We introduce propositional team-based logics expressively complete for (quasi) downward and (quasi) upward closed properties in a syntactically dual way, by using variants of the inclusion atom. In particular, the variants of the primitive inclusion atoms used in the (quasi) upward closed setting have equivalent formulas using variants of the might modality. The duality is visible in the logics' normal forms, mirroring the duality between the (quasi) upward and downward closed settings, where the quasi variants take special care of the empty and full team. Furthermore, we defined sound and complete natural deduction systems for each logic.

We study the local limits of uniform random triangulations with boundaries in the regime where the genus is proportional to the number of faces. Budzinski and Louf proved in 2020 that when there are no boundaries, the local limits exist and are the Planar Stochastic Hyperbolic Triangulation (PSHT) introduced in PSHT. We show that when the triangulations considered have size n and boundaries with total length p that tends to infinity with n and p=o(n), the local limits around a typical boundary edge are the half-plane hyperbolic triangulations defined by Angel and Ray. This provides, for the first time, a construction of these hyperbolic half-plane triangulations as local limits of large genus triangulations. We also prove that under the condition p = o(n), the local limit when rooted on a uniformly chosen oriented edge is given by the PSHT. Contrary to the proof of Budzinski and Louf, the latter does not rely on the Goulden-Jackson recurrence relation, but only on coarse combinatorial estimates. Thus, we expect that the proof can be adapted to local limits in similar models.

Motivated by classical problems in extremal graph theory, we study a chromatic analogue of Roth-type questions for linear equations over $\mathbb F_p$. Given a homogeneous equation $\mathcal L:\sum_{i=1}^k c_i x_i=0$ with $k\ge 3$, we study $\mathcal L$-solution-free sets $A\subseteq \mathbb F_p$ through the chromatic number of the Cayley graph $\mathsf{Cay}(\mathbb F_p,A)$. We introduce the \emph{chromatic threshold} $δ_χ(\mathcal L)$, the minimum density that guarantees bounded chromatic number of $\mathsf{Cay}(\mathbb F_p,A)$ among all $\mathcal L$-solution-free sets $A$, and determine exactly when $δ_χ(\mathcal L)=0$. We prove that $δ_χ(\mathcal L)=0$ if and only if $\mathcal L$ contains a zero-sum subcollection of at least three coefficients. A key ingredient is a quantitative chromatic lower bound for Cayley graphs on $\mathbb Z_p^n$ generated by Hamming balls around the all-ones vector. This is obtained by introducing a new Kneser-type graph that admits a natural embedding into $\mathbb Z_p^n$, together with an equivariant Borsuk--Ulam type argument. As a consequence, we resolve a question of Griesmer. We further relate our classification to the hierarchy of measurable, topological, and Bohr recurrence. In particular, we show that every infinite discrete abelian group admits a set that is topological recurrent but not measurable recurrent, extending the seminal examples of Kříž and Ruzsa.

We study the decoding problem for quantum Tanner codes and propose to exploit the underlying local code structure by grouping check nodes into more powerful generalized check nodes for enhanced iterative belief propagation (BP) decoding by decoding the generalized checks using a maximum a posteriori (MAP) decoder as part of the check node processing of each decoding iteration. We mainly study the finite-length setting and show that the proposed enhanced generalized BP decoder for quantum Tanner codes significantly outperforms the standard quaternary BP decoder with memory effects, as well as the recently proposed Relay-BP decoder, even outperforming generalized bicycle (GB) codes with comparable parameters in some cases. For other classes of quantum low-density parity-check (qLDPC) codes, we propose a greedy algorithm to combine checks for generalized BP decoding. However, for GB codes, bivariate bicycle codes, hypergraph product codes, and lifted-product codes, there seems to be limited gain by combining simple checks into more powerful ones. To back up our findings, we also provide a theoretical cycle analysis for the considered qLDPC codes.

