Papers

We answer negatively the 1-unconditional part of Question 6.4, by Langemets, Lõo, Martín, Perreau and Rueda Zoca, and, more generally, prove that no infinite-dimensional Banach space with a $K$-unconditional basis, for $1\leq K<3/2$, can satisfy the super Alternative Daugavet property. We also address two recent questions posed by Lõo and Perreau concerning weak topological structures and slicely countably determined phenomena in Banach spaces with unconditional bases. More precisely, we prove that the weak unit ball of every Banach space with a 1-unconditional basis admits a countable $\pi$-base, thereby giving a positive answer to Question 5.1. We further show that every bounded convex subset of a Banach space with a Schauder basis that is shrinking or boundedly complete admits a countable $\pi$-base for its relative weak topology. We complement this result with two permanence principles: one for shrinking Schauder decompositions whose finite partial sums have countable weak $\pi$-bases, and another for unconditional sums over boundedly complete Banach sequence spaces. Finally, we prove that, for every $k>1$, there exists a Banach space with a $k$-unconditional basis whose unit ball is not slicely countably determined, thereby giving a positive answer to Question 5.4.

We investigate biological pest control strategies for the sugarcane borer Diatraea saccharalis through the combined action of two parasitoid species: the egg parasitoid Trichogramma galloi and the larval parasitoid Cotesia flavipes. We describe the population dynamics with a six-dimensional host-parasitoid model in which host-parasitoid interactions are represented through a Holling Type II functional response, extending previous models by coupling egg and larval stage dynamics and incorporating parasitism saturation observed in laboratory experiments. We characterize the equilibrium structure of the model and analyze the local stability of the extinction equilibrium. Bifurcation analysis reveals that, for a wide range of Holling parameters, the pest population exceeds the economic damage threshold, motivating the design of active control strategies. We formulate and compare three biological control approaches: open-loop optimal control, State-Dependent Riccati Equation (SDRE) feedback control, and impulsive feedback control based on Lyapunov arguments. We perform numerical simulations to show that all three strategies successfully keep the pest population below the economic damage threshold. The impulsive strategy, in particular, achieves effective suppression with substantially fewer parasitoid releases than the continuous approaches, making it the most practically viable option for field implementation.

We study a Newton acceleration for sparse composite optimization on the Stiefel manifold. The main difficulty is geometric: the active manifold identified by the nonsmooth regularizer may fail to intersect the Stiefel manifold transversely, which obstructs a Riemannian Newton step on the identified manifold. In the transverse case, we prove local identification of the ManPG tangent proximal mapping. For nontransverse cases, we introduce an off-diagonally perturbed Stiefel family that generically restores the identification geometry while yielding an \(O(\|\Delta\|_F)\)-KKT guarantee for the original problem. We also derive verifiable support-level conditions for clean intersection, which cover nontransverse sparse patterns and yield the smooth moving local models used by the Newton correction. Based on these results, we propose MIX, a safeguarded ManPG/Newton-CG method on moving identified intersections. In the general clean-intersection setting, we prove global descent and KKT-residual guarantees for MIX. In the transverse or generically perturbed cases, if the sequence has an accumulation point satisfying certain regularity assumptions and the second-order sufficient condition (SOSC), then the full sequence converges to that point, with finite active-manifold identification and a local Q-superlinear rate. Numerical experiments on compressed modes and sparse PCA show that MIX substantially improves efficiency while preserving solution quality. Beyond the Stiefel manifold, we also outline how the safeguarded global-convergence mechanism of MIX extends to general smooth equality-constrained manifolds.

Conventional statistical models often struggle to fully capture the complex spatio-temporal dynamics, intermittent fluctuations, and heavy-tailed distributions characteristic of real-world air pollution data. Furthermore, existing literature frequently focuses on extreme events, overlooking the persistence of low-pollution states and temporal memory effects. To address these gaps, we apply superstatistical frameworks from non-equilibrium statistical physics to analyse a comprehensive five-year dataset (2020-2025) of hourly air pollutant concentrations across the United Kingdom. Excellent fits of experimentally measured distributions are obtained from our theoretical models. We observe large heterogeneities of the best fitting parameters depending on the locations where the measurements are performed. These parameters form characteristic patterns in the 3-dimensional parameter space and depend on the type of pollutant considered, as well as on the environmental conditions (high traffic, industrial, or rural surroundings). We also investigate autocorrelation functions and provide evidence for differences in day-time and night-time decays of the autocorrelation function. Our investigation mainly focuses onto the dynamics of NO, NO2, PM2.5, PM10, but we also report on some anomalous distributions observed for O3.

