Papers

For \(\frac12\le s<1\), we study a fractional Choquard equation with prescribed mass and nonlinearities at the lower Hardy-Littlewood-Sobolev and \(L^2\)-critical exponents. The sharp Hardy-Littlewood-Sobolev and Choquard Gagliardo-Nirenberg inequalities determine an explicit critical mass \(a_*\). For \(0<a\le a_*\), we compute the exact infimum of the constrained energy and prove that it is not attained and that no normalized solution exists. For \(a>a_*\), the energy is unbounded from below on the mass sphere, while the Pohozaev set is nonempty.

We review recent progress on a model for the relativistic electron-positron field interacting with a classical electromagnetic field at positive temperature. The ultimate goal is to advance our comprehension of how thermal backgrounds affect polarisation mechanisms of this quantum system.

Multimodal forecasting models that combine time series with text annotations promise richer prediction through textual context, but how do we know whether a text annotation meaningfully contributes to the forecasters prediction? This is an information-theoretic question, but to evaluate whether information-theoretic metrics can reliably measure the predictive value an annotation provides, a ground truth benchmark is needed, and none currently exist. We create a synthetic time series signal with annotations in three categories: semantically correct, incorrect, and irrelevant. Because the data generation process is fully controlled, ground-truth information content is known exactly, enabling principled evaluation of six complementary mutual information estimators (KSG, MINE, InfoNCE, CCA, PID and V-information). We show that all six estimators identify correct annotations as most informative, and are able to audit the quality of mixed text corpora, choosing the annotations that result in the best downstream forecasting results without the need for model training. Our benchmark identifies limitations of each estimator, and these are validated on seven real-world datasets, which show how estimator performance differs on weak signals. Finally, we establish practical rules for implementing these metrics for annotation auditing and fusion selection.

We introduce two independent numerator exponents into the Zeilberger–Zudilin integral and specialize them to \[ A_1=A_2=\frac{1857}{2785}. \] The resulting integer linear forms in $1$ and $\pi$ prove \[ \mu(\pi)<7.101862832357. \] This lowers the Zeilberger–Zudilin upper bound $7.103205334137\ldots$ by more than $0.001342501780$; the difference between the unrounded bounds is $0.0013425017806509\ldots$, approximately $0.01890\%$. The same parameter point is a strict two-dimensional local minimizer of the explicit auxiliary upper-bound function in its admissible arithmetic chamber. This is a local statement about that function, not a claim that the point is a global optimizer among all constructions.

We study early stopping for a continuous-time model of stochastic gradient descent (SGD) in ill-posed linear inverse problems. We consider an a posteriori stopping rule based on the discrepancy principle, which stops the dynamics once the residual reaches the noise level. Unlike deterministic gradient flow, the stochastic dynamics exhibits persistent multiplicative fluctuations whose amplitude scales with the residual itself. Under a source condition on the initial error and a polynomial bound on the spectrum of the empirical covariance operator, we prove that, with high probability, the error at this random stopping time achieves the minimax rate over the corresponding source class, up to a logarithmic factor. Our results therefore establish the discrepancy principle as an adaptive regularization strategy for continuous-time SGD that is minimax-optimal up to a logarithmic factor, despite the persistent fluctuations induced by stochastic sampling. To our knowledge, this is the first convergence-rate result for a continuous-time model of stochastic gradient descent in inverse problems under an a posteriori stopping rule: existing analyses, including those of variance-reduced variants, bound the stopping time but do not quantify the error attained there.

Let $F$ be an absolutely unramified mixed characteristic local field with rings of integers $O_F$. For a small affine algebra $R$ over $O_F$, we consider the logarithmic étale fundamental group $G_R$ of its generic fibre equipped with a "horizontal" log-structure, in the sense of Fujiwara–Kato. We study $p$-adic representations of $G_R$ and obtain a classification of these representations in terms of étale $(\varphi, \Gamma_R)$-modules. Moreover, we define and study the notion of log-crystalline representations of $G_R$, a generalisation of crystalline representations from the non-logarithmic/smooth case. Furthermore, in the logarithmic setting, we show that log-crystalline representations of $G_R$ are equivalent to Wach modules for $R$, extending our previous results from the (non-logarithmic) crystalline case.

