrams. We show the median value for relative
+difference as it is resilient to outliers.
+a BERT-style MLM can be in serious inconsisten-
+cies with each other. We build a testing dataset
+with 4 such plausible bigrams for each context and
+quantify consistencies in using difference of log
+probabilities:
+| log psolved − log pinferred|
+(6)
+Table 2 shows the results.
+5
+Summary
+This draft demonstrates and naively quantifies the
+inconsistencies that exist in large MLMs in the
+simple scenario of bigrams. Such inconsistencies
+originate from the fact that the number of bidirec-
+tional conditionals MLMs can learn far exceeds
+what is needed for constructing the joint distribu-
+tion. Given the recent evidence that MLM-based
+pretraining might be a powerful paradigm, we think
+that resolving the its consistency issue could be a
+necessary step for future work.
+Acknowledgements
+We would like to thank Fuzhao Xue for the useful
+discussions.
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+

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+page_content='sg  Abstract  Learning to predict masked tokens in a se-  quence has been shown to be a powerful  pretraining objective for large-scale language  models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content=' After training, such masked language  models can provide distributions of tokens con-  ditioned on bidirectional context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
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+page_content=', they can not be de-  rived from a coherent joint distribution when  considered together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content=' We empirically quantify  such inconsistencies in the simple scenario of  bigrams for two common styles of masked lan-  guage models: T5-style and BERT-style 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content=' For  example, we show that T5 models often con-  fuse its own preference regarding two similar  bigrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content='  Such inconsistencies may represent a theoreti-  cal pitfall for the research work on sampling  sequences based on the bidirectional condi-  tionals learned by BERT-style MLMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content=' This  phenomenon also means that T5-style MLMs  capable of infilling will generate discrepant  results depending on how much masking is  given, which may represent a particular trust  issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content='  1  Introduction  Pretraining objectives of large language models  can be roughly divided into two categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
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+page_content=', 2020)  aims to learn the distribution of the next token in a  sequence given the context to the left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content=' Second, the  masked language modeling (MLM) objective (De-  vlin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content=' Raffel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content=', 2020), which masks  out a portion of the tokens in a sequence and asks  the model to predict them, aims to learn the distri-  bution of one or more tokens given bidirectional  context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
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+page_content='com/tomyoung903/MLM_  inconsistencies  While the major breakthrough, aka, GPT3  (Brown et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content=', 2020) was demonstrated using  vanilla next token prediction, recent work (Tay  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
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+page_content=' Zeng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content=' Bavarian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content=', 2022)  has hinted that incorporating the masked language  modeling objective may be highly beneficial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content=' In ad-  dition, (Tay et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content=', 2022) has demonstrated that such  bidirectional conditionals provide strong infilling  capabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content='  One may notice that, unlike the unidirectional  conditional distributions that vanilla next token pre-  diction learns, the bidirectional conditionals that  MLMs learn are overly abundant in terms of rep-  resenting a coherent joint distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content=' Therefore,  they are not guaranteed to be self-consistent (see  Chapter 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content='  A very simple example for such inconsisten-  cies is shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content=' In this example, we  obtain the bidirectional conditional distributions  that the T5 model learned using two input masked  sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content=' The two similar sequences are designed  with a small difference, in order to examine if the  resulting conditionals satisfy a basic law of prob-  abilities (hold consistency).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content=' Results clearly show  otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content=' We design experiments to quantify such  inconsistencies in Chapter 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content='  One interesting line of research in the litera-  ture focused on whether and how the bidirectional  conditionals that BERT-style MLMs provide can  be used to construct the joint probability of a se-  quence in a principled manner (Goyal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content='  Ghazvininejad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content=', 2019), just  like vanilla next token prediction models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content=' But the  numerous papers on this topic have overlooked  the concern of inconsistencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content=' (Yamakoshi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content=',  2022) stated that “any deviations (supposedly) tend  to be negligible with large datasets”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content=' The experi-  ments shown in Chapter 4 demonstrate that this is  not the case at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content=' We thus posit that addressing the  consistency issue should be treated as the first step  in modeling the joint distribution with BERT-style  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content='00068v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content='CL]  30 Dec 2022  MLMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content='  2  Why inconsistencies can occur in  MLMs  For a set of conditional distributions to be self-  consistent, they need to be able to be derived from  a single coherent joint distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content='  One essential reason for the inconsistencies to  occur among the conditionals provided by a trained  MLM is that the number of conditionals it can  calculate far exceeds the number of degrees of free-  dom of a joint distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content='  Consider a sequence of length L and with vo-  cabulary V , the joint distribution of the tokens in  such a sequence is defined by |V |L probabilities  that sum to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content=' Therefore, the number of degrees of  freedom (D) of such a joint distribution is given  by:  Djoint = |V |L − 1,  (1)  Vanilla next token prediction models or MLMs  essentially learn conditionals that predict some  tokens in the sequence given others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content=' Such con-  ditional probabilities and probabilities from the  joint distribution can be linearly derived from each  other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content=' Therefore, each free conditional that the  language model is capable of specifying provides  an additional constraint on the joint distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content='  One can easily verify that a vanilla next token pre-  diction based language model provides |V |L − 1  free conditionals 2 to just exactly determine the  joint distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content='  Therefore, a vanilla next to-  ken prediction model (no matter how it is trained,  or even untrained) would never suffer from self-  inconsistencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content='  MLMs, which can provide distributions of  masked tokens given bidirectional context, could  specify far more free conditionals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content='  Even for the simplest case, where the MLM pre-  dicts the distribution of only 1 (masked) token  given L − 1 other (unmasked) tokens in the se-  quence, the total number of free conditionals (N)  is  Nmlm(1) = L × (|V |L − |V |L−1),  (2)  Just Nmlm(1) is already far larger than Djoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content='  We leave the discussions for Nmlm(k) for later  2A single softmax operation over V essentially gives |V |−  1 free conditionals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content=' Here we call conditionals free when they  can be assigned any values decided by an underlying neural  network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content='  work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content=' This fact sets up room for there to be in-  consistencies among the conditionals an MLM pro-  vides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content='  We explain our strategies and quantification  methods for diagnosing T5-style and BERT-style  MLMs in the next 2 sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content='  3  Diagnosing T5-style MLMs  T5-style MLMs are capable of modeling the dis-  tribution of segments of variable length in a given  bidirectional context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content=' Here we use the simple bi-  gram scenario to expose the inconsistencies that  exist among such distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content=' Consider two bi-  grams x1x21 and x1x22 that share a same token x1  in the first position, the conditional distributions  concerning such two bigrams should satisfy  p(x21|x1)  p(x22|x1) = p(x1x21)  p(x1x22)  (3)  The left hand side can be obtained by only mask-  ing the second token, leaving x1 in the context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content='  While the right hand side can be obtained by mask-  ing the whole bigram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content=' For the example in Figure 1,  “chicken” corresponds to x1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content=' “Salad” and “breast”  correspond to x21 and x22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content='  We automatically build such a dataset of bigram  pairs in a given context by running BART (Lewis  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content=', 2019) on a portion of the C4 dataset (Raffel  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content=', 2020) to generate another plausible bigram  alternative to an existing one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content=' We then use the two  sequences to test T5’s inconsistencies regarding  Equation 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content='  We can use relative difference (dr) of the left  and right hand side of Equation 3 to quantify the  inconsistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content='  dr = |lhs(3) − rhs(3)|  lhs(3)  (4)  dr is expected to be 0 for a self-consistent MLM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content='  Table 1 shows that dr is typically very large for  the T5 family, although scaling up the model has a  markable effect on reducing it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content='  Another way to quantify the inconsistency re-  garding the two bigrams is to count how often a  severe case happens where the MLM disagrees  with itself on which bigram it prefers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content=', some-  times lhs(3) > 1 and rhs(3) < 1, or lhs(3) < 1  3We focus on plausible bigrams in this draft because they  are most relevant in practice but Equation 3 should hold for  all bigrams in all sentences in all corpora in a self-consistent  MLM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content='  The  is a common choice of food.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content='  <mask>  option  𝑝  …  …  breast  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content='030  …  …  salad  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content='024  …  …  The  is a common choice of food.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content='  <mask>  option  𝑝  …  …  chicken salad  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content='00028  …  …  chicken breast  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content='00017  …  …  Basic law of probabilities  𝑝 salad chicken)  𝑝 breast chicken) = 𝑝(chicken salad)  𝑝(chicken breast)  𝑝 breast chicken) > 𝑝 salad chicken)  𝑝(chicken breast) < 𝑝(chicken salad)  chicken  Figure 1: A simple bigram example that exposes the inconsistencies in the T5 model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content=' The conditional probabilities  that the model learned (quoted from t5-11b fed with the shown masked sequences) contradict each other greatly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content='  Not only are the ratios unbalanced, the model confuses its own preference of the two bigrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content='  and rhs(3) > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content='  Figure 1 shows such a case,  where t5-11b prefers “chicken salad” over “chicken  breast” when considering the conditionals provided  in rhs(3), yet its preference flips when considering  lhs(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content=' Table 1 shows that disagreement on com-  parison happens with considerable frequency, but  scaling up models helps reduce it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content='  4  Diagnosing BERT-style MLMs  Ever since the success of BERT (Devlin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content=',  2018), there has been research effort (Goyal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content=',  2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content=' Yamakoshi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content=', 2022)  on sampling sequences from it by modeling its  implicitly specified joint distribution one way or  another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content=' For example, (Goyal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content=', 2021) views  it as an energy-based model defined using the  bidirectional conditionals of the masked tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content='  Such research effort is based on the intuition that  bidirectional conditionals could be more robust  than auto-regressive (unidirectional) conditionals  (Goyal, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content='  This line of research operates based on the as-  sumption that the overly abundant bidirectional  conditionals that the BERT-style MLMs provide  are self-consistent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content=' (Yamakoshi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content=', 2022) based  on (Heckerman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content=', 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content=' Neville and Jensen,  2007) stated that “any deviations (supposedly) tend  to be negligible”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content='  We demonstrate in this section that this is not  the case at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content=' There are considerable inconsis-  tencies that exist among the bidirectional condi-  tionals that a trained BERT-style model provides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content='  Figure 2 demonstrates such an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content=' Again  we use bigrams as the simplest example to expose  the inconsistencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content=' Because BERT-style MLMs  cannot directly model the distribution of multiple  tokens together (local joint distribution), we con-  sider 4 bigrams this time: x11x21, x11x22, x12x21  and x12x22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content=' x11 and x12 are two possible tokens  that the first position can take.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content=' x21 and x22 the sec-  ond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content=' One can easily verify 4 that the 8 conditional  distributions concerning such four bigrams should  theoretically satisfy  p(x21|x11)  p(x22|x11) × p(x11|x22)  p(x12|x22) =  p(x11|x21)  p(x12|x21) × p(x21|x12)  p(x22|x12)  (5)  One way to test the inconsistencies among the 8  conditionals is to try to solve one using the other  7 and compare the solved conditional with the  original (inferred by model) one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content=' We show the  solved conditionals in the example in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content=' It  clearly demonstrates that the probabilities given by  4Clue: converting each term to local joint distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content='  I had  eggs  <mask>  I had  at lunch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content='  <mask>  𝑝  Inferred  Solved  𝑝 duck eggs)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content='7 × 10−4  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content='0 × 10−6  𝑝 chicken eggs)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content='1 × 10−3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content='020  𝑝 duck soup)  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content='1 × 10−4  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content='8 × 10−3  𝑝 chicken soup)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content='17  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content='2 × 10−3  𝑝 eggs duck)  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content='7 × 10−3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content='11  𝑝 soup duck)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content='23  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content='012  𝑝 eggs chicken)  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content='8 × 10−4  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content='7 × 10−5  𝑝 soup chicken)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content='13  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content='31  𝑝(eggs|duck)  𝑝(soup|duck) ×  𝑝(duck|soup)  𝑝(chicken|soup) =  𝑝(duck|eggs)  𝑝(chicken|eggs) × 𝑝(eggs|chicken)  𝑝(soup|chicken)  at lunch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content='  I had  soup  <mask>  at lunch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content='  duck  I had  at lunch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content='  <mask>  chicken  Figure 2: An example of inconsistencies in the BERT-style MLM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content=' Each “inferred” value refers to the probability  given by the MLM (RoBERTa-large in this figure).