A numerical study of tetrahedral Raviart-Thomas mixed finite element methods is presented in the solution of model second order boundary value problems posed in a curved spatial domain. An emphasis is given to the case where normal fluxes are prescribed on a boundary portion. In this case the question on the best way to enforce known boundary degrees of freedom is raised. It seems intuitive that the normal component of the flux variable should preferably not take up corresponding prescribed values at nodes shifted to the boundary of the approximating polyhedron in the underlying normal direction. This is because an accuracy downgrade is to be expected, as shown in https://doi.org/10.1137/15M1045442 and https://doi.org/10.1051/m2an/2025028. In the former work accuracy improvement is achieved by means of a standard Galerkin formulation with parametric elements. The latter one in turn advocates the use of straight-edged triangles combined with a Petrov-Galerkin formulation, in which the aforementioned shift applies only to the test-flux space, while the shape-flux space consists of fields whose fluxes satisfy the prescribed conditions on the true boundary. The first purpose of this article is to show that the method studied in https://doi.org/10.1051/m2an/2025028 for two-dimensional problems can be extended quite naturally to the three-dimensional case. More particularly we illustrate this by carrying out numerical experimentation with such a version for the two lowest order methods of this family, as compared to the corresponding do-nothing strategy. In the case of the lowest order method this comparative study is enriched by assessing as well the performance of its Hermite analog introduced in https://doi.org/10.1016/j.cam.2012.08.027.

The wavefunction of the universe, as studied in perturbative quantum field theory, is a rational function whose singularities and factorization properties encode a rich underlying combinatorial structure. We define and study a broad generalization of such wavefunctions that can be associated to any lattice. We obtain these wavefunctions as the Laplace transform of a polyhedral fan, the universe fan, whose cones are defined by positivity conditions reflecting a notion of causality in the lattice, and we describe its face lattice. In the matroid case, the universe fan projects to the nested set fan, and the wavefunctions we define recover the matroid amplitudes introduced by Lam as residues. Moreover, in the case relevant for physics, the positivity conditions give a novel way to study the wavefunction, and we show how it is related to the cosmological polytopes of Arkani-Hamed, Benincasa, Postnikov. Finally, we study refinements of the universe fan induced by piecewise linear (tropical) functions. The resulting subdivisions project to refinements of the nested set fan and correspond dually to blow-ups of matroid polytopes, generalizing the cosmohedron polytope.

The Heisenberg Lie group $H_3$ is modeled on the differentiable structure of $\mathbb{R}^3$ but equipped with another non-commutative product operation. By fixing the usual metric on the Heisenberg Lie group, this work provides a comprehensive overview of the behavior of magnetic geodesics for any invariant Lorentz force. After writing the magnetic equations, we found symmetries that enable the explicit computation of the magnetic trajectories for any homogeneous exact and non-exact magnetic form. Finally we show that these magnetic trajectories are solutions of a variational problem: we present explicit examples of Lagrangians.

From descent theory to higher geometry, the idea of gluing has been embedded in many elegant and powerful techniques, proving instrumental for the solution of many problems. In this paper, we introduce a framework that allows to link important geometric objects, such as differentiable manifolds or schemes, to certain finite ringed spaces arising from sheaves on 2 dimensional semisimplicial sets, thus opening the door to their applications in fields such as discrete differential geometry.

In this paper we explore the link between the theory of sheaves on graphs and noncommutative geometry showing that many concepts and constructions in the latter can be generalized and enhanced using methods coming from the former. They include notions such as Laplacians and connections, important in the theory of discrete noncommutative geometry, that are here explored with sheaf theoretic methods and using the language of (semi)simplicial sets.

Pinching-antenna systems have recently emerged as a promising solution for enhancing coverage in high-frequency wireless communications by guiding signals through dielectric waveguides and radiating them via position-adjustable antennas. However, their practical deployment is fundamentally constrained by waveguide attenuation and line-installation requirements, which limit the achievable coverage range. To address this challenge, this paper investigates a wireless-fed pinching-antenna architecture that employs highly directional horn antennas to enable efficient coverage extension. Specifically, a full-duplex amplify-and-forward relay equipped with horn antennas is introduced between the base station and the waveguide input, which significantly improves the link budget in high-frequency bands while effectively eliminating self-interference. On this basis, we formulate a total power minimization problem subject to a quality-of-service constraint at the user equipment, involving the joint optimization of the pinching-antenna position, the relay amplification gain, and the base station transmit power. By exploiting the structure of the end-to-end signal-to-noise ratio, the optimal pinching-antenna position is first obtained in closed form by balancing waveguide attenuation and free-space path loss. Subsequently, closed-form expressions for the optimal relay gain and transmit power are derived. Numerical results demonstrate that the proposed scheme substantially outperforms conventional systems without relaying and relay-assisted transmission with fixed antennas in terms of total power consumption.

