Papers

The bandwidth of a timed language characterizes the quantity of information per time unit (with a finite observation precision $\varepsilon$). The asymptotic behavior of the bandwidth as $\varepsilon \to 0$ classifies timed regular languages in three classes: meager, normal, and obese. Normal timed automata have a bounded frequency of events and some non-punctual transitions, and, up to now, were the only class of timed automata for which no algorithm was available for computing their bandwidth. In this article, we compute the bandwidth of any such automaton in the form $\approx\alpha\log{1/\varepsilon}$. Our approach reduces this problem to computing the best reward-to-cost ratio in a weighted finite graph constructed from the given timed automaton.

We discuss a theoretical approach to discriminate whether two states of a many-body quantum system belong or not to different topology classes. This approach is based on expressing a strange correlator - a recently established tool for quantum topology discrimination - between the states as a function of Kirkwood-Dirac quasiprobabilities (KDQs). KDQs provide a first-principles representation of two-time quantum correlators. The link between strange correlators and KDQs allows to establish that strange correlators are weak values of an observable converting an initial trivial state into a topologically non-trivial one. We thus propose a quantum topology witness that is achievable measuring the prior and subsequent effects on a many-body system of a sudden quench transformation that realizes the transition between trivial and topological phases. The witness is evaluated on a probe quantum state whose main features are detailed within the paper. Finally, directly exploiting schemes that allows for the complete reconstruction of KDQs, we address an interferometric protocol for topology discrimination, along with a general discussion of the main lines and challenges towards its implementation.

In this article, we study cyclic phenomena on the full range of weighted Bergman spaces on the right half-plane $\mathcal{A}^{2}_{\alpha}(\mathbb C_{+}).$ We completely characterize cyclic and supercyclic composition operators induced by affine self-maps of $\mathbb C_+,$ thereby completing the characterization of all hypercyclic, supercyclic and cyclic composition operators induced by affine symbols on $\mathcal{A}^{2}_{\alpha}(\mathbb C_{+}).$

We study isolated singularities of positive solutions to a fractional Hartree equation with Riesz interaction, \[ (-\Delta)^s u = \left( \int_{\mathbb{R}^N\setminus\{0\}} \frac{u^p(y)}{|x-y|^\mu}\,dy \right)u^q \quad \text{in } \mathbb{R}^N\setminus\{0\}. \] The puncture changes the passage from the differential equation to its integral form: a fractional fundamental solution term may occur at the singular point. Under an upper bound at the scale of the fundamental solution and a weighted integrability condition on the source, we derive a Riesz decomposition with a nonnegative singular term. Under the same assumptions, if in addition the solution blows up at the puncture, \(p,q\ge1\), and both Kelvin weights are nonnegative, an off-center moving spheres argument yields radial symmetry and strict radial decrease. We also classify positive radial homogeneous singular solutions under the corresponding convergence conditions. If a nonzero positive radial homogeneous scaling limit exists and solves the limiting equation, this classification uniquely determines its coefficient.

We show that the class of complete separable metric spaces whose Lipschitz-free space is isomorphic to Pełczyński's universal basis space $\mathbf{P}$ is $\mathbf{\Sigma}^1_1$-complete.

In classical robust hypothesis testing, least favorable distributions often simultaneously maximize all Chernoff $u$-affinities and minimize all $f$-divergences. This paper identifies the structural mechanism that causes this equivalence to fail in general: the cone generated by fractional power functions $\{x^u\}_{u\in(0,1)}$ is strictly smaller than the cone of convex functions, inducing a separation between fractional-moment dominance and convex-order dominance. An explicit minimal counterexample is constructed on a three-point probability space, with convex, compact uncertainty classes and uniformly bounded likelihood ratios, for which a single pair maximizes all Chernoff functionals uniformly yet fails to minimize a convex $f$-divergence. It is further proved that no such separation can occur on a two-point space. Sufficient conditions for equivalence – including stochastic ordering of likelihood ratios – are discussed, and an open characterization problem in the geometry of moment cones is highlighted.

