Papers

We present new fully discrete Cartesian grid Active Flux methods for the two-dimensional Euler equations of gas dynamics. A key component of these methods is the evolution operator used to evolve point-value degrees of freedom. Therefore, we derive several new, third-order accurate, multidimensional evolution operators for point-value updates of the linearized Euler equations using the method of bicharacteristics. We also investigate how the reconstruction and choice of approximate evolution operator affect the accuracy and stability of the resulting methods.

Enormous fluid antenna systems (E-FAS) enable guided surface wave (SW) propagation and induce an end-to-end channel that is conditionally complex Gaussian with a random covariance scale. This paper investigates the impact of this channel structure on covert communication in the presence of a passive warden. Under an equal power effective mode representation, the random channel scale follows a Gamma distribution, leading to a Bessel-K distribution for the effective channel power. We show that the optimal likelihood ratio test at the warden reduces to an energy detector and derive the corresponding false alarm and missed detection probabilities. These results are then used to characterize the achievable covert throughput under a prescribed detection error constraint. Numerical results show that, for finite observation intervals, finite mode E-FAS channels improve covertness relative to the fully hardened Rayleigh limit with the same average channel power. They further reveal an asymmetric role of channel hardening: stronger hardening can improve the legitimate link, whereas retaining channel scale fluctuations at the warden can enhance finite observation covertness.

Let $k$ be a finite field. We consider the category of all functors $\mathcal{F}_n$ from the category of $k$-vector spaces of dimension $\leq n$ into the category of all $k$-vector spaces. We compare its derived category to the derived category of representations of the general linear group $GL_n(k)$ using the extension by zero functor procedure. We upgrade the well-known recollement of abelian categories into the recollement of the corresponding derived categories. Then we obtain some translations of the cohomology the general linear group over finite field. We also compare $\mathcal{F}_n$ with the category of polynomial functors $\mathcal{P}_{d,n}$, giving further reformulations over big finite fields.

We establish two critical one-component regularity criteria for the three-dimensional incompressible Navier–Stokes equations. First, for divergence-free initial data $u_0\in H^1(\mathbb R^3)$, we prove that a strong solution can be continued beyond a finite time $T$ provided that \[ u^3\in L^2\bigl(0,T;\dot B^{3/2}_{2,r}(\mathbb R^3)\bigr) \qquad\text{for some }2<r\leq\infty. \] Second, for every $2<p<\infty$, if $u_0\in\dot H^{1/2}(\mathbb R^3)$ and, for some fixed unit vector $\boldsymbol e\in\mathbb S^2$, \[ u\cdot\boldsymbol e \in L^p\bigl(0,T;\dot H^{1/2+2/p}(\mathbb R^3)\bigr), \] then the corresponding Fujita–Kato solution also extends beyond $T$, without any additional low-integrability assumption on the initial vorticity. These results extend the critical one-component regularity criteria of Han, Lei, Li, and Zhao in two different directions. At the time endpoint $p=2$, we combine a new vertical-vorticity estimate with coupled anisotropic energy estimates and an energy-dependent frequency decomposition to relax the dyadic square-summability condition \[ \dot H^{3/2}=\dot B^{3/2}_{2,2} \] to \[ \dot B^{3/2}_{2,r}, \qquad 2<r\leq\infty, \] with the endpoint case $r=\infty$ closed by an Osgood-type argument. In the nonendpoint range $2<p<\infty$, we use a positive-time heat-flow decomposition to separate the solution into a smooth linear part and a nonlinear remainder. The remainder has finite energy and acquires the required $L^q$-integrability of the vorticity at positive times, which allows the anisotropic energy estimates to be applied while the additional transport, stretching, and pressure terms generated by the heat flow remain integrable. This removes the extra $L^{q_0}$-integrability assumption on the initial vorticity while retaining the natural critical initial-data space $\dot H^{1/2}(\mathbb R^3)$.

