Papers

In this paper, we characterize the global dimension of a string algebra by using combinatorial methods. Moreover, we establish a necessary and sufficient condition for when the global dimension of a string algebra is infinite.

We prove that every free ergodic measure-preserving action of a countably infinite amenable group with positive Rokhlin entropy has infinite complex $L^1$-orbit multiplicity, both on $L^1$ and on its mean-zero subspace $L^1_0$. This gives a negative answer to a question of J.-P. Thouvenot recorded by Iwanik and establishes the corresponding endpoint statement at $p=1$ of Iwanik's theorem that positive entropy implies infinite $L^p$-multiplicity for every $p>1$. The proof combines Malykhin's rigidity theorem for independent random variables, Seward's Bernoulli factor theorem, and a Følner set argument.

In this paper, we establish an optimal global rigidity estimate for the eigenvalues of the Laguerre unitary ensemble. Using the central limit theorem, we first construct a random measure via the eigenvalue counting function and then prove its convergence to a Gaussian multiplicative chaos measure, which yields the desired rigidity result. To prove this convergence, we apply a sufficient condition due to Claeys et al.[7] and carry out an asymptotic analysis of the corresponding exponential moments.

For every smooth Riemannian metric on the two sphere whose Gaussian curvature is bounded below by one, we prove that each positive Laplace eigenvalue, counted with multiplicity, is no smaller than the corresponding eigenvalue of the unit round sphere. Equality at any positive position in the ordered spectrum forces the metric to be isometric to the unit round metric. We further establish a sharp finite spectral counting comparison for Alexandrov two spheres with curvature bounded below by one. At every positive spectral threshold of the unit round sphere, the number of Laplace eigenvalues below or at that threshold, counted with multiplicity, does not exceed the corresponding number for the round sphere. Equality at any such threshold forces the Alexandrov sphere to be isometric to the unit round sphere. As an application, we obtain the sharp Euclidean dimension bound for spaces of polynomial growth harmonic functions on complete three dimensional manifolds with nonnegative sectional curvature and positive asymptotic volume ratio, together with rigidity in the equality case.

Let $G$ be a two-orbit permutation group on $n > 2$ points. We show that $G$ contains either a derangement or an element of prime-power order with a unique fixed point. As a corollary, if the orbits of $G$ have length $n_1$ and $n_2$ and $\gcd(n_1, n_2-1) = \gcd(n_1-1, n_2) = 1$, then $G$ contains a derangement. The special case $n_1 = n_2$ was recently conjectured by Ellis and Harper and proved under various restrictive hypotheses. We prove our result by reducing to the case of simple groups and leveraging the classification of normal $2$-coverings of simple groups due to Bubboloni, Spiga, and Weigel.

We study tropical circuits with scalar multiplication gates, that is, algebraic circuits whose gates implement $\max$, $+$, or multiplication with a positive constant. For such circuits, we prove exponential size lower bounds for computing maximum weight directed spanning trees and maximum weight bipartite perfect matchings. As a corollary, we obtain an exponential size separation between monotone and non-monotone maxout neural networks, which generalize the popularly used ReLU neural networks. One conclusion from this is that neural network models with enforced convexity constraints, such as input-convex neural networks (ICNNs), sometimes need to be exponentially larger than their unrestricted counterparts in order to express the same functions.

We prove a fractal uncertainty principle over $\mathbb{Q}_p$ for porous sets, resolving a conjecture of Cohen.

In this work, we study the optimization problem for the first eigenvalue of mixed local nonlocal operators plus a potential $V$. We begin by investigating the existence and fundamental properties of the eigenvalues, with special emphasis on the first eigenvalue. Finally, we discuss the dependence of the first eigenvalue on the potential function and establish the existence of optimal potentials within certain admissible classes. In particular, we show the existence of a unique maximizer and a minimizer of the first eigenvalue on any bounded, closed, and convex subset of $L^q(\Omega)$. Moreover, these results enable us to characterize the maximizers and minimizers in the closed unit ball of $L^q(\Omega)$, as well as in the class of rearrangements of any $V\in L^q(\Omega)$.

