Confirming a conjecture by Erd\H os and Pomerance, we prove that there exist intervals of length $\frac{cn\log n}{\log \log n}$ that do not contain distinct multiples of $1, 2, \ldots, n$.

The evaluation of heuristic optimizers on test problems, better known as \emph{benchmarking}, is a cornerstone of research in multi-objective optimization. However, most test problems used in benchmarking numerical multi-objective black-box optimizers come from one of two flawed approaches: On the one hand, problems are constructed manually, which result in problems with well-understood optimal solutions, but unrealistic properties and biases. On the other hand, more realistic and complex single-objective problems are composited into multi-objective problems, but with a lack of control and understanding of problem properties. This paper proposes an extensive problem generation approach for bi-objective numerical optimization problems consisting of the combination of theoretically well-understood convex-quadratic functions into unimodal and multimodal landscapes with and without global structure. It supports configuration of test problem properties, such as the number of decision variables, local optima, Pareto front shape, plateaus in the objective space, or degree of conditioning, while maintaining theoretical tractability: The optimal front can be approximated to an arbitrary degree of precision regarding Pareto-compliant performance indicators such as the hypervolume or the exact R2 indicator. To demonstrate the generator's capabilities, a test suite of 20 problem categories, called \emph{BONO-Bench}, is created and subsequently used as a basis of an illustrative benchmark study. Finally, the general approach underlying our proposed generator, together with the associated test suite, is publicly released in the Python package \texttt{bonobench} to facilitate reproducible benchmarking.

We obtain strict stability inequalities for homogeneous solutions of the one-phase Bernoulli problem. We prove that in dimension $7$ and above, cohomogeneity one solutions with bi-orthogonal symmetry are strictly stable. As a consequence, we obtain a bound on the first eigenvalue and the decay rates of Jacobi fields, with applications to the generic regularity of the one-phase problem.

Given a graph $G$, the family of all independent sets of size $k$ containing a fixed vertex $v$ is called a star with centre $v$, and is denoted by $\mathcal{I}_G^k(v)$. Motivated by a generalisation of the Erdős-Ko-Rado Theorem to the setting of independent sets in graphs, Hurlbert and Kamat conjectured that for every tree $T$ and every $k$, the maximum of $|\mathcal{I}_T^k(v)|$ can always be attained by a leaf of $T$. While this conjecture turns out to be false in general, it is known to hold for specific families of trees like spiders and caterpillars. In this paper, we prove that this conjecture holds for a new family of trees, the perfect $r$-ary trees, by constructing injections from stars centred at arbitrary vertices to stars centred at leaves. We also show that the analogous property holds for every forest $\mathcal{T}$ that is the disjoint union of perfect trees with possibly varying sizes and arities, and determine the leaf that maximises $|\mathcal{I}_{\mathcal{T}}^k(v)|$.

The Isometry Theorem of Chazal et al. and Lesnick is a fundamental result in persistence theory, which states that the interleaving distance between two one-parameter persistence modules is equal to the bottleneck distance between their barcodes. Significant effort has been devoted to extending this result to modules defined over more general posets. As these modules do not generally admit nice decompositions, one must restrict attention to the class of interval-decomposable modules in order to define an appropriate notion of bottleneck distance. Even with this assumption, it is known that bottleneck distance may not be equivalent to interleaving distance, but that it is Lipschitz stable under certain, fairly restrictive, assumptions. In this paper, we consider the more basic question of stability of the Hausdorff distance with respect to interleaving distance for interval-decomposable modules. Our main theorem is a Lipschitz stability result, which holds in a fairly general setting of interval-decomposable modules over arbitrary posets, where intervals are assumed to be taken from any family satisfying certain closure conditions. Along the way, we develop some new tools and results for interval-decomposable modules over arbitrary posets, in the form of geometrically-flavored characterizations of the existence of morphisms and interleavings between interval modules.

