We study compatibility-constrained combinatorial chemical assembly through the lens of commutative algebra. Given a finite set $F$ of chemical families, a finite set $H$ of handle types, and a compatibility relation $Pairs(f) \subseteq H \times H$ for each $f \in F$, we construct an assembly ideal $I = J_{bool} + J_{sel} + K_{compat}$ in a polynomial ring $R = k[F,H,H']$ whose variety $V(I) \subseteq \{0,1\}^n$ encodes the set of feasible reaction triples. We prove that $I$ is zero-dimensional and radical, whence $R/I \cong k^{|V(I)|}$. Elimination ideals characterise handle diagnosticity (whether a handle determines its family), the toric ideal of the log-linear model on $V(I)$ measures redundancy in the compatibility relation, and a multi-step ideal $I^{(k)}$ encodes orthogonality constraints among simultaneous assembly plans; the clique number $\omega(G_\perp)$ of the associated orthogonality graph gives the maximum number of mutually compatible plans. We derive a necessary and sufficient criterion for a new family to raise $\omega$. The framework is instantiated on the bioorthogonal click-chemistry landscape ($|F|=8$, $|H|=17$), yielding $|V(I)|=30$, a toric ideal with 2 generators, ML degree 1, and $\omega(G_\perp)=4$. All computations are verified over $\mathbb{Q}$ in SymPy.

This paper presents a collaborative adaptive formation control framework for autonomous vehicles (AVs), that explicitly handles system uncertainties, input saturation, and communication delays. To overcome the inherent physical torque limits of steering and braking actuators, an input saturation compensation mechanism is introduced to render nonlinearities tractable and improve control reliability. Additionally, a delay-compensating auxiliary system is designed to mitigate the effects of communication delays and reduce tracking errors. Our framework incorporates a dynamic-threshold event-triggered control (ETC) strategy to optimize resource usage. Additionally, uncertainty observers and symmetric barrier Lyapunov functions are developed to ensure robust and safe formation maneuvers. Finally, the effectiveness of the proposed approach is validated through numerical simulations of vehicle formations, complemented by a 3D visualization video demonstrating the dynamic fleet reconfiguration process.

We describe a computational search for quadratic APN (Almost Perfect Nonlinear) functions in dimension 8 within a structured self-equivalence subspace. The search space is a 40-dimensional binary linear subspace consisting of all functions commuting with a linear automorphism of order 5 (class 22 in the taxonomy of Beierle, Brinkmann, and Leander, 2021), previously reported to contain no APN functions. Our approach combines random sampling via an explicit RREF parameterization (approximately 600 fresh APN-positive evaluations per core-hour) with Gröbner basis computation in Magma to enumerate all APN functions in a 24-dimensional hyperplane through each center (approximately 10 minutes per hyperplane). From 428 hyperplane computations, covering 0.65% of all 65,536 hyperplanes, we obtained 566 quadratic APN functions forming six CCZ-equivalence classes under the ortho-derivative invariant. Four classes, comprising 500 functions, match no entry in the 2025 database of 3,775,599 quadratic APN functions or in the pre-2020 compilation of 12,921 instances. Two classes (66 functions) are CCZ-equivalent to the Gold functions x^3 and x^9, confirming the correctness of the search pipeline. A membership analysis shows that the three new classes (B, C, D) lie entirely outside the original subspace and occur only in Gold-centered slices, demonstrating the essential role of the Gröbner basis stage. In 532 experiments using database functions as slice centers and 20 experiments with random centers, no APN neighbors were found, indicating that the gateway phenomenon is specific to the self-equivalence structure of the search space. Since the ortho-derivative invariant is a complete CCZ-invariant for quadratic APN functions, the absence of matching signatures provides a rigorous proof of CCZ-inequivalence.