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content=' Each “solved” value is obtained by passing the other 7 “inferred”  values to the equation in the red square.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content=' We see that the difference between each inferred and solved value is  significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content=' And the solved value may even be larger than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content='  Metric  T5-base  T5-large  T5-3b  T5-11b  Relative difference (dr, median, %)  47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content='5  45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content='8  44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content='7  42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content='0  Disagreement on comparison (%)  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content='64  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content='85  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content='53  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content='54  Table 1: Inconsistencies in the T5 model tested on 19399 pairs of bigrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content=' We show the median value for relative  difference as it is resilient to outliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content='  a BERT-style MLM can be in serious inconsisten-  cies with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content=' We build a testing dataset  with 4 such plausible bigrams for each context and  quantify consistencies in using difference of log  probabilities:  | log psolved − log pinferred|  (6)  Table 2 shows the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content='  5  Summary  This draft demonstrates and naively quantifies the  inconsistencies that exist in large MLMs in the  simple scenario of bigrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content=' Such inconsistencies  originate from the fact that the number of bidirec-  tional conditionals MLMs can learn far exceeds  what is needed for constructing the joint distribu-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content=' Given the recent evidence that MLM-based  pretraining might be a powerful paradigm, we think  that resolving the its consistency issue could be a  necessary step for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content='  Acknowledgements  We would like to thank Fuzhao Xue for the useful  discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content='  References  Mohammad Bavarian, Heewoo Jun, Nikolas Tezak,  John Schulman, Christine McLeavey, Jerry Tworek,  and Mark Chen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content='  Efficient training of lan-  guage models to fill in the middle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content=' arXiv preprint  arXiv:2207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content='14255.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content='  Tom B Brown, Benjamin Mann, Nick Ryder, Melanie  Subbiah, Jared Kaplan, Prafulla Dhariwal, Arvind  Neelakantan, Pranav Shyam, Girish Sastry, Amanda  Askell, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content=' Language models are few-shot  learners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content=' arXiv preprint arXiv:2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content='14165.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content='  Jacob Devlin, Ming-Wei Chang, Kenton Lee, and  Kristina Toutanova.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content=' Bert: Pre-training of deep  bidirectional transformers for language understand-  ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content=' arXiv preprint arXiv:1810.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content='04805.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content='  Marjan Ghazvininejad, Omer Levy, Yinhan Liu, and  Luke Zettlemoyer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content='  Mask-predict: Parallel  Metric  Roberta-base  Roberta-large  log-probability difference  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content='836  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content='792  Table 2: Difference of log-probabilities between inferred and solved conditionals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content=' The difference would be 0 for  self-consistent MLMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content=' Roughly a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content='8 difference means that one is 120% larger than the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content='  decoding of conditional masked language models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content='  arXiv preprint arXiv:1904.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content='09324.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content='  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content=' Goyal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content=' Characterizing and overcoming the  limitations of neural autoregressive models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content='  PhD  thesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content='  Kartik Goyal, Chris Dyer, and Taylor Berg-Kirkpatrick.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content='  2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content=' Exposing the implicit energy networks behind  masked language models via metropolis–hastings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content='  arXiv preprint arXiv:2106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content='02736.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content='  David  Heckerman,  David  Maxwell  Chickering,  Christopher Meek, Robert Rounthwaite, and Carl  Kadie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content=' 2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content=' Dependency networks for inference,  collaborative filtering, and data visualization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content=' Jour-  nal of Machine Learning Research, 1(Oct):49–75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content='  Mike  Lewis,  Yinhan  Liu,  Naman  Goyal,  Mar-  jan Ghazvininejad, Abdelrahman Mohamed, Omer  Levy, Ves Stoyanov, and Luke Zettlemoyer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content='  Bart: Denoising sequence-to-sequence pre-training  for natural language generation, translation, and  comprehension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content=' arXiv preprint arXiv:1910.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content='13461.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content='  Jennifer Neville and David Jensen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content=' 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content=' Relational  dependency networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content=' Journal of Machine Learning  Research, 8(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content='  Colin Raffel, Noam Shazeer, Adam Roberts, Katherine  Lee, Sharan Narang, Michael Matena, Yanqi Zhou,  Wei Li, Peter J Liu, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content=' Exploring the limits  of transfer learning with a unified text-to-text trans-  former.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content=' Mach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content=' Learn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content=', 21(140):1–67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content='  Yi Tay, Jason Wei, Hyung Won Chung, Vinh Q  Tran, David R So, Siamak Shakeri, Xavier Gar-  cia, Huaixiu Steven Zheng, Jinfeng Rao, Aakanksha  Chowdhery, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content='  Transcending scaling  laws with 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content='1% extra compute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content='  arXiv preprint  arXiv:2210.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content='11399.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content='  Alex  Wang,  Kyunghyun  Cho,  and  CIFAR  Azrieli Global Scholar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content='  Bert has a mouth,  and it must speak: Bert as a markov random field  language model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content=' NAACL HLT 2019, page 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content='  Takateru Yamakoshi, Thomas L Griffiths, and Robert  Hawkins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content=' Probing bert’s priors with serial re-  production chains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content=' In Findings of the Association  for Computational Linguistics: ACL 2022, pages  3977–3992.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content='  Aohan Zeng, Xiao Liu, Zhengxiao Du, Zihan Wang,  Hanyu Lai, Ming Ding, Zhuoyi Yang, Yifan Xu,  Wendi Zheng, Xiao Xia, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content='  Glm-130b:  An open bilingual pre-trained model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content=' arXiv preprint  arXiv:2210.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}
+page_content='02414.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'}

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@@ -0,0 +1,1309 @@
+Draft version January 6, 2023
+Preprint typeset using LATEX style AASTeX6 v. 1.0
+AN EQUATION OF STATE OF CO FOR USE IN PLANETARY MODELING
+M. Podolak
+Dept. of Geosciences, Tel Aviv University, Tel Aviv, 69978 Israel
+A. Levi
+Braude College of Engineering, Karmiel, 2161002 Israel
+A. Vazan
+Astrophysics Research Center (ARCO), Dept. of Natural Sciences, Open University of Israel, Raanana, 43107 Israel
+U. Malamud
+Dept. of Geosciences, Tel Aviv University, Tel Aviv, 69978 Israel
+Department of Physics, Technion – Israel Institute of Technology, Technion City, 3200003 Haifa, Israel
+ABSTRACT
+Although carbon monoxide (CO) is an abundant molecule and may have great importance for planetary interiors,
+measurements of its properties are difficult due to its extreme volatility. We calculate the equation of state for CO over
+a range of temperature and density that is applicable to the conditions in planetary interiors. Previous experimental
+and theoretical studies cover only a limited temperature-density range. Our calculations match these early results
+well, but now cover the full range of relevance. The method of calculation is based on the general-purpose quotidian
+equation of state described by More et al. (1988), which is here used in order to generate a freely downloadable look-up
+table to be used by the community.
+1. INTRODUCTION
+When modeling planetary interiors, it is necessary to have adequate descriptions for the behavior of the constituent
+materials.
+Thus equation of state (EOS) tables have been produced for the two most abundant elements in the
+universe, hydrogen and helium (see, e.g. Chabrier et al. 2019), as well as other materials expected to be of importance
+for planet models, such as water (see, e.g. Haldemann et al. 2020), various silicates such as dunite (Benz et al. 1989),
+granite (Pierazzo et al. 1997), basalt (Pierazzo et al. 2005), quartz (Melosh 2007) and important metals such as iron
+(e.g. Emsenhuber et al. 2018).
+Since both carbon and oxygen have relatively high cosmic abundances, and since CO is a very stable molecule, CO
+could be an important constituent in planetary interiors (see, e.g. Lisse et al. 2022). Yet this possibility cannot be
+properly addressed because only limited regions of the CO EOS have been studied, and there are no complete equation
+of state tables available in the literature. Empirical measurements of the density of solid (α-cubic, β-hexagonal) and
+liquid CO have been made (Boon et al. 1967; Bierhals 2001), in addition to various other physical properties such as
+viscosity, heat capacity (Rudesko & Schubnikow 1934; Tancredi et al. 1994), and elastic constants (Gammon 1978). All
+of these studies are applicable to extremely low temperature and pressure conditions, and are ill-suited for planetary
+interior applications. The behavior of CO at higher pressures and temperatures has been studied, to a limited extent
+by Nellis et al. (1981) who reported the results of shock experiments.
+More recent work by Zhang et al. (2011)
+gives a more refined hugoniot for CO. In addition, theoretical calculations by Goodwin (1985) have investigated the
+region of pressures below 100 MPa. Individual pressure-temperature-density points have been computed from quantum
+molecular dynamics calculations by Massacrier et al. (2011), Wang & Zhang (2010), and Leonhardi & Militzer (2017).
+However, all of this data is insufficient for planetary modeling, where a much larger range of pressures and temperatures
+are encountered.
+The fact that shock-derived carbon condensates have diameters of the order of a few nanometers (Titov et al. 1989;
+Viecelli et al. 2001; Kr¨uger et al. 2005), and growth timescales of 100’s of picoseconds (Armstrong et al. 2020)) make
+amitlevi.planetphys@gmail.com
+arXiv:2301.02176v1  [astro-ph.EP]  5 Jan 2023
+
+2
+direct DFT based molecular dynamics simulations of this system particularly challenging. Overcoming such immense
+difficulties often requires some synthesis between a DFT based approach and more classical force field models using
+various training models often referred to as machine learning approaches (see, e.g. Lindsey et al. 2020; Singraber et al.
+2019). These techniques are very demanding computationally. Therefore, our model which is in good agreement with
+experimental data and covers a very wide pressure-temperature domain is of merit.
+To this end we have generated an equation of state table for CO which we describe below.
+Our calculation is
+admittedly more crude, but it should be sufficiently close to reality so as to be useful in establishing model trends such
+as was done in the models of Podolak et al. (2022), for example. This paper is structured as follows: Section 2 gives a
+brief description of the method for computing the quotidian EOS (QEOS). This computation requires the knowledge
+of the density and bulk modulus at low energy. The DFT calculation of these parameters is described in section 3,
+and the results are given in section 4. The resulting EOS table and its comparison to experimental and theoretical
+work described above is given in section 5. It is hoped that this work will encourage more detailed EOS modeling for
+CO in the future.
+2. QUOTIDIAN EQUATION OF STATE
+More et al. (1988) present a general-purpose method for computing equations of state at high pressure, called the
+Quotidian Equation of State (QEOS). The QEOS is a statistical-mechanics-based method, in which thermodynamic
+quantities are derived from the Helmholtz free energy. The Helmholtz free energy term is composed of three parts: an
+ionic contribution, an electronic contribution, and a bonding correction. The ionic part is calculated by the Cowan
+model, a semi-empirical model which interpolates between known limiting physical cases (ideal gas law, Lindemann
+melting law, Dulong-Petit law, Gr¨uniesen EOS, Debye lattice). The electronic part is calculated using a modified
+Thomas-Fermi (TF) model. The TF model neglects attractive (bonding) forces between neutral atoms and therefore
+overestimates the critical point and the pressure near normal conditions. The bonding correction is used here to correct
+for the electronic part failure by calibration of the EOS with density and bulk modulus at reference conditions of zero
+(low) energy.
+This method has been used to develop EOS tables for Fe, SiO2 and H2O for use in planetary modeling which compare
+well with other EOS tables such as SESAME and ANEOS for these substances (Vazan et al. 2013, 2018, 2022). The
+QEOS input variables are: atomic number, atomic weight, and reference conditions density and bulk modulus. The
+calculated quantities are: pressure, specific internal energy, and specific entropy. The temperature-density range of
+the calculation is 11.6 < T < 1.16 × 106 K, and 2.5 × 10−13 < ρ < 100 g cm−3. The liquid-vapor phase transition is
+determined with regard to the Maxwell construction, based on finding equal Gibbs free energy on the liquid and the
+vapor sides of each isotherm (up to the critical temperature). As a result, there is no coexistence of vapor and liquid
+phases in the resulting smooth QEOS.
+In order to calculate a QEOS for CO, the method requires prior knowledge of the density and bulk modulus of the
+material at very low temperature and pressure. Unfortunately there have been no measurements of these quantities for
+the α-phase of CO. We therefore performed a first-principles calculation for this state using density-functional theory
+(DFT). This calculation described in the next section.
+3. COMPUTATIONAL METHODS
+Here we study the equation of state of α-CO at 0 K. The structure is taken from Hall & James (1976). We performed
+static total energy relaxations with the CP2K code (K¨uhne et al. 2020). We use the quickstep framework within CP2K
+with the Gaussian and plane waves mixed bases (GPW). We adopt the Gaussian basis sets from VandeVondele et al.
+(2005); VandeVondele & Hutter (2007), in conjunction with the pseudopotentials (GTH-PBE) of Goedecker, Teter,
+and Hutter (Goedecker et al. 1996; Hartwigsen et al. 1998; Krack 2005).