Explicit filters play a pivotal role in the scale separation and numerical stability of advanced Large Eddy Simulation (LES) closures, such as dynamic eddy-viscosity or Approximate Deconvolution (AD) methods. In the present study, it is demonstrated that the performance of commonly used explicit filters applicable to general unstructured grids highly depends on the grid configuration, specifically the cell aspect ratio, which can result in poor filter spectral properties, ultimately leading to large errors and even solution divergence. This study introduces a novel, efficient explicit filter for general unstructured grids, addressing this shortcoming through a combination of a face-averaging technique and recursive filtering. The filter parameters are then determined through a constrained multi-objective optimization, ensuring desirable spectral properties, including high-wavenumber attenuation, filter-width precision, filter stability and positivity, and minimized dispersion and commutation errors. The AD-LES of turbulent channel flow benchmarks using the new filter demonstrate a noticeable improvement in turbulent flow predictions on highly stretched boundary-layer-type grids, particularly in reducing the log-layer mean velocity profile mismatch, compared to simulations using conventional filters. The analyses show that this enhancement is mainly attributed to the sufficient level of attenuation near the Nyquist wavenumber achieved by the new filter in all spatial directions across various grid configurations, among others. The new filter was also successfully tested on unstructured prism grids for the 3D Taylor-Green vortex benchmark.

This work studies the following doubly degenerate parabolic-elliptic nutrient taxis system $$ \begin{cases} u_t = (uvu_x)_x -(u^2 vv_x)_x + uv, \\[1.5 ex] \hspace{0.2 cm}0 = v_{xx} - uv + f(x,t), \end{cases} $$ in a bounded interval $Ω\subset \mathbb{R}$, under no-flux boundary conditions and nonnegative initial value $u(x,0) = u_0(x) \geq 0$, where $f(x,t) \geq 0$ is known external supply of the nutrient. It is shown that for any nonnegative $u_0 \in W^{1,\infty}(Ω)$ and $f \in C^1\big(\barΩ \times [0,\infty) \big)$, $f \not \equiv 0$, a global weak solution of the problem can be constructed by means of a regularization approach. The core of the analysis lies on a Harnack-type inequality for the second that allows us to overcome the lack of uniform coercivity. Together with time regularity properties, we obtain relative compactness through a combination of the Arzelà-Ascoli theorem and the Aubin-Lions lemma.

Pinching antenna systems (PASS) have recently emerged as a promising architecture for high-frequency wireless communications. In this letter, we investigate localization in PASS by jointly exploiting the received signal amplitude and phase information, unlike recent works that consider only the amplitude information. A complex baseband signal model is formulated to capture free-space path loss, waveguide attenuation, and distance-dependent phase rotation between the user and each pinching antenna. Using this model, we derive the Fisher information matrix (FIM) with respect to the user location and obtain closed-form expressions for the Cramer-Rao lower bound (CRLB) and the position error bound (PEB). A maximum likelihood (ML) estimator that jointly considers the received signal amplitude and phase is developed to estimate the unknown user location. Given the non-convexity of the estimation problem, a two-stage solution combining coarse grid search and Levenberg-Marquardt refinement is proposed. Numerical results demonstrate that the proposed phase-aware estimator outperforms existing amplitude-only method in terms of positioning accuracy.

We apply Krönecker's approximation theorem to measure (in a topological sense) a set of constants which turn a vector field into a non-globally hypoelliptic operator. We present situations in which this set is a discrete enumerable (hence, meager) subset of the real line, and we also show that this set may be a dense $\mathcal{G}_δ$ subset of the complex numbers (hence, nonmeager), which produces a contrast to a known result stating that this set has null Lebesgue measure.

