Papers

We investigate a three-dimensional parabolic system that arises as a hyperviscous and penalized approximation of the incompressible Navier–Stokes equations. The model combines three complementary dissipative mechanisms: the classical viscous diffusion, a biharmonic (hyperviscous) regularization, and a divergence penalization. In addition, a Temam-type correction is incorporated into the nonlinear convection term to compensate for the weak compressibility effects generated by the penalization procedure. We prove the global existence of weak solutions for arbitrary initial data belonging to $L^2(\mathbb{R}^3)$. For sufficiently small initial data in $H^2(\mathbb{R}^3)$, we establish the existence and uniqueness of global strong solutions. Furthermore, for initial data in $L^1(\mathbb{R}^3)\cap H^2(\mathbb{R}^3)$, we derive optimal large-time decay estimates, showing that the solutions exhibit the same asymptotic decay rates as those of the classical heat equation. A key feature of our analysis is that all the obtained a priori estimates are uniform with respect to the positive penalization parameter $\varepsilon$. These uniform bounds provide a stable and rigorous analytical foundation for the study of the penalized approximation of incompressible flows.

Let $S_n=\{p\in\mathbb{P}:p<10^n\}$, $N_n$ denote the total number of decimal digits occurring in the primes of $S_n$, $C_n(d)$ be the number of occurrences of a digit $d\in\{0,\ldots,9\}$ among those digits, and $P_n(d)$ be the probability of occurrence of a digit, $d$ among those digits. We prove that \[ P_n(d)=\frac{C_n(d)}{N_n} =\frac{1}{10} +O\!\left(\frac{\log n}{n}\right), \qquad n\to\infty, \] uniformly for every decimal digit $d$. The argument is entirely unconditional and combines the Prime Number Theorem, the Erdős–Turán discrepancy inequality, and classical Vaughan–Vinogradov estimates for exponential sums over primes. The principal step establishes quantitative equidistribution for interior digit positions, while the logarithmically many exceptional positions near the ends of the decimal expansion are shown to have asymptotically negligible influence after averaging over all digit positions and prime lengths. Consequently, the decimal digits occurring in primes, when pooled over all positions and all primes below $10^n$, become asymptotically equidistributed. We also clarify the precise scope of the theorem by distinguishing this averaged equidistribution result from the substantially stronger and presently unresolved questions concerning pointwise digit equidistribution, normality, and higher-order digit correlations in the sequence of prime numbers.

This paper presents a symplectic Hamiltonian direct discontinuous Galerkin (DDG) method for approximating wave propagation problems, including the linear and semilinear wave equations. Within an auxiliary-variable-free DG framework, we prove that the symmetry of the numerical flux bilinear form is equivalent to the existence of a discrete Hamiltonian structure. It follows that methods such as the symmetric interior penalty, BR2, and symmetric DDG (SDDG) methods admit a discrete Hamiltonian structure, whereas the Baumann–Oden and DDG methods do not possess this property. Exploiting this structure, we construct fully discrete symplectic schemes by combining the SDDG spatial discretization with symplectic time integrators. We also develop a unified treatment of periodic, Dirichlet, Neumann, and Robin boundary conditions that preserves the discrete Hamiltonian structure of the SDDG scheme. We further derive error estimates for the SDDG method applied to semilinear wave equations under these boundary conditions, showing the optimal convergence rate for the displacement and the suboptimal convergence rate for the velocity. Numerical experiments validate the theoretical convergence rates and demonstrate that the symplectic Hamiltonian DDG method achieves superior long-time energy conservation and accuracy.