We study Bayesian model selection in colored Gaussian graphical models (CGGMs), which combine sparsity of conditional independencies with symmetry constraints encoded by vertex- and edge-colored graphs. A computational bottleneck in Bayesian inference for CGGMs is the evaluation of Diaconis-Ylvisaker normalizing constants, given by gamma-type integrals over cones of precision matrices with prescribed zeros and equality constraints. While explicit formulas are known for standard Gaussian graphical models only in special cases (e.g. decomposable graphs) and for a limited class of RCOP models, no general tractable framework has been available for broader families of CGGMs. We introduce a new subclass of RCON models for which these normalizing constants admit closed-form expressions. On the algebraic side, we identify conditions on spaces of colored precision matrices that guarantee tractability of the associated integrals, leading to Block-Cholesky spaces (BC-spaces) and Diagonally Commutative Block-Cholesky spaces (DCBC-spaces). On the combinatorial side, we characterize the colored graphs inducing such spaces via a color perfect elimination ordering and a 2-path regularity condition, and define the resulting Color Elimination-Regular (CER) graphs and their symmetric variants. This class strictly extends decomposable graphs in the uncolored setting and contains all RCOP models associated with decomposable graphs. In the one-color case, our framework reveals a close connection between DCBC-spaces and Bose-Mesner algebras. For models defined on BC- and DCBC-spaces, we derive explicit closed-form formulas for the normalizing constants in terms of a finite collection of structure constants and propose an efficient method for computing them in the commutative case. Our results broaden the range of CGGMs amenable to principled Bayesian structure learning in high-dimensional applications.

Generalizing the work of Atkin and Kaneko-Zagier in the elliptic case, we describe the non-ordinary locus of a genus-zero non-compact curve $Y$ in a Hilbert modular variety in terms of the zeros of generalized Atkin's orthogonal polynomials. The argument relies on the recent construction of lifts of partial Hasse invariants for $Y$. We further describe these orthogonal polynomials as denominators of Padé approximants to the logarithmic derivatives of solutions of the Picard-Fuchs differential equations associated with $Y$. This provides a new link between Padé approximation and the geometry of the non-ordinary locus, extending a classical observation of Igusa for the Legendre family and applying, in particular, to situations where the Picard-Fuchs equations do not admit modular solutions. As applications, we determine the three-term recurrence relations for Atkin polynomials attached to triangle curves via hypergeometric identities, and compute the supersingular locus of a double cover of the Teichmüller curve $W_{17}$. In the latter case, we conjecture that the associated supersingular polynomial is self-reciprocal, implying that supersingular points occur in pairs.

We study controlability issues for the group of volume-preserving diffeomorphisms of the torus $\mathbb T^d$ for system $\dot x=f(x)+u(t)$, where $f$ is a fixed divergence free vector field on $\mathbb T^d$ and $u(t)$ are constant vector fields which generate translations of the torus. Main results concern $d$ equals two or three.

We construct characteristic-zero lifts of partial Hasse invariants for genus zero non-compact curves in Hilbert modular varieties. The construction is based on recent results on the associated Picard-Fuchs differential equations. As an application, we relate the size of the non-ordinary locus of the modulo $p$ reduction of these curves to the dimension of spaces of (twisted) modular forms. We compute it explicitly for several Teichmüller curves, obtaining Deuring-like formulae. Moreover, we study the modulo $p$ reduction of (twisted) modular forms on not necessarily arithmetic genus-zero Fuchsian groups with modular embedding.

A cap set in projective or affine geometry over a finite field is a set of points no three of which are collinear. In this paper, we propose a new construction for complete cap sets that yields a cap set of size 124928 in the affine geometry AG(15,3). It should be noted that the constructed cap set in AG(15,3) is more powerful and exceeds at least by 4096 points than those that can be obtained from the previously known ones using the product or doubling constructions.

