Papers

We extend the “Brownian bees” model of Berestycki et al.\ (2021, 2022) to cooperative reproduction, $kA\to(k{+}1)A$, of a population of $N$ symmetric random walkers with removal, at each birth event, of the particle farthest from the origin. Working in the limit $N\to\infty$, we formulate a hydrodynamic free-boundary problem for this model. Using this formalism, we determine steady state population densities for all~$k$ and prove their linear stability for $k\le 2$ and instability for $k\ge 4$. In the marginal case $k=3$, there is a whole continuous family of steady states at a single critical ratio of the reproduction and diffusion rates. Above criticality the population undergoes an asymptotically self-similar finite-time collapse to the origin. Below the criticality the population spreads diffusively, but the reproduction remains quantitatively relevant. For $k\ge 4$, the unstable steady state separates regimes of a finite-time collapse and diffusive spreading. Here the collapse dynamics is asymptotically self-similar, and the population density exhibits a scale separation requiring a matched-asymptotic description. Our analytical predictions are confirmed by numerical solutions of the hydrodynamic free-boundary problem and by Monte Carlo simulations of the original microscopic model.

Let $X$ be a proper smooth rigid analytic variety over a complete algebraically closed $p$-adic field $\mathbf C$. Fix a continuation $\operatorname{Exp}$ of $\exp$. Faltings, in the curve case, and Heuer showed that any lifting $\widetilde X$ of $X$ over $B_{\mathrm{dR}}^+/t^2$ induces an equivalence between the category of Higgs bundles on $X_{\text{\'et}}$ and the category of $v$-vector bundles on $X_v$. Changing $\widetilde X$ is known to alter the correspondence nontrivially. This raises the natural question of how the isomorphism class of the Higgs bundle associated with a fixed $v$-vector bundle varies with $\widetilde X$, a question that has not been systematically studied. In this paper, we address this question when $X$ is a curve of genus $g\ge 2$. More precisely, we call a Higgs bundle lift-independent if it corresponds to the same $v$-bundle under the $p$-adic Simpson correspondence for every lifting $\widetilde X$. We prove the following results. (1) Every lift-independent Higgs bundle of rank $r\le g$ is nilpotent. (2) Every semistable lift-independent Higgs bundle of rank $r\le \sqrt{g}+1$ has zero Higgs field. (3) In contrast, vanishing fails even within the class singled out by Faltings' conjecture: there always exists a lift-independent semistable Higgs bundle of degree $0$ with nonzero Higgs field.

Integrated sensing, communication, and computation (ISCC) provides a promising framework for indoor human-centric applications. In these applications, short-term human pose prediction facilitates continuous human tracking and resource allocation in advance. In this paper, we propose a Cramer-Rao bound (CRB) guided resource allocation framework for indoor mmWave ISCC systems to minimize the human pose prediction error under communication, latency, and energy constraints. We characterize the impact of sensing power on range-estimation uncertainty and point-cloud perturbation based on the CRB. To capture the impact of computation resources on prediction performance, we adopt an adaptive-depth Mamba-based pose prediction model, where lightweight prediction heads are attached after every layer to enable inference with different model depths. With this unified sensing-computation modeling, we establish a quantitative relationship among sensing power, model depth, and prediction error. Furthermore, we formulate a joint resource allocation problem to minimize the pose prediction error. To solve this problem efficiently, we develop an alternating optimization (AO)-based algorithm, where closed-form solutions are derived for the sensing power and model depth update steps. Simulation results show that the proposed scheme significantly reduces pose prediction error compared with baseline methods, validating its effectiveness for resource-constrained indoor human-centric ISCC systems.

We extend the notion of mex, which is central in combinatorial number theory, to an arbitrary combinatorial structure, and we prove a general theorem to determine the generating function of the objects having fixed mex. We then study this new mex statistic for several classical combinatorial structures, by providing the mex generating function and/or a closed formula for its coefficients in each of the cases.

In this paper, we consider the hyperdissipative Navier-Stokes equations with fractional dissipation $(-\Delta)^{\beta}$ with $\beta>1$. We prove that smooth solutions of the hyperdissipative Navier-Stokes equations are non-unique with arbitrarily small initial data in ${B}^{-\beta-\alpha}_{\infty,1}(\mathbb{T}^d)$ for any $\alpha>0$. Moreover, we show the existence of a solution with arbitrarily small initial data in ${B}^{-\beta-\alpha}_{\infty,1}(\mathbb{T}^d)$ ($\alpha>0$) that grows arbitrarily large in $\dot{B}^{-s}_{\infty,\infty}(\mathbb{T}^d)$ for all $s\in\mathbb{R}$ in arbitrarily small time. It is worth pointing out that ${B}^{-\beta-\alpha}_{\infty,1}(\mathbb{T}^d)$ lies in the subcritical regime when $0<\alpha<\beta-1$. To the best of our knowledge, this is the first non-uniqueness result of the Navier-Stokes equations with initial data at the subcritical regularity. To show the sharpness of the above results, we establish the local well-posedness of the hyperdissipative Navier-Stokes equations with initial data in $\dot{B}^{-\beta-\alpha}_{\infty,\infty}(\mathbb{T}^d)$ with $\alpha< 0$.

