Papers

We study the number of $\mathbb{F}_q$-rational points on Danielewski and double Danielewski surfaces over the finite field $\mathbb{F}_q$ and show an interesting connection with the polygonal numbers.

Frank–Wolfe methods avoid projections, but over curved feasible regions the full-space linear minimization oracle (LMO) can itself become the computational bottleneck. We introduce random-subspace Frank–Wolfe (RSFW), the first Frank–Wolfe framework, to our knowledge, that replaces the ambient LMO by exact LMOs over random low-dimensional affine sections of a general feasible set, while preserving feasibility in the original space. For smooth convex objectives over compact strongly convex feasible sets, we prove a dimension-explicit approximate-oracle inequality and derive the standard \(O(1/k)\) open-loop rate, with high-probability and almost-sure counterparts. Under short steps and a gradient lower bound, the same geometric control yields linear convergence, and we extend the sublinear theory to finite-sum stochastic gradients. We also show that random sections can improve the local curvature model controlling short steps: for smooth objectives, the quadratic model along a sampled section is governed by the compressed Hessian, yielding computable \(d\times d\) curvature constants for quadratic objectives over balls and ellipsoids. These results provide a geometric theory of oracle-side randomization in projection-free optimization.

Information compression plays a central role in diverse fields of modern science and technology, from communication theory to machine learning. In condensed-matter physics, the intermediate representation (IR) basis has recently been developed as an efficient method for compressing imaginary-time Green functions, which are fundamental quantities for describing quantum many-body systems. This compression relies on the rapid decay of the singular values with the basis index and the unusually weak growth of the effective rank with inverse temperature. Because of these useful features, the IR basis is now widely used as a standard method in quantum many-body calculations. However, the analytic origin of its compression capability has remained unclear. Here we uncover a finite-Laplace-transform structure underlying the IR kernel, which reveals that the eigenfunctions of the IR kernel admit a natural expansion in terms of classical special functions, the oblate spheroidal wave functions. This finite-Laplace-transform structure also enables us to analytically clarify the compression mechanism of the IR basis. Our results provide a mathematical foundation for the compression of imaginary-time Green functions, connecting quantum many-body physics with theories of information compression and finite integral transforms.

In this paper, we apply a self-similar transformation to convert the parabolic equation with a Sobolev-Hardy term \begin{align*} u_t-\Delta u= \frac{|u|^{q-2}u}{\left|x\right|^s} & \text { in } \mathbb{R}^N \times(0, \infty), \end{align*} into the following elliptic equation \begin{equation*} -\Delta v-\frac{1}{2} y \cdot \nabla v=\alpha v+ \frac{|v|^{q-2} v}{|y|^s}, \end{equation*} where $2 < q \leq 2^*(s)=\frac{2 N-2 s}{N-2}, 0 \leq s < 2, \alpha=\frac{2-s}{2q-4}$. For this equation, we establish the weighted Hardy inequality and Sobolev inequality. Furthermore, by virtue of the variational methods, we obtain infinitely many solutions in the subcritical case, and prove the existence of solutions in the critical case. We also apply the Pohozaev identity to establish the nonexistence of solutions under certain conditions.

In this paper, we apply a self-similar transformation to convert the parabolic equation with a Hardy term \begin{equation*} \begin{cases}u_t-\Delta u-\mu \frac{u}{|x|^2}=|u|^{2^*-2} u & \text { in } \mathbb{R}^N \times(0, T), u(x, 0)=u_0(x) & \text { in } \mathbb{R}^N , \end{cases} \end{equation*} into the following parabolic equation \begin{equation*} \begin{cases} v_s-\Delta v-\frac{1}{2} y \cdot \nabla v=\beta v+\frac{\mu v}{|y|^2}+|v|^{2^*-2} v &\text { in } \mathbb{R}^N \times(0, S), \left.v\right|_{s=0}=v_0 & \text { in } \mathbb{R}^N, \end{cases} \end{equation*} where $N \geqslant 3$, $\mu\in [0,(N-2)^2 /8]$ and $2^{\ast}=2N /(N-2)$. For this equation, we establish a weighted Hardy inequality. Furthermore, by virtue of the modified potential well method and Palais-Smale sequence analysis, we investigate the long-time behavior and finite-time blow-up properties of solutions to the parabolic equation.