In the present paper, with the help of the Faà di Bruno formula and several identities of partial Bell polynomials, the author establishes explicit formulas of the Gauss hypergeometric functions \begin{gather*} {\,}_2F_1\biggl(\frac{1-n}{2},\frac{2-n}{2};\frac{3}{2}-m;z^2\biggr), \quad {\,}_2F_1\biggl(-\frac{n}{2},\frac{1-n}{2};\frac{1}{2}-m;z^2\biggr),\\ {\,}_2F_1\biggl(a,a+\frac{1}{2};\frac{3}{2}-m;z^2\biggr), \quad {\,}_2F_1\biggl(a,a+\frac{1}{2};\frac{1}{2}-m;z^2\biggr) \end{gather*} for $m,n\in\mathbb{N}$ and $a\in\mathbb{C}$, and then derives two combinatorial identities \begin{equation*} \sum_{k=0}^{m}\frac{2^k}{k!} \binom{2m-2k}{m-k} \sum_{\ell=0}^{k} \frac{(-1)^\ell}{2^\ell} \frac{(2k-2\ell-1)!!}{(n-\ell)!} \binom{2k-\ell-1}{\ell-1} =\frac{1}{n!}\binom{2m-n}{m} \end{equation*} and \begin{equation*} \sum_{k=1}^{m}\frac{1}{(k!)^2}\binom{2m-2k}{m-k} \sum_{\ell=1}^{k} \binom{k}{\ell}\ell(2k-\ell-1)! (2a)_\ell =\binom{2m+2a}{m}-\binom{2m}{m}, \end{equation*} where $m\in\mathbb{N}_0$, $n\in\mathbb{Z}$, and $a\in\mathbb{C}$. These newly-established identities extend and generalize the beautiful combinatorial identity \begin{equation*} \sum_{k=0}^{n} \frac{2^{k}}{k!}\binom{2n-2k}{n-k} \sum_{j=0}^{k}\frac{(-1)^{j}}{2^j} \frac{(2k-2j-1)!!}{(n-j)!} \binom{2k-j-1}{j-1} =\frac{1}{n!}, \quad n\in\mathbb{N}_0, \end{equation*} which was obtained in Theorem 4 of the paper "F. Qi, C.-Y. He, and D. Lim, Explicit formulas of two Gauss hypergeometric functions and several combinatorial identities, Discrete Appl. Math. Vol. 393 (2026), 215–229. DOI: https://doi.org/10.1016/j.dam.2026.06.023."

In this paper, we propose a mathematical model for class scheduling using binary linear programming with the aims to automate the tedious task and to minimize the leftover seats distributed over all the classrooms. A case study at the Department of Mathematics, King Mongkut's University of Technology Thonburi (KMUTT), Thailand, is also presented. The result from our model is shown to have improved significantly over the current manual scheduling method.

Alesandroni proved that Betti numbers of monomial ideals in at most four variables are independent of the characteristic of the base field, while characteristic-dependent Betti numbers occur in six variables. We prove that the five-variable case is characteristic-independent. More precisely, if $S=k[x_1,\ldots,x_5]$ and $M\subseteq S$ is a monomial ideal, then the multigraded, graded, and total Betti numbers of $S/M$ are independent of $\operatorname{char}(k)$. The proof reduces arbitrary monomial ideals to squarefree twin ideals and then applies Hochster's formula. The topological input is that simplicial complexes on at most five vertices have torsion-free integral homology. The six-variable example arising from the six-vertex triangulation of $\mathbb{RP}^{2}$ shows that the bound is sharp. We also record a computation of the graded Betti tables of squarefree monomial ideals in five variables up to relabeling.

We study periodic partitions of the plane into two distinct cells minimizing the anisotropic $\ell_1$-perimeter. For a rectangular lattice $G$, we compute explicitly the $(G,\ell_1)$-isoperimetric profile and classify all minimizers. When one cell has small area, the optimal tiling is generated by a square and a chipped rectangle, while in the remaining regime the tiling is generated by two adjacent rectangles. Further minimizing over all possible planar lattices, we show that the $\ell_1$-isoperimetric profile is attained by the Pythagorean double tiling of two axis-aligned squares sharing a vertex. This configuration is unique unless the two cells are assigned the same area. Finally, we prove that the limiting (non periodic) partitions obtained by sending one volume to infinity are locally $\ell_1$-isoperimetric.

A novel variant of the Jacobi-Davidson (JD) type method for Hermitian eigenvalue problems, designated as JD-V, is proposed based on a newly designed correction equation, whose solution is shown to be nearly as effective as that of the standard correction equation for subspace expansion. Rigorous convergence analysis of MINRES for solving these equations reveals that the inner iterations of JD-V are significantly more efficient than those of the standard JD method when highly clustered eigenvalues are of interest. A thick-restart JD-V algorithm with deflation and purgation is developed to compute several eigenpairs of a a large-scale Hermitian matrix. Numerical experiments confirm the theoretical results and demonstrate the considerable superiority of JD-V over standard JD in overall efficiency.

