Papers

For polarized manifolds, we show that semistability for the quantized Mabuchi energy at one fixed finite parameter is equivalent to uniform K-stability and to unique existence of constant scalar curvature Kähler (cscK) metrics. We obtain sharp comparisons between quantized Mabuchi slopes at different parameters and with the ordinary Mabuchi slope. In the presence of automorphisms, we give an equivariant criterion using torus twists under the vanishing of the Futaki character. The estimates also hold for finite-energy geodesic rays in compact Kähler classes. On a fixed ruled surface with discrete automorphism group, we construct cscK polarizations with unbounded critical parameters. Their ratios to the $\alpha$-invariant are also unbounded.

We study normalized principal Steklov eigencouples on periodically perforated domains when both the bulk energy and the boundary normalization have general, possibly nonhomogeneous, Orlicz growth. The surface term on the hole boundaries carries the critical surface-to-volume scaling. For each fixed mass level, the constrained minima converge along the whole family, while selected minimizers, fluxes and Lagrange multipliers converge along subsequences to the corresponding objects of a homogenized problem. In the limit, the bulk operator is given by a convex cell problem, and the boundary term becomes a volume term weighted by the total surface density of the reference hole. The key tool is a strong approximation of the microscopic traces by cell averages, measured in the scaled surface modular.

The majorization simplex $\mathcal{P}_n$ is the set of probability vectors of length $n$ whose entries are arranged in non-increasing order. Under the majorization order, $\mathcal{P}_n$ is a lattice, and it is the natural space for comparing distributions when the labels of the outcomes are not relevant. In this work, we study measures of dissimilarity (and, similarity) on $\mathcal{P}_n$. Our first set of results concerns the Rényi-entropy based distance $d_\alpha(p,q)=H_\alpha(p)+H_\alpha(q)-2H_\alpha(p\vee q)$ proposed by Yadav and Shkel (2026), which generalizes a metric of Cicalese, Gargano and Vaccaro (2013). We show that $d_\alpha$ is a metric on $\mathcal{P}_n$ for $\alpha\in[1,\infty)$, while for $\alpha\in[0,1)\cup\{\infty\}$, $d_\alpha$ is not a metric in general. Our main result in this part identifies the necessary and sufficient conditions for equality in the triangle inequality of $d_\alpha$ when $\alpha>1$, for which we identify the equality conditions for the supermodularity of the Rényi entropy $H_\alpha$. Our second set of results concerns the total variation distance and the Hellinger affinity (also called the Bhattacharyya coefficient) on $\mathcal{P}_n$. We obtain a number of bounds on these measures, along with some of their equality conditions. The resulting Hellinger affinity based upper bound on the total variation distance can be smaller than the bounds obtained from Pinsker's and the Bretagnolle–Huber inequalities.

Let $p>1$ and $\Omega\subset\mathbb{R}^n$ be a bounded smooth domain. We study the following overdetermined problem for the p-Laplacian: \begin{equation*} \left\{ \begin{array}{ll} -\Delta_p u=f(x)&x\in\Omega,\\ u=0&x\in\partial\Omega,\\ \frac{\partial u}{\partial\nu}=g(x)&x\in\partial\Omega, \end{array} \right. \end{equation*} where $\nu$ is the inward unit normal to $\partial\Omega$ and both data may depend on position. A boundary-local version of the moving-plane argument is developed for the possibly degenerate p-Laplace operator. The advantage of this argument is that it avoids the analysis of the degenerate set of the p-Laplacian. Under some reasonable structural conditions, we prove that the solvability of the above mentioned problem forces the domain to be a ball, the source to be radial, and the Neumann datum to be a suitable constant. We also give an explicit formula for the solution. Further consequences include one directional domain symmetry, the conclusion that full rigidity follows from reflection information in any set of directions whose linear reflections generate a dense subgroup of the orthogonal group $O(n)$, and a constant mean curvature characterization.