Many complex systems are observed only through temporally unpaired distribution snapshots, making trajectory-based dynamical learning difficult without additional assumptions. We therefore formulate the problem directly in distribution space, treating the distribution itself as the dynamical state. The challenge is that distribution space is infinite-dimensional, making compact and approximately closed representations difficult to learn from finite snapshots. We introduce DisKO, which extends deep Koopman learning to distribution dynamics by jointly learning predictive distributional observables, a finite-dimensional Koopman representation, and a generative map back to the full distribution. Across seven diverse benchmarks, DisKO achieves state-of-the-art extrapolation performance, with substantially slower error accumulation on long-horizon prediction tasks. DisKO further recovers leading Koopman eigenvalues and eigenfunctions on systems with analytic spectra, revealing meaningful dynamical structure in the learned representation.

In this work, different from previous results, the explicit expression of numerical Onsager-Machlup action functional for Euler discretized SDEs driven by fractional Brownian motion is derived provided the drift coefficient and numerical reference path satisfy some suitable conditions. Then numerical fractional Euler-Lagrange equations for numerical Onsager-Machlup action functional are also obtained. Finally, numerical experiments to illustrate and support the theoretical findings.

We develop and analyse a symmetric weighted interior penalty polytopal discontinuous Galerkin (SWIP-PolyDG) method for multi-species reaction–diffusion systems with nonlinear reaction terms of up to cubic order. Such terms arise naturally when higher-order interactions are incorporated into population models, allowing non-additive effects among multiple species to influence local growth and conversion mechanisms. The proposed framework accommodates heterogeneous and possibly anisotropic diffusion tensors on general polygonal meshes. To control the nonlinear coupling, we consider a hierarchical block structure in the reaction operator, whereby each population block depends only on its own variables and on those associated with preceding blocks. In addition, the cubic self-interactions within each block are assumed to have a dissipative diagonal structure. Under these hypotheses and proceeding recursively over the hierarchy of population blocks, we derive local-in-time stability estimates in two spatial dimensions for the semi-discrete formulation in both the $L^2(\Omega)$-and dG-norms. We further establish an a priori error estimate in a combined $L^2(\Omega)$-dG energy norm for sufficiently regular solutions for the semi-discrete formulation. Finally, numerical experiments confirm the predicted convergence behaviour and illustrate the robustness of the method under heterogeneous diffusion.

We lay out the analysis of the “Kronecker preconditioners” adopted in [24] for the linear wave equations. Importantly, we deliver the analysis justifying the claims in~[24, Section 4.2] about these preconditioners and we unpack all of the necessary numerical background for a full understanding of the observed phenomena.

For a graph \(F\), the Turán number \(\ex(n,F)\) is the maximum number of edges in an \(F\)-free graph on \(n\) vertices. Let \(q\ge 2\) be an integer and set \(n=q^{2}+q+1\). Brown and Erdős, Rényi and Sós independently proved that $\ex(n,C_{4})\ge \frac12 q(q+1)^{2}$ for every prime power \(q\), and Füredi subsequently established the upper bound $\frac12 q(q+1)^{2}$ for $\ex(n,C_{4})$ whenever \(q\notin\{1, 7,9,11,13\}\). In this article, we prove that every \(C_{4}\)-free graph \(G\) on \(n\) vertices with at most \(\frac12 q(q+1)^{2}\) edges satisfies $\tau(G)\le n^{(n-3)/2}$, where \(\tau(G)\) denotes the number of spanning trees of \(G\). In particular, for every prime power $q\notin\{7,9,11,13\}$, the above upper bound on $\tau(G)$ is attained precisely by the orthogonal polarity graphs, thereby proving London's conjecture for all such $q$.

Flow matching (FM) has recently emerged as a promising framework for generative modeling due to its conceptual simplicity and strong empirical performance. In FM, samples are transported along a vector field parameterized by a neural network, inducing a probability path that evolves from a simple noise distribution to the target data distribution, governed by an ordinary differential equation (ODE). However, existing FM approaches predominantly rely on probability paths derived from optimal transport (OT) between Gaussian distributions, which may be suboptimal for capturing complex data with inhomogeneous structures such as heavy tail or sharp contrast. In this work, we generalize FM to the broader class of location-scale families for handling data inhomogeneity and introduce a novel class of probability paths defined as geodesics on the manifold of probability distributions. We name this approach probabilistic geodesic flow matching to distinguish it from prior geodesic (Riemannian) FM methods defined in input space. We argue that Euclidean OT-based paths are not necessarily optimal in probability space and may limit modeling flexibility. Through synthetic benchmarks and scientific datasets at different scales, we demonstrate that the proposed method more effectively captures complex distributions, leading to improved or comparable performance compared with SOTA geometry-motivated generative models.

