Papers

We prove a one-variable functional inequality which is the free probability analog of Bobkov's isoperimetry inequality. The inequality involves the $L^1$ norm of the difference quotient of a function $f$ and can be viewed as a non-local isoperimetric inequality. We also prove related inequalities for subsets of an interval as well as for subsets of roots of Hermite polynomials. This paper is also an experiment in AI-based exploration of free analogs of classical probability statements.

In this paper, we introduce a suitable notion of flat solutions for the anisotropic surface diffusion equation with elasticity in three-dimensions, based on a minimizing movement scheme inspired by that introduced by Cahn and Taylor. Using this scheme, we prove the existence of classical solutions to the equation without using a curvature regularization term. Moreover we establish the consistency of the approximation method.

This paper investigates the analytic structure of the parametric harmonic zeta function $\zeta_{H}\left(s, a, b\right)$, a Dirichlet series associated with generalized harmonic numbers. We establish its meromorphic continuation to the complex plane and determine the residues at all of its poles. We then derive a Taylor expansion for $\zeta_{H}\left(s, a, b+t\right)$, leading to harmonic analogues and extensions of classical identities due to Landau, Singh-Verma, and Srivastava. We further develop a systematic theory of the associated harmonic Stieltjes constants by deriving explicit formulas, including previously unknown cases, together with a limit representation for the first-order constants. Finally, we construct the harmonic analogue of the classical digamma function, establish its principal analytic properties and its connection with the harmonic Stieltjes constants, and obtain Raabe-type formulas for the parametric harmonic zeta function, the harmonic Stieltjes constants, and the harmonic digamma function.

A pair of proper cones $(\mathsf{C}_1,\mathsf{C}_2)$ is said to have the Lorentz factorization property (LFP) if every $(\mathsf{C}_1,\mathsf{C}_2)$-positive map factors through a direct sum of Lorentzian cones, i.e., cones over Euclidean balls. Clearly, $(\mathsf{C}_1,\mathsf{C}_2)$ has the LFP if either $\mathsf{C}_1$ or $\mathsf{C}_2$ is a direct sum of Lorentzian cones, and our main goal is to find other examples. We show that such examples cannot be found for pairs $(\mathsf{C}_1,\mathsf{C}_2)$ where $\mathsf{C}_1=\mathsf{C}_2$, or in the case where both $\mathsf{C}_1$ and $\mathsf{C}_2$ are polyhedral. We also focus on the case where $\mathsf{C}_1=\mathsf{C}_\square$ is the square-based cone in $\mathbf{R}^3$. Here, we show that $(\mathsf{C}_\square,\mathsf{C})$ has the LFP whenever $\mathsf{C}$ is a symmetric cone, i.e., a direct sum of Lorentz cones, cones of positive semidefinite matrices over the real numbers, complex numbers or quaternions, and the cone of $3\times 3$ positive semidefinite matrices over the octonions. We leave open the question whether there are more examples, but we show that this list cannot be extended by any strictly convex cone $\mathsf{C}$ or for a cone $\mathsf{C}$ with $\text{dim}(\mathsf{C})\leq 5$. Finally, we discuss an application to a problem in quantum information theory.

For a Hausdorff topology on the set of ideals of the semilattice $M(X)$ of coarse equivalence classes of metrics on a set $X$, the space $I(M(X))$ of ideals is the closure of the set of principal ideals, thus allowing to view non-principal ideals as generalizations of coarse equivalence classes of metrics. Some ideals arise from coarse structures on $X$. We define a map $\Phi$ from $I(M(X))$ to the set $CS(X)$ of coarse structures on $X$, and a map $\Psi$ backwards, and show that $\Psi\circ\Phi$ is the identity map, thus allowing to identify coarse structures with some ideals of $M(X)$. We show that there are ideals that do not come from $CS(X)$. For any ideal $F$ we define the generalized uniform Roe algebra as the direct limit $C^*$-algebra of the uniform Roe algebras for the equivalence classes of metrics in the ideal, and show that it coincides with the uniform Roe algebra of $\Phi(F)$.