Let $R$ be an integral domain and let $w$ denote the $w$-operation on $R$. A nonzero GV-torsion-free $R$-module $M$ is called $w$-secondary if, for every $r\in R$, either $M/rM$ is GV-torsion or $r^nM=0$ for some $n\in\N$. A $w$-module $M$ is said to admit a $w$-secondary representation if $M=(N_1+\cdots+N_n)_w$, where each $N_i$ is a $w$-secondary submodule of $M$. Such a representation is called minimal if the ideals $P_i=\sqrt{\Ann_R(N_i)}$ are pairwise distinct, and $M$ is called $w$-representable if it admits such a representation. If the sum can be chosen direct, then $M$ is called strongly $w$-representable. We characterize $w$-secondary modules and investigate their localization and $w$-closure properties. For a nonfield domain $R$, we prove that $R$ is $t$-almost Dedekind if and only if $\wdim R=1$ and every $w$-representable $w$-module is locally strongly $w$-representable: each of its nonzero localizations at maximal $w$-ideals is strongly representable. Global strong $w$-representability is retained when all the radical annihilators in the representation are nonzero.

This paper delves into the intricate aspects of Reconfigurable Intelligent Surfaces (RIS) in the antenna design. It investigates the properties of the array factor of RIS, highlighting the significance of phase shifts in establishing of the reflect path. Additionally, the paper investigates spatial positions requiring identical phase shifts, nullification of the array factor, the count of total maxima, second maximum determination (amplitude of the second lobe), and the calculation of the 3dB bandwidth. They are very useful for RIS-assisted radio communication systems.

Resampling is the standard remedy against weight degeneracy in particle filters, yet its effect on the spectrum of the particle second-moment matrix is poorly understood. We study a single resampling step for a d-dimensional ensemble of N particles with d/N -> c. For any unbiased scheme, the resampled empirical second-moment matrix dominates its pre-resampling counterpart in conditional convex order, while the conditional mean is preserved; the excess second spectral moment equals the Frobenius energy of the resampling perturbation. For multinomial resampling, the empirical count law converges to the mixed-Poisson law determined by the limiting weight law. For residual resampling, the count law is determined by the joint limit of floor-fraction pairs: we classify all residual limits of weight sequences converging to delta_1 in law as a one-parameter family between the deterministic identity and the full multinomial count law. Each count law feeds a common covariance-side spectral map, yielding exact consequences: a closed-form zero-eigenvalue mass, a rank-and-spectral-gap transition with a universal lower edge benchmark, computable regular edges, unbounded right support, and scheme-specific second-moment costs. The effective sample size determines one bulk statistic but cannot determine the limiting spectrum. Pre-registered numerical experiments confirm the theory at every level.

The latent variable proximal point (LVPP) method solves optimization problems in function spaces with pointwise constraints by a sequence of smooth subproblems whose iterates are strictly feasible. We apply it to linear-quadratic elliptic optimal control problems. Control and observation are distributed but may be restricted to subdomains. We prove that the LVPP subproblems are well posed in appropriate function spaces, that the iteration converges for every sequence of proximity parameters with divergent sum, and that it delivers the Lagrange multiplier of the constraint as a by-product. We interpret the Newton step for the LVPP subproblem as a linear-quadratic optimal control problem with a spatially varying control cost, and we show that Newton's method converges locally quadratically. The Newton systems are solved by the minimal residual method with a block-diagonal preconditioner. We prove that the spectrum of the Newton operator with respect to the inner product defined by the preconditioner is bounded independently of the proximity parameter, the iterate, the mesh and the observation and control domains. The results are combined into a complete algorithm, and tested in numerical experiments.

Learning interaction structures from spatiotemporal event data is a key statistical problem in fields such as neuroscience, epidemiology, and seismology. We study this problem for Hawkes processes on a continuous spatial domain, where events may trigger future events at different locations. Following the framework developed in “Spatiotemporal Hawkes processes with a graphon-induced connectivity structure” by Baars, Laeven, and Mandjes (2026), we model the spatial interaction mechanism using an underlying graphon that characterizes how the intensities' excitation strength varies across locations. By expanding the baseline intensity, temporal excitation kernel and graphon in orthogonal bases, we represent the problem as a high-dimensional parametric inference problem. We propose a sparse penalized least-squares estimator that recovers the effective interaction structure while controlling model complexity. Our analysis establishes non-asymptotic error bounds in a high-dimensional regime where the number of parameters may grow with the observation horizon, without requiring the conditional intensity to be almost surely bounded. We further establish sparsistency results, showing that the latent interaction network can be consistently recovered. Simulation studies demonstrate the good finite sample performance of the proposed method and an application to large-scale neuronal spike data, comprising 1.81 million events, showcases its practical effectiveness.