We study the linearized dynamics of the Kolmogorov flow $U^{(0)}=(a\sin x_2,0)$ for the two-dimensional incompressible Navier–Stokes equations with horizontal dissipation $\nu\partial_1^2$. Unlike the fully dissipative case, this anisotropic system admits $U^{(0)}$ as an exact unforced steady state. On each horizontal Fourier mode $k$, the dissipation reduces to the scalar $-\nu k^2$ and commutes with the advection, so that the linearized semigroup factors exactly into $e^{-\nu k^2t}$ times the inviscid Euler group. For $|k|>1$, we prove two-sided bounds showing that the decay rate is exactly $\nu k^2$, uniformly in the shear amplitude $a$; hence no enhanced dissipation occurs. This reflects a fundamental mismatch: the shear transfers enstrophy to high vertical frequencies, which horizontal dissipation does not detect. At the critical modes $|k|=1$, the inviscid group grows like $\sqrt{2at}$, producing a transient amplification of size $(a/e\nu)^{1/2}$ before decay at the rate $\nu$. The horizontally independent modes form an infinite-dimensional undamped kernel. For $0<|k|<1$, the viscous spectrum is an exact translate of the inviscid one, and the mode is linearly unstable if and only if $a\Lambda(k)>\nu k^2$. In particular, when $L>2\pi$, the flow is linearly unstable for all sufficiently large $a$.

Topological signals are dynamical variables supported on higher-order structures such as simplicial or cell complexes. In this work, we investigate the impact of time-delayed interactions on the emergence of global synchronization of topological oscillators, with application to the Stuart-Landau system. We first examine the case where signals are supported on simplexes, or cells, of a given dimension and coupled through the Hodge-Laplace matrix. We derive the Master Stability Function and we prove that for suitable discrete delay values, the stability of the synchronous solution becomes analytically tractable and can be solved by using the Lambert W-function. This analysis yields explicit spectral conditions for Global Topological Synchronization to emerge and shows that admissible delays may either promote or suppress synchronization. We then study delayed interactions between signals supported on simplexes, or cells, of different dimensions coupled by the Dirac operator. Under analogous admissible time delays the variational problem, still related to the Master Stability Function, can again be simplified, however in this case we prove that Global Topological Dirac Synchronization cannot emerge. Numerical simulations on simplicial and cell complexes corroborate and complement these findings.

It was proved in [3] that on a closed manifold of dimension greater than one, every smooth weak Riemannian metric on the space of smooth positive probability densities, that is invariant under the action of the diffeomorphism group, is a multiple of the Fisher-Rao metric. Here we extend this result to compact manifolds with corners, and we determine all diffeomorphism invariant tensor fields on the space of smooth positive probability densities.

In this article, we study derived Hecke algebras with coefficients in torsion-free profinite rings regarded as discrete rings. We compare each such algebra with the degreewise inverse limit of the derived Hecke algebras with coefficients in finite rings and with its subalgebra consisting of elements with uniformly finite double-coset support. Under cohomological comparison hypotheses, the natural map to this subalgebra has image given by its degree-zero part and its positive-degree torsion. We describe the kernel in terms of continuous cohomology and prove that it is a divisible square-zero ideal annihilated by every positive-degree element. These comparisons are compatible with Yoneda products, and the action on torsion Ext classes factors through the image. We compute the resulting algebras for the additive group $\mathbb{Q}_p$ and determine the underlying graded abelian groups for $(\operatorname{GL}_2(\mathbb{Q}_p),\operatorname{GL}_2(\mathbb{Z}_p))$ with coefficients in $\mathbb{Z}_p$, for $p>3$. The examples exhibit distinct effects of the coefficient topology and the support condition.

