Papers

We present a rigorous formulation of the spin-1 Affleck–Kennedy–Lieb–Tasaki (AKLT) state within the framework of hidden quantum Markov models (HQMMs). We show that the AKLT ground state admits a natural representation as the observable output of a causal HQMM, thereby endowing it with an underlying hidden quantum memory that is fully consistent with its standard finitely correlated (matrix product state) description. This viewpoint yields a compact and structurally transparent characterization of the AKLT chain as a quantum spin system equipped with intrinsic quantum memory. Our results further indicate that the HQMM framework provides a promising setting for analyzing measurement-based quantum computation (MBQC) and related information-processing tasks.

We find novel confined states in the spin-$S$ nearest-neighbor antiferromagnetic Heisenberg model on the two-dimensional Penrose lattice. Linear spin waves have massively degenerate eigenstates strictly confined to tricoordinated sites. They contrast with the well-known itinerant analogs in the tight-binding model, where electrons are confined but extended to both tricoordinated and pentacoordinated sites. It is the site potentials in the spin-wave Hamiltonian, originating from Coulomb interactions between electrons, that confine spin waves to minimally coordinated sites only. Confined states in the tight-binding Hamiltonian consist of six types of building blocks, whereas those in the spin-wave Hamiltonian consist of only four of them. Confined spin waves are robust against $1/S$ corrections. Emergent $O(S^{0})$ interactions further confine – superconfine – spin waves into two separate groups within tricoordinated sites.

It is shown that a JB-algebra which can be generated by the union of two of its associative Jordan subalgebras is a JC-algebra, hence special. A similar refinement of Macdonald's principle for JB-algebras is obtained. Moreover, we prove that the free unital JB-algebra generated by $n$ projections is a JC-algebra if and only if $n\in \{1,2,3\}$. Finally, we give an explicit description of the free unital JB-algebra generated by two projections paralleling the Raeburn-Sinclair theorem for C*-algebras.

Comparison geometry for Bakry-Émery Ricci curvature has been extensively developed by Wei-Wylie and others. Motivated by the weighted sectional curvature framework introduced by Wylie and further developed by Kennard-Wylie-Yeroshkin, we study radial comparison geometry on manifolds with density through a modified Hessian arising from this framework. Under nonnegative weighted sectional curvature together with suitable density control assumptions, we obtain a modified Hessian estimate for the radial function $u = \frac{1}{2}r^2$. From this estimate, we derive Hessian comparison, shape operator comparison, weighted Laplacian comparison, asymptotic radial volume density estimates, and polynomial weighted volume growth bounds. We introduce a normalized weighted radial volume density satisfying a monotonicity property analogous to the radial volume density monotonicity underlying Bishop-Gromov comparison. We also study rigidity phenomena associated with these comparison estimates. Equality in the Hessian comparison theorem yields radial conformal rigidity, while equality in the modified Hessian estimate forces the metric to have an exact metric cone structure.

In this paper, we establish a higher order Morrey's inequality in the framework of $\mathsf{CD}(K,N)$-spaces for $K\in\mathbb{R}$ and $N\in\mathbb{N}$. We do so by first introducing an alternate version of second order Sobolev spaces in general metric measure spaces, which contains amply many functions even when $p>N$. These spaces contain the traditional $H^{2, p}(X)$ spaces on $\mathsf{RCD}(K, N)$-spaces.

The nudged elastic band (NEB) and Dimer methods are standard tools for computing minimum-energy paths and index-one saddle points in atomistic transition problems. They are increasingly driven by surrogate or learned force models, whose force errors are often anisotropic and spatially varying near transition states and defect cores, where saddle-search iterations are most sensitive. We introduce uncertainty-aware NEB and Dimer methods (UA-NEB, UA-Dimer) that use covariance as an optimizer-level reliability metric while preserving the mean-potential saddle-search equations: an oblique normal projection for NEB and covariance-weighted rotation and translation for Dimer. Both algorithms fit Robbins–Monro recursions; under a local Lyapunov stability hypothesis, verified explicitly for a canonical UA-NEB setting and stated as a hypothesis for UA-Dimer, the stochastic iterations converge almost surely within the corresponding local stability neighborhood. In the analytic benchmark, UA-NEB reduces mean barrier error by $21\%$ relative to stochastic NEB and UA-Dimer reduces the reflected-gradient residual by $22\%$; in the 127-atom tungsten-vacancy benchmark, full UA-NEB reduces mean barrier error by $56\%$ relative to stochastic NEB and by $23\%$ relative to diagonal covariance weighting. These results show that anisotropic uncertainty is most useful when embedded in the constrained geometry of the optimizer rather than collapsed into a scalar acquisition or trust criterion.