Finding convenient spaces in which certain hypotheses regarding an assumed sparse structure of natural signals hold true has become a desirable result in recent research, its implications being reflected in areas such as data compression, noise reduction and feature extraction. While the extensively used analytical transforms, such as DFT or DCT, already provide efficient algorithms and robust sparse representations, they assume a fixed prior about the data, failing to accurately capture the specific structure of more restrictive classes of signals. To address this, the concept of a data-adaptive, learnt transform has been introduced in the literature, allowing for the reduction of a residual term in the transform domain. More recent studies have shown that the condition number serves as a good metric in this context, where the desired outcome alternates between a generalizing tendency and one that achieves minimal approximation error. Motivated by these considerations, we introduce the learning of a structured, explicitly conditioned transform formulated as the product of a fixed canonical matrix and a refining data-adaptive sparse component. This approach seeks to preserve the advantages of fast and stable analytical transforms, while introducing controllable adaptivity to the data. No references that concern this specific formulation have been identified so far, indicating its novelty. The proposed algorithm is motivated within the framework of inexact proximal methods, leveraging a newly derived closed-form projection operator. Empirical observations demonstrate state-of-the-art results on the doubly sparse transform learning problem and comparable performance with its dense variant at significantly lower computational costs and sometimes faster convergence and better avoidance of bad local minima.

The Brownian sphere, also known as the Brownian map, is a canonical random metric measure space homeomorphic to the two-dimensional sphere $\mathbf S^2$. It can be interpreted as the uniform measure on surfaces homeomorphic to $\mathbf S^2$, in the sense that it arises as the scaling limit of many natural models of random planar maps chosen uniformly from a given class. It is also equivalent to the $\sqrt{8/3}$-Liouville quantum gravity sphere. We prove that the Brownian sphere is quasisymmetrically rigid, meaning that, almost surely, it has no nontrivial quasisymmetric automorphisms. We also show that two independent Brownian spheres are almost surely not quasisymmetrically equivalent. Our argument also gives a new proof that the conformal structure of the Brownian sphere is almost surely determined by its metric structure.

The Graph Burning Problem (GBP) is a NP-Hard combinatorial optimization problem that models the propagation of influence or contagion in a network. The propagation is represented through the metaphor of a fire spreading through the vertices of a graph. A burning process takes place in a series of discrete time-steps. At each time step, the burning process is characterized by a propagation (where burned nodes spread the fire to their neighbors), and an ignition (where one additional unburned node is chosen to become burned). The minimum number of steps required to burn all vertices of a graph defines its burning number. Literature provides integer linear programming formulations to solve the problem, but with no surprise, such approaches struggle to converge as the graph size increases. Therefore, reducing the search space explored by these formulations becomes a key point to improve performances. In this work, we study the similarities of the Graph Burning Problem with the well-known Dominating Set Problem. We propose a new formulation based on this study and apply dominance rules and symmetry-breaking techniques to reduce the search space and consequently speed up resolution time. We also introduce a perturbation of the proposed objective function, as well as a pruning rule for the perturbed model in order to further accelerate its resolution.

We discuss birational properties of actions of finite groups on algebraic varieties, linearizability, torsors, and versality.

We consider a problem to develop a framework for model identification adhering to the tenets of neuromorphic computation, without resorting to neural networks as the mathematical substrate. In particular, all computations take place in continuous time. We are naturally led to adaptive observers, where the main technical obstacle is the possible mismatch between the unknown plant order and the observer order. The key concept that informs the proposed framework is an overparameterized model, an input-output equivalent model that provides a suitable parameterization in the overmodeled case, with theoretical extensions also addressing the undermodeled case. A discrete algorithm orchestrates successive experiments to incrementally learn the model order, while a standard parameter adaptation law learns the parameters.

Let $\Sigma$ be a closed embedded minimal hypersurface in $\mathbb{S}^{n+1}$. We establish an improved lower bound for the first non-zero eigenvalue of the induced Laplace-Beltrami operator on $\Sigma$. It is slightly better than the bound of Duncan-Sire-Spruck.

We review random matrix approaches to chaotic wave scattering and transport in open systems. Starting from the effective non-Hermitian Hamiltonian formulation, we discuss the scattering matrix, reaction matrix, time delays, and complex resonances as complementary probes of open chaotic dynamics. We emphasize universal statistics governed by symmetry, openness, and channel coupling. Topics include the maximum-entropy description of fixed-energy scattering and its applications to quantum transport, energy correlations, resonance and eigenfunction statistics, and selected wave-chaotic phenomena induced by finite absorption. The focus throughout is on non-perturbative methods and universal structures underlying open quantum and wave chaotic systems.

