Papers

A mixed-precision GPU-oriented optimization framework is developed for computing the minimum enclosing ball of a collection of balls. The approach is built on an equivalent second-order cone programming reformulation and a relative-type inexact proximal augmented Lagrangian method (ripALM), which provides a high-accuracy optimization backbone while solving the inner subproblems only to a progress-dependent relative accuracy. The proximal augmented Lagrangian inherits a constraint-wise separable structure: its objective, gradient, generalized Hessian, and multiplier updates can be efficiently evaluated on GPUs as parallel maps over the input balls followed by reductions. To further improve efficiency, a mixed-precision reduction strategy is introduced. A low-precision ripALM run identifies balls near the approximate boundary, a high-precision ripALM run refines the reduced problem, and a full a posteriori feasibility check detects and reintroduces any violated discarded balls. Thus, low precision is used only for screening and warm starting, while the final feasibility is enforced against the original problem. Experiments on synthetic and real-data instances show that the proposed methods can compute accurate solutions, with GPU execution providing substantial speedups and mixed-precision reduction further reducing running times on most tested instances.

We introduce the framework of cotangent buildings to complement and refine that of Weinstein handlebodies. While Weinstein handlebodies are suitable for a “bottom-up" analysis of the Weinstein structure, cotangent buildings also enable a “top down" analysis. Further, cotangent buildings include a precise control over the interaction of any subcollection of the various building blocks, each of which is modeled on the cotangent bundle of a manifold with corners. Our main result is that any Weinstein manifold is Weinstein homotopic to one admitting the structure of a cotangent building.

Fractional Gaussian noise and $\FARIMA(0,d,0)$ have the same long-memory pole $|\theta|^{-2d}$ and hence the same leading PACF law $\alpha(n)\sim d/n$. We show that this agreement breaks at the first non-universal order. For $0<d<1/2$, the pure fGn PACF satisfies $$ \alpha_{\fGn}(n)=\frac d n+\frac{C_{\fGn}(d)}{n^2}+o(n^{-2}), \qquad C_{\fGn}(d)<d^2, $$ The proof uses the Bingham–Inoue–Kasahara representation, a phase-coefficient expansion for fGn, and a Hankel-operator perturbation argument. Thus the fGn spectral envelope is invisible at first order but visible in second-order finite prediction, explaining why short-memory order selection can differ when fGn data are fitted by FARIMA-type models.

We establish improved nonasymptotic bounds for Langevin Monte Carlo in the strongly log-concave setting, when the error is measured by the Wasserstein distance. The main result shows that the discretization error is governed by an average coordinate-wise smoothness constant, rather than by the usual global smoothness constant. The proof is short and probabilistic, and relies on a refined use of the synchronous coupling. We further show that the same ideas lead to improved bounds for variable step sizes, for potentials whose Laplacian is Lipschitz-continuous, and for finite-sum problems sampled by stochastic-gradient Langevin dynamics with fixed point control variates. In the Laplacian-smooth case, the usual Hessian-Lipschitz contribution is replaced by a weaker trace-type third-order smoothness quantity. In the finite-sum setting, the resulting SGLD bound improves the dependence on the root mean square smoothness of the component functions. Applications to generalized linear models with Gaussian design show that these refinements can yield substantial, dimension-dependent improvements over previously known bounds, especially for correlated covariates.

We prove a finite-scale comparison between lower Brin-Katok local entropy and Katok covering entropy. Let $(\mathcal{X},d,T)$ be a compact metric topological dynamical system and let $\mu$ be ergodic. Then, for every $\epsilon>0$ and every $\delta\in(0,1)$, $$ h^K_\mu(6\epsilon,\delta)\leq \underline h^{BK}_\mu(\epsilon). $$ Combining this estimate with the usual Katok-type variational principle for metric mean dimension gives the corresponding variational principle with lower Brin-Katok local entropy.

We prove a lower bound on the Weil height, the so-called Bogomolov property, for the algebraic extensions of $\mathbb Q$ cut out by the adelic Galois representations attached to certain eigenforms whose local component at a prime $p$ is supercuspidal. To this end, we give a method for constructing metric inequalities over $p$-adic Lie extensions of fields over $\mathbb Q$ that are finitely ramified at $p$.

