Papers

We study post-learning inference for structural properties of data-dependent combinatorial optimizers. The target is whether an oracle optimizer, rather than a latent parameter or smooth functional, belongs to a prescribed class, such as a category-mix, inventory, or resource-feasibility class. We focus on a high-dimensional contextual multinomial logit model with sequentially adaptive data collection, where the parameter-to-optimizer map is discontinuous and the policy induces temporal dependence. We propose a novel perturbation test based on a nonsmooth max-difference revenue statistic comparing the best null assortment with the best alternative assortment. The test perturbs the estimated terminal revenue surface on the selected support: random unit directions capture directional uncertainty, while the minimal perturbation radius captures magnitude uncertainty and yields a p-value. This localizes inference near the null–alternative boundary and avoids uniform error control over the full candidate class. The data are collected by an \(\ell_1\)-penalized online likelihood policy that performs variable selection while controlling regret. Using a new anti-concentration argument for Gaussian maxima differences and martingale Gaussian coupling, we establish uniform estimation rates, effective support recovery, and asymptotic validity of the proposed p-value under adaptive assortment selection. We prove asymptotic size control and power consistency under a localized signal condition.

Generically the rigidity of bar-joint structures admits combinatorial characterisations in the Euclidean plane and, more generally, for frameworks on the sphere and the torus. The remaining case of compact surfaces of genus at least two has remained open. Using the hyperbolic geometry of their universal covers, we develop a theory of infinitesimal rigidity for frameworks on compact surfaces of genus at least two. By the uniformisation theorem, every such surface is a quotient of the hyperbolic plane by a surface group, allowing frameworks on the surface to be represented as infinite symmetric frameworks in the hyperbolic plane. Encoding the symmetry through gain graphs, we prove that infinitesimal rigidity is determined entirely by finite combinatorial data. Specifically, a framework is generically rigid if and only if its associated gain graph contains a spanning (2,3,1,0)-gain tight subgraph. This yields the first combinatorial characterisation of generic rigidity for frameworks on compact surfaces of genus at least two.

Let $S(\infty)$ denote the group of finitary permutations of the set $\mathbb N:=\{1,2,3,\dots\}$. It is a countable group admitting a lot of different topologies compatible with the group structure. In particular, such topologies arise from partitions of the set $\mathbb N$ into blocks of infinite size. The corresponding categories of continuous unitary representations of $S(\infty)$ were studied by Nessonov (Sbornik: Mathematics, 2012). We propose a different approach to his classification results based on the so-called semigroup method. Some additional information is also obtained.

Macroscopic observables allow the recovery of intrinsic dynamics from stationary quantum states. I show that, by interpreting the squared amplitude as the probability density for each definite value of intrinsic time, a curvature emerges in the time direction. For example, from the perspective of intrinsic quantum time, the Friedmann-Lemaître-Robertson-Walker cosmological model emerges from spherically symmetric stationary solutions in four-dimensional Euclidean space, without presupposing gravity. If there is no unique direction of time, curvature emerges in all spacetime dimensions, without presupposing gravity, from the variable amplitude of the stationary wavefunction alone. This opens a new possibility that general relativity or some modification of it emerges from intrinsic time observables.

This paper addresses the problem of designing recommendation systems for social networks and e-commerce platforms from a control-theoretic perspective. We formulate recommendation design as an infinite-horizon state-feedback optimal control problem whose performance index rewards alignment/engagement while penalizing polarization, large deviations from an uncontrolled baseline, recommendation mismatch, control effort, and exposure disagreement across neighboring users. We derive explicit spectral conditions under which the reduced quadratic stage cost is strictly positive-definite, and we show that the failure of these conditions makes the resulting recommendation design exhibit pathological behaviors, such as unstable free modes, non-attainment of the infimum, or failure of the stationary affine synthesis.

We study the stability problem for Einstein manifolds with boundary with respect to the Einstein-Hilbert action. The geometric boundary conditions we are using arise naturally from studying the calculus of variations associated with Ricci flow. Upon the introduction of a boundary, the space of TTg tensors no longer arise naturally as the defining space for the stability condition. Thus we must settle for the larger subspace of tensors preserving the scalar curvature, the total volume and the Bianchi gauge condition, which we call the space of TVg tensors. As a test-case, we shall discuss stability of the Riemannian Schwarzschild anti-deSitter family of metrics; a problem known as "a Black hole in a box".