Amdeberhan, Shareshian, and Stanley recently proved that a function $\varphi$ arising in the theory of partition Eisenstein series counts the alternating permutations of $\{1,\dots,2n\}$ with a given `record' partition, and they asked whether there is a similar theory for record compositions, suggesting a role for noncommutative symmetric functions. Here we solve their open problem by showing that the number of alternating permutations of $\{1,\dots,2n\}$ with record composition $(\alpha_1,\dots,\alpha_\ell)$ is \[ \prod_{j=1}^{\ell}\binom{2s_j-1}{2\alpha_j-1}E_{2\alpha_j-1}, \] where $s_j=\alpha_1+\dots+\alpha_j$, $E_k$ is an Euler number, and the record composition of $w=a_1a_2\dots a_{2n}$ (so $a_1>a_2<a_3>\dotsb$) lists the factor lengths obtained by cutting $a_1a_3\dots a_{2n-1}$ before each left-to-right maximum other than the first. These numbers are the coefficients of a natural lift of the degree-$n$ sprout symmetric function with seed $\sec(\sqrt{t}\,)$ to noncommutative symmetric functions, expanded in products of noncommutative power sums of the first kind. An analogous refinement holds for every sprout sequence whose seed is given by the exponential formula. AxiomProver autonomously produced and verified the results in this paper in Lean.

We study diffusion equations with completely monotone memory when the instantaneous diffusion form is merely non-negative and may therefore lose coercivity. For a kernel whose Bernstein representing measure has finite total mass $M_{0}=\nu([0,\infty))$, we introduce an extended state consisting of the physical variable and its continuum of internal variables. The aggregation and constant-embedding operators are adjoint with respect to the memory energy, and the resulting cross-term cancellation makes the augmented generator $m$-dissipative. This yields a unique mild solution, Lipschitz dependence on the data, and a contraction estimate that contains no positive lower bound for the instantaneous form. The zero-prehistory trajectories form a memory graph space, in which the problem is well posed in the sense of Hadamard. If, in addition, the first Bernstein moment $M_{1}=\int_{[0,\infty)}\lambda\,\diff\nu(\lambda)$ is finite, the memory potential and first-moment field possess the regularity needed to identify the semigroup solution with an encoded weak formulation and to obtain explicit stability bounds. We further prove uniform norm-resolvent convergence and convergence of the associated semigroups when a coercive instantaneous contribution vanishes. Under an additional $L^{2}(0,\Tend;V)$-regularity assumption on the limiting solution, the convergence rate in the memory graph norm is $O(\varepsilon^{1/2})$. These results provide a continuous stability target for structure-preserving and certified discretisations of memory-dominated diffusion.

In this work, we provide a comprehensive survey of the code equivalence problem and its variants. We explain the existing results, highlighting the relationships between different problem formulations, algorithmic techniques, and hardness assumptions. In addition, we systematically review known attacks, identify the parameter regimes in which they are effective, and discuss their limitations. Lastly, we outline several open problems and research directions, with the aim of clarifying the current landscape and guiding future work toward a deeper understanding of the hardness of code equivalence.

We consider a quiz show game in which a contestant is presented with a sequence of questions. Each time the contestant answers a question correctly, she receives a prize and proceeds to the next question; the probability of answering each question correctly is given. If the contestant answers a question incorrectly, she receives a consolation prize and the game ends. The contestant's problem of determining the optimal order in which to answer questions, for known fixed parameters, is a classic one studied in Kadane (1969), and admits a simple index-based solution. We consider game-theoretic versions of this problem in which a game show host can choose how to allocate a fixed prize budget. Our models are motivated by operational search problems in national security involving reconnaissance missions and inspecting for evidence of nuclear enrichment, as well as certain scheduling problems. We study three variants of the game, corresponding to different ways in which the host can distribute the prize money. For the first variant, we provide complete closed-form solutions, including equilibrium strategies and the value of the game. We reduce the second variant to a game solved in the literature. For the third variant, we obtain partial results by analyzing a more general game with a geometric structure.