A randomly perturbed digraph is an $n$-vertex directed graph with all out- and in-degrees linear in $n$, to which a linear number (depending on the degree) of random edges have been randomly added. We show that randomly perturbed digraphs whose edges have been colored uniformly with $n$ colors have a rainbow copy of every orientation of every possible length cycle, simultaneously, with high probability. This is a common generalization of work of Araujo, Balogh, Krueger, Piga, and Treglown in the uncolored setting and Katsamaktsis, Letzter, and Sgueglia for consistently oriented spanning cycles. Our proof uses Montgomery's distributive absorption method.

We study constant horizontal mean curvature surfaces in the sub-Lorentzian Heisenberg group. We derive the first-variation formula for horizontal area under volume-preserving radial variations and show that smooth isoperimetric candidates have constant horizontal mean curvature away from the characteristic set. We then give a complete classification of smooth boost-symmetric constant mean curvature surfaces: their characteristic sets, causal behaviour, and ambient sub-Lorentzian isometry classes. From this classification, we single out a family of smooth, acausal, boost-symmetric surfaces with nonzero constant mean curvature. Written as a two-sheeted graph over the exterior of a future hyperbola, this family is a natural sub-Lorentzian analogue of the Pansu bubbles and leads us to conjecture that it gives the isoperimetric maximisers in the sub-Lorentzian Heisenberg group.

We study spectral invariants arising in the noncommutative geometry of topological Markov chains and Cuntz–Krieger algebras. Their noncommutative geometry is described by spectral triples built from log-Laplacians, which are known to have non-trivial index theory and exotic quantum symmetries. We prove statistical eigenvalue asymptotics, local heat trace asymptotics, local Weyl laws, and an analogue of Connes' trace theorem. In all cases the local asymptotics are governed by the Kubo–Martin–Schwinger state of the gauge action on the Cuntz–Krieger algebra.

Can gradient descent be accelerated by just choosing better stepsizes? Surprisingly, the answer is yes. This short expository article provides an accessible introduction to this phenomenon of stepsize hedging.

Let $G$ be a discrete group with a finite-dimensional model for the classifying space for proper actions, and let $k$ be a commutative Noetherian ring of finite global dimension. In this setting, we prove that the homotopy category of projective $kG$-modules, the stable module category of $kG$-modules as defined by Mazza-Symonds, and the derived category of permutation $kG$-modules with finite isotropy, admit descent to finite subgroups. As an application, we show that any $kG$-module of type $FP_\infty$ is a retract of a module that admits a finite resolution by finitely generated $\natural$-permutation modules with finite isotropy, generalizing a result of Balmer-Gallauer.

Let $\Sigma_g^0\subset S^4$, $g\ge3$, be the standard unknotted closed oriented surface, and let $a\subset\Sigma_g^0$ be an oriented nonseparating curve. For every nontrivial knot $J\subset S^3$, let $\Sigma_{g,a,J}\subset S^4$ be the surface obtained from $\Sigma_g^0$ by ordinary untwisted rim surgery along $a$. We compute its extendable mapping-class subgroup exactly: $$ E(\Sigma_{g,a,J}) = \operatorname{Stab}_{\operatorname{Mod}(\Sigma_g)}(q_0) \cap \operatorname{Stab}_{\operatorname{Mod}(\Sigma_g)} (\Gamma_\mu(J)\cdot[a]). $$ Here $q_0$ is the Rokhlin quadratic form of the standard embedding, $[a]\in H_1(\Sigma_g;\mathbb{Z})$ is the oriented rim homology class, and $\Gamma_\mu(J)\subset\{\pm1\}$ records whether a meridian-preserving diffeomorphism of the knot exterior can preserve or reverse the preferred longitude. Thus ordinary rim surgery cuts Hirose's unknotted extendable subgroup by the stabilizer of the rim homology class, with the only additional ambiguity coming from this peripheral symmetry of $J$. We also prove a prescribed-mapping-class classification for such ambient pairs $(S^4,\Sigma_{g,a,J})$. More precisely, given two such pairs and $f\in\operatorname{Mod}(\Sigma_g)$, we characterize when $f$ is induced by an orientation-preserving pair diffeomorphism in terms of the Rokhlin quadratic form, the rim homology classes, and the meridian–longitude symmetries of the knot exteriors.