In a previous work, we defined the complex of cluster parking functions. On one side, they encode the type-refined enumeration of faces of the cluster complex, and on the other side, they have a reduced homology which is isomorphic to (ungraded) diagonal coinvariants. The goal of this work is to take into account the underlying dihedral symmetry. We thus have a product of a dihedral group and a symmetric group (there is a precise conjecture in the case of other finite Coxeter groups, but we focus on symmetric groups because of technicalities about diagonal coinvariants beyond this case). Under the action of the product group, the reduced homology of cluster parking functions is conjecturally isomorphic to diagonal coinvariants up to tensoring by a sign character of the dihedral group. This isomorphism can be reformulated as a dihedral sieving phenomenon. The main technical contribution is the definition of the dihedral automorphism group of cluster parking functions, and we discuss various features of the reduced homology character and its conjectural connection with diagonal coinvariants.

Timely communication of information in Internet of Things (IoT) networks is critical to enhancing system performance and energy efficiency by minimizing the transmission of outdated or redundant data. Although timeliness metrics such as the Age of Information (AoI) effectively quantify information freshness, they do not account for content evolution. The Version Age of Information (VAoI) addresses this gap by tracking version lag at the receiver, thereby providing a practical content-aware metric. However, prior research has primarily focused on first-moment analyses in single-hop settings, leaving the distributional properties of VAoI in multi-hop networks, as well as the impact of feedback mechanisms, unexplored. In this study, we provide a comprehensive characterization of VAoI in multi-hop networks with transmission constraints and acknowledgment-based feedback. A bi-level optimization framework is formulated to jointly optimize the update policy of a rate-constrained source and the feedback-aware forwarding policies of the intermediate nodes, aiming to minimize communication overhead while maintaining VAoI performance at the destination. We show that the optimal source policy follows a threshold-based update strategy and derive the optimal threshold in closed form. For both the optimal threshold policy and a randomized baseline, we obtain closed-form expressions for the stationary distribution and average VAoI, along with the corresponding update rates across network nodes under feedback-aware forwarding. Numerical results corroborate the analytical findings and illustrate the advantages of utilizing VAoI and feedback to reduce redundant transmissions while preserving data freshness and informativeness in multi-hop systems.

Mass lumping enables explicit time stepping for finite-element discretisation of the wave equation if the resulting quadrature weights are positive and accuracy is preserved. New rules for elements of degree two and three on the 4-simplex are presented. Numerical quadrature for the stiffness matrix can be more efficient than exact evaluation if it requires fewer nodes. For the latter, new rules in two and four space dimensions for the lower-degree elements on the simplex were found, as well as some additional results for three dimensions.

Spectral and transport properties of high-contrast resonator systems can be described in the subwavelength regime in terms of the so-called capacitance operator. In this paper, we consider an infinitely periodic chain of high-contrast resonators in three dimensions. The first result is a precise estimate of the off-diagonal decay rate of the capacitance operator $C$. Importantly, we demonstrate that the decay rate is long-range and critical: as $|n-m|\to\infty$, \begin{equation*} C(n,m)\sim \frac{1}{|n-m|\log^2|n-m|}, \end{equation*} which is $\ell^1$ summable but slower than quadratic. This borderline decay of the off-diagonal entries makes the present proof of Anderson localisation with arbitrary disorder, which is observed numerically in this paper, out of reach; we hope that this physical example of classical wave systems with critical long-range interactions provides new insight in the field of Anderson localisation. As the second main result, based on the off-diagonal decay estimate, we prove a strong convergence of the finite capacitance operator, which corresponds to a truncated chain, to the capacitance operator as the size of the truncated chain grows to infinity. Using this strong convergence, we improve the results of [Ammari et al., SIAM J. Math. Anal., 2023 and Bull. London Math Soc., 2025] by presenting a rigorous estimate of the convergence rate of the spectrum.