We study Hartree-type nonlinear heat equations associated with fractional generalised anharmonic oscillators \(A_{k,l}\) in weighted modulation spaces. We first derive Strichartz-type estimates for the associated heat semigroup and then apply them to establish global well-posedness for small initial data. For \(s>\frac{d}{q'}\), the result is obtained via trilinear estimates exploiting the algebra property of the weighted modulation space, \(M^{p,q}_s\). We further establish refined trilinear estimates that bypass the algebra structure, thereby extending the global well-posedness theory to the wider range \(s\ge 0\).

We prove rigidity in the equality case of the sharp bottom spectrum estimate under scalar curvature lower bound. Under the same topological assumptions as in our previous work, a closed manifold $(M,g)$ with $\mathrm{Sc}_g\geq -n(n-1)$ and $\lambda_1(\widetilde M,\widetilde g)=(n-1)^2/4$ must be hyperbolic. This gives rigidity results for closed hyperbolic manifolds and for closed manifolds admitting a metric of nonpositive sectional curvature.

We consider the problem of computing the dominant eigenvector of a symmetric matrix via the power iteration algorithm subject to constraints in the computation of matrix-vector pr ucts. In particular, we focus on scenarios where the entries of matrix-vector products with the input matrix are only partially observed. Such constraints frequently arise on cloud architectures implemented via the controller-worker model where the matrix-vector products are distributed across workers on remote servers. Instead of a prolonged delay incurred by waiting for the slowest workers to return their output to the controller, a phenomenon known as straggling, a set of pre-determined values can replace the values of the delayed workers and allow the power iteration to proceed to the next iteration. In this paper, we develop two algorithms whose expected approximation converges to the true dominant eigenvector. The first algorithm relies on a probabilistic switch between two different approaches to set the omitted entries: either set them to zero or to their previous recorded value. The second algorithm relies on averaging previously generated partial power iteration approximations obtained by ignoring a set of columns of the iteration matrix. several theoretical details are discussed while numerical experiments verify the effectiveness of the two proposed schemes and demonstrate their comparative performance advantage over current state-of-the-art.

In the present paper we are dealing with the following quasilinear elliptic problem: \begin{equation*} \begin{cases} -\mathrm{div}(\rho(x_N) |\nabla u|^{p-2}\nabla u) =a|u|^{s-2}u &\mbox{in }&\ \mathbb{R}^N_+, -|\nabla u|^{p-2}\frac{\partial u}{\partial x_N}=b|u|^{q-2}u&\mbox{on }&\ \mathbb{R}^{N-1}, \end{cases}\ \end{equation*} where $a,b\in \mathbb{R}$, $p,q,s\in(1,\infty)$ and $\rho$ is a continuous positive function on $[0,+\infty)$. We first prove new and sharp embedding results that we establish for the associted weighted energy spaces. In application, we establish existence and regularity of weak solutions to the above problem. We also prove for this problem the nonexistence of nontrivial weak solutions by a new Pohozaev-type identity we obtain. The new results about existence and nonexistence highlight the role of the weight $\rho$ on the solvability of the problem contrasting strongly with those when $\rho$ is constant.

We prove a Skorokhod decomposition for the Markov processes $X^a$ and $X$ associated to the gradient Dirichlet forms with respect to the measures $\rho^a\mu^{\beta}$ and $\rho\mu^{\beta}$, respectively. Here, $\mu^{\beta}$ is the law of the standard Brownian bridge $\beta$, while $\rho^a$ and $\rho$ denote densities which are given by $\rho^a(z):= \mathbf{1}_{[0,\infty)}(\bar{z}_a)$ and $\rho(z):= \int_0^1 \mathbf{1}_{[0,\infty)}(\bar{z}_x) \, dx$, respectively, for all $z\in L^2(0,1)$ which have a (unique) continuous representative $\bar{z}$ which vanishes at zero and one. To this end, we derive infinite-dimensional integration by parts formulas (IbPFs) w.r.t. $\rho^a\mu^{\beta}$ and $\rho\mu^{\beta}$, which contain Hida distributions alongside the usual drift terms. We represent these Hida distributions by integration w.r.t. vector measures of bounded variation. The vector measures in question are constructed via an approximation argument, making use of a generalization of Prokhorov's theorem for vector measures. We further prove that, almost surely, the sample paths of $X^a$ and $X$ take values in the equivalence class of continuous functions vanishing at zero and one for all and $dt$-almost all times, respectively. The main motivation for studying $\rho^a\mu^{\beta}$ and $\rho\mu^{\beta}$ lies in the fact that the distributional terms in their IbPFs are simplifications of the distributional term in the IbPF w.r.t. the law of the reflected Brownian bridge on the unit interval $\mu^{|\beta|}$. Representing the latter by integration w.r.t. a vector measure of bounded variation is still an open problem.