In a recent paper (H. Wang, Math. Comp. 93, 2861-2884, 2024), Wang has presented a number of results concerning the convergence of approximations based on generalised Laguerre polynomials, claiming to have provided "the first rigorous proof of root-exponential convergence of Laguerre approximations for analytic functions". In this note we argue that the proofs of the main results of Wang's paper are incomplete, because they rely on taking a limit under an integral sign, and this is not properly justified. We explain how the proofs could be completed using the dominated convergence theorem, provided that certain nontrivial inequalities involving generalised Laguerre polynomials and their weighted Cauchy transforms hold. We make a related conjecture, which, if true, would complete the proofs of Wang's results. This conjecture remains unproven, but appears plausible, based on numerical experiments.

This paper investigates finite-horizon tail and ruin asymptotics for models with random-size clusters of dependent main and delayed claims under Levy investments. Under long-tailed, dominatedly varying claim distributions, a conditional, uniformly weighted Ko-Tang condition within clusters, and moment conditions, univariate asymptotics are derived for discounted aggregate claims and ruin probabilities. For two business lines sharing renewal arrivals but having independent claim marks and Levy processes, marginal and joint tail asymptotics are established as thresholds diverge, without restricting their ratio. Results provide asymptotic formulas for ruin in both lines, simultaneous ruin, and ruin in at least one line.

A three-dimensional multiphysics finite element model for the simulation of high-frequency induction welding of tubes is presented. The model couples a time-harmonic $h-\phi$ formulation with a stabilized quasi-stationary advection-diffusion heat transport equation, enabling accurate prediction of electromagnetic and thermal fields under industrial operating conditions. The model incorporates parameterized geometry generation and semi-automated, physics-tailored mesh construction and is implemented using the open-source tools GetDP and Gmsh. Validation against measurements from a commercial induction welding line for AISI 304 stainless steel tubes demonstrates good agreement with operating data. The validated model is subsequently applied to investigate the influence of impeder material by comparing a conventional FeNiZnV ferrite with the soft magnetic composite Ferrotron 559H for the induction welding of AISI 304 stainless steel tubes.

A proper conflict-free coloring of a graph is a proper vertex coloring in which every non-isolated vertex has a color appearing exactly once in its neighborhood. A graph $G$ is proper conflict-free $(\mathrm{degree}+2)$-choosable if every list assignment $L$ with $|L(v)|\ge d_G(v)+2$ for each $v\in V(G)$ admits such a coloring from the lists. We prove that every graph with girth at least $7$ and maximum average degree less than $8/3$ is proper conflict-free $(\mathrm{degree}+2)$-choosable. Consequently, every planar graph of girth at least $8$ has this property, improving the previously established sufficient girth bound of $9$.

We prove, in $\mathsf{ZFC}$, that the $\lambda$-terraced cube relation fails whenever $\lambda$ is an uncountable cardinal. The corresponding terraced relation for quadruples fails for every $\lambda$. If $\lambda$ is $\aleph_0$ then the pretinent terraced relation has consistency strength of at least one Woodin cardinal. We prove positive polarized relations at a successor and a double successor from wondrous ideals. We show, however, that there are no such ideals over two consecutive cardinals simultaneously.

We prove commutation relations between the ordinary type A crystal operators and a sign-reversing involution on semistandard set-valued tableaux of skew shape. We describe the involution by Boolean fibers with fixed maximum entries. It commutes with every raising operator except the one indexed by the entry changed by the involution. We characterize commutation at this exceptional raising operator and for lowering operators away from the two colors involving that entry using the box where the involution acts and reduced signatures. We construct a bijection from these tableaux to decorated five-vertex states that preserves weight and excess, subject to a restriction on nontrivial bumps determined by the inner partition. Rotation with alphabet reversal yields the dual statements.