Monads are a ubiquitous structure in functional programming used for modelling computational effects. For example, the state monad models the effect of a computation interacting with a memory system. Here we introduce the quantum instrument monad $\mathcal{I}_\mathcal{A}$, which models the effect of a computation interacting with a quantum system with algebra of observables $\mathcal{A}$. It can be thought of as a noncommutative generalization of the state monad. We construct this quantum instrument monad in two versions: a finitary version on the category of sets and a measure-theoretic version on the category of measurable spaces (the latter under the assumption that $\mathcal{A}$ is a type I von Neumann algebra with separable predual). Both versions are strong monads. The construction of the measure-theoretic version is based on a new notion of integral of a quantum-operation-valued function against a state-valued measure.

We determine all involutions in the Cayley-Dickson construction that extend the involution of the original $*$-algebra. We also find all algebra isomorphisms between the resulting Cayley doubles that extend the identity automorphism of the original $*$-algebra, and consequently classify the resulting $*$-algebras up to $*$-algebra isomorphism. As applications, we show that Cayley doubles without zero divisors admit exactly one additional involution, prove that the classical Cayley-Dickson involution is the unique scalar involution, and obtain a classification of the $*$-algebras arising from $\mathbb{R}$ up to dimension 4.

We construct higher derived Artin stacks parametrizing constructible sheaves on complex algebraic varieties and compact real analytic varieties. Furthermore, we show that every perversity function gives rise to an open substack of perverse sheaves, which is a 1-Artin stack locally of finite presentation that generalizes usual character stacks. As a sample application of the derived structure, we construct new examples of cohomological Hall algebras associated to punctured Riemann surfaces.

Bent partitions play a significant role in constructing bent functions and have rich connections with coding theory and combinatorics. In this paper, we introduce $s$-plateaued partitions, which generalize the bent partitions. Let $\Gamma=\{A_{i}, 1 \leq i \leq K\}$ be a partition of $V_{n}^{(p)}$, where $V_{n}^{(p)}$ is an $n$-dimensional vector space over the prime field $\mathbb{F}_{p}$ and $p \mid K$. Then $\Gamma$ is called an $s$-plateaued partition of $V_{n}^{(p)}$ of depth $K$ if each $p$-ary function $f: V_{n}^{(p)} \rightarrow \mathbb{F}_{p}$ for which every $j \in \mathbb{F}_{p}$ has exactly $\frac{K}{p}$ of sets $A_{i}$ in $\Gamma$ in its preimage set, is a $p$-ary $s$-plateaued function. By using an $s$-plateaued partition, a large number of $p$-ary $s$-plateaued functions, vectorial $s$-plateaued functions and generalized $s$-plateaued functions can be constructed. In particular, $0$-plateaued partitions are just bent partitions. In general, $s$-plateaued partitions are much more complicated than bent partitions. We analyze the possible cardinality of $A_{i}$ of an $s$-plateaued partition. We give some explicit constructions of $s$-plateaued partitions for which any generated $p$-ary $s$-plateaued function has no nonzero linear structure. We give a characterization of an $s$-plateaued partition $\Gamma=\{A_{i}, 1 \leq i \leq K\}$, where $p$ is odd, $K \geq 5$ and $-A_{i}=A_{i}, 1 \leq i \leq K$. Based on which, we show that if $p \geq 5$, then the preimage set partition of a $p$-ary $s$-plateaued function $f: V_{n}^{(p)} \rightarrow \mathbb{F}_{p}$ with $f(x)=f(-x)$ is an $s$-plateaued partition if and only if $f$ is of $(p-1)$-form, where $n+s$ is even.When $s=0$, we partially address an open problem on whether a bent partition $\Gamma$ of $V_{n}^{(p)}$ of depth $p^{\frac{n}{2}}$ must be obtained from spreads.

We analyze the asymptotics of a block-Wishart random matrix ensemble of the type ${\boldsymbol W}_k = ({\boldsymbol X}^* \otimes {\boldsymbol I}_k){\boldsymbol T}({\boldsymbol X}\otimes{\boldsymbol I}_k)$ for ${\boldsymbol X} \in\mathbb{C}^{n\times p}$ with i.i.d. rows satisfying a suitable concentration-of-measure property, and ${\boldsymbol T} := \textrm{\bf Diag}({\boldsymbol T}_i)_{i\in[n]}$ a block diagonal matrix with self-adjoint blocks ${\boldsymbol T}_i\in \mathbb{C}^{k\times k}$, under the proportional asymptotics $n/p\to\alpha$ with $k$ fixed. These matrices play a prominent role in the analysis of $k$-index models in high-dimensional statistics. By studying the matrix Stieltjes transform of this random matrix model and its inverse ($K$-transform), we derive variational formulas for two functionals of the asymptotic spectral density of ${\boldsymbol W}_k$: the left (equivalently right) edge of its support, and its logarithmic potential.