We study spaces of conformal blocks associated with line bundles over elliptic curves, with coefficients in a vertex algebra. For vertex algebras satisfying suitable finiteness and semisimplicity conditions, which are met by all admissible affine vertex algebras as well as admissible W-algebras associated with nilpotent elements of standard Levi type, we prove the holonomicity of the sheaf of conformal blocks over the moduli space of bundles. Furthermore, we show that the space of flat sections of the associated Jacobi-invariant connection is spanned by trace functions on modules. This result provides a substantial generalization of the celebrated theorem of Yongchang Zhu to quasi-lisse vertex algebras. As a special case, we deduce that for affine vertex algebras at admissible level, the dimension of the space of conformal blocks coincides with the number of admissible weights at that level.

We consider Subtraction Nim, where two players have exactly the same options, but which is partizan in the sense that at the game ending, a partizan rule is applied for the decision of the winner. We consider the following example: Let $S$ be the set of removable numbers, which is a non-empty finite subset of positive integers greater than or equal to $2$, applied for both players Left and Right. At the end of the game, Left wins if the number of remaining tokens is even, and Right wins if the number of remaining tokens is odd. We computed the outcomes for many $S$, and found surprising phenomena that in most examples of $S$ (almost $98\%$ of some samples), the outcomes are $\mathcal{L}$-positions for all large enough $n$. In comparison, $\mathcal{R}$-positions appear only occasionally. The main theorem explains why this phenomenon occurs. We prove that $n+1$ and $n-1$ are $\mathcal{L}$-positions when $n$ is an $\mathcal{R}$-position. Similarly, $\mathcal{L}$-positions appear whenever $\mathcal{P}$-positions or $\mathcal{N}$-positions appear. Only $\mathcal{L}$-positions can last forever.

The Random Gradient hyper-heuristic was recently shown to be able to learn the optimal neighbourhood size when optimizing the LeadingOnes benchmark via the Randomised Local Search (RLS) meta-heuristic. However, for this to happen, a learning period of a certain length $\tau$ had to be used, differently from classic hyper-heuristics, which change their behaviour based on the success of only the previous iteration. In this paper, we show how to automatically set this new parameter value, relieving the user from the non-trivial task of controlling this novel algorithm parameter. We prove that the resulting hyper-heuristic selects the optimal neighbourhood size in a $1-o(1)$ fraction of the iterations and, consequently, optimises the LeadingOnes benchmark in the best possible time (apart from lower-order terms) achievable with these neighborhood sizes.

Let $g$ be a smooth metric on $\mathbb R^3$ with non-negative scalar curvature. We show that if $g$ satisfies $\vert g(x)-g_{\text{euc}}(x)\vert = O(\vert x\vert^{-1-\tau})$ for some $\tau > 0$ then $g$ must be flat.

This paper investigates a multi-user indoor integrated sensing and communication (ISAC) system operating in the terahertz (THz) band, designed for adaptive communication based on gesture recognition. Leveraging gesture tracking through an extended Kalman filter (EKF), the access point (AP) dynamically adjusts resource allocation in response to detected gesture variations, thereby improving sensing accuracy. Based on the gesture recognition results, the AP further updates the communication quality requirements of different users, enabling efficient resource allocation. To this end, an adaptive joint optimization algorithm for power allocation and beamforming is developed to maximize the overall sensing signal-to-interference-plus-noise ratio (SINR) while satisfying the gesture-dependent communication quality of service (QoS) constraints. Simulation results demonstrate that the proposed method effectively responds to gesture dynamics, achieving superior sensing accuracy and communication performance compared with conventional single-variable optimization baselines.