The Fourier transform of a bounded measurable function, $f$, on the real line is shown to be the second distributional derivative of a Hölder continuous function. The Fourier transform is written as the difference of $\int_{-1}^1 e^{-ist}f(t)\,dt$ and the second distributional derivative of the integral $\int_{\lvert{t}\rvert>1}e^{-ist}f(t)\,dt/t^2$. The space of such Fourier transforms is isometrically isomorphic to $L^\infty(\mathbb{R})$. There is an exchange theorem, inversion and convolution results. The Fourier transform of the functions $x\mapsto\cos^m(a/x)$ for each natural number $m$ are computed. Also for $x\mapsto x\sin(a/x)$ and $x\mapsto\arctan(x/a)$.

We propose a new kind of localized shock capturing for continuous (CG) and discontinuous Galerkin (DG) discretizations of hyperbolic conservation laws. The underlying framework of dissipation-based weighted essentially nonoscillatory (WENO) stabilization for high-order CG and DG approximations was introduced in our previous work. In this general framework, Hermite WENO (HWENO) reconstructions are used to calculate local smoothness sensors that determine the appropriate amount of artificial viscosity for each cell. In the original version, candidate polynomials for WENO averaging are constructed using the derivative data from von Neumann neighbors. We upgrade this standard `cell-cell' reconstruction procedure by using WENO polynomials associated with mesh vertices as candidate polynomials for cell-based WENO averaging. The Hermite data of individual cells is sent to vertices of those cells, after which vertex-averaged HWENO data is sent back to cells containing the vertices. The new `cell-vertex' averaging procedure includes the data of vertex neighbors without explicitly adding them to the reconstruction stencils. It mitigates mesh imprinting and can also be used in classical HWENO limiters for DG methods. The second main novelty of the proposed approach is a quadrature-driven distribution of artificial viscosity within high-order finite elements. Replacing the linear quadrature weights by their nonlinear WENO-type counterparts, we concentrate shock-capturing dissipation near discontinuities while minimizing it in smooth portions of troubled cells. This redistribution of WENO stabilization preserves the total dissipation rate within each cell and improves local shock resolution without relying on subcell decomposition techniques. Numerical experiments in one and two dimensions demonstrate substantial improvements in accuracy and robustness for high-order elements.

Representations of the operator system determined by the canonical generators of the free product of two cyclic groups of order $2$ and $k$, or $d$ cyclic groups of order $2$, are studied for the purpose of shedding light on the noncommutative geometry of noncommutative $d$-cubes and $k$-prisms. By way of the duality of the categories NCConv and OpSys of noncommutative convex sets and operator systems, respectively, an analysis of noncommutative extreme points, exactness, the lifting property, automatic complete positivity, controlled completely positive extensions, tensor products, and operator system duality is undertaken. Of note is the pairing of two classical dilation theorems of Halmos and Mirman to give a complete description of the noncommutative triangular prism in terms of joint unitary dilations.

We introduce FedSGM, a unified framework for federated constrained optimization that addresses four major challenges in federated learning (FL): functional constraints, communication bottlenecks, local updates, and partial client participation. Building on the switching gradient method, FedSGM provides projection-free, primal-only updates, avoiding expensive dual-variable tuning or inner solvers. To handle communication limits, FedSGM incorporates bi-directional error feedback, correcting the bias introduced by compression while explicitly understanding the interaction between compression noise and multi-step local updates. We derive convergence guarantees showing that the averaged iterate achieves the canonical $\boldsymbol{\mathcal{O}}(1/\sqrt{T})$ rate, with additional high-probability bounds that decouple optimization progress from sampling noise due to partial participation. Additionally, we introduce a soft switching version of FedSGM to stabilize updates near the feasibility boundary. To our knowledge, FedSGM is the first framework to unify functional constraints, compression, multiple local updates, and partial client participation, establishing a theoretically grounded foundation for constrained federated learning. Finally, we validate the theoretical guarantees of FedSGM via experimentation on Neyman-Pearson classification and constrained Markov decision process (CMDP) tasks.