+Our system is converged for a planewave cutoff energy of 600 Ry and a REL CUTOFF of 40 Ry. We use the revised
+PBE exchange functional GGA X PBE R from Zhang & Yang (1998) and a PBE correlation functional, GGA C PBE
+(Perdew et al. 1996, 1997). These are found to be adequate choices when describing an aqueous system in conjunction
+with the non-local van der Waals correlation using the Grimme D3 method (Grimme et al. 2010), achieving convergence
+for R CUTOFF of 14. The calculations were done done on a 2x2x2 supercell consisting of 32 CO molecules. The
+derived data at 0 K is obtained using CELL OPT within CP2K and reported below.
+4. THE EQUATION OF STATE
+In table 1 and fig. 1 we give the volumes and energies derived for different pressures at 0 K. This data is fitted to a
+third order Birch-Murnaghan equation of state with a bulk modulus B= 6.556 ± 0.074 GPa, a pressure derivative for
+
+3
+Table 1.
+The volume, internal energy,
+and derived enthalpy as a function of pres-
+sure for the α-CO solid. Data is for a cubic
+supercell consisting of 32 CO molecules.
+P
+V
+U
+H
+[bar]
+[˚A3]
+[Ha]
+[Ha]
+30,000
+1012.664
+-694.3516
+-693.6548
+20,000
+1063.250
+-694.3819
+-693.8941
+10,000
+1134.213
+-694.4058
+-694.1456
+5000
+1186.408
+-694.4146
+-694.2785
+1000
+1243.957
+-694.4184
+-694.3899
+500
+1252.903
+-694.4185
+-694.4041
+250
+1257.361
+-694.4186
+-694.4114
+100
+1260.298
+-694.4186
+-694.4157
+50
+1261.144
+-694.4186
+-694.4172
+25
+1261.663
+-694.4186
+-694.4179
+10
+1261.931
+-694.4186
+-694.4183
+1
+1262.103
+-694.4186
+-694.4186
+the bulk modulus of B′ = 6.846 ± 0.120, and a zero pressure volume of V0 = 157.80 ± 0.05 ˚A3. The error bars are at
+the 2σ level. As mentioned above, the QEOS requires a knowledge of ρ and B at reference conditions of zero energy,
+and, based on this calculation, and a fit to the four lowest pressure points we take ρ = 1.179 g cm−3 and B= 2.676 GPa
+as the input parameters. Note that this value of B falls between the best fit value of 6.556 GPa given above and the
+value of 1.3 GPa measured by Gammon (1978) for β-CO.
+125
+130
+135
+140
+145
+150
+155
+160
+0
+0.5
+1
+1.5
+2
+2.5
+3
+3.5
+P [ GPa ]
+Fitted data to BM3
+Figure 1. Pressure versus unit cell volume for α-CO. The blue circles are unit cell volumes from our optimization data at 0 K, and the
+solid red curve is the fitted third order Birch-Murnaghan equation of state (BM3).
+Using the results of the DFT calculation described above in the quotidian code, we produced an equation of state
+
+4
+table giving the pressure, energy and entropy of CO for a large range of temperatures and densities.
+5. COMPARISON TO OTHER RESULTS
+0
+2
+4
+6
+8
+10
+12
+14
+16
+18
+20
+1.0E+08
+1.0E+09
+1.0E+10
+1.0E+11
+1.0E+12
+1.0E+13
+1.0E+14
+Density (g/cc)
+Pressure (Pa)
+Figure 2. Density as a function of pressure at zero temperature for the quotidian equation of state (black curve), and for the S-Z equation
+of state (blue curve). The red dots are the results of the DFT calculation.
+Salpeter & Zapolsky (1967) (S-Z) describe a semi-empirical formula for predicting the zero temperature pressure-
+density relation for materials with any average atomic number. In principle, the S-Z EOS is similar to the More et al.
+(1988) approach, since it relies on a Thomas-Fermi-Dirac model of the atom. However it does not include the effect
+of temperature, so it is not always suitable for planet modeling. Fig. 2 shows the comparison between our quotidian
+equation of state (QEOS) at zero temperature, and the S-Z EOS. As can be seen, the agreement is excellent, and
+improves at higher pressures, as expected. The red dots in the figure are the DFT calculations given in table 1 and
+fig. 1. These fall right on the QEOS curve.
+The QEOS can be compared to experimental data at higher temperatures as well.
+Goodwin (1985) gives the
+thermophysical properties of CO up to a pressure of 100 MPa. Fig. 3 shows that data for an isotherm at 1000 K (red
+dots) compared to the QEOS isotherm at that temperature (black curve). The discontinuity in the QEOS is due to
+the fact that the QEOS finds two phases in present in this pressure-temperature range and traverses this region using
+a Maxwell construction. As a result, the computed pressure remains constant over the relevant density range. The
+actual pressure, as shown by the red dots, increases along the extrapolation of the lower part of the curve, as expected.
+The exact position of the phase transition is sensitive to the choice of input parameters (zero energy density and bulk
+modulus), and the actual value may be shifted somewhat.
+
+5
+1.0E-04
+1.0E-03
+1.0E-02
+1.0E-01
+1.0E+00
+1.0E+01
+1.0E+02
+1.0E+06
+1.0E+07
+1.0E+08
+1.0E+09
+1.0E+10
+1.0E+11
+1.0E+12
+1.0E+13
+1.0E+14
+1.0E+15
+Density (g/cc)
+Pressure (Pa)
+Figure 3. Density as a function of pressure for an isotherm at T = 1000 K (black curve), compared to the data in Goodwin (1985) (red
+dots). See text for details.
+At still higher pressures and temperatures, there are the shock wave experiments of Nellis et al. (1981). In this case
+the temperatures are only inferred from the Hugoniot relations, and are different for the different pressures. More
+recently, Zhang et al. (2011) have used quantum molecular dynamics calculations to compute points along a hugoniot.
+These are shown (blue dots) together with the hugoniot calculated from our QEOS in Fig. 4. The black dots are the
+experimental points of Nellis et al. (1981). As can be seen, the agreement is quite good and is in the range of these
+works. At the highest temperatures (T ≳ 105 K) dissociation and ionization become important, and these effects are
+not directly included in our calculation. Nonetheless, the energies we compute for CO at T = 5 × 105 K for densities
+of 0.1, 1, 10, and 100 g cm−3 all fall within a factor of 1.5 or less from the values shown in fig. 9 of Massacrier et al.
+(2011).
+The full QEOS is summarized in Fig. 5. A short version for a range of pressures and temperatures that are expected
+to be important for planetary interior modeling given in table 2, while the complete table is available at the following
+site: CO EOS download.
+
+6
+0
+0.5
+1
+1.5
+2
+2.5
+3
+3.5
+4
+0
+50
+100
+150
+200
+250
+300
+350
+Density (g/cc)
+Pressure (GPa)
+Figure 4. Density as a function of pressure for a hugoniot (blue curve) corresponding to the conditions of the shock experiments of Nellis
+et al. (1981) (black dots) and the quantum molecular dynamics calculations of Zhang et al. (2011) (blue dots).
+
+7
+Figure 5. Thermodynamic properties of CO as a function of density and temperature as computed from the quotidian equation of state.
+Upper left: total pressure. Upper right: pressure divided by ideal gas pressure. This shows the region where an ideal gas approximation
+may be used. Lower left: specific internal energy. Lower right: specific entropy.
+
+co: 1
+CO: log p (Pa)
+0
+0
+log p (g/cm3)
+10
+log p (g/cm3)
+5
+5
+0og p (/pideal
+5
+4
+3
+2
+1-10
+-10
+-5
+2
+4
+6
+2
+logT(K)
+Co: log u (erg/g)
+CO: Io
+14
+0
+0
+13
+-5
+-5
+12
+11
+.10
+-10
+2
+4
+6
+2
+log T (K)0
+4
+6
+log T (K)
+gs(erg/g/K)
+9
+8
+7
+6
+5
+4
+3
+4
+6
+log T (K)8
+h
+Table 2. Equation of state for CO.
+log T
+log ρ
+log P
+log u
+log s
+[K]
+[g/cc]
+[Pa]
+[erg/g]
+[erg/g − K]
+1.06465
+0.10
+8.25060
+10.33294
+4.12858
+1.06465
+0.20
+9.33277
+10.36235
+3.94748
+1.06465
+0.30
+9.90369
+10.46041
+3.83167
+1.06465
+0.40
+10.35107
+10.63593
+3.74763
+1.06465
+0.50
+10.73560
+10.86075
+3.67754
+1.06465
+0.60
+11.08026
+11.10070
+3.61334
+1.06465
+0.70
+11.39668
+11.33587
+3.55157
+1.06465
+0.80
+11.69173
+11.55846
+3.49070
+1.06465
+0.90
+11.96991
+11.76663
+3.43006
+1.06465
+1.00
+12.23437
+11.96089
+3.36933
+1.06465
+1.10
+12.48741
+12.14250
+3.30834
+1.06465
+1.20
+12.73083
+12.31285
+3.24704
+1.06465
+1.30
+12.96602
+12.47327
+3.18540
+1.56465
+0.10
+8.25291
+10.33305
+5.29718
+1.56465
+0.20
+9.33286
+10.36239
+4.92635
+1.56465
+0.30
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+4.63269
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+0.40
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+0.50
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+0.60
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+0.70
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+0.80
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+0.90
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+11.76663
+3.94060
+1.56465
+1.00
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+3.87576
+1.56465
+1.10
+12.48741
+12.14250
+3.81236
+1.56465
+1.20
+12.73083
+12.31285
+3.74960
+1.56465
+1.30
+12.96602
+12.47327
+3.68706
+2.06465
+0.10
+8.32026
+10.33617
+6.33748
+2.06465
+0.20
+9.33736
+10.36427
+6.10876
+2.06465
+0.30
+9.90446
+10.46124
+5.82885
+2.06465
+0.40
+10.35123
+10.63621
+5.53226
+2.06465
+0.50
+10.73564
+10.86083
+5.24767
+2.06465
+0.60
+11.08027
+11.10073
+5.00266
+2.06465
+0.70
+11.39669
+11.33588
+4.80674
+2.06465
+0.80
+11.69174
+11.55846
+4.65419
+2.06465
+0.90
+11.96991
+11.76663
+4.53392
+2.06465
+1.00
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+2.06465
+1.10
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+1.20
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+0.10
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+2.56465
+0.20
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+10.64227
+6.45354
+2.56465
+0.50
+10.73692
+10.86349
+6.29013
+
+9
+Table 2. Equation of state for CO continued
+log T
+log ρ
+log P
+log u
+log s
+[K]
+[g/cc]
+[Pa]
+[erg/g]
+[erg/g − K]
+2.56465
+0.60
+11.08076
+11.10182
+6.11803
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+3.56465
+1.30
+12.96665
+12.47450
+6.60487
+4.06465
+-1.00
+8.72672
+11.12752
+7.61469
+4.06465
+-0.90
+8.84338
+11.11787
+7.60037
+4.06465
+-0.80
+8.96101
+11.10873
+7.58527
+4.06465
+-0.70
+9.07819
+11.09997
+7.56926
+4.06465
+-0.60
+9.19305
+11.09161
+7.55219
+4.06465
+-0.50
+9.30353
+11.08214
+7.53390
+4.06465
+-0.40
+9.40769
+11.07207
+7.51424
+4.06465
+-0.30
+9.50433
+11.06010
+7.49310
+4.06465
+-0.20
+9.59411
+11.04556
+7.47045
+4.06465
+-0.10
+9.68161
+11.02824
+7.44633
+4.06465
+0.00
+9.77804
+11.00927
+7.42091
+4.06465
+0.10
+9.90243
+10.99228
+7.39448
+4.06465
+0.20
+10.07443
+10.98353
+7.36684
+
+10
+Table 2. Equation of state for CO continued
+log T
+log ρ
+log P
+log u
+log s
+[K]
+[g/cc]
+[Pa]
+[erg/g]
+[erg/g − K]
+4.06465
+0.30
+10.30242
+10.99646
+7.33856
+4.06465
+0.40
+10.57225
+11.04588
+7.30967
+4.06465
+0.50
+10.86205
+11.14146
+7.28014
+4.06465
+0.60
+11.15517
+11.27981
+7.24992
+4.06465
+0.70
+11.44272
+11.44692
+7.21887
+4.06465
+0.80
+11.72104
+11.62736
+7.18684
+4.06465
+0.90
+11.98825
+11.80866
+7.14860
+4.06465
+1.00
+12.24611
+11.98690
+7.10900
+4.06465
+1.10
+12.49512
+12.15893
+7.06828
+4.06465
+1.20
+12.73600
+12.32343
+7.02608
+4.06465
+1.30
+12.96954
+12.48020
+6.98198
+4.56465
+-1.00
+9.39391
+11.85300
+7.82194
+4.56465
+-0.90
+9.49858
+11.83884
+7.80733
+4.56465
+-0.80
+9.60479
+11.82481
+7.79226
+4.56465
+-0.70
+9.71228
+11.81091
+7.77665
+4.56465
+-0.60
+9.82054
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+7.76038
+4.56465
+-0.50
+9.92889
+11.78321
+7.74336
+4.56465
+-0.40
+10.03655
+11.76908
+7.72545
+4.56465
+-0.30
+10.14277
+11.75438
+7.70651
+4.56465
+-0.20
+10.24707
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+7.68639
+4.56465
+-0.10
+10.34964
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+7.66492
+4.56465
+0.00
+10.45180
+11.70292
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+4.56465
+0.10
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+0.20
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+0.30
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+4.56465
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+0.50
+11.13044
+11.64026
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+4.56465
+0.60
+11.33652
+11.66954
+7.47311
+4.56465
+0.70
+11.56221
+11.72900
+7.44136
