We construct non-flat minimal capillary cones with bi-orthogonal symmetry groups for any dimension and contact angle. These cones interpolate between rescalings of a singular solution to the one-phase problem and the free-boundary cone obtained by halving a Lawson cone along a hyperplane of symmetry. The existence and uniqueness of such cones is proved by solving a nonlinear free boundary equation parametrized by the contact angle and obtaining monotonicity properties for the solutions. The constructed cones are minimizing in ambient dimension $8$ or higher, for appropriate contact angles, demonstrating that the regularity theory for minimizing capillary hypersurfaces can have singularities in codimension $7$ and completing the capillary regularity theory for contact angles near $π/2$. We further develop the connection between capillary hypersurfaces and solutions of the one-phase problem, consequently producing new examples of singular minimizing free boundaries for the Alt-Caffarelli functional.

A balanced colouring of a graph is one in which every colour appears the same number of times. Given a fixed graph $H$ on $r$ vertices and a balanced $k$-colouring of the complete graph $K_{nrk}$, Hollom (2025) asked the following question: can we always find an $H$-factor $F$ covering all vertices of the complete graph $K_{nrk}$ such that the inherited colouring of $F$ is almost balanced? This is known to be the case for palettes of only two colours, or when $H$ is only a single edge. We answer the above question in full, finding an $H$-factor which is at most $C_{r,k}$ edges away from being balanced, where $C_{r,k}$ depends only on $r$ and $k$. In fact, we work in the more general setting wherein our palette of colours is a subset of $\mathbb{S}^{d-1}$, and find an $H$-factor where the sum of the colours of all edges has bounded Euclidean norm.

We study the quantization of real-valued bandlimited signals on graphs, focusing on low-bit representations. We propose iterative noise-shaping algorithms for quantization, including sampling approaches with and without vertex replacement. The methods leverage the spectral properties of the graph Laplacian and exploit graph incoherence to achieve high-fidelity approximations. Theoretical guarantees are provided for the random sampling method, and extensive numerical experiments on synthetic and real-world graphs illustrate the efficiency and robustness of the proposed schemes.

General ridge estimators are widely used in the general linear model because they possess desirable properties such as linear sufficiency and linear admissibility. However, when the covariance matrix of the error term is partially unknown, estimation typically requires a two-step procedure. This paper derives conditions under which the general ridge estimator based on the covariance matrix coincides with the one that does not depend on it. In particular, we provide practically verifiable conditions for several linear models, including Rao's mixed-effects model, a seemingly unrelated regression model, first-order spatial autoregressive and spatial moving average models, and serial correlation models. These results enable the use of a covariance-free general ridge estimator, thereby simplifying the two-step estimation procedure.

Most of the contemporary literature on information freshness solely focuses on the analysis of freshness for martingale estimators, which simply use the most recently received update as the current estimate. While martingale estimators are easier to analyze, they are far from optimal, especially in pull-based update systems, where maximum aposteriori probability (MAP) estimators are known to be optimal, but are analytically challenging. In this work, we introduce a new class of estimators called $p$-MAP estimators, which enable us to model the MAP estimator as a piecewise constant function with finitely many stages, bringing us closer to a full characterization of the MAP estimators when modeling information freshness.

In this work, we design and analyze semi/fully-discrete virtual element approximations for the time-dependent Navier--Stokes-Cahn--Hilliard equations, modeling the dynamics of two-phase incompressible fluid flows with diffuse interfaces. A new variational formulation is derived involving solely the velocity, pressure, and phase field, together with corresponding a priori energy estimates. The spatial discretization is based on the coupling divergence-free and $C^1$-conforming elements of high-order, while the time discretization employs a classical backward Euler scheme. By introducing a novel skew-symmetric trilinear form to discretize the convective term in the Cahn--Hilliard equation, we propose discrete schemes that satisfy mass conservation and energy bounds. Moreover, optimal error estimates are provided for both formulations. Finally, two numerical experiments are presented to support our theoretical findings and to illustrate the good performance of the proposed schemes for different polynomial degrees and polygonal meshes.

We show that the knots $K\in\{4_1,5_1\}$ can be paired with a corresponding knot $K^\prime$ such that $u(K\#K^\prime)<u(K)+u(K^\prime)$. As a consequence unknotting number fails to be additive for these knots. We also provide a candidate knot $K^\prime$ for the knot $3_1$.