The trade-off relations between the two types of error probabilities in binary i.i.d. quantum state discrimination can be expressed by single-copy formulas in terms of the Petz-type and the sandwiched Rényi divergences of the two states representing the two hypotheses. In the non-i.i.d. setting, the error exponents can usually be expressed in terms of regularized Rényi divergences, which do not admit explicit formulas in general. Here, we consider a class of states, translation-invariant and gauge-invariant quasifree states on doubly infinite fermionic chains, and give explicit formulas for a wide range of regularized Rényi divergences between such states, including $(\alpha,z)$, log-Euclidean, geometric, measured, and the recently introduced integral Rényi divergences. We show that the case where there is a single mode at each lattice site becomes asymptotically classical, with all the different types of regularized Rényi divergences being equal, while in the case of multiple modes per site, non-commutativity persists under regularization, and for any fixed $\alpha$, the regularized Rényi $(\alpha,z)$-divergences give different regularized values for different $z$ parameters in general. We also generalize a previous construction from [Bunth, Maróti, Mosonyi, Zimborás, Lett.~Math.~Phys.~113:(7), 2023] to the case of multiple modes per lattice site to obtain a large class of states exhibiting super-exponential decay of the discrimination error probabilities.

We study the full-plane two-state fuzzy $4$-Potts model, obtained by assigning independent balanced $\{\pm1\}$ spins to the open clusters of a critical $q=4$ random-cluster configuration. Its spin field has the same law as a single-spin projection and as the polarization field of the isotropic Ashkin–Teller model at its Potts point. We prove that, after proper normalization, all even multi-point spin correlation functions converge to explicit conformally covariant Coulomb-gas type neutral charge sums. As a consequence, we prove convergence in law of the rescaled magnetization field and identify its limit in law with $2\mathopen{:}\cos(X/2)\mathclose{:}$ for a full-plane Gaussian free field $X$. The proof combines the Baxter–Kelland–Wu coupling, convergence of the six-vertex height function to the Gaussian free field, and a charge-completion mechanism: an enlarged discrete sum over charge assignments with total charge in $4\mathbb Z$ produces a combinatorial cancellation of connection patterns, while only the neutral charge sector survives in the scaling limit.

In the context of hyperbolic numbers we define the concept of negation of finite hyperbolicvalued probability distributions that is based on the partial order induced by the idempotent structure of hyperbolic numbers. Then, a hyperbolic majorization and general hyperbolic negators are introduced. For a broad class of generated negators, we prove that the original distribution majorizes its negation. This comparison yields that entropy increase for the strong hyperbolic Shannon entropy and the hyperbolic Gini-Simpson entropy, and it implies component-wise uniformization of the iterated negation. Finally, we analyze involutive property of hyperbolic negators and prove that are structurally distinct from the generated negators responsible for the entropy increase. These results show that hyperbolic probabilistic negation is not merely a component-wise copy of the real case, but a theory governed by the interaction between idempotent decomposition, partial order, and entropy measure.

We are interested in the generic simplicity of the spectrum of self-adjoint operators under bounded potential perturbations. More precisely, given a semibounded self-adjoint operator with compact resolvent and a suitable space of real-valued bounded perturbations, we study whether all eigenvalues of the perturbed operator are simple for a generic choice of the potential. In the first part of this paper we prove an abstract criterion which ensures that the set of perturbations giving only simple eigenvalues is residual. In the second part, we apply this criterion to several geometric and analytic settings, including sub-Laplacians and maximally hypoelliptic operators on compact manifolds, Laplacians on bounded domains with different boundary conditions, and Schrödinger-type operators on non-compact spaces.