+4.56465
+0.80
+11.79910
+11.82012
+7.40932
+4.56465
+0.90
+12.04056
+11.93746
+7.37731
+4.56465
+1.00
+12.28172
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+7.34534
+4.56465
+1.10
+12.51981
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+7.31336
+4.56465
+1.20
+12.75346
+12.36132
+7.28123
+4.56465
+1.30
+12.98189
+12.50572
+7.24694
+5.06465
+-2.20
+9.01221
+12.79781
+8.22267
+5.06465
+-2.10
+9.10356
+12.78341
+8.20949
+5.06465
+-2.00
+9.19496
+12.76882
+8.19617
+5.06465
+-1.90
+9.28643
+12.75405
+8.18271
+5.06465
+-1.80
+9.37804
+12.73910
+8.16910
+5.06465
+-1.70
+9.46983
+12.72399
+8.15535
+5.06465
+-1.60
+9.56187
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+5.06465
+-1.50
+9.65423
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+5.06465
+-1.40
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+5.06465
+-1.30
+9.84023
+12.66200
+8.09870
+5.06465
+-1.20
+9.93407
+12.64619
+8.08407
+5.06465
+-1.10
+10.02860
+12.63027
+8.06923
+
+11
+Table 2. Equation of state for CO continued
+log T
+log ρ
+log P
+log u
+log s
+[K]
+[g/cc]
+[Pa]
+[erg/g]
+[erg/g − K]
+5.06465
+-1.00
+10.12395
+12.61427
+8.05415
+5.06465
+-0.90
+10.22023
+12.59820
+8.03880
+5.06465
+-0.80
+10.31750
+12.58210
+8.02315
+5.06465
+-0.70
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+12.56597
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+5.06465
+-0.60
+10.51516
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+7.99078
+5.06465
+-0.50
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+12.53369
+7.97396
+5.06465
+-0.40
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+12.51752
+7.95663
+5.06465
+-0.30
+10.81818
+12.50129
+7.93872
+5.06465
+-0.20
+10.92019
+12.48494
+7.92015
+5.06465
+-0.10
+11.02242
+12.46839
+7.90082
+5.06465
+0.00
+11.12490
+12.45155
+7.88063
+5.06465
+0.10
+11.22793
+12.43435
+7.85947
+5.06465
+0.20
+11.33226
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+7.83721
+5.06465
+0.30
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+7.81373
+5.06465
+0.40
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+5.06465
+0.50
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+12.36618
+7.76260
+5.06465
+0.60
+11.79684
+12.35360
+7.73472
+5.06465
+0.70
+11.93708
+12.34707
+7.70521
+5.06465
+0.80
+12.09132
+12.35034
+7.67410
+5.06465
+0.90
+12.26000
+12.36788
+7.64155
+5.06465
+1.00
+12.44186
+12.40353
+7.60770
+5.06465
+1.10
+12.63451
+12.45973
+7.57309
+5.06465
+1.20
+12.83475
+12.53549
+7.53795
+5.06465
+1.30
+13.03937
+12.62723
+7.50218
+
+12
+6. ACKNOWLEDGEMENTS
+The authors wish to thank Gilles Chabrier and an anonymous referee for many constructive comments. M.P. is
+supported by a grant from the Pazy Fund of the Israel Atomic Energy Commission. A.L. is supported by a grant from
+the Simons Foundation (SCOL #290360 to D.S.). The computations for this paper were run on the Odyssey cluster
+supported by the FAS Division of Science, Research Computing Group at Harvard University. A.L. is grateful to the
+administrative staff for their technical support. A.V. acknowledges support from ISF grants 770/21 and 773/21.
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+

_dAyT4oBgHgl3EQfdvdN/content/tmp_files/2301.00307v1.pdf.txt

@@ -0,0 +1,834 @@
+Bending Deformation Driven by Molecular Rotation
+Pedro A. Santos-Florez,1 Shinnosuke Hattori,2 and Qiang Zhu1, ∗
+1Department of Physics and Astronomy, University of Nevada, Las Vegas, NV 89154, USA
+2Advanced Research Laboratory, R&D Center, Sony Group Corporation, 4-14-1 Asahi-cho, Atsugi-shi 243-0014, Japan
+(Dated: January 3, 2023)
+Recently, some molecular crystals have been found to be surprisingly flexible by undergoing a large extent
+of elastic or plastic deformation upon various mechanical loads. Despite the increasing experimental reports on
+mechanically flexible crystals, this phenomenon has never been reproduced in numerical simulation and thus
+there is no atomistic mechanism to explain its physical origin. Using three recently reported naphthalene diimide
+derivatives as the examples, we perform the first direct molecular dynamics simulation to model their mechanical
+behaviors from brittle fracture to elastic/plastic deformation upon mechanical bending. Our simulation reveals
+that molecular rotational freedom is the key factor to determine the crystal’s mechanical response. Furthermore,
+we propose the use of rotation-dependent potential energy surface to classify organic materials’ mechanical
+response and screen new mechanically flexible candidates in future.
+While most molecular crystals are known to be brittle, there
+exists a class of compliant organic crystals that can easily bend
+under a large mechanical stress loading1,2. Since early 2000,
+there has been a growing number of experimental identifi-
+cations of mechanically flexible crystals3–9. In general, the
+mechanical response of an organic solid depends on both the
+molecular substance and the corresponding crystal packing.
+A remarkable example is shown in Fig.
+1, three crystals,
+made of similar molecules from naphthalene diimide deriva-
+tives, were found to exhibit distinct responses from brittle
+fracture to compliant deformation with either reversible (elas-
+tic) or irreversible (plastic) characteristic10. The flexible na-
+ture of such organic materials is vital for a variety of appli-
+cations, e.g., high-performance modular organic solar cells11,
+actuators12, photochemistry13, electronics14, optics15, as well
+as drug tabulation16.
+In the recent years, various computational techniques have
+been introduced to characterize the observed mechanical
+properties on different molecular systems10,17–19.
+They in-
+clude the topological analysis, elastic properties calculation17,
+and the simulation of shear/tensile deformations10,18. These
+techniques are partially successful in identifying the brittle
+materials which usually exhibit a complex three dimensional
+packing. Within such an interlocked environment, molecu-
+lar motions are largely restricted, resulting a brittleness un-
+der bending. On the other hand, the compliant class of ma-
+terials are featured by a strong anisotropy with plausible slip
+planes17,20. Therefore, these materials become compliant over
+a broad range of applied stress along some specific crystallo-
+graphic directions. However, all available techniques fail to
+explain the difference between the elastic and plastic materi-
+als. While there have been plenty of studies on the bending of
+metals21–26, to our knowledge, no attempts have been made to
+directly simulate the bending of organic materials at the atom-
+istic level.
+Among the compliant crystals, ductile materials are often
+favored in engineering applications16. Hence, researchers at-
+tempted to use the well established dislocation theory to ex-
+plain the observed plasticity on organic materials2,3. Simi-
+lar to the plastic deformation in ductile metals, it was found
+that mechanical shearing can also occur via the slippage of
+dislocated molecular layers on the molecular crystals with a
+β
+α
+γ
+α
+β
+γ
+(a)
+(b)
+(c)
+ (degree)
+α
+x
+z
+x
+y
+z
+γ
+β
+α
+y
+y
+z
+x
+-30
+30
+0
+FIG. 1.
+The simulated bending on three different materials based
+on naphthalene diimide derivatives. (a) brittle Pr (50.3×7.0×6.8
+nm3), (b) elastic Et (50.7×6.4×6.6 nm3) and (c) elastic/plastic Me
+(50.2×6.4×6.9 nm3). These three crystals consist of very similar
+molecules that differ only in the side groups. In the left panel, the
+initial and finally deformed configurations are colored by the molec-
+ular alignment (α) along the x-axis. The corresponding molecules
+and the definition of rotation angles are shown in the right panel.
+layered packing27,28. Using these facile slip planes, a bend-
+ing model was proposed accordingly to explain the underly-
+ing mechanism3. Although the dislocation is not uncommon
+in molecular crystals29,30, there has been no direct experimen-
+tal evidence to support that the dislocation is present in the
+organic crystals under bending. Furthermore, this mechanism
+fails to explain the observed crystals that can also bend elas-
+tically to a large extent. In fact, two crystals as shown in Fig.
+arXiv:2301.00307v1  [cond-mat.mtrl-sci]  31 Dec 2022
+
+X
+z7X
+z2
+1b-c possess very similar crystal packing. Give the apparent
+similarity in both molecular structure and crystal packing, it
+is expected that the elastic crystal (Fig. 1b) should undergo
+similar molecular events like the plastic crystal (Fig. 1c) by
+following the ending mechanism. But the actual deformation
+was observed to be elastic. Clearly, our current understanding
+on the elasticity and plasticity remains limited.
+In this work, we present our efforts in questing the molec-
+ular bending mechanism with the aid of atomistic simula-
+tion.
+To achieve this goal, we start by developing a ro-
+bust simulation protocol that can directly model the bend-
+ing of organic crystals at the atomic level. Specifically, we
+employed a three-point bending model within a partial peri-
+odic boundary condition31. In our calculation, we performed
+non-equilibrium molecular dynamics simulation by applying
+the indentation on the center of molecular slab under finite
+temperature31.
+We also carefully tested the choice of slab
+models and thermal equilibration to ensure the robustness of
+our simulation set up. In order to automate the simulation,
+we developed a computational pipeline to automate the gen-
+eration of molecular force fields from the AmberTools20
+software32. Force field parameters are assigned by the Gen-
+eral Amber Force Field (GAFF) with atomic charges using
+semi-empirical (AM1) with bond charge correction (BCC)33.
+All simulations were performed on the LAMMPS package34 at
+room temperature with the strain rate of 10 m/s.
+In the following, we will focus on three naphthalenete-
+tracarboxylic diimide crystals as discussed in Fig. 1. The
+three molecules share the same backbone while differing only
+in the side chains. The brittle crystal consists of the molecules
+with the propyl group, featured by the orthorhombic space
+group Pbca with one molecule in the asymmetric unit. On the
+other hand, the elastic/plastic crystals have the ethyl/methyl
+groups, both adopting the monoclinic space group P21/c with
+half a molecule in the asymmetric unit. For convenience, we
+follow the previous literature10 to name these systems accord-
+ing to their molecular functional groups (i.e., Pr, Et, Me).
+In all three cases, the weak interaction are formed by alkyl
+groups at the (001) plane. However, the overall molecular
+packing in the brittle-Pr crystal are more complex since there
+exist eight different types of molecular alignments due to the
+mmm symmetry operations. On the contrary, there are two
+types of molecular alignments in the Et/Me crystals, and each
+(001) layer contains only one type of molecular alignment (see
+Fig. S1 and table S1).
+Fig.
+2 summarized the simulated evolution of average
+molecular potential energy as a function of indentation depth
+for all three materials. For a fair comparison, we set up the
+model size close to ∼ 50.0 × 7.0 × 7.0 nm3. Encouragingly,
+our calculations reproduced the experimentally observed brit-
+tle fracture, elastic deformation and plastic bending, respec-
+tively. First, Pr is clearly brittle as evidenced by the abrupt
+drop of energy in Fig. 2a, which is also consistent to the
+appearance of crack pattern in Fig. 1a when the indentation
+depth reaches 3.5 nm. On the other hand, Et is more com-
+plaint with a maximum indentation of 6.2 nm. Apply further
+loading would lead to the formation of crack as well. If we re-
+lease the indentation before Et reaches 6.2 nm, the simulation
+FIG. 2.
+The evolution of average molecular potential energy as
+a function of indentation depth upon (a) loading and (b) unloading.
+In (b), only two samples (Me-elastic and Me-plastic) are shown for
+clarity.