While sporadic examples of virtual resolutions with homology have been constructed, their occurrence is not well understood or controlled. Our results build a new set of tools for studying virtual resolutions of monomial ideals as arising from simplicial complexes, including characterizing them by the acyclicity of certain induced subcomplexes. Using this characterization, we give a description of minimal simplicial complexes supporting virtual resolutions as well as a technique for removing homology from simplicial virtual resolutions.

We introduce the notion of a ``sofic $\mathcal{C}$-action'' of one group on another by automorphisms, for $\mathcal{C}$ a class of groups. We show that if $\mathcal{C}$ is the class of (i) sofic, (ii) hyperlinear, (iii) linear sofic or (iv) weakly sofic groups, then the class $\mathcal{C}$ is closed under taking semidirect products with sofic $\mathcal{C}$-action. We use this to construct a wide variety of new examples of groups in the classes (i)-(iv), many of them arising as ``halo products'' in the sense of Genevois-Tessera. We have a parallel set of results producing new examples of semidirect products which are locally embeddable into finite groups. Our framework also unifies existing results in the literature, due to Hayes-Sale; Brude-Sasyk and Gao-Kunnawalkam Elayavalli-Patchell.

We show that Kähler-Einstein metrics with cone singularities along simple normal crossing (SNC) divisors define RCD spaces, both in the compact setting and in certain non-compact cases, thereby producing many examples of Einstein RCD spaces. In particular, we show the existence of smooth non-compact $4$-manifolds carrying ALE Ricci-flat RCD$(0,4)$ metrics with any space form $S^3/Γ$ as the link of the tangent cone at infinity, answering a question raised by D. Semola. Our proofs rely on the characterization of RCD spaces in the almost-smooth setting due to S. Honda and Honda-Sun.

This paper presents a novel indoor visible light positioning (VLP) system utilising one vertical-cavity surface-emitting laser installed at the ceiling centre of a space. The system offers three-dimensional localisation by sweeping through space at one-degree resolution in two dimensions (azimuth and elevation), significantly simplifying hardware. Through incorporating the angle of arrival and received signal strength, this system demonstrates excellent precision in indoor positioning. Simulation results verify that the system attains sub-centimetre precision for most test points, outperforming conventional multi-transmitter VLP schemes in cost-efficiency and simplicity.

We show that for all integers $2\le s\le t$, any $K_{s,t}$-free subset of $[N]$ with size $Ω(n^{1-1/s})$ must contain a nontrivial solution to every fixed translation-invariant linear equation in at least five variables. This extends earlier results for Sidon sets due to Conlon-Fox-Sudakov-Zhao and Prendiville to the full family of $K_{s,t}$-free sets. We also study the corresponding problem in vector spaces over finite fields. In $\mathbb F_q^n$ we obtain stronger quantitative bounds, including polylogarithmic savings, by combining Fourier-analytic transference with polynomial-method input from the arithmetic cycle-removal lemma of Fox-Lovász-Sauermann.

We consider isometrically flexible Wasserstein spaces and demonstrate that adding a single point to the underlying metric space makes these Wasserstein spaces rigid.

We introduce several new models whose common feature is to take into account effects from topological vorticity. The macroscopic unknown is driven by a dissipative anomalous diffusion (of SQG-type) and is coupled with the orientation of the crystal, moving by the gradient flow of the energy of maps. The main idea of such models is to have a better insight on the vorticity formulation of the Liquid Crystal Flow and to tackle some regularity issues in the associated conserved geometric motions. One of the advantage of the present PDEs is to capture features of the Navier-Stokes equations (or Euler) through a {\sl scalar} unknown, keeping the advection-diffusion structure of the orientation field. We obtain regularity for mild solutions under natural assumptions for the initial data, which are actually near-optimal. Along the way, we also draw some links with natural models of (anti-)ferromagnets previously investigated.