In this article, we study the existence and global regularity results for the fully nonlinear elliptic Grad–Mercier type equations with oblique boundary conditions in the context of weighted Orlicz spaces. Our approach employs an asymptotic analysis in which global regularity is transferred from a limit profile, namely, the recession operator associated with the governing operator, using topological and stability methods. In addition to the main regularity result, we derive global weighted Orlicz estimates for the Hessian and establish global Morrey-type estimates for the problem. This article extends the results established by Caffarelli–Tomasetti (Comm. Pure Appl. Math. 76 (3): 604–615, 2023), Zhang et al. (Nonlinearity 39 (2): 025011, 2026), and Bessa (J. Funct. Anal. 286 (4): 110295, 2024).

We provide a complementary result to the quasi-invariance of Gaussian measures supported on Sobolev spaces with high regularity under the dynamics of the three-dimensional periodic defocusing nonlinear wave equation from Gunaratnam-Oh-Tzvetkov-Weber (2022). Namely, given $p$ and $\sigma$ large enough, we prove the existence of dense sets of Sobolev spaces $W^{\sigma,p}(\mathbb{T}^3)$ which do not preserve the regularity $\sigma$ throughout the aforementioned dynamics. This is in sharp contrast with the propagation under the flow of almost sure properties.

In this work, we build a bridge between the Pólya–Schur program and Voiculescu's free probability theory. A cornerstone of the former is the Pólya–Benz Theorem, classifying a central family of real-root preserving operators on the space of polynomials, as those given by $f(\partial_z)$ for a Laguerre–Pólya function $f$ and the derivative operator $\partial_{z}$. We prove that any free (additive) infinitely divisible distribution can be attained as the weak limit of root distributions of Appell polynomials $f_n(\partial_z)z^n$ as $n\to\infty$, for a suitably chosen sequence $f_n$ of Laguerre–Pólya functions. Such questions on the (global) limiting distributions of real rooted polynomials belong to the active research area of finite free probability. In contrast to its standard tools, our approach allows for non-compactly supported limiting distributions, (barely) complex rooted polynomials and even provides the full microscopic description of the roots. Moreover, we extend our results to differential operators generating free multiplicative infinitely divisible distributions, to the rectangular free convolution, and to $f_n(\partial_z)p_n$ for real rooted polynomials $p_n$, implying a generalization of the recent connections between the heat flow and free Brownian motion to any free Lévy process. As corollaries, we identify free stable distributions by choosing $f_n$ to be a fixed rescaled Laguerre–Pólya function, and we prove various convergence results on the zero distributions of Jensen polynomials, e.g. the limiting root distribution of Jensen polynomial of the Riemann $\Xi$-function is given by the Cauchy distribution.

In monostatic, decision-aided, or known-waveform integrated sensing and communications (ISAC) formulations, the sensing receiver is often modeled as knowing the transmitted waveform. This assumption is not suitable for passive, bistatic, or distributed settings where the sensing receiver knows the signaling rule but not the transmitted symbols. We study such a symbol-unaware ISAC model, where sensing is measured by the unconditioned mutual information $I(S;V)$ rather than the symbol-aware quantity $I(S;V|X)$. For discrete-input memoryless channels, we characterize the capacity-sensing region through an auxiliary time-sharing variable, showing that the optimal upper boundary is the upper concave envelope of the single-mode frontier. Thus, explicit time sharing is unnecessary when the single-mode frontier is already concave, but strictly beneficial when its upper concave envelope strictly dominates the frontier. For Rayleigh-fading BPSK, we further show that the curvature of the single-mode boundary is determined by the stochastic ordering of the communication- and sensing-side effective SNR distributions. Communication-side dominance yields a concave single-mode frontier and no time-sharing gain, sensing-side dominance yields a convex single-mode frontier and a strict time-sharing gain, and equality yields a linear boundary. The result extends to SIMO-BPSK through the ordering of post-combining SNR distributions. These findings explain when symbol-unaware ISAC optimally moves from data-symbol transmission to pilot-like sensing modes.