While tensor-based methods excel at Direction-of-Arrival (DOA) estimation, their performance degrades severely with faulty or sparse arrays that violate the required manifold structure. To address this challenge, we propose Tensor Completion for Defective Arrays (TCDA), a robust algorithm that reformulates the physical imperfection problem as a data recovery task within a virtual tensor space. We present a detailed derivation for constructing an incomplete third-order Parallel Factor Analysis (PARAFAC) tensor from the faulty array signals via subarray partitioning, cross-correlation, and dimensional reshaping. Leveraging the tensor's inherent low-rank structure, an Alternating Least Squares (ALS)-based algorithm directly recovers the factor matrices embedding the DOA parameters from the incomplete observations. This approach provides a software-defined 'self-healing' capability, demonstrating exceptional robustness against random element failures without requiring additional processing steps for DOA estimation.

We present a classification of all scalar Lie point symmetries of the Standard Model with one or two real gauge-singlet scalars (SM+S and SM+2S). By analyzing the associated field equations, we identify all realizable and inequivalent Lie point symmetry algebras of these models, distinguishing strict variational, variational (including divergence symmetries), and Euler--Lagrange cases. In addition, we devise efficient algorithms that, for any given numerical instance of the models, determine the Lie point symmetry algebra in each of the three categories by a parameter-based decision procedure using affine reparametrizations and simple parameter tests, thereby avoiding explicit symmetry analysis and the need to derive and solve the determining equations. Finally, we prove several relevant general results, including a characterization of the three disjoint types of Lie point symmetry generators -- strict variational, divergence, and non-variational -- for a broad class of Lagrangians with potentials, including the SM+S and SM+2S.

We consider the minmax Ljusternik-Schnirelmann levels of the constrained $p-$Dirichlet integral, on a general open set of the Euclidean space. We show that, whenever one of these levels lies below the threshold given by the $L^p$ Poincaré constant ``at infinity'', it actually defines an eigenvalue of the Dirichlet $p-$Laplacian. We also prove an exponential decay at infinity for the relevant eigenfunctions: this can be seen as a Šnol-Simon--type estimate for the nonlinear case. Finally, we exhibit some peculiar examples of unbounded open sets to which our main result applies.

This paper investigates the regularity of stable radial solutions to semilinear elliptic equations arising in MEMS problems, modeled by the Dirichlet problem $-Δu=f(u)$ in the unit ball $B_1$, where the nonlinearity $f\in C^1([0,1))$ is nonnegative and satisfies $\int^1_0f(s)\,ds=+\infty$. We focus on the case where $f$ blows up as $u\to 1^{-}$. Micro-electro-mechanical systems (MEMS) are widely used devices in engineering and technology. Our main result establishes for dimensions $2\le n\le 6$, every stable radial solution is regular, meaning $\|u\|_{L^{\infty}(B_1)}<1$. This result gives a positive answer to an open problem posed by Bruera and Cabré concerning the regularity of stable solutions for singular nonlinearities without requiring a Crandall-Rabinowitz type condition, at least in the radial case.

This paper introduces delay-alignment modulation (DAM) for secure integrated sensing and communication (ISAC). Due to the broadcast nature of multi-user downlinks, communications are vulnerable to eavesdropping. DAM applies controlled per-path symbol delays at the transmitter to coherently align the multipath components at the intended user, enhancing the received signal power, while simultaneously creating delay misalignment at the eavesdropper (Eve). To mitigate sensing degradation caused by multipath propagation, we propose a two-stage protocol that first estimates the angle and then the delay of the line-of-sight (LoS) path after suppressing multipath interference. We derive the secrecy spectral efficiency (SSE) and the Cramer-Rao (CRB) of the target delay. Finally, we develop a path-based zero-forcing (ZF) precoding framework and formulate a max-min SSE design under CRB and power constraints. Simulation results show DAM significantly outperforms the strongest-path (SP) benchmark in terms of SSE, while meeting sensing requirements, since intentional delay alignment at legitimate users degrades reception at Eve.

In this article, we explore a natural extension of the quadratic parametrization introduced in our previous work. By replacing the integer $n$ by $n^s$ ($ s\in\mathbb{R}, s>1$) and allowing the parameters to be real, we obtain for each $n\ge 1$ a decomposition $\frac{k}{n^s} = \frac{1}{x_s(n)}+\frac{1}{y_s(n)}+\frac{1}{z_s(n)}$ with $x_s(n), y_s(n), z_s(n) \in \mathbb{R}^*+$. Summing this equality over all integers brings forth the Riemann zeta function. Subject to an analytic continuation of the quantities $x_s(n), y_s(n), z_s(n)$ to complex values of $s$, one would obtain a new function \(G_k(s)\) satisfying $G_k(s)=k\,ζ(s)$, thus establishing a deep connection between the structure of the conjecture and the zeros of $ζ$.