We study Fraïssé classes of Abelian groups with pure embeddings. We characterize Abelian groups that are universal and homogeneous for: finitely co-generated groups, finite groups, groups of size less that $\kappa$, where $\kappa$ is strongly inaccessible.

These are lecture notes for an advanced topics course in the master's programme at Bonn University. It aims to give a concise review of some of the work that has been done around the topic of adaptive dynamics from a rigorous stochastic point of view over the last 25 years and to make this area accessible to students with a good knowledge of probability in general, and the theory of Markov processes in particular. Our emphasis is on the issue of emerging scaling limits, where scaling parameters are time, population size, mutation rates, and mutation step size. These allow to exhibit within a fairly simple class of models a variety of biologically relevant phenomena. These notes are organised in three parts. Part 1 presents some historical background as well as the mathematical setting and main tools. Part 2 is the core of the notes and discusses the various scaling regimes and scaling limits. Part 3 looks at some extensions on the basic model, notably diploid models, phenotypic plasticity, and effects of environmental changes over time.

We introduce graph-compatible powers determined by the coupling graph for multicomponent weakly coupled elliptic systems and establish concavity of the associated transformed components through a concave-envelope method and a weighted viscosity comparison principle. We further establish a componentwise constant-rank theorem for the transformed Hessians, which yields strict power concavity under additional assumptions.

This paper discusses about an inverse scattering problem to the nonlocal nonlinear Schrodinger equation. We extend the previous result treated by Y. Zhao and E. Fan of inverse scattering result at the point with slowly decaying data. More precisely, we show the global existence of solution in $H^{1,s}$ for $s>1/2$ by assuming small data in $L^1$. The proof relies on the inverse scattering method based on the ZS-AKNS (Zakharov-Shabat, Ablowitz-Kaup-Newell-Segur) eigenvalue problem on the spectral theory of ordinary differential equations upon the framework of slowly decaying data.

In 2017, Keith defined and studied a multi-restricted partition function $RD^{l,t}(n)$, the number of integer partitions of n into parts that are simultaneously $l$-regular and $t$-distinct if none of the parts is divisible by $l$ and parts appear fewer than $t$ times. Thereafter, in a series of papers Buragohain et al. and Nadji et al. examined combinatorial and arithmetic properties of $RD^{l,t}(n)$. In this article, we focus on simultaneously $l$-regular and $t$-distinct partitions in 3-colors, $RD^{l,t}_3(n)$ and establish infinite families of congruences modulo powers of 3 for specific values of $(l,t)$. For instance, for positive integers $n$ and $k$, we have \begin{align*} \sum_{n=0}^{\infty}RD_3^{3,3}\left(3^kn+\frac{3^k+1}{2}\right)q^n\equiv0 \pmod{3^{k+1}}. \end{align*} Our approach involves Hirschhorn's H operator technique.

Agentic systems increasingly improve themselves by editing skills: prompts, rubrics, plans, tool contracts, examples, validators, and traces. Skill edits are not independent coordinates in a vector space: they are local repairs to structured artifacts whose effects are observed only after rollout, validation, and critique. Distinct edits can have the same immediate visible effect while differing in routing context, template state, guardrail scope, or future composability. The order of edits can matter as well: repairing a schema before a normalization rule need not be equivalent to applying the same edits in the reverse order. This paper introduces a new framework for skill optimization called LASKO, for Lie Algebroid SKill Optimization. LASKO models typed, anchored Markdown skills as the base category and available edit policies as sections of a controlled Lie algebroid with anchor $\rho$. The anchor maps an edit policy to its visible Markdown effect; the kernel $\ker(\rho)$ represents latent template, routing, or implementation structure; and the algebroid bracket measures noncommuting edit composition. As shown in the paper, LASKO achieves order-of-magnitude speedups in skill optimization in our preliminary benchmark results, primarily because it substitutes inexpensive Lie-bracket screening tests that run in microseconds, before investing in expensive validations that require running large language models. On a causal extraction from natural language task, LASKO achieved a speedup of almost $15 \times$ compared to a brute-force approach that validated all edits by running them through a DeepSeek V3.1 4-bit model with 671B parameters.