We present eighty-nine optimal degree-diameter Cayley graphs that are non-isomorphic to previously known optimal Cayley graphs with the same degree and diameter. For each graph, we provide comprehensive data on its parameters and structural invariants. Comparison with known constructions reported by Marston Conder on the Combinatorics Wiki and in related works reveals significant differences in girth, algebraic connectivity, domination number, automorphism group structure, cycle distributions, distance-related properties, and other graph invariants. These findings show that optimal degree-diameter Cayley graphs are frequently non-unique. The new examples enlarge the catalogue of known optimal graphs and provide benchmarks for investigations in degree-diameter problems, algebraic graph theory, and extremal graph theory. Since some of the graphs were identified through randomized search techniques, the collection is not exhaustive.

We study topologies on Ran spaces. In the literature, two distinct topologies frequently appear: the Hausdorff topology, and a finer one constructed as a colimit, that we call the final topology. In this work, given a metric space $M$, we construct new metric topologies on $\mathrm{Ran}(M)$, called weighted topologies. They interpolate between the Hausdorff and final topologies, the later being recovered as a limit in the category of spaces. This structure equips the final topology with a uniformity, which we show to be complete. Finally we study the Ran spaces as stratified spaces. We show that, whenever $M$ is a Riemannian manifold, the weighted topologies are conically stratified, while the final topology is only so in a weak sense.

We study efficient online methods for dynamic inverse problems with infinite time horizon. We concentrate, in particular, on problems whose forward model arises from a PDE. Our motivating application is flow monitoring with Electrical Impedance Tomography (EIT). The idea of such online methods is to take single steps of of standard optimisation algorithms, on each time index; each data frame. A predictor, based on problem dynamics, is used to transfer iterates one from time index to the next one. If we monitor a fast flow with a correspondingly fast measurement modality, such as EIT, basic methods are unable to solve the PDE before new data arrives. Our idea, then, is to not solve it, and instead, on each iteration, each time index, take single or few steps of standard iterative solvers towards the solution of both the PDE and an adjoint PDE. This is what “single loop” refers to. To the overall problem, we apply standard online optimisation methods, at the outside developed for exact gradients $\nabla E_k(x^k)$ of the iteration-dependent data fidelity $E_k$ that incorporates the PDE. We replace the gradient by a single-loop estimate $\tilde\grad E_k(x^k)$ that satisfies standard smoothness properties with summable errors. This allows standard regret proofs to go through. Our numerical experiments on dynamic EIT validate the theoretical predictions and highlight the potential of the proposed approach for the real-time solution of PDE-constrained dynamic inverse problems.

In this paper, we develop a quiver approach to coquasi-Hopf algebras with the dual Chevalley property. We introduce a modified generalized path coalgebra $\Bbbk(\mathrm{Q},\mathcal{S})$ associated with a given quiver $\mathrm{Q}$ and a collection of simple coalgebras $\mathcal{S}=\{C_i\mid i\in \mathrm{Q}_0\}$ indexed by the vertices of $\mathrm{Q}$, such that its link quiver coincides with $\mathrm{Q}$. We prove that such a coalgebra admits a graded coquasi-Hopf algebra structure with the dual Chevalley property if and only if $\mathrm{Q}$ is a generalized Hopf quiver and $\bigoplus_{i\in \mathrm{Q}_0}C_i$ forms a cosemisimple coquasi-Hopf algebra. Moreover, we provide a classification of these coquasi-Hopf algebra structures. We then study the link-indecomposable components of a coquasi-Hopf algebra with the dual Chevalley property, and give the generalized dual Gabriel's theorem for such coquasi-Hopf algebras. As an application, we apply the quiver method to classify finite integral tensor categories with the Chevalley property of finite representation type. We also give structural characterizations of coradically graded coquasi-Hopf algebras of tame corepresentation type. Furthermore, we investigate finite braided integral tensor categories with the Chevalley property via the quiver approach.