The Euclidean paradigm that spheres optimize mean curvature variational problems breaks down in the sub-Riemannian Heisenberg group: neither the Pansu sphere nor the Korányi sphere is optimal for the variational problems associated with the Minkowski and Heintze-Karcher inequalities. Motivated by this phenomenon, we develop a variational theory for geometric problems driven by the horizontal mean curvature, focusing on the total mean curvature functional and the related Minkowski inequality, introducing suitable notions of non-characteristic stationarity and stability. We fully classify critical surfaces, identifying a new one-parameter family of rotationally invariant surfaces, which we call Pansu-Minkowski spheres. We show that a distinguished member, the optimal Pansu-Minkowski sphere, emerges as the unique critical point of the Minkowski quotient, and uniquely minimizes it among rotationally invariant competitors. We prove non-characteristic stability and local minimality of Pansu-Minkowski spheres under rotationally invariant perturbations, while showing their instability under unrestricted perturbations. Finally, we show that the Minkowski quotient does not admit any positive lower bound: in the first sub-Riemannian Heisenberg group, no Minkowski inequality holds.

We generalize the classical Ceva's and Menelaus's theorems to curvilinear triangles bounded by circular arcs. We introduce trilinear coordinates associated with such triangles and develop several geometric constructions. In particular, for any proper Möbius triangle we define the incenter, excenters, and orthocenter.

The spectrum of Ramanujan's partial theta function $\theta (q,x):=\sum _{j=0}^{\infty}q^{j(j+1)/2}x^j$, $q\in \mathbb{D}_1$ (the unit disk centered at the origin), $x\in \mathbb{C}$, is the set of values of the parameter $q$ for which $\theta (q,.)$ has a multiple zero. We show that there is no spectral value in the set $\mathbb{S}\cup \mathbb{D}_{c_0}$, $c_0=0.2078750206\ldots$, where $\mathbb{S}$ is the sector $\{ 0<|z|<0.6,{\rm arg}(z)\in [\pi /4 ,7\pi /4 ]\}$. There is a single spectral value in the set $\mathbb{S}\cup \mathbb{D}_{0.31}$ which equals $0.309249\ldots$. For $q\in \mathbb{S}\cup \mathbb{D}_{c_0}$, the moduli of the zeros of $\theta$ are separated by the negative half-integer powers of $|q|$.

Contrastive learning is effective for aligning paired views or modalities, but alignment beyond two modalities remains non-trivial and comparatively underexplored. Pairwise CLIP-style losses decompose multi-modal alignment into independent two-way comparisons and therefore do not explicitly model higher-order dependencies among multiple modalities. Recent beyond-pairwise objectives approach this problem from statistical or geometric perspectives, but arbitrary-modality alignment still lacks a principled criterion for defining what each modality should preserve and compress relative to the others. We revisit arbitrary-modality alignment through the Information Bottleneck principle. In multi-modal learning, sufficiency should preserve information predictable from the remaining modalities, while minimality should compress modality-specific information not supported by them. This naturally leads to a One-vs-All view, where each modality is characterized with respect to the remaining modalities. We propose OVA-IB, an Information Bottleneck framework for arbitrary-modality alignment. OVA-IB optimizes a tractable One-vs-All contrastive lower bound for sufficiency connected to a Dual Total Correlation-style objective, uses a parameter-free geometry-aware projection score, and derives a tractable upper-bound regularizer for minimality by bounding each representation's dependence on its own input with representation distributions induced by the remaining modalities. Experiments on classification, regression, modality-agnostic evaluation, and cross-modal retrieval benchmarks demonstrate strong and robust performance.

The asymptotic Karush-Kuhn-Tucker (AKKT) optimality conditions are distinguished from other approaches in the literature by virtue of their capacity to be effectively derived through numerical methods, such as the utilization of an appropriate version of the augmented Lagrange method. These tools are of a theoretical nature, yet they possess practical utility in the identification of candidate solutions to continuous-time programming problems. While this type of optimality condition is valid without imposing any constraint qualification, it is not sufficiently robust to generate good candidate solutions. In some cases, solutions satisfying AKKT conditions are not even stationary. In this study, we investigate conditions that effectively refine this set of candidate solutions with the same precision as the classical Karush-Kuhn-Tucker (KKT) conditions. This is achieved by introducing a novel constraint qualification, designated AKKT-regularity. It has been demonstrated that, under AKKT-regularity, each local optimal solution is shown to satisfy the KKT conditions. In addition, it is demonstrated that this constraint qualification is the weakest possible to ensure such a property. Furthermore, sufficient conditions are provided for its applicability.

This paper studies a type of rank-based mean field game in which competing agents strategically switch among multiple effort regimes. We propose an entropy regularized auxiliary problem where the switching decisions are randomized to the control of transition probability for a continuous-time finite-state Markov chain. We first establish the existence of regularized equilibrium in this auxiliary problem. Assuming the convexity of reward scheme, we then prove that the equilibrium is unique and can be approximated by a fictitious play iteration scheme. Furthermore, as the entropy regularization vanishes, we establish the convergence analysis of the regularized equilibrium towards the relaxed equilibrium in the original MFG of optimal switching. The uniqueness of the population ranking distribution under the relaxed equilibrium is also obtained given a strictly convex reward scheme.