We study the recovery of the hidden vertex correspondence between two correlated dense random geometric graphs. Each graph is generated from $n$ latent unit vectors in $\mathbb R^d$; the vectors $u_i^A$ and $u_i^B$ of corresponding vertices satisfy $\langle u_i^A,u_i^B\rangle=\rho$, and the vertex labels of the second graph are permuted by an unknown permutation $\pi$. For every fixed correlation $\rho\in(0,1]$ and every edge density bounded away from zero and one, we give a randomized polynomial time algorithm that recovers $\pi$ exactly with probability tending to one whenever $d\ge(\log n)^L$, where $L$ depends only on the correlation and the density bounds. The algorithm assigns to each vertex a signature formed by counting decorated paths with a constant number of seeds. The number of such seeded templates grows exponentially with the path length, with an arbitrarily large constant base, so the resulting statistics amplify an arbitrarily small constant correlation. We control the dependencies created by the latent geometry in two ways: when $d$ is larger than $n/(\log n)^{L_0}$ for a suitable constant $L_0$, by comparing local moments with those of correlated Erdős–Rényi graphs; for smaller $d$, by spectral estimation of the latent geometry combined with a harmonic correction. We also provide low-degree evidence for a tradeoff between the correlation and the computational cost.

Assuming that $\mathcal{P}(\omega)$ is tightly $\sigma$-filtered, we prove that $C(\omega^*)$ fails the ball fixed point property. This axiom holds under the continuum hypothesis and in the $\aleph_2$-Cohen model. The proof shows that the Stone space of the countable–cocountable algebra $\operatorname{Cc}(\omega_1)$ is a continuous retract of $\omega^*$ and uses the equivalence between tight $\sigma$-filteredness of $\operatorname{Cc}(\omega_1)$ and of $\mathcal{P}(\omega)$. We also prove that the quotient of a tightly $\sigma$-filtered Boolean algebra $B$ with the countable separation property by an $\aleph_1$-generated $P$-ideal is a Boolean retract of $B$.

In the complex Hilbert space $L_2(\mathbb{R}_+)$ on the half-line $\mathbb{R}_+ = [0, \infty)$, we investigate non-symmetric Schrödinger operators \[ Su = -u'' + qu, \qquad u(0) = 0, \] with complex-valued, highly singular distributional potentials \[ q = s + Q' \in H_{\mathrm{loc}}^{-1}(\mathbb{R}_+), \qquad s \in L_{1,\mathrm{loc}}(\mathbb{R}_+), \quad Q \in L_{2,\mathrm{loc}}(\mathbb{R}_+), \] where the derivative $Q'$ is understood in the sense of Schwartz distributions. A rigorous operator-theoretic setting is established within the framework of Shin–Zettl quasi-derivatives. We prove that every accretive operator $S$ of this class is automatically $m$-accretive in $L_2(\mathbb{R}_+)$ and $C$-self-adjoint with respect to the complex conjugation involution $C : u \mapsto \bar{u}$. In particular, no additional boundary conditions at infinity are needed, meaning that a Weyl limit-point type situation always prevails for accretive realizations. Constructive local conditions on the coefficients $s$ and $Q$ are found under which the generated strongly continuous semigroup $\{e^{-tS}\}_{t \ge 0}$ is contractive or holomorphic in a sector of the complex plane.

We prove an upper bound for the error term in the higher rank hyperbolic lattice counting problem for ${\hbox{SL}(n, \mathbb{Z})}$. Blomer and Lutsko achieved recently a significant improvement of previous known results by Duke–Rudnick–Sarnak and Gorodnik–Nevo–Yehoshua. We deduce an improved upper bound for the error which holds for almost all pairs $z,w \in {\hbox{SL}(n, \mathbb{R})}$. We also derive a $\Omega$-result for the error term, in the spirit of Rudnick–Phillips.