We introduce a new tool, Express, for converting a non-causal attention approximation into a causal approximation with matching approximation guarantees. When combined with the state-of-the-art Thinformer approximation, Express improves upon the best known causal attention guarantees, delivering $\log^{3/2}(n)/s$ approximation error with only $O(s)$ memory and $O(s^2 \log^2(n))$ compression overhead for a sequence of length $n$. We pair these developments with an efficient I/O-aware Triton implementation, demonstrate substantial speedups over FlashAttention 2, and use Express to overcome four resource bottlenecks in the language modeling pipeline: long-context prefill, KV cache compression, long-form memory-constrained decoding, and long-form compute-constrained decoding.

The concept of $\rho$-frequent hypercyclicity is introduced in order to provide a refined form of frequent hypercyclicity. This is achieved by replacing the denominator in the definition of frequent hypercyclicity by an appropriately chosen calibration function $\rho$. A $\rho$-Frequent Hypercyclicity Criterion is determined and the $\rho$-frequent hypercyclicity of weighted backward shifts is investigated.

In this work, we propose a novel approach, based on the path integral formulation of Koopman spectrum, to discover the phase-space geometry of the planar Cislunar Restricted 3 Body Problem (CR3BP). In contrast to existing techniques, which use trajectory-based (usually) local analysis, we leverage the Koopman operator framework, which generates a global linear representation of the system, to reconstruct the global phase space geometry of the CR3BP. In particular, we compute the principal eigenfunctions of the Koopman operator via the path integral approach and show how the zero level curves of these eigenfunctions encode the phase space characteristics of the planar CR3BP.

We study representations of \(1/n\) as a sum of three unit fractions whose denominators are all divisible by a prescribed integer \(m\). After scaling, this is equivalent to representing \(m/n\) as a sum of three unit fractions. Our main focus is the Erdős–Straus case \(m=4\). We introduce a divisor-based function \(\mathrm{fab}(n,a,b)\), prove that its admissible parameters recover exactly the decompositions of \(1/n\) with all three denominators divisible by \(4\), and compare this parametrization with well-known Type I/II descriptions.

This paper introduces range regularization for federated learning with linear systematic components to enhance statistical accuracy and induce cross-client regularity conducive to quantization, coding, and resource efficiency. Our approach identifies features with shared weights across different clients and adaptively clusters the weights of personalized features at extreme values, a process we refer to as polar clustering. Theoretical analysis of the associated estimators poses significant challenges due to the seminorm nature and non-decomposability of the regularizer. We develop new proof techniques for the nonasymptotic analysis of statistical accuracy and faithful pattern recovery. Moreover, a fast optimization algorithm that leverages varying degrees of local strong convexity is proposed to reduce iteration complexity. Experiments support the efficacy and efficiency of the proposed approach.

For a finite dimensional algebra $A$ of finite global dimension we study the Auslander–Yoneda algebra defined as the Yoneda algebra of the direct sum of all indecomposable $A$-modules. We show that the Auslander–Yoneda algebra is an Auslander–Gorenstein algebra if and only if every indecomposable left and right $A$-module is spherical in the sense of Auslander and Bridger. This motivates the study of spherical algebras defined by the condition that every indecomposable module is spherical. We characterize spherical algebras by a certain natural pair of subcategories being a split torsion pair. Moreover, we prove that representation-finite algebras which are spherical are directed and give a full classification of spherical Nakayama algebras. Furthermore, we show that replicated algebras of hereditary algebras are spherical. As a final application of the new notion of spherical algebras, we give a negative answer to a question of Venjakob on Auslander regular algebras in general, but show that there is a positive answer when assuming that every indecomposable left $A$-module is spherical.

We consider maps between two Riemannian manifolds and study a functional given in terms of the $L^\infty$-norm of the derivative. This functional is not differentiable, but we can define critical points with the help of a subdifferential. The resulting notion includes, for example, minimisers in a given homotopy class. We derive a geometric condition equivalent to criticality in this sense. The condition is formulated in terms of a vector-valued $1$-current on the domain manifold, which encapsulates some of the key properties of the critical point. Moreover, this $1$-current is itself a critical point of a generalised mass functional.