We present a method for estimating the hydrodynamic coefficients in the Cummins equations using time-series data from a moving body, such as a floating offshore structure. The proposed data-driven method is based on incorporating the Cummins equations governing the dynamics of a structural body interacting with water waves into a physics-informed neural network (PINN), along with available motion data. The proposed method first estimates the structure's state in terms of translational and rotational degrees of freedom, and then solves the inverse problem to determine the hydrodynamic forces acting on the body, expressed in terms of added mass, damping coefficients, and/or hydrostatic restoring. The Cummins equations are formulated as a first-order system, and both state and parameter estimation are performed using PINNs. The method is verified on the free decay of a sphere and a box. The results demonstrate that it is possible to estimate the state and hydrodynamic coefficients accurately, although accuracy depends on the volume and quality of the movement data.

For $q\ge 1$, the $q$-th squarefree power $I(G)^{[q]}$ of the edge ideal of a graph $G$ is generated by the squarefree monomials supported on $q$-matchings of $G$; it is the Stanley–Reisner ideal of the complex $\Delta_q(G)=\{F\subseteq V(G):\nu(G[F])<q\}$, where $\nu$ denotes matching number. We prove a general formula for the matching number of an arbitrary graph join, \[ \nu(G\ast H) = \min\Big(\nu(G)+|V(H)|,\ \ \nu(H)+|V(G)|,\ \ \Big\lfloor\tfrac{|V(G)|+|V(H)|}{2}\Big\rfloor\Big), \] via the Tutte–Berge formula, and use it to decompose $\Delta_q(G\ast H)$ for arbitrary graphs $G,H$. Specializing to the wheel graph $\mathcal{W}_n = \mathcal{C}_n\ast\mathcal K_1$, we determine the Krull dimension and height of $R/I(\mathcal{W}_n)^{[q]}$ exactly for all $n\ge 3$, $1\le q\le\lfloor n/2\rfloor$, and – combining our matching-number computations with a recent Tutte-type Cohen-Macaulayness criterion of Ficarra and Moradi – prove that at the top squarefree power $q=\nu(\mathcal{W}_n)=\lceil n/2\rceil$, the ideal $I(\mathcal{W}_n)^{[\nu(\mathcal{W}_n)]}$ is literally the squarefree Veronese ideal, so that $R/I(\mathcal{W}_n)^{[\nu(\mathcal{W}_n)]}$ is Cohen-Macaulay with \[ {\rm dim} = {\rm depth} = {\rm reg}\big(R/I(\mathcal{W}_n)^{[\nu(\mathcal{W}_n)]}\big) = 2\Big\lceil\frac n2\Big\rceil-1. \] This resolves all four classical invariants at the top power, and confirms there the pattern depth$(R/I(\mathcal{W}_n)^{[q]}) = 2q-1$ that our computational data (now extended to $n\le13$, every valid $q$) suggests holds throughout. We prove a general depth formula for squarefree powers of cone graphs, via a Betti-splitting exact sequence, that reduces this pattern to two more tractable statements about the underlying cycle alone; both are verified computationally in every case checked but left open in general.

We study the approximation of the Weierstrass function by means of superoscillating sequences. Superoscillatory functions are band-limited functions whose local oscillation rate can exceed the highest frequency contained in their Fourier spectrum. Starting from Lagrange-type interpolation at nodes in $[-1,1]$, we construct a double-indexed family $\mathcal{W}_{N,n}(x)$ that approximates the truncated Weierstrass function $W_N(x)$ for each fixed truncation order~$N$. We prove that if the number of interpolation nodes $n_N$ grows sufficiently fast relative to the highest frequency $b^N\pi$, namely $b^N\pi/n_N\to 0$, then $\mathcal{W}_{N,n_N}$ converges uniformly to the full Weierstrass function on every compact set. We also show that the two limits in $N$ and $n$ do not commute: for any fixed~$n$ the series $\lim_{N\to\infty}\mathcal{W}_{N,n}(x)$ diverges for every $x\neq 0$, a phenomenon called the Divergence Wall.