We construct a concrete model for the measured wall of finite dinensional hyperbolic space, and construct the measure on it, so that the hyperbolic distance between two points equals to the measure of walls seperating them, up to a constant positive scalar. This offers a concrete proof for the fact that SO(n,1) is not T-group.

We establish the hydrodynamic limit of the one-dimensional Boltzmann equation with hard-sphere collisions toward Riemann solutions of the compressible Euler system. The Riemann solutions covered by our result include generic superpositions of elementary waves: either two shock waves and a contact discontinuity, or a rarefaction wave, a contact discontinuity, and a shock wave. For suitably well-prepared initial data and sufficiently small wave strength, we prove that the corresponding Boltzmann solution exists globally in time and converges, as the Knudsen number vanishes, to the local Maxwellian associated with the Riemann solution in $L^2([0,T]\times\mathbb R_x\times\mathbb R^3_\xi)$ for any $T>0$. The proof combines the macro–micro decomposition with a kinetic adaptation of the $a$-contraction method, in which the propagation speeds of Boltzmann shocks are determined by Rakine-Hugoniot speed with dynamical modulation parameters. The resulting coercive control of the shock translation modes, together with layer analysis and the uniform bound of the Shifts, allows us to pass to the Knudsen limit in the full space-time domain. To the best of our knowledge, this is the first rigorous result on the hydrodynamic limit towards generic Riemann solutions containing shocks: either the shock–contact–shock case or the rarefaction–contact–shock case, in a global space-time energy norm without removing neighborhoods of either the initial time or the shock layers. In the special case of a single shock, the argument further yields a sharp quantitative description of the kinetic shock layer, up to the dynamically selected Shift.

With the rapid proliferation of wireless and Internet of Things (IoT) devices, ensuring secure and reliable device identification has become a significant challenge. Traditional security techniques, such as IP or MAC address-based authentication, are susceptible to spoofing, whereas Radio Frequency Fingerprint Identification (RFFI) offers a more secure alternative by exploiting the unique hardware imperfections in devices' RF signals. In this paper, we propose a novel deep learning-based framework for RFFI that enhances both accuracy and reliability in challenging RF environments. The core of our approach is the Signal Inception Transformer (SinFormer), which leverages a specialized multi-scale self-attention mechanism to effectively capture both large-scale and fine-grained fingerprints in signals, significantly improving identification accuracy. To further enhance robustness and reliability, we introduce a two-stage training strategy that enables the model to learn general signal features and maintain performance under adverse conditions, such as low Signal-to-Noise Ratio (SNR) or channel variations. The effectiveness of the proposed method is validated using a real-world dataset. Experimental results show that the SinFormer framework consistently outperforms existing methods in accuracy and robustness across diverse and challenging scenarios.

We consider an initial boundary value problem of the space fractional Allen-Cahn equation with logarithmic Flory-Huggins potential. As an approximation technique, first-order weighted and shifted Grunwald difference formulae of the left and right fractional derivatives are used. The main focus of the present work is to study the spectral features of the underlying matrices and matrix sequences and to design proper preconditioners based on the spectral information. Then a computational and spectral analysis of the resulting preconditioned matrix sequences is performed. Numerical evidence and a short list of open problems complete the current study.