We solve (modulo groups) the word problem for free semigroup satisfying $xy = (xy)^n$. This enables us to show that the variety ${\cal {SCR}}_n$ given by this identity is not equal to the join of the varieties ${\cal{CR}}_n$ and ${\cal{SI}}$ defined by the identities $x = x^n$ and $xy = (xy)^2$ respectively. Therefore the result of (M.V.Volkov, T.A.Ershova, The lattice of varieties of semigroup with completely regular square, Monash Conference on Semigroup Theory in Honour of G. B. Preston, World Scientific, Singapore, 1991, 306-322) that the lattice $L({\cal {SCR}}_n)$ is modular does not follow from the earlier results concerning $L({\cal{CR}}_n)$ and $L({\cal{SI}})$. However the variety ${\cal{SO}}_n$ consisting of all semigroups from ${\cal {SCR}}_n$ whose idempotents form a subsemigroup turns out to be equal to the join of the varieties ${\cal O}_n$ of all orthogroups and ${\cal{SI}}$. We deduce from this a description of the lattice $L({\cal{SO}}_n)$ modulo groups.

We focus on systems of nonlinear equations, where some equations are non-polynomial. Building on the remarkable developments in solving polynomial systems during the recent decades, and on a few extensions to specific examples of the transcendental case, a general framework for finding isolated roots is proposed. The name paxotopy is proposed after (1) Polynomial ApproXimation of the transcendental functions, then (2) solving the approximating polynomial system by the homOTOPY continuation method, and finally (3) the roots of the polynomial systems are used as starting points of a Newton's iteration to arrive at the solutions of the original system. These steps combine the advantages of local and global methods.

We prove mass estimates for PDE-constrained measures on sets of $\sigma$-finite ${\mathcal H}^m$-measure, assuming an appropriate rank condition. As an application, we derive a Michael–Simon inequality for varifolds with bounded anisotropic first variation with respect to an integrand that satisfies the natural atomic condition (AC1).

We investigate Strong Data-Processing Inequality (SDPI) constants for Rényi Divergences on finite spaces. We study their dependence on the Rényi order $\alpha$, proving that they are non-decreasing and that their scaling by $(\alpha-1)$ is convex for $\alpha\geq1$. We also identify several support restrictions on the probability measures involved in determining these constants. In particular, in the distribution-independent setting, measures supported on a common set of at most two points suffice to evaluate the Rényi-SDPI constant. For $\alpha\in[0,1]$, we further prove equality with the $\chi^2$-SDPI constant, while at order infinity we obtain a closed-form expression. In order to link contraction over product spaces to contraction along individual coordinates, we provide tensorisation bounds for arbitrary product channels analogous to those known for $\varphi$-Divergences. At finite orders, the Rényi-SDPI constants are bounded above and below through comparisons with the $\chi^2$-Divergence and Hellinger Divergences, with sharpness established in multiple cases. At order infinity, we instead relate these constants to the contraction of Total Variation Distance. Finally, our findings are applied to local differential privacy (LDP) and the analysis of Markov chains. This yields sharp contraction guarantees for pure-LDP mechanisms, and connects Rényi-LDP to Rényi-SDPI constants. For Markov chains, we derive finite-time convergence bounds and exhibit a family of chains for which Rényi-SDPIs improve on classical $\chi^2$-based bounds by arbitrarily large factors.

Let $f(z)=\sum_{k=0}^{n}a_kz^k$ be a polynomial with positive coefficients, and define its coefficientwise reciprocal by $f^{\#}(z)=\sum_{k=0}^{n}\frac{z^k}{a_k}.$ It is known that if $f$ is palindromic and has only negative real zeros, then all zeros of $f^{\#}$ lie on the unit circle. We prove that if $p$ and $q$ are palindromic polynomials of degrees $n$ and $n+1$, respectively, with only negative real zeros, then their coefficientwise reciprocals have only simple zeros, and $p^{\#}$ strictly interlaces $q^{\#}$ on the unit circle. Our proof is based on finite Blaschke products and the comparison of their boundary phases. As an immediate consequence, we obtain strict interlacing for the reciprocal binomial, reciprocal Eulerian, and reciprocal Narayana polynomials. By combining a sign change in the $\gamma$-coefficients with coefficientwise reciprocation, we further construct strictly interlacing families from Rogers–Szegő, Poupard, and Kreweras-related polynomials.