A transitive tournament is an acyclic orientation of a complete graph. We study decompositions and packings of the transitive tournament \(TT_n\) into connected two-arc motifs. The three motifs considered are chains, colliders, and forks, which are also fundamental local configurations in directed acyclic graphs. We first construct decompositions of \(TT_n\) into mixtures of these motifs whenever such decompositions exist. We then consider the corresponding pure packing problem for each individual motif. For \(H\) equal to a chain, a collider, or a fork, we determine the maximum number of arc-disjoint copies of \(H\) in \(TT_n\). These results give a precise extremal description of two-arc motif packings in transitive tournaments and suggest further questions on motif decompositions in broader classes of directed acyclic graphs.

In the philosophical tradition of `analytic pragmatism', which attempts to account for linguistic meanings in terms of their practices of use, logical expressivism is a theory which offers a distinct perspective on logic. We shed light on Brandom and Hlobil's recent formalization of logical expressivism by reconstructing its formal semantics through the use of universal constructions. This reveals similarities with categorical logic: there is a focus on internalizing judgment structure, and connectives arise from adjunctions.

We study stochastic density control between Gaussian-mixture endpoint distributions under Brownian prior dynamics. Since the direct Schrödinger bridge between Gaussian mixtures is generally not available in closed form, we introduce a lifted path-space construction in which each trajectory is augmented with a source–target component label. Consequently, the problem decomposes into Gaussian component-to-component Schrödinger bridges with explicit marginal, drift, and cost formulas, while the mixture-level assignment reduces to a finite-dimensional entropic coupling problem with a Sinkhorn scaling form. We then analyze the projection obtained by discarding or forgetting the label. By construction, the projected law satisfies the original Gaussian-mixture endpoint constraints, but its relative entropy generally differs from the lifted relative entropy by a nonnegative conditional label-information gap. This gap reveals a path-space obstruction: the lifted optimizer cannot, in general, be identified with the direct unlabeled Schrödinger bridge after projection. We also derive the posterior-averaged Markov drift associated with the projected marginal flow, prove a kinetic-energy upper bound, and identify a common path-potential condition under which the projection gap vanishes. Several numerical illustrations showing density and shape control are recorded for a self-contained exposition.

We explain how the undecidability of the quasi-isomorphism and derived Morita equivalence problems for semi-free finitely presented dg associative algebras follows as an immediate consequence of an extension of Adams' cobar theorem to non-simply connected spaces and the triviality problem for finitely presented groups.

We prove the existence of time-periodic solutions for a driven–damped $\phi^4$ wave equation posed on a bounded spatial domain with periodic boundary conditions. The result is obtained at the level of the full partial differential equation. The analysis is based on a Lyapunov–Schmidt decomposition in a time-periodic function space. A key structural feature is the presence of linear damping, which ensures uniform invertibility of the associated linear operator on the complement of the kernel and removes the need for nonresonance conditions. After introducing a truncation to control the cubic nonlinearity, we solve the complement equation via a contraction mapping argument and reduce the problem to a scalar kernel equation. The latter is treated using a mean/zero-mean decomposition and a fixed-point argument. A period-averaged energy estimate then allows one to remove the truncation and obtain a time-periodic weak solution under explicit smallness conditions on the forcing parameters

We analyze electron dynamics in a plane electromagnetic wave using the Landau-Lifshitz equation with a quantum radiation reaction correction modelled by a Gaunt factor. In this geometry, the quantum parameter $\chi$ depends solely on the lightfront momentum, allowing the modified equation of motion to retain the integrable structure of the classical problem. We derive a semi-analytical solution for the energy evolution and the four-velocity, which reduces to the known classical result in the appropriate limit. The results provide an analytical and deterministic description of semiclassical radiation reaction in plane-wave fields.