Modern statistical learning theory and deep learning characterize generalization primarily in terms of continuous capacity control (e.g., norm-based regularization, margin maximization, low-rank bias). While highly successful in continuous domains, deep learning consistently fails to extrapolate exact algorithmic or discrete algebraic rules, reflecting a missing inductive bias toward algorithmic complexity minimization. We propose the Cayley-table completion as the canonical testbed for this missing bias, serving as the discrete algebraic counterpart to matrix completion. Just as matrix factorization combined with weight decay yields an implicit geometric bias toward low linear rank, recent results demonstrate that operator-valued tensor factorizations paired with a flatness prior yield an implicit algorithmic bias toward exact discrete associativity. We pose the open problem of establishing formal exact recovery bounds for Cayley-table completion, and challenge the community to generalize continuous flatness priors to autonomously discover broader discrete algorithmic axioms without combinatorial search.

We prove undecidability for every positive relevant logic extending the system axiomatized by hypothetical syllogism, prefixing, and suffixing and contained in the logic of the semilattice frame $(P_{\mathrm{fin}}(\mathbb{N}), \cup, \varnothing)$. This settles the longstanding decision problem for the semilattice relevant logic S in the negative, contrary to prevailing expectations of decidability. It also provides a new proof of Urquhart's (1984) undecidability theorem for R, E, and T, now by reduction from the Wang tiling problem for arbitrarily large finite isosceles right triangular regions of the plane.

Motivated by the recent work of Batkam-Tcheka on pointed multiplicative operads, we construct in this paper new chain complex algebras and two distinct bicomplex algebra structures on a free symmetric connected multiplicative differential graded operad. Furthermore, we focus on the non-differential graded case and construct a differential graded Hopf algebra structure using the odot product together with an analogue of the Alexander-Whitney homomorphism and a compatible differential. As a consequence, we extend the Malvenuto-Reutenauer result by showing that every free symmetric connected multiplicative operad naturally carries a differential graded Hopf algebra structure.

We establish the existence of solutions of the 2D incompressible non-homogeneous Euler equations with $C^{0}_{t}C^{1,\,\sqrt{\frac{4}{3}}-1-\varepsilon}_{x}\cap C^{0}_{t}L^{2}_{x}$ source terms that develop a singularity in finite time. In order to achieve this, we adapt the Boussinesq blow-up we set up in arXiv:2505.20988 to the non-homogeneous Euler setting. Furthermore, we bring the potential existence of two different types of singularities of the forced system to light.

In this work, we introduce a new class of Leibniz algebras, called quasi-Artinian Leibniz algebras, which generalizes the minimal condition on ideals. Furthermore, we provide some characterizations and give conditions under which a quasi-Artinian Leibniz algebra is Artinian. Finally, within the framework of Leibniz algebras, we establish a connection between prime ideals and the quasi-Artinian structure.

We prove that the spaces $\ell_p(C(\alpha))$ and $\ell_p(C[0,1])$ have the uniform primary factorisation property whenever $\alpha$ is an ordinal and $1<p\leq\infty$. For the case $p=1$, we establish a general criterion ensuring that $\ell_1(X)$ inherits the uniform primary factorisation property from $X$. As a consequence, $\ell_p(C(K))$ is primary for every compact metrizable space $K$ and every $1 \leq p \leq \infty$.

We study a question of Harju from 2019 regarding the existence of infinite ternary square-free words whose subsequences modulo $p$ and $q$ are also square-free for relatively prime integers $p$ and $q$. Among such pairs $(p, q)$ with $p, q \geq 3$, the only two pairs with this property known prior to this work were $(3, 11)$ and $(5, 6)$. We prove that there are finitely many pairs $(p, q)$ of relatively prime integers with $p, q \geq 3$ for which there is no infinite ternary square-free word whose subsequences modulo $p$ and $q$ are square-free. To prove our result, we combine different techniques, including the construction of words from multi-valued square-free morphisms and circular square-free morphisms. We also introduce the notion of square-free transducers, a generalization of square-free morphisms that may be of independent interest.