Approximate message passing (AMP) provides fast iterative algorithms for high-dimensional signal recovery with Gaussian design matrices, while the Convex Gaussian Min–max Theorem (CGMT) gives a static optimization framework for obtaining sharp asymptotic characterizations of convex estimators. Although these two frameworks often lead to the same scalar state-evolution equations, their connection is usually indirect. In this paper, we establish a direct connection between the two for regularized linear regression in the proportional high-dimensional regime. When the CGMT Auxiliary Optimization (AO) and Primary Optimization (PO) give the same primal-dual solution, we show that the CGMT framework recovers the AMP fixed-point equations, including the Onsager correction. We further identify the AO Gaussian vectors with the Gaussian perturbations in the primal and residual AMP channels. For regularized $M$-estimation, the same viewpoint recovers the fixed point of scalar-variance max-sum Generalized AMP (GAMP). These results show that the AMP (and GAMP) fixed-point equations can be recovered from the CGMT framework and may suggest AMP-like algorithms in settings where standard derivations are unavailable.

Optimizing functionals over the space of probability measures is now ubiquitous in machine learning. A widely used approach is to perform the optimization directly over the Wasserstein space, but many objective functionals of practical interest are non-convex along Wasserstein geodesics, making the analysis of standard first-order methods challenging. In this work, we study a class of objectives over the Wasserstein space that admit a difference-of-convex (DC) decomposition and we lift the classical convex-concave procedure (CCCP) to this setting. Under smoothness and strong convexity assumptions on the convex components of the decomposition, we prove almost stationarity along the iterates of the resulting algorithm. Our main focus is on the Maximum Mean Discrepancy (MMD) and the Energy Distance (ED) functionals, for which we develop explicit Wasserstein DC decompositions, and establish local convergence of the scheme under mild assumptions. Empirically, we show that well-chosen DC decompositions yield faster and more stable convergence than Wasserstein gradient descent on these MMD objectives.

In this paper we deal with Nil geometry, in whose projective model the geodesic curves are helix-like and fit onto geodesic Nil cylinders of revolution with fibrum axes. In this paper we investigate relations for geodesic cylinders and thus also geodesic curves, which lead to an analogous result to Pappus's hexagon theorem and provide important information about the structure of the considered space.

This article investigates a remarkable combinatorial identity involving a distinguished family of matrices whose entries are defined via binomial coefficients. Specifically, we consider a class of \( n \times n \) matrices parameterized by a positive integer \( m \), where each entry reflects a structured pattern derived from Pascal's triangle, particularly the diagonals corresponding to figurate numbers such as triangular, tetrahedral, and higher-dimensional simplex numbers. We establish, by means of a bijective argument, that the determinant of any such matrix is identically equal to \( 1 \), independent of the specific values of \( m \) and \( n \), provided that \( 2 \leq m \leq n \). This result unveils a profound connection between classical binomial identities and the enumeration of lattice paths in grid graphs.

A quantum Latin square of order \(n\), denoted by \(\operatorname{QLS}(n)\), is an \(n \times n\) square whose entries are unit column vectors in the \(n\)-dimensional Hilbert space \(\mathcal{H}_n\), such that each row and each column forms an orthonormal basis of \(\mathcal{H}_n\). The cardinality of a QLS($n$) is the number of distinct vectors up to a global phase in the array. A \(\mathrm{QLS}(n)\) whose main diagonal and anti-diagonal each forms an orthonormal basis of \(\mathcal{H}_n\) is called a diagonal quantum Latin square (\(\mathrm{DQLS}(n)\)). In this paper, we focus on the existence of the \(\mathrm{DQLS}(n)\) with maximum cardinality ($\mathrm{MCDQLS}(n)$). By employing direct constructions based on row-quantum Latin rectangle and special complete mapping, together with the recursive techniques such as the singular direct product construction, We have almost completely determined the existence of \(\mathrm{MCDQLS}(n)\), except for a few exceptional cases. This result is based on the study of the existence of idempotent \(\mathrm{QLS}(n)\) with maximum cardinality (\(\mathrm{MCQLS}(n)\)), and implies an existence result for pandiagonal quantum Latin squares with maximum cardinality (\(\mathrm{MCPQLS}(n)\)).