We generalize the definition of truncated multiple zeta values by allowing arbitrary integers as arguments. This leads to interesting identities, particularly with the argument 0. Truncated multiple zeta values satisfy the same quasi-shuffle algebraic identities as multiple zeta values, but we need to extend the algebra QSym of quasi-symmetric functions to a larger algebra. Using this algebra, we are able to sum systematically powers of harmonic and generalized harmonic numbers. This leads to summation identities such as \[ \sum_{n=1}^\infty H_n^3\bigg(\zeta(2)-\sum_{k=1}^n\frac{1}{k^2}-\frac{1}{n}\bigg)= -\frac{11}2\zeta(4)+\zeta(3)+3\zeta(2)-6. \] We also prove analogous identities involving alternating sums of harmonic numbers and their powers.

We study $k$-equivariant energy critical wave maps $\mathbb{R}^{1+2} \to \mathbb{S}^2$, for any equivariance degree $k\ge 2$. We prove that the radiation associated with any finite-energy blow-up solution cannot satisfy a certain regularity condition; in particular, it cannot be smooth. The assumption $k \geq 2$ is necessary, since for $k = 1$ such solutions are known to exist. The starting point of our analysis is the soliton resolution theorem. The key ingredient is a novel application of the modulation method, in which we compare the effects of the radiation and inner bubbles to study the dynamic behavior of the widest bubble.

The contraction coefficient of the trace distance is a central tool in quantum information, quantifying how strongly a quantum channel degrades the distinguishability of states. However, being an extremal ratio, it captures only the most optimistic behaviour of the channel and is often trivial, even for very noisy channels. Moreover, a single scalar is poorly suited to describe how contraction accumulates under channel composition. In this work we introduce the contraction and expansion values, two monotone sequences that refine the contraction and expansion coefficients in the same way singular values refine the operator norm. They arise from a min–max variational principle over subspaces of traceless Hermitian operators, admit an operational interpretation in terms of two state-discrimination games, and are shown to coincide with the Gel'fand or Bernstein numbers of the channel restricted to traceless operators. This identification places the sequences within Pietsch's theory of $s$-numbers and yields, in particular, bounds under channel composition that the contraction coefficient alone cannot provide. We establish their main structural properties and compute or estimate them for single-qubit channels, $d$-dimensional amplitude damping channels, and direct-sum channels.

We study integrability properties of the Wolff potential in context of Choquet integrals with respect to the Hausdorff content. As an application we give integrability results to the solutions of the $p$-Laplace equation in this context.

We investigate approximation of random differential equations driven by semimartingales satisfying a singularly perturbed Langevin equation with scaled mixing random force. By a diffusion approximation approach, we explore the limit of the rough path lift of this semimartingale, and a universal limit theorem is applied to identify the limit of random differential equation. A structurally parallel proof also applies to establish an iterated weak invariance principle for the mixing random force, which is itself an independent interesting result. We find that, the limit of both of the second-level processes, have the form of iterated integral of Stratonovich form plus an anti-symmetric part which is proportional to the time increment.

This is a survey of the recent works that describe theta correspondence as a C*-correspondence. The main mechanism is elucidated in the special case when one of the groups involved in the theta correspondence is compact.

Let $p$ be a fixed prime number and let $R$ be a $p$-quasisyntomic ring. We provide conditions for objects in the absolute prismatic site $R_\Prism$ to cover the final object in $\Shv(R_\Prism)$ and to admit finite self-coproducts in $R_\Prism$. More precisely, we introduce in $R_\Prism$ the so-called transversal objects, so that the coproduct of any object in $R_\Prism$ with a transversal object exists in $R_\Prism$. As an immediate generalization, we introduce the so-called relatively quasiregular semiperfectoid covers of $R$, whose prismatic cohomology (of $\delta$-pairs, in the sense of Antieau-Krause-Nikolaus) produces objects in $R_\Prism$ that cover the final object in $\Shv(R_\Prism)$ and admit finite self-coproducts in $R_\Prism$.

In this paper, we discuss the analyticity, $\Delta_{C_\phi}$- regularity, kernel condition and subnormality of Cauchy dual of $k$-quasi-$m$-isometric composition operators on directed graphs with one circuit and multiple branching vertices.

In this paper, we consider the Cauchy problem for the $3$D incompressible Navier–Stokes equations and prove the existence of unique global solutions in the framework that the horizontal component of the velocity field is small in some critical Besov spaces including the classical Fujita–Kato class, while the vertical component is large in the wide class $\dot{B}_{\infty,\sigma}^{-1}(\mathbb{R}^3)$ ($1 \leq \sigma < \infty$) where the Navier–Stokes equations are known to be ill-posed in.