This letter investigates Airy beam dispersion in near-field wideband terahertz communications. Unlike conventional focusing beams, whose dispersion mainly appears as focal-point migration, Airy beams exhibit frequency-dependent shifts of both the reference focusing point and the self-bending main-lobe trajectory. Based on the Fresnel diffraction integral, a closed-form trajectory expression is derived to characterize the dispersion behavior across subcarriers. Furthermore, a true-time-delay (TTD)-assisted Airy beamforming structure is developed to actively control the trajectory dispersion. By properly designing the time delay parameters, the proposed scheme can either generate frequency-dependent curved trajectory clusters for sensing-oriented scanning or suppress trajectory drift for reliable communication.

This work investigates a nonlinear two-point boundary value problem arising in diffusion–reaction processes in catalyst slabs with power-law kinetics and fractional reaction order. For sufficiently large Thiele modulus, the solution develops a dead-core region separated from the active region by an unknown free boundary. We propose a structured Physics-Informed Neural Network (PINN) framework that incorporates the asymptotic behavior at the dead-core interface into a hard-constrained trial solution and treats the interface location as a trainable parameter. The concentration profile and free boundary are therefore approximated simultaneously without explicit interface tracking or penalty-based enforcement of the interface conditions. The method is validated against the exact solution for power-law kinetics and a high-precision numerical shooting method. Numerical experiments covering near-critical and strongly supercritical regimes demonstrate accurate recovery of both the concentration profile and dead-core location, together with robustness to random initialization and collocation sampling. While classical shooting is more efficient for the present one-dimensional benchmark, the proposed formulation provides a flexible framework for extensions to multidimensional geometries and problems for which analytical solutions are unavailable.

For $p,q\in\mathbb{N}$ and $\alpha,\beta\in\mathbb{R}$, we investigate the family of improper integrals \[\int_0^\infty\frac{(\cos\alpha x-\cos\beta x)^p}{x^q}dx.\] We establish a complete classification of the parameter ranges $(p, q; \alpha, \beta)$ for which the integrals converge or diverge, and we derive explicit closed-form evaluations in all convergent cases. The analysis also reveals a family of combinatorial identities arising naturally from coefficients in the trigonometric power expansions. As a further application of the same method, we study an analogous class of integrals involving powers of sine differences. This extends the work of Laoharenoo and Boonklurb in 2022.

The subgradient method is a classical and foundational approach in non-smooth convex optimization; its simplicity, robustness, and role as a conceptual and algorithmic starting point have made it the backbone of many significant optimization algorithms. Motivated by classical Euclidean results and recent advances in first-order Riemannian optimization, we study the convergence of the subgradient method on Hadamard manifolds with lower bounded curvature. Assuming a nonempty solution set and employing a corresponding non-summable diminishing step-size condition, we establish convergence of the generated sequence $\{x^k\}$ to a minimizer whenever at least one of the following holds: (a) the sequence $\{x^k\}$ is bounded; (b) the solution set $S$ is bounded; or (c) the step-sizes are square-summable ($\sum_{k=1}^{\infty}\lambda_k^2<\infty$). Additionally, we prove that if $\operatorname{int}(S)\neq\emptyset$, the method achieves finite termination. Our main contribution provides a Riemannian counterpart to Shepilov's Euclidean analysis [Cybernetics, 12 (1976), pp. 544-548], thus complementing existing literature on convex minimization over manifolds with lower bounded curvature.

Submodular optimization has become a fundamental paradigm for data selection, retrieval, summarization, and representation learning due to its ability to model coverage, diversity, and representativeness. However, classical submodular objectives optimize only the selected subset and do not explicitly preserve structural information between the selected subset and the remaining data. In many modern machine learning applications, including train/validation/test splitting, benchmark construction, and robust subset selection, the quality of a selection depends critically on preserving balanced structure across both the selected subset and its complement. In this work, we introduce Complement Submodular Information (CSI), a new class of complement-aware submodular objectives that quantify shared structural information between a subset and its complement. Our framework induces complement-aware variants of several classical submodular functions including Facility Location, Graph Cut, LogDet, Saturated Coverage, Set Cover, Probabilistic Set Cover, and Feature Based Functions. We analyze the theoretical properties of CSI objectives and show that they exhibit approximate monotonicity under bounded curvature conditions, leading to near-$(1-1/e)$ greedy approximation guarantees. Empirically, CSI objectives consistently outperform standard submodular objectives on robust hidden-slice-aware subset selection. In particular, CSI objectives significantly improve preservation of coherent rare/tail semantic structure while simultaneously suppressing noisy and isolated outliers, leading to substantially improved downstream predictive performance. Synthetic experiments further illustrate how different CSI instantiations capture complementary notions of representativeness, diversity, connectivity, and balanced neighborhood preservation.