A new alternative numerical procedure to the Szegő quadrature formulas for the estimation of integrals with respect to a positive Borel measure $μ$ supported on the unit circle is presented. As in many practical situations, we assume that the values of the integrand $F$ are only known at a finite number of points, which we will assume to be uniformly distributed on the unit circle (although this does not actually constitute a restriction). Our technique consists of obtaining an approximating Laurent polynomial $L$ to $F$ by interpolation in the Hermite sense in a collection of these points that mimic the zeros of a para-orthogonal polynomial with respect to $μ$, and to use the values of $F$ at the remaining nodes to improve the accuracy of the approximation by a process of simultaneous complex regression. Some numerical examples are carried out.

We develop a theory of weights for a quantum analogue of the symmetric pair (gl4,gl2 x gl2) realised as a quantum symmetric pair subalgebra. Based on Letzter's triangular decomposition we define Verma modules. Using magical operators that are compatible with weight spaces, we classify weight Verma modules and characterise their irreducible finite dimensional quotients. We then prove the existence of weight bases in tensor products by explicitly constructing some highest weight vectors. These constructions allow us to mimic the important aspects of the classical finite dimensional representation theory. Applications include a definition of rational representations, the BGG resolution, a Clebsch--Gordan formula, the Harish-Chandra isomorphism and central characters, as well as a classification and description of all irreducible polynomial representations.

Dynamic Low Rank (DLR) methods are a promising way to reduce the computational cost and memory footprint of the high-dimensional thermal radiative transfer (TRT) equations. The TRT equations are a system of nonlinear PDEs that model the energy exhchange between the material temperature and the radiation energy density; due to their high dimensionality, solving the TRT equations is often bottleneck in multi-physics simulations. DLR methods represent the solution in terms of time-evolving SVD-like factors of angle and space. Although previous work has explored DLR methods for TRT, most of the methods have limitations that make them impractical for realistic scenarios and uncompetitive with current non-DLR production codes. Here we develop new PN-like and SN-like Dynamic Low Rank (DLR) methods for TRT. In the SN-like DLR method, we use the time-evolving angular basis functions to select time-evolving angles; this DLR formulation enables us to use the highly optimized SN transport sweep as our main computational kernel, and results in a practical way of leveraging low-rank methods in production TRT codes. In contrast, our PN-like DLR method uses an even-parity formulation and results in positive-definite linear systems to solve for each time step. We demonstrate the methods on several challenging, highly heterogenous problems in two spatial dimensions $(4$D) that these DLR schemes can give significant reduction in angular artifacts (``ray effects'') with the same cost as gold-standard SN methods.

Partial differential equation (PDE) models for infectious diseases, while less common than their ordinary differential equation (ODE) counterparts, have found successful applications for many years. Such models are typically of reaction-diffusion type, and model spatial propagation as a diffusive process. However, given the complex nature of human mobility, such models are limited in their ability to describe airborne infectious diseases in human populations. Recent work has advocated for the inclusion of an additional chemotaxis-type term as an alternative; spatial propagation of infection fronts is assumed additionally to flow from low-to-high concentrations of susceptible populations. The present work extends the study of such models by providing an epidemiologically interpretable analysis, directly connecting model behavior to information readily available to policymakers. In particular, we derive a spatially-aware basic reproduction number, which accounts for spatial heterogeneity in population density. Furthermore, we discuss several important aspects concerning the numerical solution of the model, including the introduction of a stabilization scheme. Finally, we perform a series of simulation studies in the Italian region of Lombardy (severely affected by the COVID-19 outbreak in 2020) and in the US state of Georgia, in which we demonstrate the model's potential to better capture important spatiotemporal dynamics observed in real-world data compared to pure reaction-diffusion models.

We use tori and Hilbert schemes of K3 surfaces to construct explicit bases for the real, complex, and quaternionic versions of rational symplectic and rational Spin bordism. The key input to our work is a theorem of Oberdieck, Song, and Voisin on the Milnor genus of Hilbert schemes of K3s.

We provide a Poisson approximation result for dependent thinnings of Gibbs point processes as well as qualitative and quantitative central limit theorems for geometric functionals of Gibbs point processes in increasing observation windows. The present paper extends prior work on finite-range Gibbs processes to processes with repulsive pairwise interaction of unbounded interaction range as well as processes on marked Euclidean space. The proofs rely on coupling different Gibbs processes using the disagreement coupling technique, which we generalize to infinite-volume domains under a suitable non-percolation condition. For the case of repulsive pairwise interactions, we introduce a version of disagreement coupling that constructs the Gibbs process by thinning a random connection model thus making previous approximation methods more flexible.