In this paper we investigate the uniqueness of enhancements of the natural subcategories of weakly approximable triangulated categories. The main idea is to enhance at the level of $\infty$-categories the recently developed theory of excellent metrics. The applications of our results include a vast generalization of the known results about the (strong) uniqueness of enhancements in the linear and nonlinear setting, providing positive answers to some open questions. In addition we prove that, under some natural assumptions, the equivalences between such subcategories can be lifted through their natural inclusions. This completes the picture started in our previous paper arXiv:2505.10374 and extends the known results about Margolis Uniqueness Conjecture for the homotopy category of spectra.

In spatially structured populations, rare neutral mutations can spread through large regions during a range expansion, a phenomenon known as gene surfing. Whether deleterious mutations can also surf remains poorly understood. To address this question, we study a deterministic version of the spatial Muller's ratchet, given by an infinite system of reaction-diffusion equations describing an asexual population subject to mutation, migration, and density-dependent reproduction and death. After establishing that the system of PDEs is well-posed, we analyse the distribution of deleterious mutations within the population. In the monostable regime, we derive quantitative bounds on the ratio between the density of individuals carrying a given number of mutations and the density of mutation-free individuals. Under a Fisher-KPP condition, we further determine the spreading speed of the population into an empty habitat, confirming non-rigorous computations of Foutel-Rodier and Etheridge. Finally, using a tracer dynamics approach, we show that deleterious mutations cannot surf deterministic waves: although they are present at the expansion front, they only arise as recent descendants of the wild type.

History-dependent entropic variational formulations require a calibrated terminal differential for accumulated power and a spatial power split that remains meaningful for weak diffusion fields. Endpoint calibration and cocycle additivity determine a unique oriented spending increment, while independent local selector fields give its distributional channel form. The diffusion identity for a potential-weighted flux is established at $H(\Div)$ regularity and extended to the finite-energy class $H^1\times L^2$ through a distributional balance component. For regular diffusion models, the weighted species channel yields the local balance whenever persistent zero-potential states are dynamically isolated in the admissible state class. An independent multiplier extends the same balance to the natural weak space. These ingredients are assembled in one synchronized thermo-diffusion functional whose directional stationarity yields energetic and thermal conjugacy, species balance, flux closure, the entropy equation, and both terminal routing rules. A Cahn–Hilliard specialization verifies the construction at finite-energy regularity.

In this paper, we mainly focus on the fractional GJMS operators $P_{2\gamma}$ which are defined on the conformal infinity of a Poincaré-Einstein manifold. We derive two comparison inequalities of the fractional Yamabe constants associated to the fractional GJMS operators. One is between $P_{1}$ and $P_{2\gamma}$ for $\gamma\in (1/2,1)$, and the other is between $P_{2}$ and $P_{2\gamma}$ for $\gamma\in (1,2)$. They both imply the rigidity theorems by characterizing the equalities. Together with the result in \cite{WZ1}, we partially provide some evidence for the monotonicity of the fractional Yamabe constants.

Nonparametric permutation tests are widely used for statistical analysis. However, exact computation of test p-values can be algorithmically challenging, particularly for custom tests with complex test statistics. In contrast, Monte Carlo sampling can be easily applied to any test statistic, but it suffers from poor relative accuracy when estimating small p-values, interfering with multiple hypothesis testing correction and leading to other issues. In this work, we present a hash-augmented adaptive multilevel splitting Monte Carlo algorithm that enables accurate estimation of arbitrarily small p-values in two-sample permutation tests. Using the Kolmogorov-Smirnov and the Mann-Whitney U tests as examples, we highlight potential pitfalls related to the discreteness of the test statistic distribution and show how to address them. By comparing with an exact algorithm, we demonstrate the accuracy of the p-value estimates provided by the proposed algorithm and the validity of the associated confidence intervals. We provide a reference implementation of the proposed algorithm in the Python package hamstest, which allows p-value estimation for a user-defined statistic.