In this work, we study restricted Laistrygonian algebras over a field of positive characteristic $p$. We compute their Drinfeld double $\mathtt{D_p(q,a)}$ and give explicit presentations by generators and relations. We prove that $\mathtt{D_p(q,a)}$ is a pointed Hopf algebra. For $\mathtt{D_p(1,a)}$, we describe an abelian exact sequence of Hopf algebras and identify one of its quotients as a restricted enveloping Lie algebra. This exact sequence enables us to classify the finite-dimensional simple $\mathtt{D_p(1,a)}$-modules. We also introduce an infinite-dimensional Hopf algebra covering the restricted Drinfeld double and study its relation with the corresponding finite-dimensional case. Using this covering together with a Frobenius map, we construct a $3\times 3$ commutative diagram whose rows and columns are exact sequences of Hopf algebras. In particular, our approach provides a unified treatment of the Drinfeld doubles associated with restricted Laistrygonian algebras in arbitrary positive characteristic.

In this paper we show there does not exist any arrangement of degree $7$ consisting of $k\geq 1$ conics and $d\geq 1$ lines admitting certain naturally chosen ${\rm ADE}$-singularities.

In this note we show that all interval exchange transformations of three intervals (3-IETs) are spectrally disjoint from all mixing and disjoint from lightly mixing systems. Until now, it was known that almost every 3-IET is rigid, which implies this property. However, not all 3-IETs are rigid, hence our result is a strict improvement of current knowledge. We then use this result, among many others, to prove that although some 3-IETs are mildly mixing, they are always disjoint in the sense of Furstenberg from lightly mixing systems.

The sums of the absolute values of higher-order cumulants are classical contrast functions for independent component analysis (ICA) and lead naturally to robust approximate diagonalization of higher-order cumulant tensors using $\ell_1$-based diagonalization over the orthogonal or unitary group. We propose Jacobi-SubG, a Jacobi-type framework that selects index pairs using Riemannian subdifferential information and performs plane rotations with a refinement step that preserves nonzero diagonal contractions. For the general tensor model, we prove global convergence of the entire iterate sequence from any initialization with nonzero diagonal contractions, provided that the subproblem solvers satisfy a uniform sufficient-ascent estimate. For real and complex third- and fourth-order cumulant tensors, which are widely used in ICA, we develop specialized subproblem solvers satisfying these conditions. The resulting algorithms thus enjoy global convergence under the simple initialization condition that all diagonal contractions are nonzero. Numerical experiments on ICA and tensor diagonalization show that the proposed Jacobi-SubG variants achieve better source recovery and higher final objective values in several settings, particularly at the higher tested noise levels.

Let $\Phi(G)$ denote the number of perfect matchings of a graph $G$. Goedgebeur et al. posted two questions about the perfect matchings of planar 3-connected graphs: does there exist an infinite family of matchable planar 3-connected graphs of girth 5 with a constant number of perfect matchings? Whether there exist infinitely many planar 3-connected graphs with minimum degree 5 which have a constant number of perfect matchings. We present answers to those questions. First, we construct, for every even $n\ge196$, a simple planar 3-connected graph of order $n$ and girth five with exactly $c$ perfect matchings, where $c$ is a fixed positive integer. Second, for a simple planar 3-connected graph $G$ with a perfect matching and minimum degree at least five, we show that $\Phi(G)\ge2^{\lceil(|V(G)|+20)/22\rceil}$. This implies that, for each positive integer $t$, only finitely many such graphs, up to isomorphism, have at most $t$ perfect matchings.

The deep hole problem is a fundamental problem in coding theory, and it has many important applications in code constructions and cryptography. The deep hole problem of Reed-Solomon codes has gained a lot of attention. As a generalization of Reed-Solomon codes under Hamming metric, we study the covering radius and deep holes of Hyperderivative Reed-Solomon (HRS) codes under the Niederreiter-Rosenbloom-Tsfasman (NRT) metric. We determine the NRT covering radius of HRS code and characterize its NRT deep holes via Maximum Distance Separable (MDS) extensions and polynomial remainder conditions. We further relate the enumeration of NRT deep holes to the characteristic polynomial of an associated hyperplane arrangement. We also provide three classes of explicit NRT deep-hole constructions, propose a Cheng-Murray type conjecture for HRS codes under the NRT metric and obtain a conditional low-layer description. Using these characterizations, we derive exact enumeration formulas in several special cases.