Singular statistical models-including mixtures, matrix factorization, and neural networks-violate regular asymptotics due to parameter non-identifiability and degenerate Fisher geometry. Although singular learning theory characterizes marginal likelihood behavior through invariants such as the real log canonical threshold and singular fluctuation, these quantities remain difficult to interpret operationally. At the same time, widely used criteria such as WAIC and WBIC appear disconnected from underlying singular geometry. We show that posterior tempering induces a one-parameter deformation of the posterior distribution whose associated observables generate a hierarchy of thermodynamic response functions. A universal covariance identity links derivatives of tempered expectations to posterior fluctuations, placing WAIC, WBIC, and singular fluctuation within a unified response framework. Within this framework, classical quantities from singular learning theory acquire natural thermodynamic interpretations: RLCT governs the leading free-energy slope, singular fluctuation corresponds to curvature of the tempered free energy, and WAIC measures predictive fluctuation. We formalize an observable algebra that quotients out non-identifiable directions, allowing structurally meaningful order parameters to be constructed in singular models. Across canonical singular examples-including symmetric Gaussian mixtures, reduced-rank regression, and overparameterized neural networks-we empirically demonstrate phase-transition-like behavior under tempering. Order parameters collapse, susceptibilities peak, and complexity measures align with structural reorganization in posterior geometry. Our results suggest that thermodynamic response theory provides a natural organizing framework for interpreting complexity, predictive variability, and structural reorganization in singular Bayesian learning.

Acoustic scattering by vehicle surfaces can have significant effects on overall noise levels. In this paper, we present a space-time Galerkin time-domain boundary element method (TDBEM) that offers several distinct advantages over contemporary scattering methods for prediction of acoustic scattering and shielding of complex aeroacoustic sources such as propellers and rotors. The time-domain approach allows efficient simulation of transient, rotating, and broadband noise sources, while the Galerkin formulation is robust and unconditionally stable without any tuned numerical parameters. The main challenge of the Galerkin approach, namely the numerically difficult double space-time integration, is resolved through an efficient decomposition-based quadrature procedure. We present three cases with analytical solutions to validate the method and study its numerical properties, demonstrating excellent agreement for scattering and shielding by a variety of different geometries. We then apply the TDBEM to a trailing edge-mounted propeller case, comparing the numerical predictions with experimental measurements. The results demonstrate good agreement between predicted and measured scattering and shielding in a practical application case.

We establish a free analogue of Obata's rigidity theorem. More precisely, Cheng and Zhou (2017) proved that on a weighted Riemannian manifold, the sharp spectral gap (Poincaré constant) is achieved only when the space splits isometrically off a one-dimensional Gaussian factor, providing an infinite-dimensional counterpart of Obata's rigidity theorem. We obtain the corresponding phenomenon in free probability, extending it beyond the setting of analytic self-adjoint potentials: Assume a self-adjoint $n$-tuple $X=(X_1,\dots,X_n)$ admits Lipschitz conjugate variables in the sense of Dabrowski (2014). Under a suitable non-commutative curvature-dimension condition, we show that any non-zero saturator of Voiculescu's free Poincaré inequality must be an affine function of the generators. Consequently, we deduce that the von Neumann algebra $M=W^*(X_1,\dots,X_n)$ necessarily splits off a freely complemented semicircular component $W^*(Y_1)\simeq L^{\infty}([-2,2],μ_{\rm sc})$, which is also maximal amenable in $M$. More generally, whenever the first eigenspace of the free Laplacian $Δ=\partial^*\bar\partial$ is finite-dimensional of rank $r\ge 1$, our rigidity argument shows that these $r$ extremal directions form a free semicircular family, yielding a free product decomposition with an $L(\mathbb{F}_r)$ factor. This provides a free-probability analogue of the classical Gaussian splitting phenomenon and reveals a rigidity mechanism under non-commutative curvature.

We study equilibrium concepts in non-cooperative games under uncertainty where both beliefs and mixed strategies are represented by non-additive measures (capacities). In contrast to the classical Nash framework based on additive probabilities and linear convexity, we employ capacities and max-plus integrals to model qualitative and idempotent decision criteria. Two equilibrium notions are investigated: Nash equilibrium in mixed strategies expressed by capacities, and equilibrium under uncertainty in the sense of Dow and Werlang, where players choose pure strategies but evaluate payoffs with respect to non-additive beliefs. For games with compact strategy spaces and continuous payoffs, we establish existence results for both equilibrium concepts using abstract convexity techniques and a Kakutani-type fixed point theorem.