We introduce variants of the Maker-Breaker and Waiter-Client games, which we call stotting, in which a player grants a slight advantage to the opponent. We prove that a winning strategy in either stotting variant yields winning strategies for both Maker and Waiter in the classical setting. Several existing Maker strategies in the literature in fact win with stotting, and therefore automatically provide both classical winning strategies (and similarly for stotting Waiter). Knox previously disproved a conjecture of Beck asserting that whenever Maker wins the Maker-Breaker game, Waiter also wins the corresponding Waiter-Client game; in this sense, our framework may be viewed as a way of repairing Beck's conjecture.

The term Gibbons conjecture is widely used in connection with symmetry results for the Allen-Cahn equation. However, its origin is less transparent than its frequent citation suggests. In this note, we revisit its emergence, tracing it to a 1995 paper by Carbou and to subsequent developments in the literature. We argue that the attribution likely arose from informal communication rather than from a formally stated conjecture, illustrating how mathematical terminology may develop through transmission and collective usage.

Distributed online stochastic optimization has received extensive attention in large-scale distributed learning and other related fields due to its unique advantage in processing streaming data. However, information exchange through the communication network during the optimization process may lead to privacy leakage. To address this issue, this paper proposes a locally differentially private distributed online stochastic optimization algorithm that employs an elaborately designed dynamic stochastic quantizer to mask the exchanged information prior to communication. Theoretical analysis shows that the proposed algorithm not only converges almost surely to the optimal solution but also achieves $(0,\delta^i)$-local differential privacy for each agent $i$ even when the number of iterations tends to infinity. Furthermore, the algorithm is fully distributed and applicable to scenarios where the interaction network among agents is a directed graph. To the best of our knowledge, this is the first work on distributed online stochastic optimization that simultaneously achieves exact convergence and rigorous local differential privacy over a directed graph by exploiting quantization effects. Numerical experiments of distributed online training on the mushroom classification dataset, handwritten digits recognition dataset, and brain-computer interface dataset verify the effectiveness of the proposed method.

This paper presents a distributed nonlinear model predictive control that uses alternating direction method of mul tipliers for district heating networks. Exploiting a graph-based modeling of the thermal dynamics, our controller optimizes the mass flow absorption of buildings in a distributed cooperative scheme that mediates between the superior performance of the centralized control and the privacy preservation of the decentralized schemes. A benchmark three-building network simulation is used to compare the performance of the proposed solution with a decentralized model predictive control scheme.

We introduce the Topological Stability Index (TSI), a variance-based scalar measure for persistence barcodes that quantifies the dispersion of persistence lifetimes. Unlike persistent entropy, which depends only on normalized weights, the TSI captures absolute variability and is sensitive to heterogeneous feature scales. We establish fundamental properties of the TSI, including its scaling behavior, invariance under lifetime translation and explicit update formulas under insertion and deletion of bars. We also consider a complementary first-moment-type quantity, the Topological Signal Index (TSigI), which captures the typical scale of persistence lifetimes and provides additional interpretability alongside the TSI. We further introduce a normalized version, $cv\text{TSI}$, which is scale invariant and admits an explicit algebraic relation to the Rényi entropy of order two. In particular, $cv\text{TSI}$ is an affine function of the collision probability $\sum_i p_i^2$, and therefore a monotone reparametrization of the Rényi entropy, providing a direct link between variance-based and entropy-based summaries in topological data analysis. Numerical experiments on synthetic data and stochastic time series demonstrate that the TSI captures structural variability complementary to entropy: it is relatively insensitive to deterministic trends, while responding strongly to stochastic fluctuations and variations in persistence magnitude.

We show that groups with a mild form of non-positive curvature (a navigable path system) satisfy the weak rank rigidity conjecture: they either have linear divergence or a Morse element. This class includes discrete groups of projective automorphisms of open convex cones, Helly groups (answering a question of Genevois), Coxeter groups, weak Garside groups (in particular Deligne's groups and fundamental groups of Salvetti complexes of oriented matroids), hierarchically hyperbolic groups, and other examples. Along the way, we show that those groups satisfy the Morse local-to-global property, providing a unified proof for the whole class. In the metric setting, the same condition of non-positive curvature allows to provide a local definition (that is, in a sense, optimal) of rank one/Morse geodesics, mirroring the one using parallel Jacobi fields from Riemannian geometry; to deduce linearity of divergence from linearity on a sequence; to obtain new cases in which Morse geodesics are strongly contracting. The main new tool introduced is the generalised contraction space, a hyperbolic space that encodes the negative curvature of a given space.

In this note, we provide a family of $2\times 2$ tetrablock contractions that have tetrablock isometric dilation, but the corresponding fundamental operators do not commute. This answers a question raised by Bhattacharyya [Indiana Univ. Math. J. 2014] about the necessity of commuting fundamental operators for a tetrablock contraction to have rational dilation in a negative direction.