In this paper, we prove fibration theorems for manifolds with almost nonnegative Ricci curvature and certain extra regularity assumptions. We show that a closed $n$-manifold $M$ satisfying $\mathrm{diam}(M)^2\mathrm{sec}_M \geq -\kappa$ and $\mathrm{diam}(M)^2\mathrm{Ric}_M \geq -\delta$, where $\delta>0$ is sufficiently small depending only on $n$ and $\kappa$, fibers over a $b_1(M)$-torus. This removes the upper sectional curvature bound required in the earlier result of Yamaguchi \cite{Y88}. As a corollary, we obtain a refinement of Yamaguchi's smooth fibration theorem (\cite{Y91}), showing that the fiber itself (rather than a finite cover of it) fibers over a $b_1$-torus. Our results extend to manifolds satisfying a generalized Reifenberg condition introduced in \cite{HH24}, which encompasses both a lower bound on sectional curvature and the local rewinding Reifenberg condition. In the nonsmooth setting, a similar result also holds for a non-collapsed $\mathrm{RCD}(-\epsilon(D,r,n),n)$ space whose diameter is bounded by $D$ and which satisfies the $(r,\delta(n))$-local rewinding Reifenberg condition. The proofs rely on an equivariant regularity theorem for almost submetries under a lower Ricci curvature bound. In addition, we study the stability of rank of Abelian actions along equivariant Gromov-Hausdorff convergence in this paper.

In this article, motivated by a problem asked by Allison and Panagiotopoulos, we study a problem concerning the complexity of group extensions within a hierarchy (denoted by $\alpha$-CLI and L-$\alpha$-CLI) on the class of non-archimedean CLI Polish groups: Given a non-archimedean Polish group $G$ and one of its closed normal subgroup $N$, suppose $N$ and $G/N$ are $\alpha$-CLI and $\beta$-CLI, respectively. Is $G$ always $(\alpha+\beta)$-CLI? We provide a positive answer under a certain additional assumption. We then construct two examples yielding negative answers: for each countably infinite ordinal $\alpha$, there exists a group $G$ that is not $\alpha$-CLI, but $G$ has a $1$-CLI normal subgroup $N$ such that $G/N$ is proper $\alpha$-CLI; there exists a proper $3$-CLI group $U$ that has an abelian normal subgroup $N$ such that $U/N$ is also abelian. These examples also provide negative answers to the original problem raised by Allison and Panagiotopoulos. Finally, we show that if $N$ and $G/N$ are $\alpha$-CLI and $\beta$-CLI with $\beta>0$, respectively, then $G$ is $\beta\cdot(\omega\cdot\alpha+1)$-CLI, which gives an upper bound on the complexity of the extended group.

This work explores the quantum propagator $K(x,t)$ as a solution of the system's dynamical equation. We develop a generalized propagator framework in which the propagator is written in the form $K=\exp\!\left(R+\frac{i}{\hbar}S\right)$, where $S$ governs the semiclassical phase structure and $R$ governs the amplitude transport and weighting of configurations. Starting from nonrelativistic quantum mechanics, the classical Hamilton–Jacobi equation emerges in the semiclassical limit, while $R$ reduces to the logarithm of the Van Vleck determinant. The formalism is then extended to relativistic field theory and minisuperspace quantum cosmology using functional methods and generalized Hamiltonian constraints. Then the resulting semiclassical equations recover the corresponding classical dynamics along characteristic flows. In minisuperspace models, the formalism yields coupled equations for geometric and matter sectors analogous to semiclassical Wheeler–DeWitt systems. In semiclassical gravitational settings, the gravitational contribution may admit an entropy-like interpretation, consistent with thermodynamic weighting factors appearing in Euclidean gravitational path integrals. The resulting framework suggests a unified semiclassical description in which propagator phases encode dynamical evolution while amplitude functionals encode transport and configuration weighting. This perspective motivates viewing mechanics as a relation between the differential structure of spacetime and the differential structure of the space of dynamical configurations.