We study canonical basis elements in higher-level Fock spaces associated with the quantum group $U_q(\mathfrak{gl}_\infty)$, which are conjecturally related to Calogero-Moser theory for complex reflection groups. We generalize the Leclerc-Miyachi formula to arbitrary levels by introducing new explicit constructions based on symbols, including a column removal theorem and closed formulas in several cases. These results provide explicit descriptions of canonical basis elements with applications to Calogero-Moser cellular characters and to the decomposition matrices of Ariki-Koike algebras.

A new notion of thickness for subsets of $B[0,1]\subset \mathbb{R}^n$ called affine thickness is defined; this notion of thickness is a generalisation of Falconer-Yavicoli thickness and is adapted to be used in the study of certain sets with affine cut outs. Thick sets are proven to be winning for the matrix potential game introduced in (arXiv:2508.11577) and as an application we can prove that for a thick set, there exists $M\in\mathbb{N}$ depending on the thickness of the set, such that the set contains a homothetic copy of every finite set with at most $M$ elements. Additionally, the author provides a counter-example to the gap lemma in $\mathbb{R}^n$ ($n\geq 2$) for Falconer-Yavicoli thickness, stated in (Math. Z., 2022) proving this result does not hold in the generality stated. We go on to provide a gap lemma for affine thickness in $\mathbb{R}^n$ (for $n\geq 2$) under additional conditions to the classical Newhouse gap lemma.

Many applications, such as systems of interacting particles in physics, require the simulation of diffusion processes with singular coefficients. Standard Euler schemes are then not convergent, and theoretical guarantees in this situation are scarce. In this work we introduce a Lyapunov-tamed Euler scheme, for drift coefficients for which the weak derivative is dominated by a function that obeys a certain generic Lyapunov-type condition. This allows for a range of coefficients that explode to infinity on a bounded set. We establish that, in terms of Lp-strong error, the Lyapunov-tamed scheme is consistent and moreover achieves the same order of convergence as the standard Euler scheme for Lipschitz coefficients. The general result is applied to systems of mean-field particles with singular repulsive interaction in 1D, yielding an error bound with polynomial dependency in the number of particles.

We construct an explicit isomorphism between the HOMFLY-PT homology of $(n,n+1)$ torus knots and the direct sum of hook isotypic components of the space of diagonal coinvariants. As a consequence, we compute the action of tautological classes in HOMFLY-PT homology of $(n,n+1)$ torus knots and prove that it extends to an action of the Lie algebra of Hamiltonian vector fields on the plane. We also compute the action of differentials $d_N$ in Rasmussen spectral sequences from HOMLFY-PT to $\mathfrak{gl}(N)$ homology of $(n,n+1)$ torus knots.

We generalize a result of Popa-Schnell and show that the isogeny class of the Picard variety is twisted derived invariant. Using this, we prove that any twisted Fourier-Mukai partner of an abelian variety is an abelian variety. We then provide a necessary and sufficient isogeny-based condition for two abelian varieties to be twisted derived equivalent.

We study virtual retracts in groups acting on rooted trees. We show that finitely generated branch groups do not have the local retraction (LR) property. Furthermore, we specialize to iterated monodromy groups of post-critically finite quadratic complex polynomials and show that the (LR) property characterizes, among post-critically finite quadratic complex polynomials, those with a euclidean orbifold, i.e. the powering map and the Chebyshev polynomial. Lastly, we show that periodic quadratic complex polynomials provide new examples of pro-$2$ groups with complete finitely generated Hausdorff spectrum.

We consider a three-dimensional domain occupied by a homogeneous, incompressible, non-Newtonian, heat-conducting fluid with prescribed nonuniform temperature on the boundary and no-slip boundary conditions for the velocity. No external body forces are assumed. The constitutive relation for the Cauchy stress tensor is assumed in a general form that includes, in particular, the power-law and Ladyzhenskaya models with the power-law exponent in the range where neither regularity, uniqueness, nor the validity of the energy equality is known to hold. Nevertheless, we introduce a novel concept of solution suitable for this setting, which enables us to establish the existence of global-in-time solutions for arbitrary physically relevant initial data. A remarkable feature of this formulation is that the steady-state solution is nonlinearly stable: every such solution converges, in a suitable sense, to the steady state as time tends to infinity. This provides the first result that combines existence with long-time stability in this physically relevant yet mathematically challenging regime.