Sequential extremality and stationarity properties are discussed with the emphasis on those corresponding to fixed sequences of translations. Exact quantitative characterisations of the properties are provided. We show, in particular, that the sequential (as well as conventional) extremality and approximate stationarity properties possess certain stability, while the (non-approximate) stationarity does not. Dual necessary conditions for the sequential extremality and stationarity properties with fixed and non-fixed sequences of translations are established. A version of the sequential extended extremal principle is formulated. In the statements, we employ certain generalised separation conditions $(GS)$ and $(GS_\alpha)$ as well as a complementary primal-dual condition $(PD)$. To illustrate the model, we prove dual optimality/stationarity conditions for a constrained minimisation problem in which the minimal value is not necessarily attained.

We consider a pinned harmonic chain perturbed by random velocity flips, which conserve the total energy. Previous results have established the diffusive hydrodynamic limit for the energy profile at the level of expectations. In this article, under suitable fourth-moment bounds on the initial data, we prove for the first time a law of large numbers for the empirical energy distribution. Our approach is based on the Wigner distribution and shows that its random fluctuations vanish in the macroscopic limit. Consequently, the empirical energy profile converges in probability to the deterministic solution of the corresponding heat equation.

We study the expressivity of spiking neural networks, which provide a natural framework for asynchronous, event-driven computation complementary to conventional feedforward neural networks. We consider the time-to-first-spike model in a setting for which the input-output map is continuous and piecewise linear, with affine pieces governed by causal feasibility constraints that determine which presynaptic spikes occur before a neuron fires. We first show that each neuron's firing time admits a maxout-like representation with exponentially many, highly constrained affine pieces. We then formalize causal regions as polyhedral regions with fixed causal sets and derive upper and lower bounds on the maximal number of causal regions in both shallow and multilayer feedforward spiking networks. Our theoretical and experimental results show that spiking networks can generate richer partitions of the input space than conventional feedforward ReLU networks.

Graphical causal inference supplies a complete theory of efficient covariate adjustment for the average treatment effect: one adjustment set, computable from the graph, is optimal under every compatible distribution. We show that this is a property of the average treatment effect's inverse-prevalence weights, not of causal estimands in general. For the average treatment effect on the treated we index the efficiency bound by the adjustment set and derive exact identities for its change under treatment-side and outcome-side extensions of a valid set. Covariates that predict only the treated-arm outcome are exactly efficiency-neutral, and covariates that predict the control-arm outcome can strictly increase the bound when the propensity is below one half – a reversal of the supplementation lemma whose source is an arithmetic-geometric-mean inequality that holds for the average treatment effect and fails for the treated-population estimand. A construction with two faithful distributions on one graph proves that no graphical optimality criterion exists for the treated-population estimand; under no effect modification the ATE-optimal set is nonetheless optimal among the graphically valid sets, with an exact expression for its advantage. The results extend to weighted average treatment effects with propensity-dependent weights, yielding symmetric thresholds for overlap weights, an estimand-drift phenomenon under instrument adjustment, and a characterization of constant weights as the only smooth positive weights for which outcome-side supplementation never increases the bound. Simulations and the LaLonde data provide illustrations.

Let $S=\kk[x_1,\ldots,x_n]$ with its standard $\mathbb N^n$-grading, and let $\mathfrak m=(x_1,\ldots,x_n)$. We study the essential images of the functors $M\mapsto H^i_{\mathfrak m}(M(-\mathbf1))$ on categories of positively determined $S$-modules, where $\mathbf1=(1,\ldots,1)$. For a fixed determining bound $\mathbf t\in\mathbb N^n$ and $0\le i<n$, we identify the image in degree $i$ with the category of finite-dimensional representations of the subposet of $[\mathbf0,\mathbf t]$ consisting of multidegrees with at least $n-i$ positive coordinates. These images are Serre subcategories of the category of negatively $\mathbf t$-determined modules. The top-degree image consists precisely of second cosyzygies in that category and is extension-closed, but is not abelian when $\mathbf t\ne\mathbf0$. For the non-top image categories, we determine attached primes from projective covers and compute the Bass numbers at $\mathfrak m$ and the injective dimensions over $S$ of projective and injective objects. In every non-top degree, we construct an exact functorial section selecting, for each nonzero target, a source of minimal projective dimension $n-i$ and maximal depth $i$ among all its sources. If $n\ge2$ and $\mathbf t\ge\mathbf1$, no fully faithful section exists in degrees $1$ and $n-1$, even without requiring exactness. In top degree, graded Matlis duality followed by $S$-duality gives a fully faithful functorial section. We also describe the global images obtained by varying the determining bound. The non-top global categories have enough injectives and no nonzero projective objects, while the top category corresponds to second cosyzygies in the category of finite-support representations of $\mathbb N^n$.