In this note, we give a characterization of all projectable and divisible-projectable reduced $f$-rings satisfying the first convexity property and admitting elimination of quantifiers, in the language of lattice-ordered rings with the divisibility relation, the radical relation associated to the minimal prime spectrum, and the local divisibility relation.

We prove that every non-Archimedean Polish group is topologically isomorphic to the outer automorphism group of a countable discrete group. Furthermore, our construction is Borel. This implies that determining whether two countable discrete groups have topologically isomorphic outer automorphism groups is at least as complicated as classifying non-Archimedean Polish groups up to topological isomorphism, and so in particular not classifiable by countable structures. Our construction is based on the theory of right-angled Coxeter groups. Along the way, we prove results of independent interest on the topological group $\mathrm{Aut}(W)$, for $W$ a countable right-angled Coxeter group. This has purely group-theoretic applications; in particular, we give a graph-theoretic characterization of when the group $\mathrm{Spe}(W)$ of special automorphisms of $W$ coincides with $\mathrm{Inn}(W)$. Combined with previous work, this gives a characterization of when $\mathrm{Aut}(W)$ is inner-by-graph, for $W$ countable.

We give characterizations of sharp minimizers that emphasize their geometric properties. These include tilt invariance and weak upper gradient conditions. We relate sharp minimality to cusps in nonsmooth manifolds when interpreted locally as graphs and connect the rolling of tangent planes to the tilting of functions.

We propose several linear, fully decoupled numerical schemes with first- and second-order temporal accuracy for a novel Q-tensor-based two-phase hydrodynamic model describing the coupling of active nematic liquid crystal solutions with isotropic solid substrates. The model is derived from the generalized Onsager principle and includes nontrivial terms that contribute zero to the total free-energy dissipation. We prove that the proposed decoupled linear schemes are thermodynamically consistent at the discrete level. In the passive limit, the SGE-BDF1 and SGE-PDG schemes are unconditionally energy stable, while the SGE-BDF2 scheme is energy stable with respect to a modified energy under a standard boundedness assumption and a sufficiently large stabilization parameter. We perform extensive numerical simulations to investigate how activity and other model parameters affect active nematic fluid-solid interactions. Finally, we analyze the physical mechanisms underlying the observed behaviors, providing deeper insight into the dynamics of soft confined active nematic fluids.

We study learning-to-sample – a basic algorithmic task underlying generative modeling – for Ising models, a standard testbed for algorithmic ideas in both theoretical computer science and machine learning. Given i.i.d. samples of an unknown target distribution, the goal of learning-to-sample is to learn a computationally efficient generation procedure that produces new samples following approximately the same distribution. We construct a family of Ising models of constantly bounded-width which lie just beyond the spectral threshold $\lambda_{\max}(J)-\lambda_{\min}(J)=1$, and show that learning-to-sample for this family is computationally hard under standard cryptographic assumptions, even when the learner is given both polynomially many i.i.d. samples from the model and explicit access to its parameters. Combined with results of [AJKPV24,KLV25] showing tractability of learning-to-sample below the spectral threshold, this establishes a sharp computational phase transition at the spectral threshold. Moreover, combined with prior results on parameter learning for bounded-width Ising models [KM17,WSD19,VML20], this shows that learning-to-sample can be more difficult than parameter learning. Finally, we show that any efficient learner for these hard instances exhibits a natural memorization-hallucination dichotomy: the learner must either output configurations that, after a simple transformation, match the (transformed) training data or place substantial mass on configurations of negligible probability under the target distribution.