The first and third authors recently proved that for each knot $K\subset S^3$ there are only finitely many hyperbolic fibered knots which are ribbon concordant to $K$. In this paper, we remove the hyperbolic constraint, proving that every knot in $S^3$ has only finitely many fibered predecessors under ribbon concordance. The key new input is an inequality relating the knot Floer homology of a generalized satellite knot with that of its companion, proved via the immersed curves formulation of bordered Heegaard Floer homology, which should be of independent interest. Our work, together with results of Kojima--McShane, also leads to an explicit upper bound on the Gromov norm of the complement of any fibered predecessor of a knot $K \subset S^3$, in terms of the arc index and genus of $K$.

In this work, we investigate the large-scale transport properties of a passive scalar advected by a turbulent fluid, modelled as a superposition of divergence-free vector fields, each weighted by an independent symmetric $α$-stable-like process. Motivated by recent works showing that complex small-scale spatial structures often lead to Brownian dispersion, we study if this principle persists when the driving noise exhibits heavy-tailed jump statistics. Our numerical results show a clear dichotomy linked with the tail behaviour of the noise. When considering standard $α$-stable processes, very large jumps survive the interaction with the spatial complexity and yield anomalous, super-diffusive transport. In contrast, when the $α$-stable noise is either truncated or exponentially tempered, suppressing extremely long jumps, the transport undergoes a transition to a classical diffusive regime.

We consider decision-making problems that are formulated as non-convex optimization programs where uncertainty enters the constraints through an additive term, independent of the decision variables, and robustness is imposed using a finite data-set, according to the scenario robust optimization paradigm. By exploiting the structure of the constraints, we show that both a priori and a posteriori distribution-free probabilistic robustness certificates for a possibly sub-optimal solution to the resulting data-driven optimization problem can be obtained with minimal computational effort. Building on these results, we also discuss a one-shot and an incremental procedure to determine the size of the data-set so as to guarantee a user-chosen robustness level. Notably, both the a posteriori robustness assessment and incremental data-set sizing do not require to solve the non-convex scenario program. A comparative analysis performed on the unit commitment problem using real data reveals a limited increase in conservativeness with a significant computational saving with respect to the application of scenario theory results for general, non necessarily structured, non-convex problems.

We investigate a q-deformation of the arc length metric on the unit circle. This q-deformation arises naturally from the Dirac operator by replacing the standard integers by their q-deformed analogues. Nonetheless, we show that the corresponding metric structure only makes sense at the level of quantum metric spaces as introduced by Marc Rieffel. This means that the quantum metric we obtain on the continuous functions on the circle does not arise from a classical metric on the circle. In the special case where q equals one we recover the usual arc length metric and we show that our family of quantum metric spaces depend continuously on the deformation parameter with respect to David Kerr's complete Gromov-Hausdorff distance.

We introduce a general notion of $J$-tribe, and construct the $J$-tribe of $J$-frames in a given tribe $\mathcal{T}$, where $J$ a suitable generalized direct category. This construction applies to semi-cubical diagrams for a category of semi-cubes with symmetries and reversals.

In this work, we study compact Riemannian manifolds with boundary satisfying V-static-type equations. By combining a generalized Reilly formula with Steklov-type boundary value problems, we derive integral inequalities for geometric quantities associated with the boundary. These inequalities lead to rigidity results, including characterizations of geodesic balls in space forms. In particular, our results offer new insights into several known rigidity theorems in the literature.

We develop minimal slicing via capillary hypersurfaces to understand positive scalar curvature metric on manifolds with boundary. The method provides rigidity statements once the regularity of minimizers of capillary area functional holds. In particular, in dimension $4$, we prove following comparison and rigidity statement: given a compact Riemannian $4$-manifold $(M^4,g)$ with a mean convex boundary whose boundary is diffeomorphic to boundary of a connected convex domain in $\mathbb R^4$, if the scalar curvature is non-negative and the scaled mean curvature comparison holds along the boundary, then $M$ is isometric to the Euclidean domain.