Parameter space symmetries and conservation laws play an important role in understanding the loss landscapes and implicit biases of neural networks. Inspired by Noether's theorem in physics, prior works have sought to derive conservation laws under gradient flow from parameter symmetries, but the scope and limitations of this connection remain unclear. We develop a unified geometric framework that clarifies the precise relationship between the two notions, including the conditions under which symmetries correspond to conservation laws. We introduce a notion of compositional identifiability and use it to establish a general inheritance principle for complete characterizations of symmetries and conservation laws in multilayer networks. We apply the framework to multi-head and grouped-query attention, polynomial neural networks, and square deep linear networks.

We propose a Bayesian nonparametric factor model that infers the number of factors, induces row-wise sparsity, and merges redundant dictionary elements via exact clustering. A Dirichlet process prior is placed on the columns of an overcomplete loading matrix and fully marginalized to an exact Pólya urn, avoiding stick-breaking and auxiliary variables. A spike-and-slab base measure allows entire factors to be exactly zero. The model uniquely combines exact zeros, exchangeability over columns, and exact merging within a single marginalized Dirichlet process, unlike CUSP, MGP, or the beta process. An exact Gibbs sampler with canonical relabeling and parallel C/MPI implementation is developed. We prove posterior contraction at rate $\sqrt{M s_0 \log n / n}$ for the covariance matrix, and in the fixed-dictionary setting obtain the minimax optimal rate $\sqrt{s_0 \log n / n}$ plus underfitting consistency; the overfitting direction is an open conjecture. The spike-and-slab is essential: without it the effective dimension scales as $pM$, yielding a slower rate. Simulations show the method is the only fully adaptive approach to recover the true rank, achieving the smallest covariance, loading, and signal-reconstruction errors, beating an oracle baseline. On van 't Veer breast cancer data ($n=97$, $p=1213$), the posterior concentrates on eight interpretable programmes; seven pass coherence and two pass Bonferroni-corrected Hallmark enrichment.

This paper presents a permutation XOR cellular automaton, a simple digital dynamical system composed of an elementary cellular automaton of the XOR rule and a partial-shift permutation connection characterized by a shift-part parameter. The proposed system is suitable for precise analysis and simple hardware implementation. Depending on the partial-shift connection and system dimension, the automaton can generate long binary periodic orbits (LBPOs) with strong stability. As a principal result, we identify the partial-shift parameters and dimensions that enable the generation of LBPOs. In our analysis, these LBPOs are characterize using half-primitive polynomials over $\mathrm{GF}(2)$. Furthermore, we present a simple FPGA-based hardware implementation. The hardware transforms typical LBPOs into electrical signals for potential engineering applications.

In this paper, we study bounded self-similar solutions for the exponential semilinear heat equation via the $F$-functional and entropy. Motivated by recent developments in mean curvature flow, we prove that every non-constant bounded solution of the self-similar equation is entropy unstable. We also show that the trivial profile $w\equiv0$ is quantitatively isolated among bounded solutions. These results provide both a variational instability theorem for nontrivial profiles and a rigidity theorem for the trivial profile.

Using methods similar to Morel's stable connectivity theorem, we prove that, over an arbitrary field, the Tate truncation functors $f_{0/n}$ preserve motivic connectivity of $ S^1$-spectra. An analogous result for complexes yields a Hurewicz theorem for the $L^{p,n}$- and $L_{bir}^n$-localizations over perfect fields. Over such fields, we establish a stronger connectivity property for categories of correspondences, showing that $f_n^\mathcal{C}$ preserves connectivity of motivic spectra with $\mathcal{C}$-transfers, whenever $\mathcal{C}$ satisfies cancellation. We then use the motivic reconstruction theorem to deduce that $f_n$, and consequently $s_n$ and $f_{0/n}$, also preserve connectivity for effective (and thereby, $\mathbb{P}^1$-) motivic spectra. Along the way, we also establish the slice analog of the motivic (effective) reconstruction theorem, as well as the slice analog of motivic $S^1$- and $\mathbb{P}^1$-recognition theorems.

We define the classical BV pushforward and establish its two main properties. First, we prove that it maps solutions of the classical master equation (CME) on the total space to solutions of the CME on the smaller space. Second, we prove that, for isolated critical points, it maps BV canonically equivalent solutions to BV canonically equivalent solutions. We illustrate the construction with examples and discuss its applications.