We show that for any finite partition of $\mathbb{N}$ there is an infinite sequence whose finite sums are monochromatic and such that infinitely many of the products with a fixed number of factors are monochromatic – though not necessarily belonging to the same color class as the finite sums. We are able to build these infinite configurations in parallel by refining arbitrary partitions of $\mathbb{N}$. We apply these techniques to prove that many complex infinite sum-product configurations are guaranteed to be monochromatic for arbitrary finite colorings of $\mathbb{N}$.

We study the sample complexity of robust binary hypothesis testing under three standard contamination models: $\varepsilon$-additive (Huber), $\varepsilon$-subtractive, and $\varepsilon$-total variation (TV), denoted by $n^*_{\mathrm{Hub}}(\varepsilon)$, $n^*_{\mathrm{Sub}}(\varepsilon)$, and $n^*_{\mathrm{TV}}(\varepsilon)$, respectively. For subtractive contamination, we show that least favourable distributions exist and provide explicit formulas for the same, bringing this model in line with the classical Huber and TV models. Next we show that in all three models, sample complexity may be highly unstable in the contamination parameter $\varepsilon$, increasing by polynomial factors even for $o(\varepsilon)$ perturbations. Similarly, there may be polynomial factor gaps between the sample complexities when $\varepsilon$ is known exactly versus when it is known up to $o(\varepsilon)$ error. Despite the instability of the sample complexity in all models, we show that the sample complexities across models are comparable up to constant-factor rescaling of $\varepsilon$. Specifically, for any fixed $\delta_0>0$, the following hold for all distributions $p$ and $q$: (i) $n^*_{\mathrm{Hub}}(\varepsilon) \lesssim n^*_{\mathrm{TV}}(\varepsilon) \lesssim n^*_{\mathrm{Hub}}(2\varepsilon)$, (ii) $n^*_{\mathrm{Sub}}(\varepsilon) \lesssim n^*_{\mathrm{TV}}(\varepsilon) \lesssim n^*_{\mathrm{Sub}}((2+\delta_0)\varepsilon)$, and (iii) $n^*_{\mathrm{Sub}}(\varepsilon) \lesssim n^*_{\mathrm{Hub}}(\varepsilon) \lesssim n^*_{\mathrm{Sub}}((1+\delta_0)\varepsilon)$, and the scaling constants are tight. Finally, we extend our results to adaptive versions of the contamination models.

Identifying the most influential nodes in a network, typically using centrality measures, is a central task in applied network analysis. However, real-world networks are often constructed from noisy or incomplete data, which can distort rankings and lead to errors in identifying the true top-$k$ nodes. In this paper, we study how network noise affects the recovery of the true top-$k$ node set based on degree centrality. Specifically, we consider a noisy network observation in which edges are randomly added or removed according to a probabilistic noise model, and analyze the resulting empirical top-$k$ set. We show that top-$k$ recovery under network noise is governed by the relationship between the degree gap and the noise magnitude, which separates recoverable and unrecoverable regimes. To quantify ranking stability, we derive upper and lower bounds on the expected discrepancy between the empirical and true top-$k$ sets in a general framework and for specific network models. We also extend the analysis to eigenvector centrality, showing that similar noise-gap tradeoffs arise in spectral rankings. Simulation studies support our theoretical findings and illustrate the practical impact of network noise across a range of settings.

We disprove a long-standing open conjecture due to Simon stating that all skeleta of simplices are extendably shellable. In particular, for every $d \geq 3$ we provide a pure $d$-dimensional shellable simplicial complex which is not shelling completable.

While large-scale models such as LLMs and diffusion models have achieved practical success, public institutions have emphasized the importance of explainability in AI. Existing methods for explaining AI, however, are not designed to provide completely faithful explanations of the behavior of large-scale AI systems. Although a completely faithful and interpretable explanation of the behavior of an AI system might be useful for AI governance, it has not been known whether providing such an explanation is theoretically possible. In this paper, we mathematically prove a fundamental quadrilemma in explaining AI, stating that AI and its explanation cannot satisfy the following four conditions simultaneously: 1) the complexity of the operation environment, 2) the goodness of the AI's performance, 3) the interpretability of the AI's explanation, and 4) the complete faithfulness of the AI's explanation. This quadrilemma suggests that, in most applications where we cannot change the environment or sacrifice good AI performance and an interpretable explanation, we should give up complete faithfulness of explanations and should instead aim to explain only the parts that are important for applications. As a consequence, the quadrilemma implies that AI governance should be designed on the premise that the faithfulness of AI explanations is always incomplete.