We propose a framework for computing multiphotonic scattering probabilities in a lossless multiport interferometer for arbitrary photon numbers and degrees of indistinguishability. By exploiting a toroidal expansion of multiphotonic states in tensor powers of single-particle states, the framework defines a map from a torus of relative phases to the probability simplex that governs the asymptotic behavior of scattering probabilities in the large-photon limit. Specifically, the probabilities concentrate on the "classically allowed region" defined by the map, and the slowly-varying part of the multiphotonic distribution reproduces a classical measure induced by the map. As a result, we are able to establish a new asymptotic formula for the multiphotonic probabilities in a general scenario of partially indistinguishable photons, while also providing a single-particle picture to explain the asymptotics of known multiphotonic transition amplitudes in the fully indistinguishable case. More broadly, our framework yields new, directly testable consequences in relation to asymptotic photon bunching patterns: it translates features of the classical map – such as caustics and voids – into direct predictions about regions of large or exponentially suppressed photon-distribution probability.

The Token Jumping and Sliding Token problems are fundamental reconfiguration problems defined on the independent sets of an undirected graph. Given two independent sets $I$ and $J$, each of size $k$, these problems ask whether there exists a sequence of elementary operations transforming $I$ into $J$ such that every intermediate configuration is also an independent set of size $k$. Suppose a token is placed on each vertex of $I$: in Sliding Token, an operation moves a token from a vertex $u \in I$ to an adjacent vertex $v \notin I$; in Token Jumping, the token may instead move to any vertex $v \notin I$. While both problems are $\mathsf{PSPACE}$-complete on general graphs, polynomial-time algorithms for one or both variants have been developed for several graph classes, including trees, block graphs, bipartite permutation graphs, cographs, $P_4$-tidy graphs, and interval graphs. In this paper, we prove that both problems are solvable in polynomial time on threshold signed graphs, also known as Dilworth-2 graphs. A graph $G=(V,E)$ is a threshold signed graph if there exist a mapping $a:V\to\mathbb{R}$ and positive real constants $S,T>0$ such that $|a(v)|< \min\{S,T\}$ for all $v \in V$, and for any distinct vertices $u,v\in V$, $\{u,v\}\in E$ if and only if $|a(u)+a(v)|\ge S$ or $|a(u)-a(v)|\ge T$. More generally, we also show that Token Jumping can be solved in time $n^{O(\mathcal{D}(G))}$, where $\mathcal{D}(G)$ denotes the Dilworth number of $G$. Thus, Token Jumping belongs to $\mathsf{XP}$ when parameterised by the Dilworth number. This graph class is a subclass of permutation graphs, for which the complexity of these problems remains open, and is incomparable with the class of bipartite permutation graphs studied by Fox-Epstein et al. (ISAAC, 2015).

In this paper, we first introduce and study the class $\mathcal{P}_{\mathcal{H}_n^0}(M)$ of normalized pluriharmonic mappings, characterized by a specific bound on the sum of their second-order partial derivatives. We prove a one-to-one correspondence between this pluriharmonic class and a class of holomorphic functions, extending the known result of Ghosh and Vasudevarao \cite{Ghosh-Allu-2020} to the setting of several complex variables. Finally, we provide sharp coefficient bounds and growth estimates for functions in the class $\mathcal{P}_{\mathcal{H}_n^0}(M)$.

In this paper, we construct the quantum double $D(\mathbf{\overline{u}}_q(\mathfrak{sl_2}))$ of the restricted quantum group $\mathbf{\overline{u}}_q(\mathfrak{sl_2})$. We describe the algebraic structure of $D(\mathbf{\overline{u}}_q(\mathfrak{sl_2}))$ by generators and relations. Moreover, we give the comultiplication $\Delta$, the counit $\varepsilon$ and the antipode $S$, respectively. Finally, we classify all irreducible representations of $D(\mathbf{\overline{u}}_q(\mathfrak{sl_2}))$ when $p=2$.