We distinguish two Casimir observables associated with a conformal surface defect. A circular cylinder embedded in flat space has a logarithmic contribution to its Casimir energy, $-d_1\log(R/\epsilon)/(24R)$, fixed by the displacement anomaly. A great cylinder obtained by radial quantization of the ambient theory instead has Casimir energy $-b/(12R)$ in the conformal defect vacuum with a fixed local anomaly convention. Free-field and holographic calculations provide checks of the proposed relations in both embeddings.

We consider the three-strand pretzel knot $P_a=P(a,-a-2,-(a+1)^2/2)$ and prove that, when $a\equiv1\pmod4$, its Alexander polynomial fails the Fox–Milnor condition. The same remains true after adding any number of opposite odd parameter pairs $q,-q$ and reordering the parameters. Combined with work of Kim–Lee–Song, this gives the same conclusion for every odd $a\ge3$. In particular, no knot in Lecuona's exceptional family is algebraically or topologically slice. Together with earlier work of Lecuona, Miller, and Kim–Lee–Song, this completes the slice–ribbon and topological-sliceness classifications for three-strand pretzel knots and removes the nonexceptional hypothesis from the Lecuona–Wand classification of prime fibered ribbon pretzel knots.

Let X be a Tychonoff space and B_1(X) be the space of real-valued Baire 1 functions. Using the classical Lebesgue's characterization of Baire 1 functions and the strong Choquet game we prove that if X is locally compact or a topological sum of sigma-compact spaces, the space B_1(X) equipped with the topology \tau_UC of uniform convergence on compacta, is a strong Choquet space. For the Sorgenfrey line S, the space (B_1(S); \tau_UC) is of the first Baire category in itself.

For integers $n>k\geq0$ and $1\leq p,q\leq\infty$, let $\lambda_3(n,k,p,q)$ denote the optimal constant in the one-dimensional Sobolev inequality \[ \|u^{(k)}\|_{L^q(0,1)} \leq \lambda_3(n,k,p,q) \|u^{(n)}\|_{L^p(0,1)}, \qquad u\in\mathring W_p^n(0,1). \] Nazarov and Shcheglova \cite{NazarovShcheglova} conjectured that \[ \lambda_3(n,1,p,1) = 2\lambda_3(n,0,p,\infty), \qquad n\geq2,\quad 1\leq p\leq\infty, \] and that the corresponding extremal functions coincide and are symmetric about the midpoint of the interval. We prove this conjecture for $(n,p)=(3,2)$ and, in particular, obtain \[ \lambda_3(3,1,2,1) = \frac{1}{32\sqrt5}. \] The proof is based on a reduction to an operator norm problem with an additional moment constraint. After rescaling to $(-1,1)$, this constraint leads to orthogonality with respect to quadratic polynomials. We extend the inverse of the second derivative through the corresponding orthogonal projection and analyze the adjoint operator by an odd–even decomposition. The odd component is controlled by a positive Gram kernel, while the even component is treated by a weak-$L^2$ estimate. The equality cases yield the characterization of all extremal functions.

Let $R$ be a countable ring. Given an $R$-module $A$, we call a decomposition $A = \bigoplus_{i \in I} N_i$ a Kaplansky decomposition if each $N_i$ is countable. We characterize the uncountable Polish $R$-modules that admit a Kaplansky decomposition: they are exactly the modules of the form $B \oplus M^{\omega}$, where $B$ and $M$ are countable and $M$ is $\Sigma$-algebraically compact. The countable summand $B$ may moreover be taken to be an elementary submodule satisfying a closure condition, which makes $M^{\omega}$ unique up to isomorphism, and hence an invariant of $A$. We use this to characterize the countable rings admitting a free uncountable Polish $R$-module, generalizing results of Shelah and Solecki. This class of rings has a purely ring-theoretic description: it consists exactly of the countable left perfect and right coherent rings, i.e. the rings identified by Chase's theorem on products of projective modules. We observe that these are also exactly the countable $F$-rings, i.e. those countable rings $R$ for which $R^{\omega}$ is free. Finally, we give a ring-theoretic characterization of the countable rings $R$ for which there exists an uncountable projective Polish $R$-module.