We consider the problem of finding sparse solutions of an underdetermined linear system $Ax=b$. In contrast to conventional approaches based on greedy algorithms or convex relaxation, we reformulate sparse approximation as a structured system of polynomial equations and connect with the literature on tensor methods. We develop an eigenvalue decomposition based method that formally guarantees recovery of all sparse solutions if there is more than one. We also develop two optimization-based methods achieving favorable computational complexity. The new methods allow explicit control of the target sparsity. Numerical experiments illustrate the performance and compare to basis pursuit (denoising) and orthogonal matching pursuit.

Optimal scheduling of combined chemo-immunotherapy is often formulated as a control-affine optimal control problem, which generically yields boundary-selected (bang-bang-type) protocols unless singular arcs occur. This structure, while convenient, neglects saturating pharmacodynamics at high dose rates. We incorporate Michaelis-Menten saturation directly into the therapy channels, making the dynamics non-control-affine and the Hamiltonian nonlinear in each control. With strictly positive exposure penalties, the Hamiltonian admits an explicit three-regime pointwise minimization law: each input is chosen at the lower bound, the upper bound, or as a unique interior minimizer. On interior intervals the strict Legendre condition holds, so continuously modulated dosing arises as a regular interior extremal rather than a singular-arc or smoothing artifact. The resulting transcription is nonconvex; we therefore use a hybrid Quantum Particle Swarm Optimization (QPSO)-Sequential Quadratic Programming (SQP) pipeline, where QPSO provides a constraint-aware warm start and SQP enforces feasibility and Karush-Kuhn-Tucker (KKT) optimality on a collocation grid. Costate reconstruction corroborates the Pontryagin Minimum Principle (PMP) structure and the predicted boundary/interior regime transitions.

The discovery of the EML operator, sufficient to evaluate the standard explicit purely transcendental elementary functions, has led to considerable interest and discussion across multiple scientific disciplines. However, most authors have focused on the binary EML itself, while numerous similar variants with slightly different properties are now known. This article attempts to close this gap by enumerating and classifying them. We also take this opportunity to clarify common misconceptions related to the EML operator. The principal goal, symbolic regression within an architecture as close as possible to proven neural networks which combine matrix multiplication with a single univariate non-linear activation function, remains beyond reach. Instead, we propose a Möbius layer, with rational functions replacing matrix operations, and showcase the recently discovered activation function eml(x,1/x), which allows exp(x) and ln(x) to be recovered separately, and hence all elementary functions to be evaluated within a rational generalization of the neural network.

Convex Optimization with Nested Evolving Feasible Sets (CONES)} was introduced in \cite{CONESVaze} where the objective function \(f\) remains fixed but the feasible region evolves over time as a nested sequence \(S_1 \supseteq S_2 \supseteq \cdots \supseteq S_T\). The goal of an online algorithm is to simultaneously minimize the regret with respect to hindsight static optimal benchmark and the total movement cost $M_\cA(T)$ while ensuring feasibility at all times. CONES is an optimization-oriented generalization of the well-known nested convex body chasing (NCBC). In this paper, we extend CONES to allow for loss functions $f_t'$s to also change over time. When all loss functions are convex, we show that the projected proximal algorithm achieves $O(T^{1-\beta}), O(T^\beta)$ simultaneous regret and movement cost, respectively, for any $\beta \in [0,1)$, over a time horizon of $T$. We also show that any weakly adaptive online algorithm with $O(T^\beta)$ regret has a movement cost of $\Omega\left(T^{\frac{1-\beta}{2}}\right)$ for any $\beta \in [0,1)$. When all loss functions are strongly convex, we show that the projected proximal algorithm simultaneously achieves $O(1)$ regret and a movement cost of $O(\log T)$. To complement this, we show that any online algorithm with sublinear anytime regret has a movement cost of $\Omega\left(\log T\right)$.