We prove that, on finite graphs of rank-width at most two, VR-recognizability and counting monadic second-order definability coincide. This advances the recognizability-versus-definability problem from bounded linear clique-width to the first nontrivial bounded rank-width level beyond the rank-width-one split-decomposition case. The proof first treats split-prime graphs. The maximal partial-tree theory of Clark and Whittle organizes the non-sequential cut-rank-two separations, while a single strong separation orients all strong equivalence classes and yields a CMSO-definable laminar family of canonical cores. Although the auxiliary partial tree is not itself transduced, it proves that every canonical local piece has a port-contiguous layout of uniformly bounded linear rank-width. The width argument uses partition atoms and the branch-width-three display theorem of Hall, Oxley, Semple, and Whittle and does not assume that graph torsos remain prime. Coherent ordered rank-two frames then permit a finite-state bottom-up evaluation whose local transitions are definable by the bounded-linear-clique-width theorem of Bojańczyk, Grohe, and Pilipczuk. Finally, the CMSO-transducible canonical split decomposition lifts the result from prime graphs to arbitrary graphs of rank-width at most two.

In contrast to most studies on neural network approximation theory that characterize results through a single parameter, such as the total number of network parameters, \cite{shen2020deep} pioneered the characterization of approximation rates as a joint function of the width parameter $N$ and the depth parameter $L$, thereby granting greater architectural flexibility. Existing works using the $(N,L)$-characterization focus on function classes with finite smoothness $s$, establishing a typical approximation rate of $\mathcal{O}\left(N^{-2s/d}L^{-2s/d}\right)$ with $d$ denoting the input dimension, which indicates that network depth and width play symmetric roles for these classes. In contrast, this paper establishes upper bounds for the approximation of analytic functions, which possess infinite smoothness, via ReLU networks under the $(N,L)$-characterization. Specifically, we derive approximation rates of $\mathcal{O}\left(N^{-C L^{\tau}}\right)$, where $C>0$ is some constant and $\tau>0$ is a parameter influenced by the relation between $L$ and $N$. In particular, $\tau=1$ if $N$ scales roughly as $L^d$. Our findings reveal that depth plays a more critical role than width in the context of analytic function approximation. The main technical difficulty of obtaining such upper bounds lies in the trade-off between the smoothness parameters and the approximation accuracy. To overcome this difficulty, we employ refined constructions of several ReLU networks to approximate power functions, multivariate multiplication, and polynomials, which may be of independent interest.

If $X$ is a smooth projective variety of dimension $\geq 2$ and Picard rank $1$ on which every ample line bundle has at least $2$ linearly independent sections, we prove that syzygy bundle of any nontrivial globally generated line bundle is stable. We apply this to show stability of syzygy bundles in non-general type complete intersections in weighted projective spaces, and all complete intersections in projective spaces. This extends earlier results of Jiang-Ren, Coandă and others. We also show that in any dimension $\geq 2$, there is a smooth projective variety of Picard rank $1$ with an unstable syzygy bundle. This completely answers a question of Fulger-Langer.

We study three closely related questions arising from the recent work of Andrews and El Bachraoui on the two-color partition series \[ S_1(q)=\sum_{n\ge0}s_1(n)q^n=\sum_{a\ge0}q^a(-q^{a+1};q)_\infty^2 \] and its odd companion, denoted by $T_o(q)$. First, for the eta-normalized companion \[ C(q)=(q;q)_\infty T_o(q)=\sum_{n\ge0}c(n)q^n, \] we prove a strong form of the Andrews–El Bachraoui sign conjecture that $\limsup c(n)=+\infty$ and $\liminf c(n)=-\infty$. Second, we construct an involution using the Franklin-type involution of Chen and Liu to combinatorially explain Andrews–El Bachraoui congruence for $s_1(n)$ modulo 4. Third, we introduce a two-variable refinement of $S_1(q)$ which records the difference of the lengths of the two colored components. Using Ramanujan's partial theta identity and heat-type equations, we obtain a closed formula for all of its even moments, and we show that the eta-normalized moments satisfy termwise congruences modulo $p^{2k}$ for every prime $p\equiv5\pmod6$.