+will roughly return to the original state. Therefore, the defor-
+mation is elastic. Interestingly, Me can survive under more
+than 10 nm indentation without breaking with two different
+setups. For the slab after a full isobaric-isothermal equilibra-
+tion, it bends elastically, as evidenced by the reversible energy
+versus indentation depth relation (denoted as Me-elastic in
+Fig. 2b). When the slab has a small strain in the initial config-
+uration (see Table S2), the corresponding energy curves upon
+loading and unloading are no longer reversible. Compared to
+the Me-elastic, this sample achieves lower energy stable when
+it approaches the maximum indentation depth upon loading.
+When the indentation is released, it does not return to the orig-
+inal states, but maintains a relatively higher energy. Therefore,
+the whole deformation process is irreversible and plastic. The
+sample will be referred to Me-plastic from now on. It is also
+important to note that the deformation is strongly anisotropic.
+For the same Me sample, the deformations are brittle if the
+indentation is applied on other directions. Such a direction-
+dependence has also been observed in recent experiments16.
+Although several recent computational studies attempted to
+explain the observed mechanical properties, they were lim-
+ited to indirect simulations such as pure tensile and shear
+tests10,17–19. Here, our results provide the first direct evidence
+from atomistic modeling and reproduce the experiment obser-
+vations on their mechanical responses upon the bending de-
+formation. Compared to the simulation results, the elastic and
+plastic samples are found to bend under larger deformations in
+real experiments10. This is because that the material’s length
+on x-axis under the actual bending test can shrink to release
+the tensile stress. However, our simulation model still obeys
+the periodic boundary condition along the x-axis. Hence we
+expect that the degree of bending from our simulation is un-
+
+(a) Loading
+Pr: brittle
+0.4
+Et: elastic
+△E (kJ/mol)
+Me: elastic
+Me: plastic
+0.2
+0.0
+0
+4
+6
+8
+10
+(b) Unloading
+Me: elastic
+△E (kJ/mol)
+0.2
+Me: plastic
+0.0
+0
+2
+4
+6
+8
+10
+Indentation Depth (nm)3
+derestimated as compared to the real situation. We also tried
+to vary the strain rate. According to our attempts, it seems that
+increasing the strain rate by 10 times does not qualitatively
+change the results. However, an ultrafast strain rate (>200
+m/s) is likely to trigger some unrealistic phase transition thus
+changes the nature of deformation significantly. Regardless
+of these restrictions on parameter choices, our simulations are
+robust in capturing the main physics.
+0.0
+0.1
+Pr: brittle
+Et: elastic
+Me: plastic
+0.0
+0.1
+40
+20
+0
+20
+40
+0.0
+0.1
+Distribution
+Rotation (degree)
+FIG. 3. The simulated distribution of accumulated rotational angles
+(with respect to the initial configurations) for all materials upon the
+bending loads. For clarity, the Me-elastic data was omitted.
+While analyzing the dynamic trajectories, we observed that
+molecules rotate strongly upon bending. Fig. 1 defines the
+alignments (α, β, γ) for each molecule that can rotate along
+the x, y, z axes in the Cartesian coordinates. Fig. 3 plots
+the distribution of molecular rotations for all three directions.
+Given that indentation direction acts on the z-axis and the
+setup of three bending points aligns along the x-axis, we ex-
+pect that the rotational mode along y axis (β) is the primary
+motion under the loading. Indeed, Fig. 3 reveals that the rota-
+tion in β is more pronounced that other directions for all three
+molecules. According to the computed moments of rotational
+inertia in Table I, the molecules with smaller size are easier to
+rotate more. Therefore, Me has overall more rotational flexi-
+bility than Et and Pr in all directions.
+TABLE I. The computed moments of rotational inertia (Da· ˚A2) for
+each system.
+System
+Number of atoms
+Ixx
+Iyy
+Izz
+Pr
+44
+1707.95 4124.74 5606.78
+Et
+38
+2332.63 2311.36 3610.28
+Me
+32
+1911.78 1710.74 2854.21
+In Figs. S3-S531, we provided the detailed analysis on each
+simulation trajectory. Among them, it is mostly interesting
+to note that there is an obvious asymmetric distribution of β
+for the plastic deformation as shown in Fig. 3. To quest its
+origin, we plot a few representative structures from the cor-
+responding trajectory in Fig. 4. Unlike the elastic deforma-
+5.0
+7.5
+10.0
+Indentation depth (nm)
+ (degree)
+β
+-30
+30
+0
+-15
+15
+FIG. 4.
+The list of representative snapshots from the simulation of
+Me-plastic deformation. The molecules are colored by the β angle
+values from red to blue. The domains of the secondary phase are
+highlighted by the red dotted eclipses. The red dotted arrows indicate
+the slip direction. The grey colored shapes represent the contacting
+locations in the three-point bending test.
+tion that all molecules are symmetrically aligned at the cen-
+tered yz plane, we found that the region near the indenter
+tip undergoes a phase transition through molecular rotation.
+This region is also evident from non-zero rotations of α and
+γ as shown in Fig. S5. This new domain, consisting of re-
+aligned molecules (denoted as the red dotted eclipse), can
+easily slip along its interface with the parent domain. Upon
+indentation, the molecules in the secondary domain, located
+on the upper surface of the slab, do not gain enough momen-
+tum to go downward as compared to other molecules due to
+the compressive stress from the bending forces. Therefore,
+the relative slipping direction of the secondary domain is up-
+ward and we observe the appearance of a bump near the in-
+denter tip. As the tip continues to go down, the secondary
+domain keeps climbing up until the bump reaches its maxi-
+mum. Upon further compression, the molecules at the bottom
+region are nearly flattened due to a large tensile stress, thus
+creating much empty space along the z-axis. Thus, the sec-
+ondary domain slips down to push the neighboring molecules
+down to fill the empty space. Clearly, this secondary domain
+serves as a buffer zone to help the system maintain a rela-
+tively low energy state and postpone the formation of crack.
+When the indentation is released, the process is supposed to
+be irreversible at low temperature since triggering the back
+transformation requires some energy barrier. Therefore, it is
+a plastic deformation. However, it is driven by the molecu-
+lar rotation, which is different from the plastic phenomenon
+in the metals that requires the migration of dislocations. Due
+to the phase transition driven by molecule rotation, the do-
+main of new phase may appear near the indenter and coexist
+
+4
+20
+10
+0
+10
+20
+30
+40
+R1 (degree)
+20
+10
+0
+10
+20
+30
+40
+R2 (degree)
+GM
+LM
+(a) Brittle
+20
+10
+0
+10
+20
+30
+40
+R1 (degree)
+20
+10
+0
+10
+20
+30
+40
+GM
+LM
+(b) Elastic
+20
+10
+0
+10
+20
+30
+40
+R1 (degree)
+20
+10
+0
+10
+20
+30
+40
+GM
+LM
+(c) Plastic
+10
+1
+100
+101
+102
+103
+104
+E (kJ/mol)
+FIG. 5. The potential energy surface as a function of molecular rotation for three crystals with different mechanical response: (a) Pr-brittle,
+(b) Et-elastic, and (c) Me-elastic/plastic deformations. The while region in (a) denotes the rotations leading to energy exceeding 104 kJ/mol.
+with the parent phase via a low-energy interface. The newly
+formed secondary phase can freely slide along the interface
+due to the external stress conditions. In the early stage, the
+upward movement of new phase results in a bump shape near
+the indenter. We note that such a bump has actually been
+found in the bending experiment10, but it was not discussed
+in the literature. Our simulation here suggests that the forma-
+tion of bump is a key characteristic of the plastic deformation
+driven by molecular rotation. If the external temperature is
+sufficiently high to cross the phase transition barrier, the pro-
+cess may become reversible, similar to the previously reported
+superelastic organic crystals4.
+So far, we have established the relation between molecular
+rotation and the observed mechanical responses. Clearly, the
+degree of freedom of molecular rotation is the key factor that
+determines the mechanical flexibility of organic crystals under
+bending. However, we are still unclear why some materials
+are more compliant than others and why we observed two dif-
+ferent deformation behaviors on the Me crystal with slightly
+different initial configurations. To quest their physical origins,
+it is necessary to examine the potential energy surface (PES)
+with respect to the molecular rotations. Therefore, we use the
+relaxed crystal structure as the reference and then systemat-
+ically rotate two groups of symmetrically-related molecules
+(colored in red and blue in Fig. S1) along the y-axis in the
+unit cell. For the Pr-crystal, each group has four molecules
+with the same alignment in β. For both Et and Me crystals,
+each group contains only one molecule. The computed poten-
+tial energy maps as the function of the rotation angles (R1 and
+R2) are summarized in Fig. 5.
+As shown in Fig.
+5a, Pr has a very stiff global mini-
+mum (GM) at (0, 0). This indicates that even a slight rota-
+tion can lead to a high energy penalty. The energy basin of
+GM is aligned diagonally, suggesting that the low energy rota-
+tion modes are synchronous due to the crystal symmetry con-
+straint. In this energy basin, the total energy of the whole sys-
+tem increase about 500 kJ/mol, when it reaches the (10, 10).
+However, such high energy penalty would eventually lead to
+the generation of crack. In addition, there is a local minimum
+(LM) centered around (20, 20). But this state is nearly impos-
+sible to reach due to a high energy barrier up to 104 kJ/mol.
+Overall, Pr has a rather limited rotational freedom, which is
+consistent with the fact that each molecule in Pr is surrounded
+by multiple types of molecular alignments.
+Compared to Pr, the Et sample (Fig. 5b) has more spreads
+around the GM (0, 0). Therefore, the molecules can rotate
+more under the mechanical load. As shown in Fig. 3, two
+rotational peaks are symmetrically distributed at ±20 degrees
+when the system reaches the elastic limit. According to Fig.
+5b, the rotation around (20, 20) would lead to a penalty energy
+of 500 kJ/mol. Therefore, the Et molecules can rotate more
+than Pr before the crack event starts. Similarly, Et has another
+LM around (30, 30), but it is unreachable due to a high energy
+barrier.
+On the other hand, the Me has a even flatter energy spread
+around the GM basin (Fig. 3c). Using 500 kJ/mol as the
+threshold, the computed area ratios are roughly 0.14 (Pr), 0.84
+(Et), 1.00 (Me). Hence, the Pr can sustain more elastic de-
+formation than other materials. These values are qualitatively
+consistent with our computed critical indentation depth val-
+ues as shown in Fig. 2, and even fits the experimental values
+better (given that Me is found to be significantly more elastic
+than Pr). In addition, Me is remarkable because there exists
+a low energy pathway that connects its LM at (30, 30). Under
+the mechanical load, there exist two scenarios. One is to con-
+tinue to expand in the GM basin and the system bends elasti-
+cally, as we found in our simulation starting with the perfectly
+equilibrated Me single crystal sample. Alternatively, it is also
+possible to reach the neighboring LM basin. While the latter
+case requires crossing a barrier on its PES map, it may be fa-
+cilitated by the pre-existing structural defects or activated due
+to kinetic reason. Indeed, we observed such a phase transition
+when the initial configuration is strained. And this eventually
+led to a plastic deformation as shown in Fig. 4. Correspond-
+ingly, the existence of molecules at the LM (30, 30) region
+resulted in a stronger peak around 30 degree as compared to
+the peak at -30 degree for the distribution of β in Fig. 3, In
+the real experiment, the latter scenario is more likely to occur
+since the defects are unavoidable. Although the deformation
+process is irreversible at low temperature upon the release of
+
+5
+indentation, it may become reversible at an elevated tempera-
+ture when it is sufficient to cross the barrier between LM and
+GM.
+In summary, we perform the first molecular dynamics sim-
+ulation to directly model the mechanical bending of organic
+crystals. Using three recently reported naphthalene diimide
+derivatives as the examples, our simulation successfully re-
+produced the experimentally observed mechanical behaviors
+from brittle fracture to elastic/plastic deformation upon me-
+chanical bending. By analyzing their atomistic trajectories,
+we found that molecular rotational freedom is the key factor
+to determine whether or not the materials are bendable. This
+phenomenon originates from the subtle interplay between ge-
+ometry packing and intermolecular interaction. Furthermore,
+we found the use of rotation-dependent potential energy sur-
+face map can be used clearly explain the origin of different
+mechanical responses for organic materials. Together with the
+recently proposed crystal packing screening model35, our re-
+sults can be used to guide the search for new mechanically
+flexible candidates with improved functionality for future de-
+vice applications.
+This research is sponsored by the NSF (DMR-2142570)
+and Sony Group Corporation. The computing resources are
+provided by ACCESS (TG-DMR180040).