We identify an exact spectral phase space for the two-sided Toda lattice. Let $q=\{a_n,b_n\}_{n\in\mathbb Z}$ be coefficients of the right and left half-line Jacobi operators and denote their spectral measures by $\sigma_{\pm}^{q}$. Define a phase space \[ \mathcal Q=\left\{ \begin{array} [c]{c}% q=\{a_n,b_n\}_{n\in\mathbb Z}: a_n>0,\ b_{n} \in \mathbb{R} \text{ and} \int_{\mathbb R}e^{c|\lambda|}\sigma^q_\pm(d\lambda)<\infty \text{ for every }c>0 \end{array} \right\} . \] The integrability condition makes the representing measures unique. We prove that $q\in\mathcal Q$ if and only if the Toda lattice with initial datum $q$ admits a classical solution for all positive and negative times. Moreover, the solution remains in $\mathcal Q$, is unique, and depends continuously on the initial datum, uniformly on compact time intervals.

For the Grassmannian embeddings defined by the spaces $H^0(X,L^p\otimes E)$, where $L$ is a positive line bundle and $E$ is a holomorphic vector bundle over a compact complex manifold, we prove a complete asymptotic expansion of the induced Grassmannian Chern forms and compute the first coefficients explicitly. As an application of the first-order asymptotics and of the theory of meromorphic transforms by Dinh and Sibony, we prove that on a compact Kähler manifold, the normalized currents of integration over the loci where several random sections become linearly dependent converge almost surely to the corresponding power of the curvature form of the positive line bundle, with a quantitative estimate for the speed of convergence. Moreover, in the determinant case, we additionally present an alternative method based on the Wishart distribution, together with variance estimates.

Long range-frequency hopping spread spectrum (LR-FHSS) is a promising uplink physical layer for massive low Earth orbit satellite Internet of Things, where low power terminals report short packets from wide area regions with limited terrestrial infrastructure. However, satellite IoT links are exposed to external interference, and the coexistence of multiple interference components can severely degrade receiver reliability and complicate interference mitigation. Existing recognition methods either focus on single interference scenarios or treat each compound interference combination as an independent class, leading to limited generalization or poor scalability. To address this problem, this paper formulates LR-FHSS uplink compound interference recognition as a multi-instance multi-label learning problem and proposes a multi-domain instance fusion method. The proposed method fuses local instances from the time-frequency and frequency domains and aggregates their predictions for bag-level multi-label recognition. A dataset construction pipeline is developed based on the US915 LR-FHSS configuration and incorporates shadowed-Rician fading and time-varying Doppler to emulate practical satellite communication conditions. Considering the difficulty of obtaining labeled compound interference samples in practice, single-to-compound generalization and few-shot compound interference adaptation are investigated as two practical receiver deployment scenarios. Experimental results show that the proposed method improves the overall exact accuracy over the strongest baseline by 14.71 percentage points in single-to-compound generalization and by 14.81 percentage points in few-shot compound interference adaptation for $r=1$.

We study self-similar translation flows on $\mathbb Z^d$-covers of compact translation surfaces. Our main goal is to investigate their ergodic properties with respect to general Maharam measures. To this end, we develop a renormalization approach based on a family of twisted transfer operators associated with the renormalizing pseudo-Anosov map acting on anisotropic spaces of distributions. We describe the discrete spectrum of these operators in terms of the action of the pseudo-Anosov on suitable twisted cohomology groups. We further show that the resonant states of the dual operator corresponding to peripheral eigenvalues give rise to Maharam distributions which are invariant under the translation flow. Motivated by this correspondence, we refer to these eigenvalues as Maharam-Pollicott-Ruelle resonances. As applications, we derive asymptotic formulas for ergodic integrals of smooth observables at Maharam-generic points, prove a central limit theorem for the associated Frobenius cocycle, and compute the Hausdorff dimension of Maharam measures.