We show that for Schrödinger operators in a connected domain, the Hohenberg-Kohn theorem holds within the class of Laplace form-bounded external potentials if and only if the single-particle density is strictly positive quasi-everywhere. Furthermore, we show that this condition is satisfied for non-interacting Schrödinger operators whenever a ground-state exists. Consequently, we establish the Hohenberg-Kohn theorem for non-interacting systems, and thereby the uniqueness of the Kohn-Sham potential, within the maximal class of Laplace form-bounded distributions. The main ingredient to establish these results is a characterization of regular states, whose proof relies on tools from classical potential theory. Moreover, this characterization reveals that, in the continuum setting, the fundamental mechanism underlying the Hohenberg-Kohn theorem is the (quasi)-unique continuation of the density rather than of the many-body wavefunction.

We identify a common convexity structure for three exponential Dirichlet problems on smooth uniformly strictly convex domains: the Liouville equation $\Delta u=e^u$, the real equation $\sigma_2(D^2u)=e^{2u}$, and its complex counterpart $\sigma_2(u_{i\bar j})=e^{2u}$. In each case $u<0$ in the domain and $u=0$ on the boundary. We prove that \[ w=-\operatorname{arcosh}(e^{-u/2}) \] is strictly convex in the underlying real variables. The argument combines domain deformation, constant-rank theory, inverse-convexity estimates, radial ball models, boundary strict convexity, and local $C^2$ stability.

Collective dynamics in multi-agent systems provide a powerful framework for understanding how coherent group-level patterns can emerge from simple interactions between individuals. Such phenomena are observed in many natural and artificial systems, including animal groups, robotic swarms, and distributed decision-making processes. In many situations, agents are not only characterized by their spatial motion, but also by internal states, e.g., opinions or preferences, which evolve through interactions with peers. Understanding how these internal states influence collective motion, and how spatial organization in turn affects internal dynamics, remains an important challenge. In this work, we propose a model of coupled collective motion and opinion dynamics. The spatial dynamics are governed by attraction–repulsion interactions, while the internal dynamics are described by a Deffuant-type opinion model. Our results show that the confidence threshold of the opinion dynamics plays a key role in controlling the number of opinion clusters, whereas the strength of the opinion-dependent spatial attraction determines whether these clusters spatially merge or remain separated. In addition, for the full-consensus state, we derive the expression for the radius of the stationary swarm distribution when a nonlinear attraction kernel is used, using a semi-analytical approach. The proposed framework may be useful for studying collective decision-making, animal group behavior, and coordination strategies in swarm robotics.

An introduction to Alexandrov spaces with curvature bounded below. Topics include various comparison conditions, the globalization theorem, tangent spaces and spaces of directions, gradient flows, the splitting theorem, dimension and volume, Gromov's selection theorem, the boundary and the doubling theorem, and quotient spaces. We also give a brief overview of the two-dimensional theory, the main precursor to modern Alexandrov geometry.

We determine the exact region $\Omega_{\beta,\phi,\tau}= \{(\beta(C),\phi(C),\tau(C))\mid C\in\mathcal{C}\}$ of possible joint values of Blomqvist's beta, Spearman's footrule and Kendall's tau over the class $\mathcal{C}$ of all bivariate copulas. The region consists precisely of all triples $(\beta,\phi,\tau)$ satisfying $-1\le \beta\le 1$, $\frac{3}{16}(1+\beta)^2-\frac12\le \phi\le 1-\frac38(1-\beta)^2$ and $\frac43 \phi-\frac13\le \tau\le \frac23 \phi+\frac13$. In other words, the known exact $(\beta,\phi)$- and $(\phi,\tau)$-regions already characterize the joint region, so that, once the value of Spearman's footrule is fixed, Blomqvist's beta imposes no additional restriction on the possible values of Kendall's tau. The proof is constructive: two $2$-parameter families of shuffles of $M$ realize all the points in $\Omega_{\beta,\phi,\tau}$ with $\phi = \frac{3}{16}(1+\beta)^2-\frac12$, and ordinal sums spread realizations of these two families over the whole region $\Omega_{\beta,\phi,\tau}$.