In this article, convex optimization is introduced as a promising tool to study Eshelby based inverse micromechanics problems. The focus is on inverse micromechanics using the Eshelby-Mori-Tanaka model given the dielectric constants of the composite material and of all of its components. The model is exactly the same for the conductivity properties as well. This choice of model is made since the model is fairly simple and has a closed form analytical solution for the case of spheroidal inclusions as well. The forward or direct micromechanics problem deals with the determination of effective properties of a composite material given the properties of its components and microstructural information. The focus is on isotropic composites and the distribution of inclusions is assumed to be such that this holds. The inverse micromechanics problem considered in this paper deals with the determination of microstructural information given the properties of the composite material and all of its components. Since in this paper, isotropy of the composite and only spherical inclusions are considered, the goal is to determine just volume fractions of the components of the composite material. The inverse problem is formulated as a Linear Programming problem and is solved. Before this, the inverse problem and certain important variants of it are examined through the lens of convex optimization. Lastly, promising results regarding the relationship between dispersive materials, noise in measurements, and quality of obtained volumetric splits are showcased. The scope of the use of convex optimization in inverse micromechanics is discussed.

We consider a two-dimensional, pure capillary drop of nearly-circular shape, having constant vorticity. We write the Craig-Sulem equations on the unit circle, then on the flat torus. We show their Hamiltonian structure and we then observe symmetries and we derive constants of motions. After showing linear stability for rotating circles, we prove the existence of rotating waves by combining a bifurcation-theoretical approach together with critical point theory. Finally, by exploiting the Hamiltonian structure, we show that whenever volume and barycenter are fixed to be the same as those of rotating circle, this solution is also conditionally energetically stable. This holds in the irrotational case as well, in agreement with the stability analysis of rotating cylinder jets in Rayleigh [25].

Two subsets $A$ and $B$ of a ground set $X$ are \emph{crossing} if none of the four sets $A\setminus B,B\setminus A,A\cap B, X\setminus (A\cup B)$ are empty. Almost fifty years ago, Karzanov and Lomonosov conjectured that every family of subsets of an $n$-element ground set with no $k$-pairwise crossing members has size $O(kn)$. We prove the bound $O_k(n)$, settling (arguably) the main problem about the growth rate of such families.

In this work, we consider a multi-operator mmWave network implementing secondary license sharing (SLS) where a primary license holder leases secondary licenses to secondary users, allowing them to access its licensed spectrum under some pre-defined transmission constraints. The highly directional nature of mmWaves, along with their sensitivity to blockages, naturally confines the interference to/from devices to narrow angular sectors within a certain range around themselves. This motivates us to consider a spatially-aware SLS that determines a secondary link's activity based on its distance/orientation relative to the primary link, as well as blockages around it. By leveraging the tools of stochastic geometry, we develop an analytical framework to design and study such spatially-aware SLS in mmWave networks. Our analysis quantifies the transmission opportunities available to secondary users and the resulting coverage probabilities for both primary and secondary links. We characterize the effect of directionality and blockage conditions, along with transmission restrictions and secondary users' density, on the performance of both operators. Via numerical investigation, we derive various insights. We show that blockage conditions can change the shape of coverage plots and thus affect key conclusions. Further, blockage and directionality can increase the transmission opportunities for secondary users, improving the feasibility and gains of SLS.

In this article we study the structured distance to singularity for a nonsingular matrix $A\in\mathbb{C}^{n\times n}$, with a prescribed linear structure $\mathcal{S}$ (for instance, a sparsity pattern, or a real Toeplitz structure), i.e., the norm of the smallest perturbation $Δ\in \mathcal{S}$, such that $A + Δ$ is singular. This is an example of structured matrix nearness problem: a family of problems that arise in control and systems theory and in numerical analysis, when characterizing the robustness of a certain property of a system with respect to perturbations that are constrained to a certain structure (for example the structure of the nominal system). We start by highlighting the parallelism between two main tools which have been proposed in the literature: a gradient system approach for a functional in the eigenvalues, which requires the solution of certain low-rank matrix differential equations (see [Guglielmi, Lubich, Sicilia, SINUM 2023]), and a two-level optimization approach in which the inner linear least-squares problem is solved explicitly (see [Usevich, Markovsky, JCAM 2014] and [Gnazzo, Noferini, Nyman, Poloni, FoCM 2025]). In particular, these articles underline the remarkable property that $Δ$ is (at least generically) the orthogonal projection onto the structure $\mathcal{S}$ of a rank-1 matrix $uv^*$. This property and the parallelism suggest a new reformulation of the problem into a system of nonlinear equations in the two vector unknowns $u,v \in\mathbb{C}^n$. We study this new formulation, and propose an algorithm to solve these nonlinear equations directly with the multivariate Newton's method. We discuss how to avoid the singularity of such system of nonlinear equations, and how to ensure monotonic convergence. The resulting algorithm is faster than the existing ones for large matrices, and maintains comparable accuracy.