We consider nonlinear diffusion equations on an interval where the diffusion coefficient in a small region near the center is scaled such that it approximates a transmission condition at a membrane. While the limiting behavior of the solutions is well-understood, we study the convergence in the sense of the energy-dissipation principle (EDP) of associated gradient structures given in terms of a free energy and a dissipation potential. EDP-convergence provides a uniquely specified limiting gradient structure that reformulates the transmission condition in terms of an effective kinetic relation for the membrane, which relates the jump of the chemical potential and the flux through the membrane. We show how \AAA properties of the chosen free energy and the mobility of small-scale diffusion migrate to the effective kinetic relation. A surprising result is that starting from the linear Onsager relation of Otto's gradient structure for the linear diffusion equation, one obtains an exponentially growing kinetic relation, the so-called Marcelin-De Donder kinetic.

Aardal and Lenstra systematically studied hard knapsack problems of the form $a_1x_1+\cdots+a_nx_n=b$, where $a_i=p_iM+r_iN$, $(M,N)$ is a coprime pair of positive integers, and the integers $|p_i|, |r_i|$ are small relative to $M$ and $N$. We investigate the corresponding challenging denumerant problem (i.e., counting the number of nonnegative integer solutions) and present a polynomial-time algorithm. This eliminates the computational bottlenecks caused by large values of $M$, $N$ and $b$. The proposed algorithm achieves a time complexity of $O(n^4\Delta^2\log n\log\Delta)$, which depends solely on the parameters $n$ and $\Delta=\max_{i,j}|r_i p_j - r_j p_i|$. Moreover, we consider the problem of expressing a general vector $(a_1,\dots,a_n)$ in the above form using the LLL algorithm.

We prove a Breuer-Major-type Donsker's invariance principle for stationary Gaussian sequences under the natural finite-variance assumption on the test function. This result, which we call the Breuer-Major-Donsker principle, or simply the BMD principle, removes the additional moment assumption imposed in the functional Breuer-Major theorem of Nourdin and Nualart (Probab. Theory Related Fields, 2020). Our method does not rely on the Malliavin-calculus estimates used by Nourdin and Nualart, in particular Meyer's inequality. Instead, it is based on a predictable-martingale decomposition of the partial-sum process, which is of independent interest. We also make systematic use of non-determinism, a central notion in Gaussian prediction theory. In the non-deterministic case, the martingale part is handled by the martingale functional central limit theorem, while the predictable remainder gains integrability above order two through Ornstein-Uhlenbeck smoothing. In the deterministic case, the martingale part vanishes, and the smoothing mechanism is no longer available along the full sequence. Nevertheless, under an additional mild assumption on the covariance function, a suitable decimation recovers non-determinism and reduces the proof to the non-deterministic case.

We introduce a notion of pairing between essentially bounded tensor fields with divergence measure and vector-valued functions of bounded variation, extending the classical theory to the tensorial setting. This naturally leads to an adaptation of the definition of normal trace for tensor fields with measure divergence even on a rectifiable set. As a consequence, we establish tensorial Gauss-Green formulas that remain valid on sets with low regularity, including sets of finite perimeter. These results yield a unified and robust framework for integration by parts in the presence of irregular tensor fields and domains.

We prove the Harnack inequality (with tails) and local Hölder regularity for the fractional logarithmic $p$-Laplace operator, which is derived by differentiating the fractional $p$-Laplace operator with respect to its order. To be more precise, for a suitable function $u,$ the operator reads as the first order derivative \begin{align*} (-\Delta_p)^{s+\log} u:= \frac{d}{dt}(-\Delta_p)^t u \Big|_{t=s} \end{align*} at any arbitrary order $s\in (0, 1).$ The kernel of this operator involves a logarithmic factor. As a consequence, it changes sign at large scales and, near the diagonal, is more singular than the kernel of the fractional $p$-Laplacian. To achieve our regularity estimates, we adopt the classical De Giorgi-Nash-Moser techniques in this setting. We also construct an example showing that the Harnack inequality fails without tail terms. Our results are new even in the linear setup $p=2$.

We consider constrained problems in which input data are affected by errors. In such settings, Wasserstein distributionally robust optimization provides a principled framework to mitigate model risk by optimizing against worst-case data distributions within Wasserstein ambiguity sets. However, the numerical resolution of the resulting problems remains challenging, especially in constrained settings. In this paper, we provide a general, practical way to solve Wasserstein distributionally robust formulations in the presence of constraints. Our approach only requires a linear minimization oracle for the feasible set, and combines two key ingredients: (i) an entropic regularization of the distributionally robust value function, which makes it possible to compute stochastic gradient estimators, and (ii) a stochastic Frank-Wolfe algorithm, which minimizes the regularized robust objective while naturally handling constraints. We illustrate the method, its tractability, and its interests against empirical risk minimization, on two problems: the traffic assignment and the minimum quadratic spanning tree.