A locally conformally Kähler (LCK) manifold is a manifold $M$ which admits a Kähler structure on its universal cover $\tilde M$, in such a way that the monodromy acts conformally on $\tilde M$. Let $M$ be an $n$-dimensional compact LCK manifold of algebraic dimension $n-1$. We prove that $M$ is bimeromorphic to the total space of an isotrivial elliptic fibration. Morever, there exists an alteration of $M$ which dominates bimeromorphically a manifold admitting a free action of an elliptic curve.

Assume $n \geq 2$ and $\ell = (r_{1}, \ldots, r_{k}) \in [0,1]^{k}$ is an increasing sequence of real numbers. Let $G_{n,\ell}$ denote the group of orientation-preserving piecewise linear homeomorphisms $h$ of $I = [r_{1}, r_{k}]$ such that: (i) $h'(x)$ is a power of $n$ where it is defined; (ii) if $h'(x)$ is undefined, then $x$ is an $n$-adic rational number, (iii) $h$ fixes each entry of $\ell$, and (iv) $h(\mathbb{Z}[1/n] \cap I) = \mathbb{Z}[1/n] \cap I$. We prove that $G_{n,\ell}$ is a diagram group $D(\mathcal{P}_{n,\ell}, \omega_{n,\ell})$ for all integers $n \geq 2$ and for all finite sequences $\ell$. The semigroup presentation $\mathcal{P}_{n,\ell}$ and the word $\omega_{n,\ell}$ can be computed from the $n$-ary expansions of the numbers $r_{i}$. If all entries in $\ell$ are rational, then $G_{n,\ell}$ has type $F_{\infty}$. Otherwise, $G_{n,\ell}$ is not finitely generated.

For level one spherical automorphic forms on the upper half-plane, we prove directly that every automorphic form is a sum of a cusp form and a linear combination of Laurent coefficients of the standard Eisenstein series. This is the simplest instance of Franke's general theorem, which asserts that automorphic forms on a reductive group are spanned by Laurent coefficients of Eisenstein series induced from cuspidal automorphic forms on Levi subgroups. Unlike Franke's general argument, ours does not invoke Langlands' construction of the discrete automorphic spectrum from cuspidal Eisenstein series. It rests instead on basic analytic properties of automorphic forms and Green's identity.

We construct examples of uniformly quasiregular mappings (uqr) acting on a sphere and having Fatou set consisting of infinitely many components. In particular we construct a uqr mapping providing a higher dimensional counterpart for the polynomial $z \mapsto z^2 - 1$ acting on the extended complex plane $\hat{\mathbb{C}}$.

We prove explicit upper bounds on the propagation velocity of one-dimensional quantum walks with periodic coins of arbitrary period. We treat two complementary settings. First, in a perturbative regime where one transmission parameter is small, we show that the corresponding almost reflecting coin acts as a bottleneck for transport: the velocity is bounded linearly in this parameter, with an explicit leading order estimate. Second, for arbitrary nonzero transmission parameters, we prove a general a priori bound in terms of their harmonic mean, together with a refined version that detects the spatial variation of neighboring coins. Moreover, we prove a general lower bound on the velocity. These bounds apply directly to the corresponding CMV setting.

Programming adaptive behaviors at the cellular level is a long-standing goal that raises the question of how probabilistic computation can be implemented in biochemical systems. Chemical reaction networks (CRNs) provide such a substrate and have been shown to realize probabilistic models, including hidden Markov models and factor graphs, with dynamics reproducing Bayesian inference and belief propagation. However, encoding these algorithms typically requires prohibitively large reaction networks, and classical CRN reduction techniques do not directly apply. By recovering the factor graph structure encoded in Napp–Adams-compiled CRNs, we transport recent factor-graph reduction results to their chemical implementations, obtaining significantly smaller CRNs while preserving the belief-propagation fixed points on surviving variables.