There are precisely four arrangements of two point-hyperplane anti-flags. We consider the corresponding relations on the set of such anti-flags and show that each of them can be recovered from any other except in one special case. If the field consists of two elements, then one of the relations cannot be used to recover each of the remaining three. This is related to a bijection between anti-flags and exterior points of the hyperbolic polar space which exists in this case.

We study the Fourier approximation $\mathcal{F}_N$ of the sign function by the Krawtchouk polynomials. We give numerical evidence that the Gibbs phenomenon of the approximation differs from the classical Gibbs constant; this is in contrast to other families of orthogonal polynomials. We also show that the steepness $\mathcal{F}_N'(0)$ of the approximation is bounded by explicitly proving $\lim_{N \to \infty} \mathcal{F}_N'(0) = \log 4$. This is also in contrast to approximations by classical orthogonal polynomials, where the steepness has been shown to be unbounded as the degree increases.

We analyze a linear parabolic equation with homogeneous Dirichlet boundary conditions posed in domains whose evolution may involve topological transitions. The domains are described as sublevel sets of a smooth space-time level set function, allowing for transitions such as domain splitting and merging and the creation or vanishing of islands and holes. We introduce anisotropic space-time function spaces that extend the classical Bochner spaces used in cylindrical domains and establish key functional-analytic properties of these spaces, including the density of compactly supported smooth functions. This framework enables the application of the Babuška-Banach theorem, yielding existence, uniqueness, and a priori estimates for weak solutions. The analysis applies to domain evolutions generated by level set functions with isolated nondegenerate critical points, which correspond to the generic topology changes classified by Morse theory in two and three spatial dimensions.

We investigate the Couette-Taylor problem for a steady incompressible viscous fluid in a 3D cylindrical annulus, where one of the two cylinders is still, under both Dirichlet and boundary conditions involving the vorticity that naturally appear in the weak formulation. The outcome of this study is twofold. First, we explicitly determine all the solutions with a specific geometric \emph{partial invariance}, which coincide with the so-called spiral Poiseuille or Poiseuille-Couette flows depending on the boundary conditions. Second, for small boundary data, we provide stability of such solutions, that is, no steady finite-energy perturbations are admissible. To achieve this result in presence of vorticity boundary conditions, we find a substantial analytical difference depending on which cylinder is still.

We develop a calculus of space-time controlled fields for rough stochastic systems. This approach provides a unified composition rule for evaluating random fields along rough semimartingales and yields a rough stochastic Itô-Wentzell formula under natural and verifiable regularity assumptions. Our motivation comes from works of Hudde et al. (2024) and, independently, Del Moral and Singh (2022) where the authors established, respectively, Itô-Alekseev-Gröbner, backward Itô-Wentzell, and diffusion interpolation formulas.

We establish a new global endpoint Sobolev inequality for measures that extends the classical theorem of Meyers-Ziemer by placing a maximal function on the right-hand side. This result has several significant consequences. It extends naturally to functions of weighted bounded variation and yields corresponding capacity and isoperimetric inequalities. The inequality is also closely connected to endpoint estimates for fractional operators, including bounds for fractional maximal functions and Hardy space endpoint estimates for the Riesz potential. Our main inequality yields a family of endpoint inequalities, characterized in terms of subrepresentation formulas, Lorentz space improvements, and isoperimetric inequalities for measures and bounded open sets. When one moves away from the endpoint to $p>1$, the analogous inequalities no longer hold in general; however, we identify a sharp bumped maximal function for which the corresponding non-endpoint inequality is valid. Finally, we show that this framework yields new $(p,p)$ two-weight Sobolev inequalities.