We recall the Lounesto classification of 1/2-spin spinor fields, based on the vanishing of spinorial bilinear quantities: the classes are the regular spinor fields (i.e. the Dirac field), as well as singular spinor fields, also known as flag-dipole spinor fields, admitting two limiting sub-classes, given by the dipole spinors (i.e. the Weyl spinor) and the flagpole spinors (i.e. the Majorana spinor). We discuss each class in terms of its representatives, with particular emphasis upon the flag-dipole spinor fields.

For a finite metric graph $X=(V,E,\ell)$, where $V$ is endowed with the shortest path metric, we consider the transportation cost problem associated with the distance $d$ on $V$. Namely, for $f$ a function with total sum 0 on $V$, write $f=\sum_{a,b\in V}P(a,b)(δ_a-δ_b)$ where the transportation plan $P$ satisfies $P(a,b)\geq 0$ for $(a,b)\in V\times V$. The cost of $P$ is $W(P):=\sum_{a,b\in V}P(a,b)d(a,b)$ and the transportation norm of $f$ is $\|f\|_{TC}=\min_P W(P)$ where $P$ runs over all transportation plans for $f$. In this semi-survey paper, we give short proofs for the following statements: 1)There always exists an optimal transportation plan supported in $V_+\times V_-$ where $V_+=\{x\in V: f(x)>0\}$ and $V_-=\{x\in V: f(x)<0\}$. If $X$ is a metric tree, we may moreover assume that this plan involves at most $|Supp(f)|-1$ transports. 2) There always exists an optimal transportation plan supported in the set of edges of $X$. 3) Better, there always exists an optimal transportation plan supported in some spanning tree of $X$. We use this to reprove known formulae for the transportation norm when $X$ is either a tree or a cycle.

This work presents a stochastic analysis of fifth-order KdV soliton momentum distribution in a damping regime. An explicit representation of the soliton momentum associated with amplitude variation is derived in terms of a random time function in the presence of dissipation. Statistical interpretations of soliton propagation modes, amplitude fluctuations, and amplification are analyzed within a $δ$-correlated Gaussian random framework. Graphical results obtained using Python illustrate the physical insight into amplitude fluctuation and energy flow. Finally, under a dominant approximation, the nonlinear momentum evolution equation is shown to reduce to the Painlevé second equation, a well-known integrable model appearing in diverse physical systems.

Edge intelligence enables AI inference at the network edge, co-located with or near the radio access network, rather than in centralized clouds or on mobile devices. It targets low-latency, resource-constrained applications with large data volumes, requiring tight integration of wireless access and on-site computing. Yet system performance and cost-efficiency hinge on joint pre-deployment dimensioning of radio and computational resources, especially under spatial and temporal uncertainty. Prior work largely emphasizes run-time allocation or relies on simplified models that decouple radio and computing, missing end-to-end correlations in large-scale deployments. This paper introduces a unified stochastic framework to dimension multi-cell edge-intelligent systems. We model network topology with Poisson point processes, capturing random user and base-station locations, inter-cell interference, distance-based fractional power control, and peak-power constraints. By combining this with queueing theory and empirical AI inference workload profiling, we derive tractable expressions for end-to-end offloading delay. These enable a non-convex joint optimization that minimizes deployment cost under statistical QoS guarantees, expressed through strict tail-latency and inference-accuracy constraints. We prove the problem decomposes into convex subproblems, yielding global optimality. Numerical results in noise- and interference-limited regimes identify cost-efficient design regions and configurations that cause under-utilization or user unfairness. Smaller cells reduce transmission delay but raise per-request computing cost due to weaker server multiplexing, whereas larger cells show the opposite trend. Densification reduces computational costs only when frequency reuse scales with base-station density; otherwise, sparser deployments improve fairness and efficiency in interference-limited settings.