For every prime power $r\ge 2$, we show that any Sperner family $\mathcal F\subseteq 2^{[n]}$ with $|\mathcal F|\ge (r-1)n+1$ contains $r$ pairwise disjoint nonempty subfamilies whose unions are all equal and whose intersections are all equal. For $r=2$, this confirms a conjecture of Hegedüs, with the sharp threshold $n+1$. In this purely combinatorial problem, our proof combines a multilinear polynomial method, a continuity argument, and the topological Tverberg theorem.

A classical result, already observed by Coble, asserts that a genus 2 Jacobian can be embedded in $\mathbf P^8$ as the singular locus of a unique cubic hypersurface; similarly, the Kummer of a genus 3 Jacobian is embedded in $\mathbf P^7$ as the singular locus of a unique quartic hypersurface. We present a precise analogue of these results in the context of hyperkähler fourfolds: for the general member in the 20-dimensional locally complete families of polarized hyperkähler fourfolds of $\mathrm{K3}^{[2]}$-type with squares 4 or 6 and divisibility 1, we establish the existence of a unique Coble type hypersurface. As a consequence, we describe several geometric aspects of the corresponding moduli spaces.

In quantum image processing, a fundamental step is encoding classical image data into quantum states. This can be achieved using methods such as Flexible Representation of Quantum Images (FRQI), Quantum Probability Image Encoding (QPIE), and Novel Enhanced Quantum Representation (NEQR). However, on real quantum hardware, these encodings can quickly lead to circuits with many gates, large circuit depth, and high qubit usage, which is a problem for Noisy Intermediate-Scale Quantum (NISQ) devices. In this work, we investigate whether low-rank state approximation, formulated via Schmidt decomposition, can help reduce this complexity. The method keeps only the most significant parts of a quantum state's entanglement structure, making state preparation more efficient while preserving most of the image information. We compare the three encoding techniques in their original form and with low-rank approximation, evaluating metrics such as circuit depth, CNOT count, MSE, and visual quality of reconstructed images. The results reveal meaningful trade-offs between accuracy and resource efficiency, with the FRQI model achieving a 97 percent reduction in circuit depth while maintaining a near-perfect reconstruction (MSE of about 0.27). This demonstrates the potential of low-rank techniques for advancing practical quantum image processing on near-term hardware.

This paper shows that the Alexander-Briggs code of a knot gives rise to a new invariant that distinguishes prime alternating knots. The restriction to prime alternating knots precisely follows from the fact that our approach relies on Tait s flyping theorem. We also provide examples where the new invariant succeeds in separating knots that the well known invariants, such as some knot polynomials, fail.

We give a structural account of the finite free convolutions of Marcus, Spielman, and Srivastava in terms of reproducing kernel inner products on polynomial spaces and a multilinear model over the squarefree algebra. In this model, additive convolution becomes algebra multiplication, and the nilpotent logarithm linearizes it, recovering the finite free cumulants of Arizmendi and Perales. This perspective leads to a class $\mathcal{LC}_n$ of multilinear polynomials characterized by nonpositivity of higher-order cumulants, closed under additive convolution and satisfying several key permanence properties associated with negatively dependent measures. We show that every graph Laplacian pencil belongs to this class, with higher-order cumulants given by Hamiltonian cycle counts in induced subgraphs.

This article studies the high-frequency behavior of eigenstates of perturbations of the Laplace-Beltrami operator on the two-sphere $\mathbb{S}^{2}$ by a measure supported on an equator. We are interested in understanding to what extent this behavior can be described in terms of the geodesic flow of the sphere. This is done by analyzing quantum limits and semiclassical measures of sequences of high-frequency eigenfunctions, which describe how their $L^2$-masses concentrate in phase space. When the Laplacian on $\mathbb{S}^{2}$ is perturbed by a bounded potential, it is known that the family of all possible semiclassical measures is contained in the set of positive measures on the unit cosphere bundle $S^*\mathbb{S}^{2}$ that are invariant under geodesic flow (with equality in the unperturbed case). In this article, we show that the presence of a singular delta potential on a closed geodesic results in the existence of sequences of eigenfunctions whose semiclassical measure is not invariant under geodesic flow. In particular, one can find a sequence of eigenfunctions whose energy asymptotically concentrates on the hemisphere bounded by the equator on which the potential is concentrated.