Free boundary problems (FBPs) are differential equations where the domain depends on the solution, with applications in diverse fields such as flame propagation, financial mathematics, and tumor growth. Classical examples include the Bernoulli problem, the obstacle problem, and the Stefan problem. The field has advanced significantly through variational and geometric approaches, notably by Alt, Caffarelli, and collaborators, who developed monotonicity formulas and viscosity solutions to address free boundary regularity. Extensions have included nonlinear operators, inhomogeneous problems, and lower-dimensional cases like the thin Bernoulli problem. Evolution problems, such as the Hele-Shaw flow, also connect to FBPs. To celebrate these developments and honor Sandro Salsa's pioneering contributions, a special session on FBPs was held at the AMS-UMI joint meeting in Palermo (2024). This is the editorial of a volume dedicated to Sandro Salsa on the occasion of his 75th birthday.

We show that Jacobi-finite potentials need not exist on finite $2$-acyclic quivers. Our main tool is a matrix-valued Golod–Shafarevich–Vinberg inequality for quotients of completed path algebras by finitely many, possibly nonhomogeneous, topological relations. Applied to cyclic derivatives, it yields a potential-dependent obstruction to the finite-dimensionality of completed Jacobian algebras. We then construct a purely quiver-level criterion excluding every Jacobi-finite potential on a given quiver, and exhibit a family of quivers for which every potential has an infinite-dimensional Jacobian algebra.

Optimal control in cancer chemotherapy is challenged by tumor heterogeneity and mutations, which complicate treatment effectiveness. Traditional methods, such as Pontryagin's maximum principle (PMP), are often hindered by their reliance on an initial guess for the costate equation, affecting accuracy and convergence. To address these limitations, this work introduces a hybrid quantum particle swarm optimization (QPSO) method based on regularization. QPSO is employed for global exploration to approximate the optimal control trajectory, followed by a regularization-based refinement to ensure smoothness and consistency with optimality conditions. The Hamiltonian function is used for first- and second-order optimality checks, verifying solution quality. Numerical case studies explore various drug effectiveness functions, demonstrating the role of periodic and localized drug delivery in achieving robust tumor suppression and providing insights into the impact of different drug combinations on optimal chemotherapy strategies.

Our main result is a $3\sqrt{2\pi}$ bound for the Komlós signing problem: every finite family of real vectors of Euclidean norm at most one admits a signed sum of $\ell_\infty$-norm less than this constant, independently of the dimension and the family size. For any $\kappa\ge0$, if a bounded open convex set supports a probability density with directional total variation at most $\kappa$ in every unit direction, then its open-set Banaszczyk transform supports another such density with the same $\kappa$, provided the translation vector $v$ satisfies $\kappa\|v\|_2\le1/3$. As a consequence, every finite set system in which each element belongs to at most $t$ sets, where $t\ge1$ is an integer, admits a two-coloring whose imbalance in each set is less than $3\sqrt{2\pi t}$. This gives the square-root dependence predicted by the Beck-Fiala conjecture. The proof was discovered by the Odin Automatic AI Research Agent.

This paper is the second in a two-part investigation of the physics of information geometry. While Part I develops a physical foundation for distributional motion on the probability simplex, the present paper studies how that framework manifests in active inference. The treatment is fully self-contained and does not require familiarity with Part I. We focus in particular on active inference through small distributional steps and the geometric structure induced by such local motion. Starting from an initial distribution, an agent evolves its belief state toward a final target distribution through a sequence of constrained updates. We define a relative free energy functional with respect to the preferred distribution and extend it to a relative potential energy analogous to the Helmholtz/Gibbs free-energy decomposition. The evolution is subject to a per-step kinetic constraint expressed through the Kullback-Leibler (KL) divergence between consecutive distributions, which serves as a discrete kinetic energy on the probability simplex. Using the information-geometric Pythagorean theorem on KL balls, we show that sufficiently small local moves dominate large direct jumps, and that greedy maximization of free-energy reduction is globally optimal under the kinetic constraint. This leads to a sequential variational principle in which the optimal trajectory minimizes the associated Lagrangian of the optimization problem. Similar to classical mechanics, the Lagrangian takes on the form as the difference between the kinetic and potential terms, establishing a least-action principle for distributional motion on the simplex. The resulting optimal update admits a closed form as an exponentially tilted version of the current distribution toward the preferred distribution, parametrized by an inverse-temperature-like multiplier. We further extend the framework to incorporate state-dependent geodesic...