The distinguishability quantified by information measures after being processed by a private mechanism has been a useful tool in studying various statistical and operational tasks while ensuring privacy. To this end, standard data-processing inequalities and strong data-processing inequalities (SDPI) are employed. Most of the previously known and even tight characterizations of contraction of information measures, including total variation distance, hockey-stick divergences, and $f$-divergences, are applicable for $(\varepsilon,0)$-local differential private (LDP) mechanisms. In this work, we derive both linear and non-linear strong data-processing inequalities for hockey-stick divergence and $f$-divergences that are valid for all $(\varepsilon,δ)$-LDP mechanisms even when $δ\neq 0$. Our results either generalize or improve the previously known bounds on the contraction of these distinguishability measures.

We find explicit solutions and singularities of the Ricci-harmonic flow of $\mathrm{G_2}$-structures, the Ricci-like flows of $\mathrm{G_2}$-structures studied by Gianniotis-Zacharopoulos in arXiv:2505.06872 (J. Geom. Anal. 36.2 (2026)) and of the negative gradient flow of an energy functional of $\mathrm{G_2}$-structures, on $7$-dimensional contact Calabi-Yau manifolds and the $7$-dimensional Heisenberg group. We prove that the natural co-closed $\mathrm{G_2}$-structure on a contact Calabi-Yau manifold as the initial condition leads to an ancient solution of the Ricci-harmonic flow with a finite time Type I singularity, and it gives an immortal solution to the Ricci-like flows with an infinite time singularity which are Type III if the transversal Calabi-Yau distribution is flat, and Type IIb otherwise. The same ansatz gives ancient solution to the negative gradient flow of $\mathrm{G_2}$-structures. These are the first examples of Type I singularities of the Ricci-harmonic flow and Type IIb and Type III singularities of the Ricci-like flows. We also obtain similar solutions for all the three flows on the $7$-dimensional Heisenberg group.

We give analytical results for propagation of uncertainty through trained multi-layer perceptrons (MLPs) with a single hidden layer and ReLU activation functions. More precisely, we give expressions for the mean and variance of the output when the input is multivariate Gaussian. In contrast to previous results, we obtain exact expressions without resort to a series expansion.

We revisit noisy twenty questions estimation and study the privacy-resolution tradeoff for adaptive query procedures. Specifically, in twenty questions estimation, there are two players: an oracle and a questioner. The questioner aims to estimate target variables by posing queries to the oracle that knows the variables and using noisy responses to form reliable estimates. Typically, there are adaptive and non-adaptive query procedures. In adaptive querying, one designs the current query using previous queries and their noisy responses while in non-adaptive querying, all queries are posed simultaneously. Generally speaking, adaptive query procedures yield better performance. However, adaptive querying leads to privacy concerns, which were first studied by Tsitsiklis, Xu and Xu (COLT 2018) and by Xu, Xu and Yang (AISTATS 2021) for the noiseless case, where the oracle always provides correct answers to queries. In this paper, we generalize the above results to the more practical noisy case, by proposing a two-stage private query procedure, analyzing its non-asymptotic and second-order asymptotic achievable performance and discussing the impact of privacy concerns. Furthermore, when specialized to the noiseless case, our private query procedure achieves better performance than above-mentioned query procedures (COLT 2018, AISTATS 2021).

This paper studies the existence of multiple non-trivial stationary solutions of a partial differential equation (PDE) model introduced in [3], motivated by collective ant behavior. Previous work suggested the presence of two types of non-trivial stationary solutions for this PDE system: spot and lane solutions. In this paper, we establish the existence of these families of solutions along a bifurcation sequence as the interaction strength grows, with progressively increasing numbers of clusters and parallel lanes, respectively. Finally, we show that, for small values of the anticipation parameter, the first bifurcating spot solutions are locally dynamically stable, while the lane solutions are unstable.