We show that the cohomology of the finite element Stokes complex consisting of piecewise polynomials spaces on an Alfeld split mesh from Fu, Guzmán, & Neilan (2020, Math. Comp., 89, 1059–1091) is isomorphic to the cohomologies of the continuous Stokes and de Rham complexes. We also construct novel "minimal" conforming finite element complexes where the $H^1$-conforming space is the lowest-order space from Guzmán & Neilan (2018, SIAM J. Numer. Anal., 56, 2826–2844) and the $L^2$-conforming space is piecewise constants. These minimal complexes also have cohomologies isomorphic to the continuous Stokes and de Rham complexes. We further construct local, bounded, cochain projections for the minimal complexes. All the results hold for strongly Lipschitz domains with nontrivial topologies and in the presence of mixed boundary conditions.

Zeroth-order (black-box) optimization is applied when gradients are unavailable and objective evaluations rely on expensive simulations. In many such applications, the oracle fidelity is tunable: higher-accuracy queries reduce noise but incur higher computational costs. To capture this trade-off, we study an accuracy-aware wall-clock model where each query with fidelity $\delta$ has a cost $c(\delta)$, and we minimize the total time $T_{\mathrm{total}} = \sum_{k=1}^{N} c(\delta_k)$, subject to a target accuracy constraint. We show how the choice of oracle type, noise model, and optimization scheme induces explicit wall-clock-optimal choices for the algorithmic parameters. For instance, we demonstrate that accelerated methods can be wall-clock inferior to non-accelerated schemes. Furthermore, we characterize the conditions under which a constant fidelity strategy is optimal in the Big-O sense. Our framework provides a unified methodology to translate convergence guarantees into practical fidelity and batching recommendations.

In this paper, we consider a large family of real-valued spin models on general transitive graphs. We show that, in the subcritical regime $\beta<\beta_c$, the correlations of the model decay exponentially fast. To prove this result, we consider the random cluster (a.k.a. FK percolation) representation of the model and obtain an inequality that generalises the OSSS inequality to monotonic measures with random connection probabilities, thus extending the inequality of Duminil-Copin, Raoufi, and Tassion. Our results apply in particular to the Blume–Capel model and general $P(\varphi)$ models, going beyond the cases of the Ising and $\varphi^4$ models treated by Aizenman, Barsky, and Fernández.

This paper establishes foundational analytic and geometric results for a broad class of reasonable flows of Spin($7$)-structures. We first prove Shi-type derivative estimates, showing that a uniform bound on the quantity \[ \Lambda(x,t)=\left(|\mathrm{Riem}(x,t)|_{g(t)}^2+|T(x,t)|_{g(t)}^4+|\nabla T(x,t)|_{g(t)}^2\right)^{1/2} \] implies bounds on all covariant derivatives of the Riemann curvature tensor $\mathrm{Riem}$ and the torsion tensor $T$. We show further that $\Lambda(x,t)$ must blow up at any finite-time singularity, and we establish a lower bound on the blow-up rate. We also prove a compactness theorem for solutions of such flows and apply these results to the analysis of finite-time singularities. These results provide a general analytic framework for studying flows of Spin($7$)-structures; once a proposed flow is shown to satisfy the reasonable condition, our estimates, compactness theorems, and singularity analysis apply.

Morier-Genoud and Ovsienko introduced q-deformations of continued fractions, Farey labelings, and Conway–Coxeter friezes, and established relationships among them in restricted settings associated with triangulations having exactly two exterior cells. In this paper, we extend these correspondences to arbitrary subsequences of quiddities arising from general triangulations. We show that the numerator and denominator polynomials of q-deformed continued fractions coincide with entries of q-deformed Conway–Coxeter friezes, while the corresponding polynomials in q-Farey labelings agree with them up to explicit powers of q. These powers are described combinatorially in terms of the number of diagonals in the triangulation, or equivalently, the number of entries equal to 1 in the associated frieze. Furthermore, we determine the minimum and maximum degrees of these polynomials in terms of the same combinatorial data.