We investigate sharp inequalities for ratios of Riesz capacities with negative exponents by combining computational experiments with rigorous analysis. For finite subsets of the line, we prove positivity of equilibrium masses when $-1<p<0$, enabling numerical tests of conjectured extremal ratios. In the plane, comparisons of the disk with regular polygon vertex sets reveal a cascade of transitions among the tested competitors and suggest a precise conjecture for the equilibrium measure of odd polygons, for which we give a partial proof. Numerical intersections of equality curves show that the regions where these sets outperform the disk are not simply nested. Similar numerical intersections occur in three dimensions between the regular-simplex equality curve and those of explicit five-point and six-point configurations. Motivated by the dimensional dependence of these comparisons, we prove that for each fixed $p<-2<q<0$, the regular simplex has a larger capacity ratio than the ball in all sufficiently large dimensions. Accompanying Python and Mathematica code supports reproduction and further testing of the conjectures.

Bilevel optimization has recently attracted growing attention, particularly in the development of efficient numerical methods. Despite substantial progress on optimistic bilevel optimization, pessimistic bilevel optimization (PBO) remains much less explored, especially the design of fully first-order, single-loop gradient-based methods. To address this gap, we propose a smooth approximation of PBO through reformulation, penalization and regularization, and establish convergence guarantees in terms of both minimizers and stationarity. Building on this framework, we then develop two single-loop algorithms for deterministic and stochastic PBOs, respectively. Both use only first-order gradient information and avoid second-order derivatives and inner-loop subproblem solves. Non-asymptotic convergence rates for the proposed algorithms are established to provide theoretical guarantees. Through a systematic empirical study of both synthetic and practical problem instances, we demonstrate that our algorithms are highly effective and efficient. In particular, our results on spam classification and Smart Predict-then-Optimize further illustrate PBO's advantages over its classical optimistic bilevel counterpart, highlighting its strong potential for practical modeling and the delivery of robust solutions.

Let $X$ be a smooth scheme of dimension $d$ over a field $F$. We study the abelian-group structure of the higher Chow groups $CH^{d+i}(X,j)$. For a prime $l$ different from the characteristic of $F$, we prove divisibility and torsion-freeness results when $i\ge$ the $l$-cohomological dimension of $F$. If $X$ is smooth proper and geometrically irreducible, we also study the kernel of the push-forward $CH^{d+i}(X,j)\to CH^i(F,j)$. We apply these results to finite fields, local fields, and global fields.

We propose a sequential smoothing majorant stochastic approximation (SMSA) method for nonconvex-nonconcave minimax optimization problems. To overcome the nonconcavity of the inner maximization problem, we introduce a new majorant stochastic approximation with an $O\left(\beta_N^2\right)$ accuracy bound, where $\beta_N$ is the sample coverage radius. The accuracy bound significantly improves upon the $O\left(\beta_N\right)$ approximation bound for the standard stochastic approximation. Moreover, we establish nonasymptotic bounds for both global optimal values and minimizer sets, and prove consistency for the Clarke stationary points, as $\beta_N \downarrow 0$ almost surely. We show that the generated sequence by the SMSA method is bounded, and that the returned point is an approximate Clarke stationary point of the majorant stochastic approximation models. Numerical experiments on a synthetic toy example and robust logistic regression on two UCI datasets demonstrate improved approximation fidelity and lower mean robust test losses relative to the standard sampled approximation.