The $\theta=\infty$ conjecture asserts that the mollified second moments of the Riemann zeta function remain bounded for mollifiers of arbitrary polynomial length. We investigate an analogue of this conjecture for automorphic $L$-functions associated with cuspidal representations of $\text{GL}_m(\mathbb{A}_{\mathbb{Q}})$, exploring its implications for the distribution of their nontrivial zeros. Extending the framework of Bettin and Gonek, we prove that if the mollified second moments of these $L$-functions remain suitably bounded for mollifiers of arbitrary polynomial length, then the $L$-functions are non-vanishing in corresponding regions of the critical strip. Furthermore, we establish a version of this criterion for families of $L$-functions, demonstrating that the $\theta = \infty$ conjecture for a family of $L$-functions implies a quasi-Riemann Hypothesis for that family.

We prove global-in-time Strichartz estimates for Schrödinger equations with multipole Aharonov–Bohm Hamiltonians on $\mathbb{R}^2$. As intermediate steps, we prove global-in-time local smoothing estimates for multipole Aharonov–Bohm Hamiltonians.

In this paper, we mainly investigate profound interconnections between combinatorial designs, linear codes, and Boolean functions.

This paper studies generalized Pólya conversion problems for the $q$-permanent. We establish a sharp threshold governing the transition from low-dimensional algebraic flexibility to higher-dimensional combinatorial rigidity. For $n \ge 3$ and $q \ne \pm 1$, we prove that the $q$-permanent is not linearly convertible to the determinant or permanent. Conversely, we completely classify the space of Schur multiplier preservers for $n=2$. Focusing on Schur multipliers, we characterize the preserver exponents as a $(2n-2)$-dimensional space of additive matrices. We show that for lower Hessenberg matrices, the general geometric obstruction disappears, yielding an explicit determinantal reduction and an $\mathcal{O}(n^3)$ evaluation algorithm. Furthermore, we classify permutational converter exponents, proving that for $n \ge 4$, admissible symmetries are strictly constrained to the dihedral group. Finally, we resolve a mixed conversion problem, showing the solution space is nonempty only for $n \le 4$, which provides a direct algebraic characterization of the $q$-permanent's zero locus in low dimensions.

We study generalized splines from the perspective of the representation theory of the category of graphs with contractions. Our main theorem proves a kind of finite generation, which in turn implies the existence of a “universal generating set” for the module of splines over any graph with fixed combinatorial genus. This theorem holds over any Noetherian commutative ring with a chosen finite list of ideals for edge-labels. We then give several applications of this theorem, including showing that a particular generating function associated to splines on trees is algebraic when the base ring satisfies certain finiteness conditions. We illustrate our technical theorems explicitly by giving a classification of splines on graphs with combinatorial genus one and two.

We are concerned with the semilinear biharmonic problem under Dirichlet boundary conditions \begin{equation*} \begin{cases} \Delta^2 u=(u^+)^{p} &{\text{in}~\Omega},\\[0.5mm] u \not\equiv 0 &{\text{in}~\Omega},\\[0.5mm] u=\dfrac{\partial u}{\partial \nu } = 0 &{\text{on}~\partial \Omega}, \end{cases} \end{equation*} where $\Omega\subset \mathbb{R}^4$ is a smooth bounded domain, $u^+ = \max\{u,0\}$, $\nu$ denotes the outward unit normal and $p>1$ is sufficiently large. The basic concentration behavior of solutions satisfying the uniform energy bound is known. In this work, we first obtain refined asymptotic estimates for the peak heights, concentration scales and rescaled profiles of multi-spike solutions. Based on these estimates, we further analyze the generalized linearized eigenvalue problem and derive sharp bounds for the Morse and augmented Morse indices. Consequently, if the limiting $k$-spike configuration ${\mathbf{x}}_\infty$ is a nondegenerate critical point of Kirchhoff--Routh function $\Psi_k$, then \[ m(u_p)=k+n_+\big(D^2\Psi_k({\mathbf{x}}_\infty)\big) \] and $u_p$ is nondegenerate for all sufficiently large $p$ along the selected concentration subsequence. The absence of the maximum principle for the Dirichlet biharmonic operator on general domains poses main difficulties in both refined asymptotic analysis and the study of the generalized linearized eigenvalue problem. To address these difficulties, we employ Green's representation together with refined blow-up and spectral analysis.