We study hp approximation and additive Schwarz decompositions for variable-order cubical finite element spaces on one-irregular meshes. For fitted homogeneous diffusion interface problems on one-irregular hexahedral meshes, we prove an hp-optimal energy-norm estimate for the interior penalty DG method. The interpolation input is a conforming hp interpolant obtained from fitted conforming closures of one-irregular vertex patches. We also derive stable decompositions for conforming and DG spaces. On one-irregular quadrilateral meshes the bounds allow locally comparable variable polynomial degrees and are independent of the mesh size, the local degrees, and, under a local coefficient quasi-monotonicity condition, the coefficient contrast. On one-irregular hexahedral meshes the conforming decomposition has the corresponding polylogarithmic loss; the DG-to-conforming reduction is used there for uniform-degree DG spaces. Numerical experiments illustrate the p-optimal DG error estimate and the robustness of the DG Schwarz preconditioner.

We analyze an often used closure model for multi-material hydrodynamics where pressure-temperature equilibrium (PTE) is assumed for every state; emphasis is placed on tabular equations of state. This multi-material model is often referred to as the four-equation model. The identification of the admissible set is presented and is proven to be convex, setting the foundation for development of invariant-domain preserving methods for this model. A novel numerical method is presented for solving the highly nonlinear system for the equilibrated pressure and temperature with an arbitrary number of materials. This new method is compared with some traditional iterative solvers through a collection of different tests. Additionally, we provide a detailed analysis of the thermodynamics of the mixture model for general equations of state and prove existence and uniqueness of the pressure-temperature equilibrium solution under some thermodynamic assumptions.

For any complete Riemannian manifold $M^n$ with nonnegative Ricci curvature and sublinear diameter growth, we establish a dimensional constraint $n\ge 4s(s-1)+k+1$ if the fundamental group $\pi_1(M)$ contains a torsion-free nilpotent subgroup of rank $k$ and step $s\ge 2$. As a consequence, if such a manifold $M$ has dimension $n<12$, then $\pi_1(M)$ is almost abelian. The proof is based on a dimensional estimate for $\mathrm{RCD}(0,N)$ spaces admitting $\mathbb{R}$-orbits of large Hausdorff dimension.

This paper introduces a control-theoretic perspective on unconstrained optimization algorithms using the backstepping methods. We model the optimization process as an augmented strict-feedback system given by $\dot{x}_1 = x_2$, $\dot{x}_2 = u$, and $\dot{z} = q(x_1,z)$, with a regulated output $y = \nabla f(x_1)$. This formulation recasts the development of unconstrained optimization algorithms as a feedback control problem, where the goal is to design the input $u$ to ensure $y(t) \to 0$. By employing backstepping, we recursively synthesize the actual feedback law $u$ after initially selecting a virtual control for $x_1$. For convex objective functions, we develop a general synthesis framework for augmented strict-feedback systems and specialize it to the standard strict-feedback case. This unified framework successfully recovers the constant-parameter Nesterov flow and the proportional-integral-derivative (PID) accelerated optimizer as direct corollaries. We further establish that, given a fixed virtual control, the universal second-step law is inverse optimal with respect to an induced outer-tracking problem. This reveals that the optimality of the control law is conditionally dependent on the target manifold prescribed by the virtual control, rather than holding globally across all possible backstepping designs. Finally, we formulate a formal optimal-backstepping theorem that elevates this optimality principle to the virtual-control stage by solving a reduced Hamilton–Jacobi–Bellman problem. These contributions collectively yield a robust and general backstepping-driven paradigm for the analysis and design of continuous-time unconstrained optimization algorithms.

We study small-amplitude solitary waves for two-dimensional capillary–gravity flows with arbitrary vorticity on the equatorial $f$-plane. The steady free-boundary problem is formulated as a reversible Hamiltonian spatial-dynamics system in which rotation enters through the speed-dependent effective gravity $g_*=g-2\Omega c$. A center-manifold reduction reduces the local bifurcation problem to finite-dimensional Hamiltonian systems governed by the low-frequency spectrum of a Sturm–Liouville problem with an eigenvalue-dependent boundary condition. We identify the Hamiltonian $0^2$, real $1{:}1$, and Hamiltonian–Hopf resonance curves and obtain corresponding families of symmetric solitary waves under standard non-degeneracy assumptions. We also show that the weak-effective-gravity threshold $g_*=0$ is separated from the $0^2$ and local Hamiltonian–Hopf resonances for uniformly non-stagnant laminar flows, and can be approached only in a near-stagnation regime.