This paper studies the topological entropy of real-analytic Liouville integrable Hamiltonian systems. We prove that the Hamiltonian flow on a compact connected regular energy hypersurface has zero topological entropy whenever a complete family of commuting real-analytic first integrals is generically independent in a connected neighborhood of that hypersurface. The proof uses finite normal expansions along minimal analytic carriers and an ergodic version of Ziglin's leading-term argument to obtain independent invariant normal polynomials, whose Poisson geometry makes all tangent Lyapunov exponents of the carriers vanish. In particular, we prove that a real-analytic Liouville integrable geodesic flow on a closed real-analytic Riemannian manifold has zero topological entropy, giving a positive answer to the conjecture proposed by Paternain in 1992.

Non-exchangeability and radial asymmetry have got substantial attention in the literature, but relation between the two has not been studied yet. We propose the question: if a copula has a given degree of radial asymmetry, what is the range of possible values of the measure of non-exhangeability. We determine the set of all points whose first coordinate is the radial asymmetry of a copula and the second coordinate is its non-exchangeability. We show that this set is a non-convex hexagon.

Adaptive modulation and coding (AMC) improves the spectral efficiency of underwater acoustic (UWA) links, but its feedback is inevitably stale: sound propagates slowly, and by the time channel state information (CSI) arrives, the fast time-varying multipath channel has moved on. Predictive link adaptation therefore needs more than a point forecast: it needs a distribution over each candidate action's consequences. We propose PRB-WM, a physics-regularized Beta world model for predictive per-subcarrier link adaptation on real time-varying UWA channels. A persistence-anchored stochastic latent-space dynamics model learns how the channel evolves; a counterfactual outcome head evaluates, for every candidate modulation and coding scheme (MCS), a Beta-binomial distribution of bit errors on shared imagined futures; and a chance-constrained receding-horizon model predictive controller maximizes expected spectral efficiency subject to a bounded probability of violating the bit error rate (BER) target. To the best of our knowledge, PRB-WM is the first world model for UWA communications. In closed-loop replay of six measured sea-trial links from the public UWA Channel Library against a 24-method pool spanning the full methodological spectrum, PRB-WM attains a mean goodput 87% above the strongest baseline, ranks first on all six time-varying links at the longest lookahead, and is the only method whose realized frame risk decreases systematically with lookahead, meeting the BER-violation contract where no baseline does. Ablations confirm that rollout sampling dispersion is essential to the chance constraint and that the physics anchor is necessary for trainability.

We give a higher unit-torus lift from q-linearized permutation polynomials over Fqn to generally nonlinear sparse permutation polynomials over Fqns. The construction preserves compositional inverses: if a linearized polynomials and its inverse have coefficients (ai) and (Ai), then the lifted polynomial and its inverse have the same coefficients. As a consequence, we obtain nonlinear binomial lifts whose inverses are again binomials in the two-dimensional case, including an explicit inverse for a binomial family over Fq4. The three dimensional specialization yields permutation trinomials whose compositional inverses are again trinomials, together with explicit involution criteria and examples.

Stable bound states living on a discrete one-dimensional N-point lattice are assumed generated by a certain two-parametric PT symmetric potential $V=V(A,B)$. At any $N$, the coordinates $A=A^{(EP)}$ and $B=B^{(EP)}$ of the Kato's exceptional point (marking the phase-transition instant of the loss of stability) are defined, exactly, via roots of a certain elementary quartic polynomial.

We prove that the singular locus of the geodesic flow on the unit tangent bundle of a nonpositively curved surface enjoys an additional regularity property: its characteristic function belongs to the geometric anisotropic Sobolev space $H^{5/8-}_xH^{-13/16}_\theta$. In particular, this implies that if the geodesic flow is not ergodic with respect to the Liouville measure, then the zero set of the curvature must be thick in the sense that it supports a nonzero, nonnegative function in $H^{5/8-}$. The proof relies on semiclassical analysis and a new radial estimate for parabolic flows.