Active feedback between geometry and physics is woven throughout the study of Nature at its fundamental level, and is of key importance in string theory. Methods of complex algebraic geometry in particular have brought about an unrivaled abundance of solutions, counted well into hundreds of orders of magnitude, reciprocated by the discovery of the wholly unexpected mirror symmetry. However, recent developments demonstrate that there are rich additional possibilities, made possible by certain generalizations that, at first glance, appear to be non-algebraic. Nevertheless, they are remarkably well-aligned within an overall mirror-symmetric framework, are amenable to almost as comprehensive quantitative analysis, and hint at a deeper relationship with symplectic geometry.

Refutation calculi are formal systems developed to derive the invalid formulas of a given logic. While the notion of refutation calculi has played a key role in the development of tableaux calculi, a refutation approach to display calculi has not yet been attempted. In this paper, we introduce refutation display calculi for basic LE-logics, i.e., those logics canonically associated with basic normal lattice expansions of any signature. In particular, we prove soundness and completeness via proof-analysis results on derivable sequents. Finally, we obtain terminating tableaux calculi from these refutation display calculi.

In this paper, we study age of information (AoI) optimization for status updating in an integrated sensing and communication (ISAC) system. We consider a discrete-time architecture in which a base station interacts with a physical environment and a remote monitor, and at each time slot can operate in one of three modes: sensing, communication, or joint sensing and communication. Each mode is unreliable and incurs a different operational cost. The objective is to minimize a discounted infinite-horizon cost that combines the AoI at the monitor with action-dependent sensing and communication costs. For the single source scenario, we formulate the problem as a Markov decision process with a two-dimensional AoI state and prove that the optimal stationary policy admits an ordered threshold structure in the AoI state space. Since the AoI evolves over an infinite space, we truncate the state space to reduce complexity and rigorously bound the resulting error. The analysis analytically determines the truncation size needed to keep the error below a given threshold. For the multi-source scenario, we formulate the scheduling problem as a restless multi-armed bandit. We develop both a Whittle index policy and an approximate Whittle index policy for scheduling under two different regimes, one where indexability is guaranteed, and one where it is not. Numerical results illustrate the structure of the optimal policy in the single-source case and show that the proposed approximate Whittle index policy performs comparably to the Whittle index policy in the indexable regime, while remaining effective beyond it.

We establish four structural results for feature learning in wide two-layer neural networks under the Maximal Update Parametrization ($\mu$P). First, we prove global existence and uniqueness of the mean-field limit of noisy gradient descent under $\mu$P, identifying the maximal admissible weight $w^*$ on the moment sequence of the initialization as the reciprocal parameter-moment-growth boundary, and hence the largest weighted moment class propagated by the flow. The finite-particle approximation has uniform-in-time squared-Wasserstein rate $O(N^{-1})$. Second, we characterize identifiability of the mean-field limit: two admissible parameter measures induce the same network function in $L^2$ exactly when their active components agree modulo the finite-rank realization symmetry of the architecture. The orbit depth $D^*_{\mathrm{orb}}$ is separated from the moment-variety depth $D^*_{\mathrm{var}}$. Third, under the Barron-Hermite target condition the active support of the long-time limit measure admits a sparse-dictionary decomposition: it is supported on at most $S^*$ atoms modulo finite-rank realization symmetry, with $S^*$ bounded by an explicit coefficient-threshold number. Fourth, we derive the total feature-learning-error decomposition into statistical, optimization, propagation-of-chaos, and sparse-residual components, with a target-dependent Hermite/Barron tail replacing any initialization-only residual. The four results are tied together by an architectural identity: the triple $(w^*, D^*_{\mathrm{orb}}, S^*)$ – the maximal admissible weight, the orbit identifiability depth, and the sparse-dictionary depth at which the target is realizable – is the natural learning cell of the architecture-data pair $(\sigma, \rho)$. The proofs are self-contained except for standard results from $\mu$P and mean-field Langevin theory.