In this paper, we study pinned $k$-star distance sets associated to compact subsets of $\mathbb{R}^n$, $n\geq 2$. For pins $x_1,\dots,x_k\in E$, the pinned $k$-star distance set is \[ \Delta_{x_1,\dots,x_k}^{k\text{-star}}(E) = \{(|x_1-x|,\dots,|x_k-x|):x\in E\}\subset\mathbb{R}^k. \] We obtain improved Hausdorff-dimension thresholds on $E$ guaranteeing that pinned $k$-star distance sets have positive $k$-dimensional Lebesgue measure. The main analytic input is a reformulation of the connection, first observed in \cite{IPPS22}, between $k$-stars in $\mathbb{R}^n$ and pinned dot products on the paraboloid in $\mathbb{R}^{n+1}$. In our framework, $L^2(\mathbb{R}^k)$ estimates for the densities of pinned $k$-star distance measures are reduced to a weighted Fourier extension estimate for the paraboloid whose weight is defined explicitly in terms of Frostman measures on $E$. For $1\leq k<n$, this yields the threshold \[\dim(E)>\alpha_{+}(n,k):=\frac{n^2+nk+k}{2n+1}=\frac{n+k-1}{2}+\frac14 +\frac{2k+1}{4(2n+1)}.\] Using the graph-building machinery of \cite{BFOPR2026}, our positive-measure results for $k$-stars can be used as building blocks for finite distance graph configurations with prescribed pins. As a consequence, we improve the best-known positive-measure thresholds for pinned $k$-simplices in every dimension $n\geq 3$ and for necklace graphs (cycles) in every dimension $n\geq 3$. We further prove nonempty interior results for $k$-stars. In the special case $k=1$, corresponding to the pinned nonempty interior of the distance set $\Delta_{x}(E)=\{|x-y|\colon y\in E\}$, we use a sharper argument to improve the pinned nonempty-interior thresholds of \cite{BFOP2026} in all dimensions $n\geq 4$.

Mutual correlated agreement captures whether a random linear combination of received words can create a new large agreement with a code, a property relevant to the soundness of batched proximity testing. We show constructively that list-decoding counterexamples yield lower bounds on the mutual correlated agreement error. Given an explicit counterexample to the $(p,L)$-list-decodability of a linear code over $\mathbb{F}_q$, we construct a related code $C'$ of the same length and dimension such that $\operatorname{err}_{\mathrm{MCA}}(C',p)\ge\frac{1}{q}\left\lceil\frac{(L+1)q}{q+L}\right\rceil$, while decreasing its minimum distance by at most one. The construction also produces an explicit pair of words witnessing this error. We further give a structure-preserving version for code families whose coordinates are indexed by a finite set $\Omega$, with each index determining a generator-matrix column through a map $v:\Omega\to\mathbb{F}_q^k$. The construction changes at most one coordinate index and ensures that the output code remains in the same indexed family. As applications, we instantiate this principle for algebraic-geometry (AG) evaluation codes and Reed–Solomon codes. For AG codes, if $G$ is the divisor defining the underlying Riemann–Roch space and $N$ is the number of rational places outside $\operatorname{supp}(G)$ available for evaluation, the resulting code remains over the same function field and Riemann–Roch space, with a modified set of evaluation places. Its mutual correlated agreement error is at least $\frac{1}{q}\left\lceil\frac{(L+1)N}{N+L\mathrm{deg} G}\right\rceil$. The Reed–Solomon conclusion follows as the Vandermonde-column specialization.

We investigate whether or not one can hear at what point (up to symmetry) a drum is struck for some special planar triangles. We prove that one can hear where the equilateral and isosceles right triangles are struck. We also prove that the 30-60-90 triangle, surprisingly, has two audibly indistinguishable points. This is the first known topologically connected such example.

Which discrete symmetry groups can arise from strategic interaction? We tile the plane with copies of a bimatrix game's support complex, joined by controlled boundary rules, and show that all seventeen wallpaper groups act on the resulting covers: explicit generators, each a machine-verified graph automorphism, every realization certified as the exact toroidal quotient, with types identified by a crystallographic recognizer in exact rational arithmetic and cross-validated in GAP. A three-line lemma turns the classical symmorphic/non-symmorphic distinction into a lattice classification: realizations whose translations contain the full tile lattice exist precisely for the thirteen symmorphic groups, and the four non-symmorphic groups are realized at translation-lattice index exactly two, the minimum possible: the tile is the glide's half-step. Two computational tracks accompany the construction. On the graph track, quotienting a straight cover by its translations recovers the tile exactly, $\beq(M/\calT)=\beq(K)$, and swap boundaries add exactly $\binom m2$, independent of payoffs and of cover size. On the game track, detecting a duplicated-strategy cover is a linear-time payoff scan, one tile solution folds to a full translation orbit of cover equilibria, and the tiled correlated-equilibrium system has dimension exactly $r(d-q)+q$, with expansion impossible. The polymatrix cover then carries the symmetry outright: every wallpaper action, glides included, is a group of genuine game automorphisms, equilibria collapse along any symmetry subgroup to a folded fixed-point problem, and a decorated refinement has game automorphism group exactly the toroidal wallpaper group.