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+6
+Supplementary Online Materials:
+Bending Deformation Driven by Molecular Rotation
+A. Crystal structures
+In this study, we focused on three systems consisting of naphthalene diimide derivatives as shown in Fig. S1. The three molecules
+share the same backbone while differing only in the side chains. The brittle crystal consists of the molecules with the propyl
+group (Pr), featured by the orthorhombic space group Pbca with one molecule in the asymmetric unit. On the other hand, the
+elastic/plastic crystals have the ethyl/methyl groups, both adopting the monoclinic space group P21/c with half a molecule in
+the asymmetric unit. In all three cases, the weak interaction plane formed by alkyl groups is (001). In Fig. S1, each molecule in
+the unit cell is colored by the alignment along the y-axis. Clearly, the overall molecular packing in the brittle-Pr crystal are more
+complex. Since there exist eight different types of molecular alignments due to the mmm symmetry operations, the Pr crystal
+has molecules aligned in different ways within the same (001) layer. On the contrary, there are only two types of molecular
+alignments in the Et/Me crystals. And the (001) layer in Et/Me crystals has all molecules aligned in the same direction.
+FIG. S1. The crystal structures of (a) Pr, (b) Et (c) Me systems.
+Table S1 summarizes the crystallographic information of three molecular crystals. Among them, Pr denotes the brittle crystal
+with the CSD refcode of DAHLOQ; Et is the elastic crystal with the CSD refcode of BIYRIM01; and Me is the plastic crystal
+with the CSD refcode of DAHMUX. In addition to the experimental cell parameters, the equilibrium cell parameters from our
+Amber force field are also shown in the parentheses for a comparison. The excellent agreement between experiment and theory
+warrants the use of Amber force field in our following simulations.
+TABLE S1. The crystallographic information of three molecular crystals.
+System
+CSD Refcode
+Space Group
+Number of molecules
+a ( ˚A)
+b ( ˚A)
+c ( ˚A)
+β (◦)
+Pr
+DAHLOQ
+Pbca
+8
+6.96 (7.30) 17.24 (17.40) 27.58 (27.90) 90.0 (90.0)
+Et
+BIYRIM01
+P21/c
+2
+4.84 (5.07)
+7.74 (7.79)
+18.32 (19.07) 90.1 (90.3)
+Me
+DAHMUX
+P21/c
+2
+4.62 (4.58)
+8.02 (8.28)
+17.02 (18.40) 94.0 (93.9)
+
+(a)
+(b)
+(c)7
+B. Simulation Setup
+To enable the direct simulation of bending, we created the slab model as shown in Fig. S2. Both x and y-axes are under the
+constraint of periodic boundary conditions, while the c-axis is not periodic. To reproduce the experimental results10, we rotated
+the crystal structures with the matrix of [[0,0,1], [0,-1,0], [1,0,0]], and then built the super cell slab models according to Table
+S2. In each case, we added the vacuum to allow the materials bend sufficiently. The slab correction was applied to remove the
+slab-slab interactions from the periodic images. Due to the non-triclinic box restriction on the computation of slab correction,
+the β angles for the slabs of Et and Me were to be set to 90◦, which are slightly different from the ideal values. However, this
+compromise should not change the results largely.
+For Me, two models were considered, including (i) the supercell after the isobaric-isothermal equilibration; and (ii) the
+supercell with the experimental cell parameters. Although these two initial configurations only differ slightly, it has been found
+they led to different elastic/plastic deformation processes in the subsequent bending simulation.
+FIG. S2. The schematic setup of a bending simulation model.
+TABLE S2. The details of models used in the bending simulation.
+System
+Deformation
+Supercell
+Number of molecules
+a ( ˚A)
+b ( ˚A)
+c ( ˚A)
+Vacuum ( ˚A)
+Pr
+brittle
+18 × 4 × 5
+5760
+503.2
+69.9
+70.6
+120.0
+Et
+elastic
+27 × 4 × 5
+6480
+508.5
+63.6
+74.7
+120.0
+Me
+elastic
+29 × 8 × 15
+6960
+501.6
+65.2
+86.5
+120.0
+Me
+plastic
+30 × 8 × 15
+7200
+510.6
+64.2
+85.1
+120.0
+Along the non-periodic z-axis, a cylinderical indenter with the radius of 30 ˚A is applied on top of the slab center in the unit
+cell. To mimic two other contacting points in the three-points bending simulation, the last one layer of molecules in the bottom
+region were frozen in the entire simulation. In addition, the first columns of molecules on both left and right side of the unit
+cell are defined as the border. The rest atoms not belonging the frozen and border groups are set to the moible group that can
+move freely. To ensure a sufficient heat bath, we first perform Langevin thermostat on both mobile and border groups, followed
+by a second thermal equilibration on only the border atoms. The fully equilibrated sample will be used to perform three-points
+bending simulation with only the border atoms being under the Langevin thermostat to mimic the external temperature reservoir.
+Upon bending, the indenter will be used to push into the simulation slab in a flow with the rate of 10 m/s. When the system
+reaches the maximum indentation depth, the indenter will be kept for 300 ps to allow the system achieves thermal equilibrium.
+Afterwards, the indenter will move upward with the previous rate to mimic the release of indenter process.
+
+Z
+Border
+Frozen
+Mobile8
+C1. Deformation Analysis on Pr-Brittle
+To quest the origin of Pr-Brittle, we plot a few representative structures from the corresponding trajectory in Fig. S3. Upon
+deformation, we found that the sample continuously to bend from 0 to 3.0 nm (the first row of Fig. S3) and 4.0 nm (the second
+row of Fig. S3). The molecules barely rotate around the x (α) and z (γ) axis. However, the rotation on y-axis is more pronounced
+and it symmetrically distributed around the central indenter. When the indentation depth exceeds 4.2 nm (the last row of Fig.
+S3), the lower surface cracks due to a large tensile stress.
+2.5
+3.5
+4.5
+Indentation depth (nm)
+ (degree)
+α
+-30
+30
+0
+-15
+15
+ (degree)
+β
+-30
+30
+0
+-15
+15
+ (degree)
+γ
+-30
+30
+0
+-15
+15
+Pr-brittle
+FIG. S3. The list of representative snapshots from the simulation of Pr-Brittle deformation.
+
+9
+C2. Deformation Analysis on Et-Elastic
+To quest the origin of Et-Elastic, we plot a few representative structures from the corresponding trajectory in Fig. S4. At a
+small indentation depth (4.0 nm as shown in the first row of Fig. S4), the molecules barely rotate around the x (α) and z (γ) axis,
+while the rotation on y-axes (β) is more pronounced and it symmetrically distributed around the central indenter. However, it
+is clear that the molecules around the center of y-axis do not rotate. Upon further indentation at 5.0 nm (the second row of Fig.
+S4) and 6.2 nm (the last row of Fig. S4), the molecules at the center of lower surface undergo a large rotation around the x and
+z due to a large compressive stress, but do not rotate around y. This suggests that molecules upon tension prefer a rotation on
+α and γ, rather than the primary rotation mode at β due to the anisotropic behavior of its potential energy landscape. Since the
+rotations are symmetrically distributed around the indenter, it is still an elastic deformation. When the indentation is released,
+the process is supposed to be reversible.
+4.0
+5.0
+6.0
+Indentation depth (nm)
+ (degree)
+α
+-30
+30
+0
+-15
+15
+ (degree)
+β
+-30
+30
+0
+-15
+15
+ (degree)
+γ
+-30
+30
+0
+-15
+15
+Et-elastic
+FIG. S4. The list of representative snapshots from the simulation of Et-Elastic deformation.
+
+10
+C3. Deformation Analysis on Me-Plastic
+To quest the origin of Me-Plastic, we plot a few representative structures from the corresponding trajectory in Fig. S5. At
+the depth of 5.5 nm, we found that the molecules near the indenter tip (in the first row of Fig. S4) have alternative changes of
+α and γ angles, which is similar to that in Fig. S4. However, these molecule has non-zero β angles. Therefore, it is no longer
+symmetric and signals a phase transition trigger by the large compressive stress in the upper surface due to bending. This domain
+of new phases, consisting of realigned molecules (denoted as the red dotted eclipse), can easily slip along its interface with the
+parent domain. Upon indentation, the molecules in the secondary domain do not gain enough momentum to go downward as
+compared to other molecules. Therefore, the relative slipping direction of the secondary domain is upward and we observe the
+appearance of a bump near the indenter tip (in the second row of Fig. S3 at the indentation depth of 6.7 nm). As the tip continues
+to go down, the secondary domain keeps climbing up until the bump reaches its maximum. In the mean time, the the molecules
+at the center bottom region are nearly flattened, which can trigger another phase transition to form a new phase domain. Upon
+further compression, the flattened molecules at the center bottom region create much empty space along the z-axis. Thus, the
+secondary domain slips down to push the neighboring molecules down to fill the empty space (see the third row of Fig. S3 at
+the indentation depth of 9.5 nm). When the indentation is released, the process is supposed to be irreversible at low temperature
+since triggering the back transformation requires some energy barrier. Therefore, it is a plastic deformation.
+5.0
+7.5
+10.0
+Indentation depth (nm)
+ (degree)
+α
+-30
+30
+0
+-15
+15
+ (degree)
+β
+-30
+30
+0
+-15
+15
+ (degree)
+γ
+-30
+30
+0
+-15
+15
+Me-plastic
+FIG. S5. The list of representative snapshots from the simulation of Me-plastic deformation.
+

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@@ -0,0 +1,889 @@
+Safety Filtering for Reinforcement
+Learning-based Adaptive Cruise Control
+Habtamu Hailemichael ∗ Beshah Ayalew ∗ Lindsey Kerbel ∗
+Andrej Ivanco ∗∗ Keith Loiselle ∗∗
+∗ Automotive Engineering, Clemson University, Greenville, SC 29607,
+USA (hhailem, beshah, lsutto2)@clemson.edu.
+∗∗ Allison Transmission Inc., One Allison Way, Indianapolis, IN,
+46222, USA (andrej.ivanco, keith.loiselle)@allisontransmission.com
+Abstract: Reinforcement learning (RL)-based adaptive cruise control systems (ACC) that
+learn and adapt to road, traffic and vehicle conditions are attractive for enhancing vehicle
+energy efficiency and traffic flow. However, the application of RL in safety critical systems such
+as ACC requires strong safety guarantees which are difficult to achieve with learning agents
+that have a fundamental need to explore. In this paper, we derive control barrier functions as
+safety filters that allow an RL-based ACC controller to explore freely within a collision safe
+set. Specifically, we derive control barrier functions for high relative degree nonlinear systems to
+take into account inertia effects relevant to commercial vehicles. We also outline an algorithm
+for accommodating actuation saturation with these barrier functions. While any RL algorithm
+can be used as the performance ACC controller together with these filters, we implement the
+Maximum A Posteriori Policy Optimization (MPO) algorithm with a hybrid action space that
+learns fuel optimal gear selection and torque control policies. The safety filtering RL approach
+is contrasted with a reward shaping RL approach that only learns to avoid collisions after
+sufficient training. Evaluations on different drive cycles demonstrate significant improvements
+in fuel economy with the proposed approach compared to baseline ACC algorithms.
+Keywords: Adaptive cruise control, Safe reinforcement learning, Safety filtering, Control
+barrier functions
+1. INTRODUCTION
+Adaptive cruise control (ACC) systems are one of the in-
+creasingly prevalent driver assistance systems for modern
+vehicles. An ACC system uses radar, computer vision, or
+laser to understand the vehicle’s surrounding and make
+control decisions. When another vehicle or object is not in
+the sensing range, ACC compensates for the road grade,
+friction, and aerodynamic resistances to maintain a speed
+set by the driver. When another car or object is in front,
+the ACC makes decisions to prevent collision and follow
+the preceding vehicle as close as possible to avoid cut-ins.
+ACC has been shown to decrease a driver’s workload, and
+make traffic flows efficient and safer (Marsden et al., 2001;
+Lang et al., 2014).
+An effective ACC system should balance the traffic condi-
+tion of the road, the vehicle performance, and the driver’s
+demanded velocity. Currently available PID-based ACC
+systems (Canale and Malan, 2003; Chamraz and Balogh,
+2018) and proposed MPC-based approaches (Naus et al.,
+2008; Yang et al., 2021) are often tuned to balance this
+trade-off for various operating environments. Although
+’adaptive’ or gain-scheduled versions (Radke and Iser-
+mann, 1987) can be sought, the fixed structure of these
+approaches limits full adaptation throughout the lifetime
+of the vehicle. Furthermore, MPC-based ACC also has
+to find a reliable way of predicting the motion of the
+leading vehicle for the future horizon. On the other hand,
+data-driven reinforcement learning (RL) approaches offer
+a mechanism to continuously customize to traffic, road
+and vehicle conditions without a predefined control archi-
+tecture (Li and G¨orges, 2020). In this work, we consider
+applications of RL-based ACC to commercial vehicles. In
+addition, while traditional ACC is primarily about the two
+tasks of speed tracking and maintaining a safe gap, we
+consider RL-based ACC (RL ACC for short) to explicitly
+optimize fuel economy via gear selection and torque control
+policies.