Circular data, representing angles or directions, are frequently encountered in computer vision, biology, geology, and meteorology. Traditional regression targets the conditional mean, which is often geometrically misleading for circular responses under multimodal, skewed, or asymmetric data structures. To address these limitations, a lightweight deep generative framework, namely ANGLE, is introduced for non-parametric distributional regression on the circle. The full conditional distribution of an angular response, given Euclidean and circular covariates, is learned through a generative map optimized via a generalized circular energy score (GCES) loss. Desirable theoretical properties, including the strict propriety of the loss and the rotational equivariance of the estimators, are established. Furthermore, both pre- and post-additive noise models are accommodated. A unified toolbox is provided for advancing previously underexplored challenges in circular statistics: extrapolation, sufficient dimension reduction, and conditional distribution equality testing. The framework's efficacy is demonstrated through extensive simulations and real-world applications. Specifically, the proposal is utilized for object pose estimation from imagery and wind direction prediction, which are integral to surveillance, autonomous vehicles, and energy systems, respectively. Superior predictive performance and robust uncertainty quantification of the proposed method in these tasks are revealed.

Conditional independence testing (CIT) is fundamental to modern statistical inference in areas related to causal discovery and variable selection. While marginal independence is relatively well-understood, despite multiple advances, no existing non-parametric CIT provides a unified, efficient, and statistically guaranteed solution across heterogeneous data. We introduce a graph-based test statistic comparing kernel similarities of the response within composite neighborhoods that use exact matching on discrete components and $k_n$-nearest-neighbor matching on continuous ones. The raw statistic, related to prior constructions, suffices under fully discrete conditioning. However, when at least one conditioning variable is continuous, we instead use a local-polynomial debiased variant that cancels the local smoothing bias. We rigorously establish its asymptotic null distribution across all data-type combinations. We further prove a dimension-free $n^{-1/4}$ detection threshold under local alternatives, eliminating the phase transition that affects geometric estimators in high dimensions. Finally, we develop efficient algorithms with near-quadratic complexity and analytic graph-based calibration, bypassing the cubic bottlenecks of global kernel methods.

For any irrational number $ \alpha>\frac{3+\sqrt{5}}{2}\approx2.618$ and given a positive $ n\in\mathbb{N} $, we use elementary number theory to introduce a necessary and sufficient condition for a natural number $ x $ to be in the $n$th iterate of the Beatty sequence of modulus $\alpha$.

We characterize stability and its robustness for a class of boundary controlled, boundary observed hyperbolic partial differential equations. In particular, we show that asymptotic and exponential stability coincide for this class. Furthermore, we introduce the real and complex stability radii for which we give explicit formulas. In addition, we consider the dual system and its application to controllability and observability. Our main results are illustrated with two examples.

In this paper, we investigate the convergence in the $L^{p}$-norm and certain shape-preserving properties of the max-product generalized sampling operators. More precisely, we establish quantitative estimates for the approximation error in the $L^{p}$-norm, for $ 1 \le p < +\infty$, in the case of non-negative and bounded functions defined on $[-1,1]$. These estimates are derived by means of the so-called $\tau$-modulus, an averaged modulus of smoothness introduced by Sendov and Popov. As a direct consequence, we prove that the max-product generalized sampling operators $L^{p}$-converge to non-negative functions that are measurable, bounded and Riemann integrable on the interval $[-1,1]$. In the final section, we extend several shape-preserving results of Coroianu and Gal, originally established for specific kernels (such as the sinc/Whittaker and Fejér kernels), to the broader class of smooth centered bell-shaped kernels. Under suitable assumptions on the kernel, we prove that the max-product generalized sampling operators partially preserve the monotonicity of any function $f:[0,1] \rightarrow \R_{0}^{+}$ that is either non-decreasing or non-increasing on $[0,1]$.

This paper presents a least squares weak Galerkin (LS-WG) finite element method for a class of second order elliptic equations of Fokker-Planck type. To address the numerical challenges arising from non-smooth diffusion tensors, the proposed method utilizes a least-squares formulation that yields a symmetric positive definite (SPD) discrete system. The numerical scheme is designed by employing locally constructed weak second order partial derivatives and the weak divergence commonly used within the weak Galerkin framework. A rigorous theoretical foundation is provided, establishing the uniqueness of the discrete solution and deriving optimal-order error estimates in a discrete energy norm. Finally, extensive numerical experiments are reported to validate the theoretical findings and demonstrate the robustness and performance of the numerical scheme.