Accurate geometric calibration is essential for fluoroscopy-guided spinal imaging, digitally reconstructed radiograph (DRR) generation, and 2D–3D vertebral registration. Although calibration quality is typically evaluated using reconstruction-based metrics such as reprojection error, its influence on projection-domain consistency remains poorly understood. This study presents a synthetic framework for evaluating how intrinsic calibration perturbations affect vertebral fluoroscopic projections and downstream registration performance. CT-derived vertebral models and controlled cone-beam imaging geometry were used to generate DRRs with both ground-truth and perturbed intrinsic calibration parameters while maintaining identical anatomy and acquisition pose. Projection-domain changes were quantified using anatomical landmark displacement, contour distance, silhouette overlap, image similarity, and landmark-based 2D–3D registration accuracy in anterior–posterior (AP) and lateral (LAT) views. Results show that even small intrinsic calibration perturbations produce measurable changes in vertebral projection geometry, contour morphology, landmark localization, and DRR appearance. Sensitivity is strongly view dependent, with LAT projections exhibiting substantially greater deformation and anatomical displacement than AP projections. These projection inconsistencies also degrade downstream 2D–3D registration, particularly rotational alignment accuracy. The findings demonstrate that projection-domain consistency complements conventional reconstruction-based calibration metrics and provides a practical framework for assessing calibration robustness. This approach may improve the reliability of DRR generation and fluoroscopy-guided vertebral registration in image-guided spinal applications.

We introduce a Walsh–quotient obstruction to study Fourier-frame existence for symmetric two-branch Bernoulli convolutions \[ \mu_{\rho,d} =\ast_{j=1}^{\infty} \frac12\bigl(\delta_{-d\rho^{j}/2}+\delta_{d\rho^{j}/2}\bigr), \qquad 0<\rho<1,\quad d>0. \] Suppose that $0<\rho<\frac12$ and $\rho^{-m}=B$ for some integer $m\ge1$ and odd integer $B\ge3$. We prove that $L^2(\mu_{\rho,d})$ admits no Fourier frame. For $m=1$, our argument proves the nonexistence of Fourier frames for odd-integer-base Cantor measures and hence resolves Strichartz's long-standing open problem for the middle-third Cantor measure. A contemporaneous independent proof of the case $m=1$ was obtained by Pont, Liehr and Taylor [arXiv:2607.08656v1]. For $m>1$, our theorem includes the non-integer reciprocal-power contraction ratios $\rho=B^{-1/m}$, which fall outside the classical integer-base Cantor-measure setting. Our proof is self-contained. It uses finite-coordinate Walsh packets to transform the frame inequalities into incompatible tangent-quotient estimates, while the identity $\rho^{-m}=B$ supplies the exact $m$-step scale relation leading to the contradiction.

Let $X_1,X_2,\ldots$ be independent $\mathrm{Bernoulli}(\theta)$ random variables, and let $\bar X_n = n^{-1}(X_1 + \cdots + X_n)$. We prove that, for every real $p \geq 1$, the sequence $\{\mathsf{E}(\bar X_n^p)\}_{n \geq 1}$ is log-convex. This proves the Bernoulli case of a conjecture of Lamkin and Tkocz [Canad. Math. Bull., 65(2):271-278, 2022]. The proof conditions on the total number of successes among $2n$ trials and reduces the desired inequality to a convex-order comparison between two normalized quadratic functions of hypergeometric random variables. All the log-convexity inequalities are strict for $p > 1$ and $0 < \theta < 1$.

Let $X/K$ be an abelian variety defined over a number field and let $f:X\to X$ be a dominant morphism defined over $K$. We show that $(f,K)$ is strongly dense orbit transversal. That is, up to replacing $K$ with a finite extension, every set of representatives for grand $(f,K)$-orbits is Zariski dense in $X$.