For any $n$-dimensional compact minimal hypersurface of the $(n+1)$-dimensional sphere, we have that the coordinate functions of the Gauss map are eigenfunctions of the stability operator associated with the eigenvalue $-n$. In this paper we consider minimal immersions from $\mbS^{k}\times\mbS^{\ell}\times\mbS^{1}$ to $\mbS^{k+\ell+2}$ of the form $\phi(y,z,t)=\left(f(t) y, f_2(t) z, f_1(t)\right)$ and we explicitly show new eigenfunctions for the stability operator associated with the same eigenvalue. We also show that the stability index of these minimal immersions is at least $k\ell+3k+3\ell+8$.

The paper discusses the classification of three-dimensional gl-regular Nijenhuis operators and their normal forms. We also prove A.~Bolsinov's conjecture.

Let $\pi$ be a generic representation of ${\mathrm{GL}}_r(F)$, where $F$ is a $p$-adic field. We show using global methods that the $\varepsilon$-factors associated to the exterior square of $\pi$ via the Jacquet-Shalika integrals and Langlands-Shahidi methods coincide. Along the way we also give a new proof of the local functional equation in the $p$-adic case following the techniques in [Beuzart-Plessis, Relative trace formulas, Simons Symp.(2021), pages 1-50].

Based on Geroch's process of dropping a system into a black hole from the vicinity of the horizon, we investigate in this paper the influence of deformation on the Bekenstein entropy bound both for (3+1) and (2+1) dimensions in the context of a generalized uncertainty principle (GUP). While providing a coherent framework that sets an upper limit on the entropy across dimensions we show, within a semiclassical treatment, that while a negative GUP deformation yields a universal relaxation of the bound, a positive deformation tightens it. Our results may be interpreted as a response to Planck-scale modifications of the near-horizon redshift.

In this paper we present an equivalence result between a randomized version of Anderson Acceleration and of randomized GMRES for linear problems. Namely, we extend the classical result of Walker and Ni (2011) to the case in which the least-squares problem in Anderson Acceleration is solved in a sketched space defined by a random projection. This equivalence suggests potential avenues for further research in the design and analysis of randomized acceleration methods.

We show that given completely multiplicative functions $f_1,\dots,f_d$ taking values in the unit circle, there exist Pythagorean triples (i.e., integer solutions to $x^2+y^2=z^2$) with $f_i(x),f_i(y),f_i(z)$ all arbitrarily close to $1$ for all $i$. This is a new special case of the conjecture that any finite colouring of $\mathbb{N}$ has a monochromatic Pythagorean triple. Our proof combines vanishing averages for aperiodic functions with concentration estimates for pretentious functions. A similar proof is applied to obtain the analogous statement for more general equations of the form $ax^2+by^2=cz^2$ whenever $a,b,c$ are perfect squares satisfying the Rado's condition.

Direct Multisearch (DMS) is a class of direct-search algorithms designed for multiobjective derivative-free optimization. Its framework consists of an optional search step and a poll step, the latter ensuring the corresponding theoretical convergence properties. Recently, a search strategy based on the minimization of quadratic polynomial models, constructed from previously evaluated points, was proposed to improve the numerical efficiency of the method. While the construction of these surrogate models has been investigated, considerably less attention has been devoted to how they should be jointly minimized, with min-max scalarization typically being adopted. This work investigates how different model-minimization strategies influence the performance of DMS. To this end, alternative strategies for exploring the quadratic polynomial models within the search step of DMS are proposed and numerically assessed, including one based on the recently proposed Improved Front Steepest Descent algorithm.

We develop thermodynamic formalism for random Anosov maps and uniformly Hölder random potentials. We assume uniform fibre hyperbolicity given by deterministic invariant cone fields, a one-dimensional stable direction, and a fibrewise mixing condition whose mixing time may depend on the base point. To do so, we construct adapted projective cones for the random Perron–Frobenius cocycle and prove that the cocycle contracts the associated Hilbert projective metrics. This allows us to construct a $\mathbb P$-relative equilibrium state, prove its uniqueness, and establish quenched exponential decay of correlations.