Institutional incentives are widely used to promote cooperation among autonomous, self-regarding agents, from human societies to multi-agent and AI systems. Existing work typically treats incentive design as a bi-objective problem: minimise institutional cost while achieving a high long-run frequency of cooperation. Whether such schemes also maximise social welfare - total population payoff net of institutional expenditure - has remained largely unexplored. We develop a welfare-centric framework for institutional incentives in finite, well-mixed populations playing a social dilemma (Donation Game and Public Goods Game), considering both rewards for cooperators and punishments for defectors. For each mechanism, we derive explicit expressions for expected social welfare and characterise how it depends on incentive efficiency and selection intensity. Analytically, we identify parameter regimes where social welfare has a single optimal incentive level and regimes with qualitative phase transitions, in which welfare becomes non-monotonic with multiple local optima. We prove that any welfare-maximising incentive is either zero or concentrated around a simple closed-form target, and we provide an efficient algorithm to compute these optima. Comparing reward and punishment, we further derive close-formed conditions under which reward outperform punishment in terms of social welfare for any given budget. Overall, our results reveal a systematic gap between incentives optimised for cost or cooperation frequency and those that maximise welfare.

This article investigates a coupled viscoelastic Navier–Stokes–Biot system describing the interaction between an incompressible viscous fluid and a poro–viscoelastic medium in three spatial dimensions. The coupling between the fluid and the porous medium is realized through Beavers–Joseph–Saffman type interface conditions. Using spectral analysis, it is proved that the coupled system admits a unique, strong, global solution for small initial data. In addition, a Serrin–type blow-up criterion is established.

Networked AI systems increasingly rely on multiple agents that collaboratively learn and adapt models over communication networks. In such systems, bilevel formulations naturally arise in hyperparameter optimization, data cleaning, and meta-learning, but the repeated evaluation of gradients, Jacobians, and Hessians can impose a substantial computational burden on individual agents. To address this challenge, we propose Snapshot-SLDBO (S$^3$LDBO), an efficient single-loop decentralized bilevel optimization algorithm that enables agents to intermittently skip expensive derivative evaluations through a snapshot mechanism. This mechanism can be interpreted as an autonomous computation-adaptation strategy for networked AI, where agents selectively perform costly local updates while maintaining global collaborative learning. We establish the ergodic iteration complexity and the high probability nonergodic iteration complexity of the proposed algorithm within a deterministic setting. Experimental results on hyperparameter optimization with synthetic and MNIST datasets, data hyper-cleaning on Fashion-MNIST, and decentralized meta-learning on miniImageNet demonstrate that the proposed algorithm improves computational efficiency while maintaining competitive learning performance.

We survey Lyapunov-based techniques for the finite-time analysis of stochastic iterative algorithms, also known as stochastic approximation (SA) algorithms, for solving fixed-point equations $\bar{F}(x)=x$, where the operator $\bar{F}(\cdot)$ can only be accessed through a noisy oracle. We first focus on the standard setting in which $\bar{F}(\cdot)$ is contractive with respect to some norm and the noise is i.i.d., and explain how generalized Moreau envelopes serve as universal Lyapunov functions, regardless of the underlying norm. We then show how this framework yields mean-square convergence guarantees and applies to stochastic gradient descent, linear SA, and value-based reinforcement learning algorithms such as Q-learning and temporal-difference learning. Finally, we discuss extensions to Markovian noise, seminorm-contractive operators, dissipative operators, and high-probability bounds, and conclude with open problems. The goal is to present a unified and self-contained roadmap for the finite-time analysis of SA and its applications, especially in reinforcement learning.

We introduce the notion of worst-case posterior and worst-case likelihood sensitivity. These measure, respectively, the sensitivity of the expected cost to worst-case perturbations of the posterior distribution and worst-case perturbations of the likelihood of a Bayesian model. Each defines a quantitative measure of robustness. A decision maker concerned about the sensitivity of the out-of-sample expected cost to deviations from her assumptions will want a decision for which both sensitivities are small. We derive posterior and likelihood sensitivities for uncertainty sets defined in terms of deviation measures. Posterior sensitivity vanishes when the posterior variance shrinks to zero, which occurs when parameter uncertainty is eliminated from learning. Parameter learning does not eliminate likelihood sensitivity. A distributionally robust formulation of a Bayesian optimization problem makes a near-Pareto-optimal tradeoff between performance (expected cost) and robustness (posterior and likelihood sensitivity).