In this article we prove that toroidal 3-manifolds have circularly-orderable fundamental groups by showing that they admit finite cyclic covers with left-orderable fundamental groups. These covers are also not L-spaces and often admit co-orientable taut foliations, as predicted by the L-space conjecture. As a consequence, we verify a conjecture of Ba and Clay, which characterises graph manifolds with circularly-orderable fundamental groups.

We study the nonisotropic fractional analogs of Laplace, Helmholtz and Yukawa equations. We provide a Duffin correspondence for the Yukawa equation and a Feynman–Kac reconstruction for the Helmholtz equation. The foundation of our analysis is the fact that the nonisotropic fractional Laplace equation is related to a symmetric $\alpha$-stable Lévy process with independent identically distributed components. By using this relation we provide a Walk-on-Cubes algorithm that simulates the solutions of Helmholtz and Yukawa equations.

Let $G$ be a finite group acting on a finite tensor category $\mathcal{C}$. We classify fiber functors on the equivariantization $\mathcal{C}^G$ in terms of equivariant exact module categories over $\mathcal{C}$, indexed by subgroups of $G$. The data are a subgroup $H\subseteq G$ and an $H$-equivariant $\mathcal{C}$-module category $\mathcal{M}$ whose underlying $\mathcal{C}$-module category is indecomposable, exact, and semisimple; they give a fiber functor precisely when $H$ acts transitively on the simple objects of $\mathcal{M}$ and the stabilizer cocycle of one, hence every, simple object is non-degenerate. Through Tannaka-Krein reconstruction this describes realizations of $\mathcal{C}^G$ as the representation category of a finite-dimensional Hopf algebra, with no semisimplicity hypothesis on $\mathcal{C}$. As applications, for odd primes $p$ we determine the fiber functors on $\mathrm{Rep}(H_p)$, where $H_p$ denotes Nikshych's semisimple Hopf algebra of dimension $4p^2$: there is one equivalence class if $p\equiv 3\pmod 4$ and two if $p\equiv 1\pmod 4$. We also use the classification for gaugings to determine which non-pointed entries in the small-dimensional list of Green and Nikshych are representation categories of semisimple factorizable Hopf algebras.

Sparsity is inherently exploited in sparse code multiple access (SCMA) and sparse vector coding (SVC), yet the interaction between these two has not been explored before. It is intriguing to ask if one can be used to improve the other, and vice versa. In this work, we present a novel SCMA inspired SVC scheme, called SCMA-SVC, for enhanced ultra-reliable low-latency communications. Our key idea is to exploit the sparse pattern and multidimensional constellation nature of SCMA, with which one is able to further enlarge the minimum Euclidean distance (MED) of the corresponding SVC codebooks. Such an innovation allows us to harvest the multiuser coding gain and the constellation shaping gain which are pertinent to SCMA. Moreover, by applying random phase rotations to the sparse vectors, it is shown that the proposed SCMA-SVC achieves full diversity order over Rayleigh fading channels. Under maximum likelihood (ML) decoding, the proposed SCMA-SVC demonstrates remarkable error rate performances over both Gaussian and Rayleigh fading channels. Additionally, we develop a low-complexity decoder that exploits the structural sparsity of SCMA-SVC while maintaining near-ML performance. Simulation results demonstrate that the proposed SCMA-SVC achieves significantly improved reliability over the existing SVC variants.

A large literature in econometrics proposes decision rules with optimality guarantees based on ex ante criteria, such as minimax regret. We develop a framework for analyzing the dynamic consistency of such rules and show that, in many empirically relevant settings, the researcher may wish to deviate from the interim prescription of ex ante optimal rules after observing the data realization. To address this problem, we propose and axiomatize two classes of optimality criteria that yield dynamically consistent decision rules.

In this paper, we study the strong uniqueness problem for the mutually catalytic super-Markov chain, which is a two-dimensional degenerate stochastic differential equation with Hölder continuous coefficients. The key step is to find a process which is a function of two coupled processes and satisfies an autonomous one-dimensional stochastic differential equation; uniqueness for this equation follows from a Yamada-Watanabe argument. A large deviation principle is then established, in the irreducible two-state case, by applying the weak-convergence approach of Budhiraja, Dupuis and Maroulas to the controlled equations.