+Despite the potential benefits of adaptability and im-
+proved performance, RL ACC faces critical safety chal-
+lenges. These derive from the needs of RL algorithms to
+explore in order to learn the optimal policies. RL learns
+how good the given state-action pair is after experiencing
+it, but for applications like vehicle control, exploration in
+an unsafe domain is unacceptable even during (on-road)
+training of the RL algorithms. However, thanks to recent
+progress in safe RL, different approaches are suggested
+to encourage or limit the exploration only in the safe
+domain. We briefly mention a few of them. Reward shaping
+approaches put large penalties into the performance objec-
+tive function if constraints are violated. On the other hand,
+constrained Markov decision process (CMDP) approaches
+assign safety constraint costs to each state-action pair and
+limit the total safety constraint cost of a trajectory to
+be lower than a certain threshold (Altman, 1999). The
+arXiv:2301.00884v1  [eess.SY]  2 Jan 2023
+
+reward shaping and CMDP approaches are implemented
+on the performance controller itself to encourage respect-
+ing safety constraints but they do not guarantee safety.
+Another set of approaches involve the use of safety filters
+that impose hard constraints. Such approaches separate
+the performance-oriented RL controller, whose only aim is
+to optimize the system’s performance objective function,
+from the safety filters, which project the unsafe actions
+proposed by the performance controller into the safe set.
+The safety filters determine the safety condition of the
+given state-action pair using the dynamical model of the
+system, or they use offline data to learn constraints (Dalal
+et al., 2018) and safety indexes (Thananjeyan et al., 2021;
+Srinivasan et al., 2020). In this paper, we pursue dynamical
+model-based safety guarantees to construct the safe set in
+such a way that gives the RL performance controller the
+freedom to explore within the safe boundaries. As its train-
+ing progresses, the RL performance controller eventually
+learns the safety boundaries and ceases to demand unsafe
+actions (Thananjeyan et al., 2021). Note that even though
+it does not interfere with the inner workings, the safety
+filter affects control performance by dictating where the
+performance controller can operate.
+Of the model-based approaches to designing safety filters,
+control barrier functions (CBFs) offer light computation
+and scalability (Li, 2021). A CBF guarantees safety by
+making the controller work in the invariant safe-set defined
+by a superlevel set of a continuously differentiable function
+h(x) : Rn → R. The actions selected by the performance
+controllers are projected into the safe set in such a manner
+that the proposed actions are modified minimally (Ames
+et al., 2019), and no unsafe actions are passed to the
+controlled system. Different approaches could be pursued
+to specify CBFs with their pros and cons. The intuitive
+one is to come up with a handcrafted CBF considering the
+dynamics of the system and the action bounds associated
+with it (Xu et al., 2018; Ames et al., 2014; Cheng et al.,
+2019). In collision avoidance problems, for instance, the
+CBF can be derived by considering the maximum decel-
+eration that the system could exert to close a distance
+gap. When possible, it is also desirable to progressively
+widen the safe set to get the maximal safe domain, a task
+currently possible with polynomial plant dynamics and
+polynomial CBFs via sum-of-squares (SOS) programming
+(Chamraz and Balogh, 2018). Another approach that is
+tailored to high relative degree nonlinear dynamical sys-
+tems such as those involving inertia effects is the use of
+exponential CBF (ECBF) (Nguyen and Sreenath, 2016).
+In this work, we derive ECBFs to work as safety filters with
+our RL-ACC controllers, thereby taking explicit consider-
+ations of inertia effects that are important for commercial
+vehicles that experience large changes in loading.
+The main contributions of this paper are then the deriva-
+tion and demonstration of CBF-based safe RL-ACC ap-
+proach for commercial vehicles that optimizes fuel econ-
+omy. While we derive ECBFs for safety certification, we
+note that straight ECBFs (or CBFs in general) assume
+unbounded actions, and in their natural form, they might
+request actions that are not feasible for the vehicle’s pow-
+ertrain to meet. We therefore put forward a method to
+provide a safety guarantee for a given parameters of ECBF
+within the vehicle action limits. Our performance RL-ACC
+coordinates traction torque control and gear decisions
+considering fuel consumption optimization objectives. The
+RL ACC augmented with the safety certificate is trained
+and evaluated on different driving cycles, and the vehicle
+performance is compared with an RL ACC with reward-
+shaping approach to safe RL, as well as with a conventional
+PID-based ACC.
+The rest of the paper is organized as follows. Section
+2 describes our derivation of the ECBF as safety filters
+for ACC and detail how we address actuation constraints
+within them. Section 3 describes the algorithmic details of
+our performance RL-ACC. Section 4 discusses results and
+discussions, and Section 5 concludes the paper.
+2. SAFETY FILTER FOR ACC
+We briefly review the definition of CBFs as follows. Details
+are given in Hsu et al. (2015). Consider a nonlinear control
+affine system:
+˙x = f (x) + g (x) u,.
+(1)
+where f and g are locally Lipschitz, x ∈ Rn is the system
+state, u ∈ Rm is the control inputs. Assume a safe set
+defined by C = {x ∈ Rn|h (x) ≥ 0}, where h : Rn →
+R is a continuously differentiable function. Then h is a
+control barrier function (CBF) if there exists an extended
+class κ∞ function α such that for all x ∈ Int (C) =
+{x ∈ Rn : h (x) > 0} :
+sup
+u∈U
+[Lfh (x) + Lgh (x) u] ≥ −α (h (x)).
+(2)
+For high relative degree nonlinear affine systems, feedback
+linearization could be used to develop exponential CBFs
+(ECBF) as detailed in Nguyen and Sreenath (2016). This
+is accomplished by transforming (input-output linearizing)
+the high relative degree nonlinear systems into a virtual
+linear system with new state variable ηb := [h(x), ˙h(x), · ·
+·, hr(x)]T , input µ and output h (x):
+˙ηb = Fηb (x) + Gµ,
+h (x) = Cηb
+(3)
+where F and G are matrices representing an integrator
+chain, and C = [1, 0, · · · , 0]. A state feedback controller
+can be designed for the transformed system as: µ = −Kαηb
+with a suitable gain vector Kα that makes F − GKα
+Hurwitz. For a system with relative degree r, µ is also rth
+derivative of the output h(x), µ = Lr
+fh(x)+Lg�Lr−1
+f
+h(x)u.
+If there exists a state feedback gain Kα that makes µ ≥
+−Kαηb (x) for all states, then one can show that h(x) is
+an exponential control barrier function (see Nguyen and
+Sreenath (2016)).
+The ACC part of the present problem is modelled with the
+state variables of separation distance z, the velocity of the
+host vehicle vh and the velocity of the leading vehicle vl.
+The corresponding state equations are:
+˙z = vl − vh
+(4a)
+˙vl = al
+(4b)
+˙vh =
+Tt
+rwmv
+− Fr (vh, mv, θ)
+mv
+(4c)
+Fr = ρAcdv2
+h
+2
++ mvgf cos θ + mvg sin θ
+(5)
+
+where Fr is the total resistance force including gravita-
+tional, rolling and aerodynamic resistances, and Tt is the
+traction torque at the wheels. The parameters cd, f, θ, mv ,
+ρ, Av, rw, al are aerodynamic coefficient, rolling resistance
+coefficient, road grade, mass of the vehicle, density of
+air, frontal area of the vehicle, radius of the wheels, and
+acceleration of the leading vehicle, respectively.
+We observe that the above model can be readily put in the
+control affine form (1). Given a collision safety objective,
+we seek the separation distance z to always be above a
+specified minimum inter-vehicle distance z0. To this end,
+we define the control barrier function (CBF) as the output
+h (x) = z − z0. Considering that the control actuation is
+the traction torque Tt, we have a control affine system of
+relative degree two. In physical terms, the safety objective
+is on position while traction torque directly manipulates
+acceleration. Inertia effects come into play and must be
+accounted for. The input-output linearization into the
+form (3) then gives:
+˙h(x) = vl − vh,
+(6)
+µ = ¨h (x) = Fr (vh, mv, θ)
+mv
++ al −
+Tt
+mvrw
+,
+(7)
+−Kαηb (x) = −kα1 (z − z0) − kα2 (vl−vh)
+(8)
+We now compute some bounds for the given control input
+µ considering actuation limits on the traction torque (Tmin
+and Tmax). For a given acceleration of the preceding
+vehicle (al) and velocity of the host (vh), the feasible
+bounds of µ are given as
+µTmin/max = al + Fr (vh, θ, mv)
+mv
+− Tmin/max
+mvrw
+(9)
+For a given gain vector Kα = [kα1, kα2], ECBF guar-
+antees safety if the proposed state feedback control,
+−kα1 (z − z0) − kα2 (vl − vh), is within the virtual linear
+system action bound [µT max, µT min]. In general appli-
+cation cases, however, this bound may not be respected.
+Nevertheless, if Kα is chosen so that the poles are placed
+sufficiently to the left in s-plane, the above ECBF could
+still bound the safe set. Safety assurance for such pole
+selections could be achieved by investigating the evolution
+of the CBF control term h (x) in worst-case situation where
+the linear virtual model is initialized with extreme possible
+η0,xrm, and then the possible limiting torque actions are
+applied. For a given minimum separation distance target
+and maximum downhill road grade, this is equivalent to
+applying the maximum possible traction torque output
+of the performance RL-ACC agent, with the host vehicle
+model (of largest loading) initialized in with the maximum
+possible velocity while the preceding vehicle is under its
+maximum deceleration. This extreme conditions gives the
+feasible µ bounds as µT min−xrm and µT max−xrm using
+equations (9).
+To capture the evolution of h (x) under these extreme
+conditions, a simulation rollout is discretized into timestep
+∆t, and the action µ (saturated with µT min−xrm and
+µT max−xrm) held piecewise constant. Algorithm 1 shows
+how this is implemented by integrating the virtual system
+(3). If the h (x) from this simulation is positive at infinity
+(or after some finite time), the selected Kα guarantees
+safety. Otherwise, the Kα needs to be changed until this
+is satisfied.
+Algorithm 1 An algorithm to enforce system bounds on
+a virtual linear system
+η ← η0
+µ ← µ0
+while t ≤ t∞ do
+t ← t + ∆t
+if µ < µT max−xrm then
+µ ← µT max−xrm
+else if µ > µT min−xrm then
+µ ← µT min−xrm
+end if
+h (x(t)) ← C(eF ∆tη0 + eF ∆t � ∆t
+0
+e−F τGµd(τ))
+µ ← −kα1h (x) − kα2 ˙h (x)
+η0 ←
+�h (x)
+˙h (x)
+�
+end while
+Once the suitable gain vector Kα is selected, the ECBF
+safety constraint enforces safety by projecting the action
+proposed by the outputs of the RL controller’s actor
+network Ta (s) (see next section) to the control traction
+torque Tt in a way that introduces minimal changes to it.
+This is done by posing and solving the quadratic program:
+T ∗
+t = arg min
+Tt
+1
+2 ∥Tt − Ta(s)∥2
+s.t.
+al + Fr (vh, mv, θ)
+mv
+−
+Tt
+mvrw
+≥ −kα1 (z − z0)
+− kα2 (vl − vh)
+(10)
+3. VEHICLE ENVIRONMENT AND RL ACC
+The powertrain controller is modeled as Markov decision
+process (MDP) consisting of states s, actions a, a reward
+function r (s, a), and discounting factor γ. The probability
+of action choices is policy π(a|s, θ) where θ denotes the
+parameters of the deep neural network used to approxi-
+mate the policy. The host vehicle velocity vl, the relative
+velocity between the preceding and host vehicles vrel,
+the separation distance between the vehicles z, the gear
+ng, the mass of the vehicle mv, the road grade θ, the
+driver demanded velocity vset and a flag to show if the
+vehicle is in ACC sensor range f constitute the states
+of the RL agent, s = {vl, vrel, z, ng,mv, θ, vset, f}. The
+RL performance controller is designed to perform both
+traction torque Ta control and gear change selection ∆ng,
+i.e. a = {Ta, ∆ng}. As shown in Fig.1, the proposed Ta is
+filtered by the ECBF safety layer to safe traction torque
+demand Tt (10). The engine torque and engine speed that
+brings about this wheel traction torque are then calculated
+utilizing transmission ratios of the selected gear and the
+final drive, and the associated fuel rate is read from the
+fuel map. Notice that while the RL controller’s actions are
+Ta and ∆ng, the ECBF safety filter does not use ∆ng in
+the safety constraint. However, taking into account that
+gear selection is crucial for fuel economy and driver ac-
+commodation, it is an integral part of the RL performance
+controller.