For complex Monge-Ampère equations with moving big cohomology classes and prescribed model singularities of positive Monge-Ampère mass, we prove that, under total variation convergence of the right-hand side non-pluripolar positive Radon measures, convergence of the prescribed model potentials in Monge-Ampère capacity is equivalent to convergence in capacity of the associated normalized solutions. We further prove that the ceiling operator coincides with the singularity envelope for potentials associated to a big $(1,1)$-class, regardless of their Monge-Ampère mass, thereby resolving a conjecture of Darvas-Di Nezza-Lu. Consequently, the singularity envelope is idempotent without the positivity assumption on the mass.

Non-stationary time evolution of interacting particle systems is in general a rather difficult topic however, exceptional examples are known where hidden processes within the model make the description manageable. These hidden processes often take the form of a finite number of interacting random walks and in some cases, rather than being hidden, were very explicitly revealed as second class particles associated with the model. The examples of asymmetric exclusion and exponential bricklayers model are known in this context, where such distributional structure was demonstrated in infinite volume. Here we find boundary mechanisms that save this remarkable structure in finite volume of the model with semi-open boundaries that let ordinary particles, but not the shocks, through. This finding also allows to characterise nontrivial two-species, non-reversible stationary distributions subject to our special boundary rates.

We give an explicit algebraic description for the infinitesimal invariants associated to the Griffiths-Pirola cycle on the universal genus 4 curve and its self intersection. We apply this to refine a result of Griffiths which recovers from these invariants the equations for the general genus 4 curve, and give a new proof (only in genus 4) of a nonvanishing result of Green and Griffiths.

Snir and Yuster [Discrete Appl. Math. 347 (2026) 160–171] asked for the least number $h(k)$ such that $k$ unrooted binary phylogenetic trees on the same $h(k)$ leaves always share a common quartet. We give a new upper bound for the $k$-tree version of the Maximum Agreement Subtree problem, namely an upper bound for the number of leaves, on which $k$ unrooted binary phylogenetic trees always share a common induced binary subtree on $n$ leaves, which is a four-times iterated exponential function. For $h(k)$, this implies a four-times iterated exponential upper bound. We also set an exponential lower bound for $h(k)$.

In this article, we investigate the Bhor phenomenon for holomorphic functions defined on a general simply connected domain in $\mathbb{C}^n$. We improve the existing results Evdordis et al. (Improved Bohr's inequality for shifted disks, Results in Mathematics, 76, 14 (2021)) for a broader class of holomorphic functions in $\mathbb{C}^n$. Furthermore, we consider pluriharmonic mappings defined on a polydisk containing the unit polydisk $\mathbb{P}\Delta(0_n, 1_n)$ and establish a Bohr-type inequality for this class of mappings.

Scenario-based model predictive control (SBMPC) is a variant of model predictive control (MPC) that explicitly accounts for uncertainty by optimizing control actions over multiple predicted scenarios. However, its computational complexity increases rapidly with the number of scenarios and prediction horizon, limiting its applicability to real-time planning and control. This paper presents a learning-accelerated Alternating Direction Method of Multipliers (ADMM) algorithm for efficiently solving SBMPC problems by leveraging parallel computing and Moreau envelope learning, while maintaining high solution accuracy. We reformulate the SBMPC problems into consensus forms that can be decomposed via ADMM, separating the scenario-dependent dynamics from non-anticipativity constraints and enabling parallel updates across scenarios and time steps. Building on this decomposition, we utilize a learning-to-optimize scheme that leverages Moreau envelope learning of the cost function to accelerate the primal update in ADMM, thereby reducing computation time. The proposed framework is evaluated on a microgrid energy management problem subject to load and renewable generation uncertainties. Comparisons with IPOPT and MadNLP, two popular and modern nonlinear programming solvers, demonstrate substantial computational speedups while maintaining reliable closed-loop control performance.