In this paper, we give error bounds for the distance distribution of Reed-Muller codes, extending prior work on the distance distribution of Reed-Solomon codes. This is equivalent to the problem of counting multivariate polynomials over a finite field with prescribed degree, coefficients, and number of zeroes. We provide a solution to this problem using the character sum method, which offers a new unified framework applicable to a broad class of polynomial enumeration problems over finite fields that involve prescribed evaluation vectors. This work effectively makes the first systematic attempt to study the coset weight distribution problem for Reed-Muller codes of fixed degree over large finite fields, which was proposed in MacWilliams and Sloane's 1977 textbook \emph{The Theory of Error Correcting Codes}.

We study function estimation in the empirical Bayes setting for Poisson and normal means. Specifically, given observations $Y_i\sim f(\cdot; θ_i)$ with latent parameters $θ_i\sim π$, the goal is to estimate $\mathbb{E}_π[\ell(θ)|X = x]$. This task lies between classical deconvolution (recovering the full prior $π$), and standard empirical Bayes mean estimation. While the minimax risk for estimating $π$ in the Wasserstein distance is known to decay only logarithmically, we show that estimating certain smooth functions admits dramatically faster rates. In particular, for polynomial functions of degree $k$ in the Poisson model, we establish a tight bound of $Θ(\frac{1}{n}(\frac{\log n}{\log \log n})^{k+1})$ and $Θ(\frac{1}{n}(\log n)^{2k+1})$ for bounded and subexponential priors, respectively, attainable by estimators mimicking those that achieve optimal regret for the mean estimation problem (Robbins, mininum distance, ERM). Our analysis identifies the approximation-theoretic origin of this improvement: smooth functions can be well-approximated by low-degree polynomials, whereas Lipschitz functions require dense polynomial approximations, incurring a $\frac{1}{k}$ loss for degree $k$ polynomial approximation. The results reveal a sharp hierarchy in the difficulty of empirical Bayes problems: ranging from slow, logarithmic deconvolution to near-parametric convergence for smooth posterior functionals, and establish new connections between nonparametric empirical Bayes theory, polynomial approximation, and statistical inverse problems. Finally, we complement our analysis with a lower bound of $Ω(\frac 1n (\frac{\log n}{\log \log n})^{k+1})$ (bounded priors) and $Ω(\frac 1n (\log n)^{k + 1})$ (subgaussian priors) for the normal means model.

The Prisoner's Dilemma is used as a model in processes involving reciprocity; however, its classical setup can be insufficient in settings where the symmetry of the simultaneous decision making is broken -- for example, in donor and recipient processes. In the alternating Prisoner's Dilemma model the two players take turns choosing their strategy. Assuming a finite memory setup, we establish the mathematical aspects of the adaptive dynamics of the alternating Prisoner's Dilemma, paying particular attention to the case of memory 1.

In this letter we study renormalization group (RG) flows between 2d conformal field theories enjoying extended higher-spin $\mathcal{W}$-symmetry. We propose a new class of RG flows between the diagonal minimal models of $\mathcal{W}_N$-algebra that take the form $\mathcal{W}_N(p,q)\to\mathcal{W}_N(p,kp-q)$. These are obtained by matching the anomalies of the non-invertible symmetry ${\mathrm{Rep}}[SU(N)_{p-N}]$ (and its discrete quotients) that is preserved by special relevant primary fields. This large non-invertible symmetry includes the familiar $\mathbb{Z}_N$ symmetry of the minimal models. Our new flows furnish a significant generalization of the ones recently found in the case of Virasoro algebra, and include all previously known RG flows of $\mathcal{W}_N$. They have the remarkable property of being uniform in the rank $N$ of the $\mathcal{W}$-algebra.