Let $\mathbb{B}$ be the unit ball in $\mathbb{R}^2$, $W_0^{1,2} \left( \mathbb{B} \right)$ is a standard Sobolev space. Suppose a function $h_{\epsilon}(x)$ is radially symmetric, nonnegative, continuous on $\overline{\mathbb{B}}$ and satifies $\underset{x \rightarrow 0}{\lim} h_{\epsilon}(x) |x|^{- 2 \epsilon} =1 $, with $h_{\epsilon} (x) >0$ on $\overline{\mathbb{B}} \setminus \{0\}$. In \citep{26}, Zhang proved that the supremum in the following inequality can be attained by some function $u_{\epsilon}$, i.e. , \begin{align} \int_{ \mathbb{B} } h_{\epsilon} (x) e^{ 4 \pi \left(1 + \epsilon \right) {u_{\epsilon}}^2 } dx = \underset{u \in W_0^{1,2} \left( \mathbb{B} \right) \cap \mathcal{S} \setminus \{0\} , ~ \int_{ \mathbb{B} } |\nabla u|^2 dx \leq 1}{\sup} \int_{\mathbb{B}} h_{\epsilon} (x) e^{4 \pi (1 + \epsilon) u^2 } dx, \label{eq: 0.1} \end{align} where $4 \pi$ is the best constant in the classical Moser-Trudinger inequality, and $\mathcal{S}$ is the set of radially symmetric functions. In this paper, we consider the compactness of the sequence $\{ u_{\epsilon} \}_{\epsilon} $ and prove that the limit of this sequence is a function $u_0 \in C^1 (\overline{ \mathbb{B}} )$. Moreover, the $u_0$ is an extremal function of the supremum \begin{align*} \underset{u \in W_0^{1,2} \left( \mathbb{B} \right) \cap \mathcal{S} \setminus \{0\} , ~ \int_{ \mathbb{B} } |\nabla u|^2 dx \leq 1}{\sup} \int_{\mathbb{B}} e^{4 \pi u^2 } dx. \end{align*}

Let $g$ be the Lie superalgebra $p(3)$ of rank 2 over an algebraically closed field $K$ of characteristic $p=3$. We classify all irreducible modules of $g$, and give the character formulae for irreducible modules.

Negami's Planar Cover Conjecture asserts that a connected graph has a finite planar cover if and only if it can be embedded on the projective plane. While this statement has already been proven for rotation compatible planar covers, namely covers equipped with a certain condition on the rotation system, the existing proof relies on advanced algebraic and topological methods. In this paper, we provide another proof of this result, focusing primarily on combinatorial arguments based on a structural analysis with respect to a spanning tree in the base graph.

Slater's list of Rogers-Ramanujan type identities remains a central source of striking series-product formulas in the theory of partitions and basic hypergeometric series. Although many of these identities admit elegant analytic proofs through Bailey pairs, Bailey chains, or transformations of basic hypergeometric series, the partition-theoretic meaning of their series sides is often much less apparent. In this paper, which continues the program initiated in arXiv:2603.14179, we apply Andrews' theory of separable integer partition classes to further identities from Slater's list. We construct a strict overpartition class and two families of overpartitions with positional gap conditions, in which overlining is permitted only at alternating positions. Their multivariate generating functions give natural refinements of the series sides of Slater's identities (12), (28), (29), (47), (48), (50), and (51). We then use Heine-type transformations, a limiting form of Heine's transformations, Watson's $q$-analogue of Whipple's theorem, and classical theta-product identities to obtain alternative series representations and recover the associated products. In addition, we derive a new companion identity to a Slater identity and a signed companion formula. Our results further demonstrate that SIP classes provide a flexible framework for converting basic hypergeometric series into structured partition generating functions, while simultaneously producing refinements, transformations, and new Rogers-Ramanujan type identities.