A linear one-dimensional singularly perturbed convection-diffusion problem is solved numerically after its solution is decomposed as $u_0+w$, where $u_0$, the corresponding reduced solution, is treated as a function known exactly or approximately. The component $w$ is then calculated using the exponentially fitted Allen-Southwell-Il'in (ASI) scheme on the Shishkin mesh and its asymptotic version. We prove that this numerical method is highly accurate, with errors that diminish when the discretization parameter increases, and, in some cases, even when the perturbation parameter decreases. This is a theoretical confirmation of earlier numerical results showing that the ASI scheme outperforms the general class of Samarskii-type schemes to which it belongs. Even higher accuracy is proved when $u_0$ is linear, in which case, the decomposition is not needed. New numerical experiments are provided to illustrate all this.

The aim of this paper is to apply Hirota's bilinear method to the integrable discrete Manakov system in the focusing dispersion regime in order to construct and analyze soliton and breather solutions. After deriving the general bilinear form of the system, we show how to obtain fundamental solitons, as well as fundamental and composite breathers. We then obtain solutions exhibiting 2 solitons and 2 breathers and combinations of a soliton and a breather, and discuss all “two-body” interactions properties, with particular emphasis on explicit formulas, visualization, and long-time asymptotic behavior, thus rigorously confirming the highly nontrivial interaction properties of these coherent structures.

The Hellinger-Kantorovich (HK) space provides a natural geometry for nonnegative measures with varying total mass, but its differential-geometric structure is less well understood than that of the closely related Wasserstein space of probability measures. In this paper, we take a step toward resolving this issue. We show that the cone representation of the HK geometry via the Wasserstein metric preserves the local Riemannian geometry along a class of lifted geodesics. Specifically, we give a constructive procedure that produces a Wasserstein geodesic on the cone along which the HK Riemannian geometry is preserved pointwise, yielding an explicit isometry of tangent spaces between HK geodesics and their Wasserstein lifts. This connection makes many Wasserstein-geometric tools available for HK computations. Concretely, we use it to approximate parallel transport on HK space by lifting to the cone and applying recently developed Wasserstein parallel transport tools, circumventing the high-dimensional PDE arising from the HK covariant derivative. We also derive closed-form expressions for the covariant derivative and parallel transport on Euclidean metric cones, using the theory of warped-product manifolds. Finally, we present simulations illustrating the behavior of parallel geodesics in HK space, which reveal that the HK geometry couples spatial and mass variation through the geometry of the cone – a feature with nontrivial implications for applied use of the framework.

Framing triangulations of unit flow polytopes have received a great deal of recent study with rich connections to various generalizations of Catalan and Cambrian combinatorics as well as volume and h*-polynomial formulas. This story has largely been restricted to unit flow polytopes, with only two recent works giving descriptions of framing triangulations on classes of non-unit flow polytopes. In this article we introduce the first theory of (unimodular) framing triangulations for arbitrary integer flow polytopes. We will observe some pathologies in general examples which are impossible in the unit case, and propose in response a class of "well-ordered" framing triangulations which we expect to inherit key properties from the unit case while still containing all settings of framing triangulations existing in the literature.

In the study of flow polytopes, a directed acyclic graph (DAG) with a choice of framing gives a regular unimodular triangulation on its space of unit nonnegative flows. In representation theory, a gentle algebra has recently been equipped with a space of unit flows admitting triangulation and subdivision results capturing its tau-tilting theory. These theories from different areas of mathematics overlap: flows on gently framed DAGs are the same as flows on paired representation-finite gentle algebras. In this article we develop the common generalization of these two theories by defining (framed) turbulence charts, which may be thought of as analogs of (framed) DAGs without the conditions of (D)irectedness and (A)cyclicity. The space of unit flows on a turbulence chart is its turbulence polyhedron. We give presentation, subdivision, and triangulation results on turbulence polyhedra which restrict to known results in the settings of framed DAGs and gentle algebras.

We establish an Arakelov-type inequality for a morphism $f \colon (X,\Delta) \to S$, where $(X,\Delta)$ is a simple normal crossing semi-log canonical pair and $S$ is a smooth projective variety. As a consequence, we derive a bound on the Iitaka volumes of algebraic fiber spaces whose geometric generic fiber admits a good minimal model.