Let $p$ be a prime and let $L$ be a quadratic $\mathbb{Z}_p$-lattice with quadratic form $Q$. For $t\neq 0$ the local representation density $α_p(t;L)$ is the stable normalised growth of the congruence counts of solutions to $Q(v)\equiv t\pmod{p^m}$. We compute these counts and densities explicitly for the hyperbolic plane $H_0$ over $\mathbb{Z}p$, uniformly in $p$, and at $p=2$ for the basic dyadic blocks (rank-$1$ Type I blocks and the even binary planes $2^aH\varepsilon$), together with the anisotropic ternary lattice $L_3=\langle 2\rangle^{\oplus 3}$. At the dyadic prime the usual Jacobian/Hensel lifting mechanism breaks down in the bilinear-lattice convention $Q(v)=\langle v,v\rangle$. The main new input is an explicit half-lift involution for diagonal sums of squares, which yields a stable lifting recursion with factor $2^{d-1}$ under the primitivity hypothesis $4\nmid a$. As applications we obtain closed forms for the three-squares congruence counts (hence $α_2(t;L_3)$) and a prime-uniform formula for the densities of the scaled hyperbolic planes $p^eH_0$ in the standard normalisation $q=\langle\cdot,\cdot\rangle/2$.

We characterize linear lexicographic $p$-ary codes. Using this characterization, when $p \ge 3$, we determine the dimensions of linear lexicographic codes obtained from several bases including the standard basis, except for those of certain minimum distances. In these excluded cases, we may obtain linear codes of higher dimensions; for instance, when $p = 3$ and $d = 6$, the ternary Golay code is obtained.

Let $p$ be a prime number, $n$ an integer $\geq 2$, and $L$ a finite extension of $\mathrm{Q}_p$. Let $ρ_L$ be an $n$-dimensional (non-critical but not necessary generic) potentially crystalline $p$-adic Galois representation of the absolute Galois groups of $L$ of regular Hodge-Tate weights. By generalizing the previous results and strategy for the crystabelline case of Ding and the recent work of Breuil-Ding, we construct an explicit locally analytic representation $π_{1}(ρ_L)$, and describe explicitly the information of Hodge filtration of $ρ_L$ it determines. When $ρ_L$ comes from a patched $p$-adic automorphic representation, we show that $π_{1}(ρ_L)$ is a subrepresentation of the $\mathrm{GL}_n(L)$-representation globally associated to $ρ_L$, under some mild hypothesis.

We prove that the category of commutative Hopf algebras over a field $k$ is co-semi-abelian. Consequently, the category of affine group $k$-schemes is semi-abelian. We establish coregularity by identifying the orthogonal factorization system of surjections and faithfully flat injections, and we deduce coexactness from Takeuchi's correspondence between normal Hopf ideals and Hopf subalgebras of commutative Hopf $k$-algebras.

We establish a class of Oppenheim--Schur-type inequalities for the convolutional Jury product of positive semidefinite matrices. These results extend to a causal convolutional setting the classical Schur and Oppenheim inequalities associated with the Hadamard product. Our approach highlights structural parallels between entrywise and convolution-based matrix operations, revealing how positivity constraints interact with causality. Building on this perspective, we introduce a broader family of causal matrix products and prove unified inequalities that simultaneously recover the classical Schur and Oppenheim bounds as well as their convolutional Jury counterparts. These results provide a common framework for understanding positivity-preserving matrix products and suggest further connections between classical matrix analysis and causal operator structures.

In many applications, weighted networks are constructed based on time series data: each time series is associated to a vertex and edge weights are given by pairwise correlations. The result is a network whose edge dependency structure violates the assumptions of most common network models. Nonetheless, it is common to analyze these "correlation networks" using embedding methods derived from edge-independent network models, based on a belief that the edges are approximately independent. In this work, we put this modeling choice on firm theoretical ground. We show that when the time series are expressible in terms of a small number of Fourier basis elements (or in some other suitably-chosen basis), correlation networks correspond to latent space networks with dependent edge noise in which the vertex-level latent variables encode the basis coefficients. Further, we show that when time series are observed subject to noise, spectral embedding of the resulting noisy correlation network still recovers these true vertex-level latent representations under suitable assumptions. This characterization of embeddings as learning Fourier coefficients appears to be folklore in the signal processing community in the context of principal component analysis, but is, to the best of our knowledge, new to the statistical network analysis literature.

We study polynomials with no zeros on the unit ball in complex Euclidean space with a view toward characterizing when a rational function is bounded on the ball. We give a complete local description of such polynomials in two variables near a boundary zero. In higher dimensions, we give a partial characterization of a simple boundary zero. Several applications are given including boundedness of rational functions with boundary singularities and constructions of examples with prescribed local properties.