We study sweeping processes in a Hilbert space driven by time-dependent uniformly prox-regular sets, allowing the moving constraint to exhibit discontinuities of bounded variation. We introduce a new integral formulation for bounded-variation trajectories, given by a global variational inequality tested against continuous admissible trajectories, and we compare it with the standard differential-measure formulation, in which the differential measure of the trajectory is constrained by the proximal normal cone. In the prox-regular (generally nonconvex) framework, the variational inequality necessarily includes a quadratic correction term reflecting the hypomonotonicity of proximal normal cones. Under mild regularity assumptions on the moving set, including lower semicontinuity in time, uniform prox-regularity of the values, and a selection-extension property guaranteeing a rich class of test trajectories (satisfied, for instance, in the convex case and for bounded prox-regular sets), we prove that the new integral formulation is equivalent to the differential-measure formulation. This yields a unified bounded-variation notion of solution for prox-regular sweeping processes. We further establish a Brézis-Ekeland-Nayroles-type variational characterization via a prox-regular variational residual: the residual is nonpositive along every admissible trajectory, and solutions are exactly those trajectories for which this residual attains its maximal value, namely zero. As a consequence, we prove a stability result: a uniform limit of admissible trajectories with vanishing residual is a solution of the limit sweeping process. The resulting variational framework provides a robust tool for stability and approximation analyses in the prox-regular, nonconvex setting.

We establish a trace formula for signatures of hermitian forms over Azumaya algebras with involution, extending Knebusch's work on symmetric bilinear forms over finite étale extensions of commutative base rings. As an application when the base ring is semilocal, we obtain an exact sequence for total signatures, related to Pfister's local-global principle and the notion of stability index.

We apply the idea of using a combination of virial inequalities and Kato smoothing, previously applied to NLS and generalized KdV pure power equations to Euler-Poisson: we assume that a solution remains very close for all times to a soliton in an appropriate space and then we prove an asymptotic convergence to a soliton for $t\to +\infty$.

We study the resolution of discontinuous singularities in gas dynamics via multi-dimensional rarefaction waves. While the mechanism is well-understood in one spatial dimension, the rigorous construction in higher dimensions has remained a challenging open problem since Majda's proposal, primarily due to the characteristic nature of rarefaction fronts which leads to derivative losses in linearized estimates. In this paper, we establish the nonlinear stability of multi-dimensional rarefaction waves for the compressible Euler equations with ideal gas law. We prove that for initial data being small perturbations of the planar rarefaction wave in $H^s$ ($s > s_c$), there exists a unique global solution that converges asymptotically to the background rarefaction wave as $t \to \infty$. Our proof relies on a novel Geometric Weighted Energy Method (GWEM), which yields stable energy estimates without loss of derivatives in standard Sobolev spaces, overcoming the limitations of previous Nash-Moser schemes. A key ingredient is a detailed geometric description of the rarefaction wave fronts via the acoustical metric, where we identify a hidden extra vanishing structure in the top-order derivatives of the characteristic speed. This is the first paper in a series, providing the crucial a priori energy bounds. The existence of solutions and applications to the multi-dimensional Riemann problem will be addressed in the forthcoming companion paper.

The Teichmüller space of a closed set in the Riemann sphere is a simply connected complex Banach manifold. Its complex structure follows from Lieb isomorphism. In this paper, we show the conformal naturality of Lieb isomorphism. We then study Douady-Earle section for these Teichmüller spaces. In particular, we study the real-analyticity of Douady-Earle section for classical Teichmüller spaces. We give two explicit examples of maximal holomorphic motions over simply connected complex Banach manifolds. As an application of the real-analyticity of the Douady-Earle section for the classical Teichmüller spaces of Riemann surfaces, we prove a new result showing that a family of Jordan curves varies real-analytically over a simply connected complex Banach manifold and as quasiconformal images of the one at the basepoint, provided that a finite number of marked points on the Jordan curves vary holomorphically over the same parameter space.