We introduce a new notation for $\tau$-rigid pairs called “balanced pairs”. The notation $\frac AB$ allows for simplification of some formulas, for example the Jasso category and its dual. Using this balanced notation we show that the $g$-vector $g(\frac AB)$ has nice geometric and algebraic implications: It defines “lower” and “upper” chambers and we prove that this corresponds to generalized Bongartz and co-Bongartz completions of balanced pairs. We also show that the balanced notation agrees the wall labels for the semi-invariant picture in the hereditary case and, in the general case, we describe the relation between balanced notation and the wall labels.

Elliptic quantum algebra is the algebraic structure characterized by the elliptic solution of the Yang-Baxter equation. In this paper, we construct a family of central elements \( \mathfrak{z}(z) \) for the elliptic quantum algebra \(\mathcal{A}_{q,p}(\widehat{\mathfrak{gl}}_{N})\) and show that they can be expressed as quantum determinants, yielding an elliptic analogue of the Liouville formula. In addition, we establish determinantal identities, including Jacobi's ratio theorem and Sylvester's theorem.

In this article, we develop a robust optimization framework for obstacle-aware aerial directional sensor networks operating under positional uncertainty using a sector-based sensing model. The monitoring area is discretized into grid points and a unified geometric formulation is developed to model directional sensing while explicitly accounting for obstacle-induced visibility loss and exclusion regions. To enhance sensing performance under uncertainty, three progressive optimization strategies, namely robust aerial grid coverage (RAGC), robust aerial target coverage (RATC) and robust aerial target scheduling (RATS), are proposed. The framework employs robust orientation optimization based on the radius of robust feasibility to maximize effective grid and target coverage under sensor perturbations, followed by a target-aware sleep scheduling mechanism that minimizes the number of active sensors without compromising target monitoring. Extensive simulations under varying target distributions, obstacle configurations, uncertainty levels and sensor failure scenarios demonstrate that the proposed framework achieves robust and energy-efficient sensing while consistently outperforming representative approaches from the literature in terms of coverage quality, target monitoring and sensor utilization.

We show that the range of a contractively decomposable projection on a noncommutative Haagerup $L^p$-space, $L^p(\mathcal{M},\varphi)$, for $1<p<\infty$, is completely isometrically isomorphic to a corner of a noncommutative $L^p$-space, that is $eL^p(\mathcal{N},\psi)(1-e)$, with $e\in\mathcal{N}$ a projection. In the setting of Schatten spaces, we obtain a more precise description: the range of a contractively decomposable projection on $S^p(K,H)$ is isometric to an $\ell^p$ direct sum of subspaces of the form $S^p(K',H')$. Furthermore, we show that contractively 1-pseudo decomposable projections on Schatten spaces are automatically contractively decomposable, establishing the equivalence between these two notions in this setting.

We prove the minimisers of the Lieb-Oxford bound (if any) for a fixed (and finite) number of particles $N \geq 1$ are necessarily compactly supported, extending a result proved by E. H. Lieb and S. Oxford in the one-particle case.

We study Nakao's problem in two and three space dimensions, a weakly coupled system consisting of a semilinear damped wave equation and a semilinear undamped wave equation. We establish the global in-time existence and uniqueness of small data mild Sobolev solutions in new admissible ranges, together with time-dependent estimates matching those for the corresponding linearized problems. The proof combines diffusion-type $L^m-L^r$ estimates for the damped component with the Poisson and Kirchhoff formulas for the undamped component. Dimension-dependent solution spaces are introduced to incorporate the weaker wave decay and logarithmic $L^2$ growth in two space dimensions. In three space dimensions, a two-level fixed point argument avoids an artificial restriction on $p$ by establishing the self-map property in a strong space and the contraction in a weaker metric. Furthermore, the undamped component scatters to a free wave in $H^2\times H^1$.