We study uniform computability properties of PAC learning using Weihrauch complexity. We focus on closed concept classes, which are either represented by positive, by negative or by full information. Among other results, we prove that proper PAC learning from positive information is equivalent to the limit operation on Baire space, whereas improper PAC learning from positive information is closely related to Weak Kőnig's Lemma and even equivalent to it, when we have some negative information about the admissible hypotheses. If arbitrary hypotheses are allowed, then improper PAC learning from positive information is still in a finitary DNC range, which implies that it is non-deterministically computable, but does not allow for probabilistic algorithms. These results can also be seen as a classification of the degree of constructivity of the Fundamental Theorem of Statistical Learning. All the aforementioned results hold if an upper bound of the VC dimension is provided as an additional input information. We also study the question of how these results are affected if the VC dimension is not given, but only promised to be finite or if concept classes are represented by negative or full information. Finally, we also classify the complexity of the VC dimension operation itself, which is a problem that is of independent interest. For positive or full information it turns out to be equivalent to the binary sorting problem, for negative information it is equivalent to the jump of sorting. This classification allows also conclusions regarding the Borel complexity of PAC learnability.

We study a nonlinear decomposition of a positive definite matrix into two components: the inverse of another positive definite matrix and a symmetric matrix constrained to lie in a prescribed linear subspace. Equivalently, the inverse component is required to belong to the orthogonal complement of that subspace with respect to the trace inner product. Under a sharp nondegeneracy condition on the subspace, we show that every positive definite matrix admits a \emph{unique} decomposition of this form. This decomposition admits a variational characterization as the unique minimizer of a strictly convex log-determinant optimization problem, which in turn yields a natural dual formulation that can be efficiently exploited computationally. We derive several properties, including the stability of the decomposition. We further develop feasibility-preserving Newton-type algorithms with provable convergence guarantees and analyze their per-iteration complexity in terms of algebraic properties of the decomposed matrix and the underlying subspace. Finally, we show that the proposed decomposition arises naturally in exponential utility maximization, a central problem in mathematical finance.

In distributed computing with untrusted workers, the assignment of evaluation indices plays a critical role in determining both privacy and robustness. In this work, we study how the placement of unreliable workers within the Numerically Stable Lagrange Coded Computing (NS-LCC) framework influences privacy and the ability to localize Byzantine errors. We derive analytical bounds that quantify how different evaluation-index assignments affect privacy against colluding curious workers and robustness against Byzantine corruption under finite-precision arithmetic. Using these bounds, we formulate optimization problems that identify privacy-optimal and robustness-optimal index placements and show that the resulting assignments are fundamentally different. This exposes that index choices that maximizes privacy degrade error-localization, and vice versa. To jointly navigate this trade-off, we propose a low-complexity greedy assignment strategy that closely approximates the optimal balance between privacy and robustness.

In this paper, we study various classes of partition functions such as those related to the parity of the number of parts, to differences of partition numbers, and to partitions with a repeated smallest part. We establish identities connecting these various classes of partitions. Moreover, our identities help us to extend the Euler's partition theorem. An analogue of Legendre's theorem of the partition-theoretic interpretation of Euler's pentagonal number theorem is also derived. Both combinatorial and $q$-series proofs are given for our results.

We propose a quantum rotation diversity (QRD) scheme for optical quantum communication using binary phase-shift-keying displaced squeezed states and homodyne detection over Gamma-Gamma turbulence channels. Consecutive temporal modes are coupled by a passive orthogonal rotation that redistributes the displacement amplitude between slots, yielding a diversity order of two under independent fading and joint maximum-likelihood detection. Analytical expressions for the symbol-error rate performance, along with asymptotic results for the diversity and coding gains, are derived. The optimal rotation angle and energy allocation between displacement and squeezing are obtained in closed form. Furthermore, we show that when both the displacement amplitude and the squeezing strength scale with the total photon number, an effective diversity order of four is achieved. Numerical results validate the analysis and demonstrate the super-diversity behaviour of the proposed QRD scheme.

Motivated by a conjecture that doubled four-dimensional Chern-Simons produces new integrable models, we perform its Hamiltonian analysis and find the theory's Poisson algebra. This requires carefully accounting for a set of boundary conditions that identify two gauge fields. Two methods for doing so are given, one of which is based on edge-modes and the other on a recharacterisation of the boundary conditions as constraints. We find that the Poisson algebra is that of an affine Gaudin model subject to a constraint, generalising the Goddard-Kent-Olive construction (from conformal field theory) to the world of integrable models. We also conjecture the existence of extended quantum groups and a generalisation of the affine Harish-Chandra Isomorphism.