Symmetries are fundamental to both turbulence and differential equations. The large-eddy simulation (LES) equations inherit these symmetries provided the LES closure respects them. Classical LES closures based on eddy viscosity or scale similarity preserve many of the original symmetries by design. Recently, data-driven neural network closures have been applied to LES to improve accuracy, but stability and generalizability remain challenges, as symmetries are not automatically enforced. In this work, we compare approaches for constructing symmetry-preserving data-driven LES closures, including tensor-basis neural networks (TBNNs) and group-convolutional neural networks, alongside unconstrained convolutional networks. All three data-driven closures outperform classical models in both the functional sense (producing the right amount of dissipation) and the structural sense (stress tensor prediction). While unconstrained networks achieve comparable prediction accuracy, symmetry-preserving models produce more physically consistent velocity-gradient statistics, suggesting that enforcing symmetries improves the quality of the learned closure beyond what aggregate error metrics such as relative tensor prediction errors capture.

Zabrodin recently proposed a generalization of Dyson Brownian motion to a setting where the particles are confined to a smooth Jordan curve in the plane. In this paper, we discuss a rigorous construction of such a process on a rectifiable Jordan curve and study some of its basic properties. Under further smoothness assumptions, we derive the associated Fokker-Planck-Kolmogorov equation, prove convergence towards the stationary Coulomb gas distribution, study large deviations at low temperature, and derive the limiting mean-field McKean--Vlasov equation in the many-particle limit.

We investigate the extreme values of a sparse and equicorrelated Gaussian field on a triangle: the correlations on every vertical or horizontal line are all equal to a parameter $r \in [0,1/2]$ and are zero everywhere else. This problem is closely linked with various problems in high-dimensional statistics and extreme-value theory. We identify the threshold for $r$ at which the standard Gumbel law breaks down. Our result is based on a subtle application of the Chen-Stein method for Poisson approximation. As applications, we discuss the implication of our results on multiple testing and resolve several questions that were left open in \cite{heiny2024maximum}, \cite{tang2022asymptotic} and \cite{Jiang19}.

In this paper, we use the particle motion bijection introduced by Warnaar and developed by the two authors, Jouhet and Konan, to study q-series and partition identities of the Andrews--Gordon type with parity restrictions. These restrictions are of the type ``even (resp. odd) parts appear an even number of times". We prove $q$-series identities where a multisum equals a sum of products, which generalise identities of Andrews and Kim--Yee in a similar way that Stanton's identities generalised the Andrews--Gordon identities. As a consequence of our results, we obtain a simple proof of a recent identity of Chern--Li--Stanton--Xue--Yee related to Ariki--Koike algebras.

We study the Dirichlet problem for a class of fractional $p$-Laplacian operators of order $s \in (0,1)$ defined through the Riesz fractional gradient, which differs fundamentally from the standard fractional $p$-Laplacian. Our analysis combines the framework of Lions-Calderón spaces, Besov embeddings, and an adaptation of Nirenberg's difference quotient method, originally developed by Savaré, to the fractional Riesz setting. As a main result, we establish global Besov regularity estimates for weak solutions. Concretely, in the superquadratic regime $p \geq 2$, we prove $u \in \dot{B}_{p,\infty}^{s+1/p}(Ω)$ for $s \in [\frac{1}{p'},1)$, and $u \in \dot{B}_{p,\infty}^{s+\frac{s}{p-1}}(Ω)$ for $s \in (0,\frac{1}{p'})$. In the subquadratic case $1<p<2$, we show $u \in \dot{B}_{p,\infty}^{s+1/2}(Ω)$ for $s \in [\frac{1}{2},1)$, and $u \in \dot{B}_{p,\infty}^{2s}(Ω)$ for $s \in (0,\frac12)$, with quantitative bounds depending on the source data.

The notion of a tropical vector bundle on a toric variety was recently introduced by Khan-Maclagan and Kaveh-Manon. In this paper, we study the Euler characteristic and rank of global sections for tropical vector bundles. We associate a convex chain (a finite integer linear combination of indicator functions of convex polytopes) to a tropical vector bundle encoding its Euler characteristic. We then see that the Khovanskii-Pukhlikov theory of convex chains gives a combinatorial Hirzebruch-Riemann-Roch theorem for tropical vector bundles. This, in particular, applies to toric vector bundles. Also, we extend Klyachko's resolution of a toric vector bundle by split toric vector bundles to tropical vector bundles. As shown by Kaveh-Manon, every matroid comes with a tautological tropical vector bundle. We answer positively a question posed by Kaveh-Manon about equality of Euler characteristic with rank of space of global sections (in other words, vanishing of higher cohomologies) for the tautological bundle of a matroid.