We study the stability of $p$-area minimizing surfaces in the Heisenberg group under perturbations of the weight function and the drift vector field in generalized least gradient problems of the form \[ \inf_{w\in BV_0(\Omega)} \int_\Omega \left(a(x)|Dw+F(x)|+H(x)w\right)\,dx. \] Owing to the lack of strict convexity, establishing stability of minimizers is challenging. We derive quantitative stability estimates for minimizers. In particular, under suitable nondegeneracy and geometric assumptions, we obtain $L^1$ and $W^{1,1}$ stability estimates with respect to perturbations of the weight function $a$ and the drift vector field $F$. We further establish unified quantitative stability estimates under simultaneous perturbations of all principal parameters, namely $a$, $F$, and $H$. Numerical simulations illustrating the stability theory are also presented.

This paper investigates the boundary stabilization of an Euler-Bernoulli beam under constant axial tension and subject to an internal time-delay. First, the well-posedness of the system is established using semigroup of linear operators theory. Then, by constructing an adequate Lyapunov functional, we establish the exponential stability of the system within a region of the $(\alpha, \xi)$-plane, explicitly characterized by a system of inequalities involving the physical parameters. This work extends the results of Han et al. (Journal of Difference Equations, 2016) by incorporating the effect of axial tension and an internal delay term without sign constraints.

The permutation groups of cyclic codes are widely applicable in determining the weight distribution of codes, decoding theory and various other areas. In this paper, by employing two distinct matrix representations, we can relate cyclic codes with very long lengths and special generator polynomials to those with prime lengths. Consequently, we mainly determine the permutation groups of certain cyclic codes over $\mathbb{F}_{r^\alpha}$ with lengths $hp$, $r^mp^n$ and $pq$ and special generator polynomials where $h$ is a positive integer and $p$, $q$ and $r$ are distinct prime numbers. For length $pq$, we manage to provide the permutation groups of cyclic codes with generator polynomials $Q_{pq}(x)$(the $pq$-th cyclotomic polynomial) or others, which seems to be the first work about permutation groups of cyclic codes with generator polynomials that are factors of $x^{pq}-1$ but not factors of $x^p-1(\text{or }x^q-1)$.

This is a survey of recent works on the germ-equivalence problem of minimal rational curves on uniruled projective manifolds. Our main interest is when the associated varieties of minimal rational tangents form an isotrivial family of projective varieties. In this case, there is a natural G-structure on a Zariski-open subset of the underlying uniruled projective manifold, which leads to an interaction of algebraic geometry of minimal rational curves with differential geometry of geometric structures. We also discuss the related question of the formal principle for the germ-equivalence of minimal rational curves.

Restless multiarmed bandits are Markov decision process models for allocating a scarce resource among projects whose states evolve under active or passive actions. Whittle's index policy is widely used for such problems, but its application to a given model requires both a proof of indexability and a means of computing the index, two analytically challenging tasks. This paper develops a performance-metric framework for proving threshold-indexability and computing Whittle indices for binary-action projects with real interval state spaces. The framework extends discounted partial conservation law (PCL) methods to a criterion-agnostic setting and works directly with reward and resource metrics of threshold policies, rather than first proving threshold optimality and then monotonicity of optimal thresholds in the resource price. The main theorem is a verification and characterization result: under marginal-resource positivity and a marginal integration-by-parts identity, threshold-indexability is equivalent to monotonicity and continuity of the marginal productivity (MP) index, which then equals the Whittle index. The framework is specialized to the discrete-time long-run average criterion by a vanishing-discount transfer of discounted threshold metrics and includes exceptional states where the MP marginal-resource denominator vanishes, handled by continuous extension or vanishing-discount limits. Applications to web crawling and noisy-channel transmission recover known long-run average Whittle indices. For scalar Kalman-filter bandits, it proves a regular-part average-cost result and reduces the remaining indexability question to explicit exceptional-state metric-limit conjectures.