+The filtered traction torque Tt and the gear change ∆ng
+actions are implemented in the vehicle environment, and
+
+Fig. 1. Training RL agent for ACC
+the suitability of the actions is measured by the reward
+function. The reward is designed to accomplish the in
+range and out of range tasks, and different performance
+objectives within each task are tuned by reward weights
+(w). When there is not a vehicle present in the sensing
+range (z > zsr), as shown in (11), the reward structure
+requires the vehicle to maintain the driver-set velocity and
+concurrently balances the fuel consumption and smooth
+torque change considerations. When there is a vehicle in
+the sensing range, on the other hand, the reward aims to
+maintain a close distance from the preceding vehicle, as
+shown in (12). In such proximity, in addition to smooth
+torque change and fuel consumption considerations, the
+reward ros discourages the host vehicle from overspeeding
+beyond the driver demanded velocity (vset). Gear hunting
+and the associated rough vehicle operation are mitigated
+by including a gear reward term weighted by wg.
+r = wv0.1
+|vh−vset|
+Vrel,max + wf0.1
+˙
+mf
+mf,max + wT 0.1
+|∆Te|
+Te,max +
+wg0.1
+|∆ng|
+ng,max
+(11)
+r = wz0.1
+Z
+Zsr + wf0.1
+˙
+mf
+mf,max + wT 0.1
+|∆Te|
+Te,max +
+wg0.1
+|∆ng|
+ng,max + ros
+(12)
+where ros = wos if vh ≤ vset, else : ros = wos0.1
+vh−vset
+vrel,max ,
+˙mf is the fuel rate and Te is the engine torque.
+To accommodate the continuous traction torque and the
+discrete gear selection, Hybrid Maximum A Posteriori
+Policy Optimization (HMPO) is found to be a good fit
+for the RL training algorithm (Kerbel et al., 2022; Neunert
+et al., 2020; Abdolmaleki et al., 2018). In addition to being
+scalable and robust like state of the art Proximal Policy
+Optimization (PPO) (Schulman et al., 2017) and Trust-
+Region Policy Optimization (TRPO) (Schulman et al.,
+2015) algorithms, the fact that it is off-policy makes it
+data efficient to apply it to the real world RL ACC
+trainings. The RL agent comprises of an actor (parame-
+terized by θ) and a critic (parameterized by φ) networks,
+in which the former determines the control policy for
+a given state π (s|θ) and the latter evaluates these ac-
+tions by providing the associated action values Q (s, a|φ).
+The actor network outputs the mean and variance of
+a Gaussian distribution, from which traction torque is
+sampled (13). In addition to that, it uses softmax ac-
+tivation at the output layer with three choices for the
+gear change decision, analogous to the available gear
+changes ∆n = {1, 0, −1}(upshift, nochange, downshift).
+Categorical sampling is then used to obtain the gear
+change policy (14). Assuming independence between the
+continuous πT
+θ (Ta|s) and discrete πg
+θ(∆ng|s) policies, the
+total policy could be factorized as (15) for combine action
+a = {Ta, ∆ng}.
+πT
+θ(Ta|s) = N
+�
+µθ (s) , σ2
+θ (s)
+�
+(13)
+πg
+θ(∆ng|s) = Cat(αθ(s)), ∀s
+3
+�
+k=1
+αk,θ (s) = 1
+(14)
+πθ (a|s) = πT
+θ (Ta|s) πg
+θ(∆ng|s))
+(15)
+In the policy improvement phase, MPO samples from the
+Q-function for different actions and update the actor-
+network parameters to output actions that maximize the
+action values Q(s, a). This is accomplished by optimiz-
+ing the likelihood function of acting optimally using the
+expectation-maximization algorithm ( see Neunert et al.
+(2020); Abdolmaleki et al. (2018)). The policy evaluation
+phase of the training fits the Q-function Qθ (s, a, φ) of
+the critic network, with parameters φ, by minimizing the
+square loss of the current Qθ (s, a, φ) and a target defined
+by retrace sampling Qret
+t
+(Munos et al., 2016).
+min
+φ L (φ) = min
+φ E(s,a)∼R
+�
+Qθ (s, a|φ) − Qret
+t
+�2
+(16)
+4. RESULTS AND DISCUSSIONS
+The above RL ACC with the ECBF safety filter is applied
+to a model of medium duty truck in urban and highway
+driving conditions. The actor and critic networks are con-
+structed with three hidden layers, and each layer consists
+of 256 nodes. The simulation uses a 10-speed automated
+manual transmission (AMT) truck that has a 5 to 10
+tons weight range. The preceding vehicle follows Federal
+Test Procedure (FTP-75) drive cycle for the urban driv-
+ing training, while for highway driving, a combination of
+Highway Fuel Economy (HWFET) and ArtMw130 cycles
+are used in succession (Barlow et al., 2009). Once trained,
+we will use different drive cycles for evaluation as will be
+described below.
+In each simulation step, as shown in Fig.1, the actor
+network proposes the torque and the gear actions for a
+given state which will be filtered by the ECBF safety
+layer. The vehicle environment then executes the safe
+actions, and the associated rewards are calculated. To
+accommodate the different objectives of each task, the
+reward is structured with weights of [wv = 0.675, wf =
+0.175, wT = 0.075, wg = 0.075] for in range, and [wz =
+0.325, wf = 0.175, wos = 0.35, wT = 0.075, wg = 0.075]
+for out of range conditions. The state, action and rewards
+are stored in the memory buffer, and afterward, batches
+of these data are used to train the networks using the
+HMPO algorithm. In order to prevent RL from learning
+the specific drive cycles, the vehicles are initialized in
+random separation distance along with the addition of
+noise to the velocity profile of the preceding vehicle. The
+weight fluctuations are considered by varying the truck
+weight within and between training episodes.
+
+m
+ECBF filter
+Memory buffer
+Ta
+Load actor
+parameters
+RL
+Training
+△ng
+π(s) ={Ta,△ng}
+Load critic
+Q(s,a)
+parameters
+Actor
+network
+π(s)
+Critic
+networkDuring training, because of the careful choice of the gain
+vector Kα = [0.2, 5] as per section 2, the vehicle never
+crashes nor comes within safe distance z0. As the training
+progresses, the RL learns to operate near the driver set
+velocity when it is out of range and follows the preceding
+vehicle more and more closely when it is in range. Even
+though it is not provided with the engine efficiency map,
+as exhibited by the improvement of MPG with training,
+the RL network eventually learns the fuel optimal gear and
+torque actions.
+Table 1. Vehicle environment and RL hyperpa-
+rameter setting
+Vehicle Parameters
+MPO Hyperparameters
+Mass
+5 - 10 tons
+Actor, critic learning rate
+10−4, 10−5
+Au
+7.71m2
+Dual constraint
+0.1
+Cd
+0.08
+Retrace steps
+15
+rw
+0.498
+KL constraints ϵµ, ϵσ, ϵd
+0.1, 0.001, 0.1
+f
+0.015
+αd, αc
+10
+zsr
+350
+γ
+0.99
+Even if it is not practical for safety critical systems, a
+reward shaping approach of safeguarding safety is consid-
+ered to compare against the ECBF-based safety filtering.
+A penalty of rs = −1 is added to the reward function
+when the host approaches closer than the minimum safe
+distance limit z0 and, in the situation of a crash, the
+penalty is enlarged to rc = −10. Due to these safety
+violation penalties, unsafe actions reduce with training,
+and eventually, the agent learns to maximize the reward
+safely. In addition to the reward shaping approach, the
+conventional PID ACC is used as a baseline which, like
+in the case of RL, is designed by dividing the control into
+phases for the in range and out of range conditions (Canale
+and Malan, 2003). The traction torque Tt request is given
+by PID controller and an optimal gear is chosen based on
+the gear with the lowest fuel rate given the desired traction
+torque and vehicle velocity (Yoon et al., 2020; Kerbel et al.,
+2022).
+After the RL ACC with ECBF is trained, its performance
+is evaluated and compared with PID ACC and RL ACC
+with reward shaping counterparts on a 9-ton truck in
+urban and highway driving conditions. For the urban case,
+the preceding vehicle follows the ArtUrban drive cycle,
+and the driver demanded velocity vset is set to be 15 m/s.
+Similarly, a vset of 25 m/s is used for highway driving, and
+to better capture different velocity profiles in the highway
+situation, the preceding vehicle follows a combination of
+ArtRoad and ARTMw150. The initial separation distance
+between the vehicles is 1500 m in both cases.
+In both driving conditions, the RL ACC successfully
+meets the in range as well as out of range objectives
+and, most importantly, safety constraints are respected.
+Fig.2 shows the RL ACC has a similar velocity profile
+to its PID ACC counterpart for the most part of the
+simulation. However, when it comes to gear selection, the
+RL ACC tends to operate at higher gears. As summarised
+in Table 2, for highway driving, the RL ACC exhibited
+an MPG improvement of 8.3%, whereas, in the case of
+urban driving, it has 7.9% higher MPG than the PID
+ACC baseline. When the preceding vehicle is in range,
+the RL ACC is less susceptible to cut-in as it follows the
+preceding vehicle closer, shown by the lower mean in range
+Fig. 2. Simulation of separation distance, velocity, and gear
+profiles of RL and PID ACC controllers in a highway
+driving.
+separation distance zir. Moreover, it is possible to see that
+the RL ACC with ECBF filter and the RL ACC with
+reward shaping arrangements achieve equivalent levels of
+fuel economy and in range car following performances.
+Table 3 shows the performance comparison with weight
+fluctuation in which the vehicle’s weight ranges from 5 to
+10-tons. The RL ACC maintains higher MPG than the
+PID ACC throughout the given weight range, and the
+separation distance is not significantly influenced.
+Table 2. Performance comparison between PID
+ACC, RL ACC with ECBF and RL ACC with
+reward shaping
+Highway driving
+Urban driving
+ACC
+PID
+RL
+RL
+PID
+RL
+RL
+Safety
+layer
+-
+ECBF
+Reward
+shaping
+-
+ECBF
+Reward
+shaping
+MPG
+8.6
+(-)
+9.3
+(8.31%)
+9.31
+(8.37%)
+6.8
+(-)
+7.35
+(7.9%)
+7.38
+(8.4%)
+Zir(m)
+95
+74
+73
+42
+39
+38
+5. CONCLUSION
+In this paper, an exponential control barrier function-
+based safety filter is employed to instill safety into RL
+based ACC system by projecting the learning exploration
+to a safe set. Since practical systems operate with bounded
+actions, we proposed an approach to verify the safety of a
+given ECBF design by forward simulating in consideration
+of worst case scenarios. After being filtered by this ECBF,
+the traction torque and gear change actions proposed by
+the RL-based ACC are implemented on a simulated vehi-
+cle environment and the associated rewards are observed.
+The RL networks are trained using Hybrid Maximum A
+Table 3. Perandomizedof PID ACC and RL
+ACC with vehicle mass fluctuation
+Weight
+(tons)
+5
+6
+7
+8
+9
+10
+RL
+with
+ECBF
+MPG
+10.58
+(10.9%)
+10.38
+(11.6%)
+9.99
+(9.6%)
+9.61
+(8.3%)
+9.3
+(8.31%)
+8.95
+(7.6%)
+Zir(m) 67
+69
+73
+75
+74
+77
+PID
+MPG
+9.54
+9.3
+9.11
+8.87
+8.6
+8.32
+Zir(m) 95
+95
+94
+95
+95
+96
+
+RL with Reward-shaping
+PID
+Sensing range
+RL with CBF
+Distance (m)
+5000
+0
+25
+0
+10
+ear
+5
+G
+11
+0
+250
+500
+750
+1000
+1250
+1500
+1750
+2000
+Time (s)Posteriori Policy Optimization (HMPO) algorithm that
+accommodates the continuous traction torque and discrete
+gear change actions. Evaluation on a medium-duty truck
+shows that the RL ACC fulfilled the velocity objectives
+and, most importantly, respected the safety constraints.
+Compared to PID ACC, the RL ACC augments MPG by
+8.3% in highway driving conditions when the preceding
+vehicle follows a combination of ArtRoad and ARTMw150
+drive cycles, and by 7.9% in urban driving conditions when
+the preceding vehicle follows ArtUrban drive cycle. More-
+over, the RL ACC learns to handle weight fluctuations
+and maintains high performance throughout the vehicle’s
+weight range.
+The current algorithm training and evaluations are per-
+formed on standard driving cycles. Future work will focus
+on using randomized traffic data and measurement noise to
+assess the performance and robustness of RL ACC in even
+more realistic driving conditions. In addition, future work
+will also look at less conservative methods of accounting
+for uncertainties (not worst-case) in ECBF design.
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