In this paper we construct a one-parameter family of periodic Airy line ensembles $\mathcal{A}^{\mathrm{per},R}$ for $R>0$ as weak scaling limits of Borodin's periodic Schur processes. We establish several properties of $\mathcal{A}^{\mathrm{per},R}$, including a determinantal point process structure with an explicit correlation kernel. We also show that, after a parabolic shift and normalization, $\mathcal{A}^{\mathrm{per},R}$ satisfies, on its domain $[-R/2,R/2]$, the same Brownian Gibbs property enjoyed by the usual Airy line ensemble. A main technical contribution of the paper is a general tightness criterion for determinantal line ensembles, which extends the argument used by Johansson in [Comm. Math. Phys. 242 (2003), 277-329] to prove a functional limit theorem for the discrete polynuclear growth model.

While the robustness of collective dynamics in static networks is well established, how agent mobility and network topology jointly shape dynamical robustness remains largely unexplored. In this letter, we investigate the robustness of mobile Stuart-Landau oscillators performing random walks on networks and interacting diffusively when co-located. In the rapid-mobility regime, we reduce the dynamics to a mean-field framework linking network degree statistics to an effective global coupling strength. We show analytically that increased degree heterogeneity in the network enhances the effective coupling strength, and strictly lowers the critical threshold of functional degradation for amplitude death. Numerical simulations confirm the predicted heterogeneity-dependent ordering of the transition thresholds, showing an earlier collective collapse in scale-free networks than in random or regular networks. Our results establish mobility and network heterogeneity as key determinants of robustness in mobile oscillator populations, with implications for resilient biological and engineered collectives.

We compute the group of based homotopy classes of self-homotopy equivalences of the non-orientable $4$-manifold $\RP^2\times S^2$. It is an elementary abelian $2$-group of rank $5$. The subgroup of classes whose based lift to the universal cover $S^2\times S^2$ has degree $+1$ has rank $4$ and is isomorphic to the group of free homotopy classes. The group maps onto the group of self-equivalences of the algebraic $2$-type, which has order $8$, with kernel $\pi_4(\RP^2\times S^2)\cong(\Z/2)^2$ generated by pinch-fold maps; that these maps are independent is shown by an adjunction argument. We then construct a non-orientable analogue of the Hambleton–Kreck braid of exact sequences in dimension $4$, for non-orientable manifolds with spin universal cover, stated for finite fundamental group. We apply the braid to $\RP^2\times S^2$ and to $\RP^4$, and for the Hambleton–Hillman manifold with fundamental group $(\Z/2)^2$ we determine the normal $1$-type and show that self-equivalences act on $\pi_2$ through diagonal sign matrices. For $\RP^2\times S^2$ the pinch-fold maps are detected by twisted spin bordism, and, assuming facts from surgery theory stated as hypotheses, exactly one of the three non-trivial pinch-fold maps is homotopic to a homeomorphism.

A set $S$ of positive integers is $2$-large if every two-coloring contains arbitrarily long monochromatic arithmetic progressions with common difference in $S$. We prove that every such set meets every nil-Bohr neighborhood of $0$. This proves a conjecture of the first author and gives a sharp answer to a question of Host, Kra, and Maass. We also show that, for every fixed nil-Bohr neighborhood \(B\) of \(0\) and every \(\eps>0\), restricting the common difference to \(B\) increases the extremal number for \(k\)-term arithmetic progressions by at most a factor of \(1+\eps\), provided \(k\) is sufficiently large.

Let $\mathbf{E}$ be the ring of values taken at algebraic points by $E$-functions. The exponential function and the Bessel function $J_0$ are classical examples of $E$-functions. We prove that $\mathbf{E}$ is not a field. More precisely, we prove that it is neither noetherian nor factorial, and construct non-units of $\mathbf{E}$ explicity, including $e+1$, $J_0(1)$, and ${}_1F_1(a;b;1)$ for rational parameters $a$ and $b$ such that $0