We establish a complete structural classification for continuum-wise hyperbolic surface homeomorphisms. Specifically, we prove that a surface homeomorphism is cw$_F$-hyperbolic if, and only if, it is a pseudo-Anosov homeomorphism whose singularities consist exclusively of spines (1-prongs). Furthermore, we classify these systems up to topological conjugacy, showing that every such homeomorphism is conjugate to either an Anosov automorphism on the torus $\mathbb{T}^2$ or to its standard hyperelliptic quotient on the sphere $\mathbb{S}^2$. As a rigid consequence of this classification, we show that such dynamics are strictly obstructed on surfaces of genus greater than one, the Klein bottle, and the projective plane.

We propose and analyze a polytopal discontinuous Galerkin method for the numerical approximation of a coupled non-Newtonian Stokes-Darcy system modeling the interaction between a non-Newtonian free-flow fluid and a non-Newtonian flow through a porous medium. Due to its geometric flexibility and arbitrary-order accuracy, the proposed discretization scheme is well-suited to configurations with complex geometries. We provide a complete a-priori analysis that considers shear-dependent and velocity-dependent non-Newtonian viscosity models for the free-flow and porous media regions, respectively. The well-posedness, stability, and error bounds of the method are established in the framework of generalized inf-sup theory. Error estimates are confirmed by numerical results.

The article deals with the problem of testing conditional axial symmetry within a~multivariate linear heteroscedastic regression framework. A new test based on integrated rank scores is introduced and its asymptotic distribution is derived. The proposed method extends a similar procedure developed for multivariate data to the regression setting. The test may also be employed to assess specific hypotheses concerning distributional properties of the error term. Its performance and application is illustrated in a small simulation study and with real economic data. The article also contains a few theoretical results regarding axial symmetry that may be of independent interest.

In this paper, we study $S$-prime elements and $S$-primary elements within the framework of multiplicative lattices. Furthermore, we define and explore weakly $S$-prime elements and weakly $S$-primary elements, which generalize weakly prime elements and weakly primary elements in multiplicative lattices respectively. We show that the weakly $S$-prime ideals (weakly $S$-primary ideals) of a commutative ring $R$ with $1$ correspond precisely to the weakly $S_L$-prime elements (weakly $S$-primary elements) of the ideal lattice $Id(R)$ of $R$, where $S_L = \{(s) \mid s \in S\}$.

Given $n \in \mathbb{N}$, an Ahlors--David $n$-regular set $S \subset \mathbb{R}^{n+1}$, and a weight $\gamma$ satisfying the local Muckenhoupt $A_{1}$-condition, we present a complete intrinsic description of the trace-space $W_{1}^{1}(\mathbb{R}^{n+1},\gamma)|_{S}$ of the weighted first-order Sobolev space $W_{1}^{1}(\mathbb{R}^{n+1},\gamma)$ to $S$. Furthermore, we construct a new family of nonlinear bounded extension operators acting from $W_{1}^{1}(\mathbb{R}^{n+1},\gamma)|_{S}$ to $W_{1}^{1}(\mathbb{R}^{n+1},\gamma)$. Finally, we find conditions on $\gamma$ that sufficient for the existence of a bounded linear extension operator from $W_{1}^{1}(\mathbb{R}^{n+1},\gamma)|_{S}$ to $W_{1}^{1}(\mathbb{R}^{n+1},\gamma)$.

We establish the Kazhdan isomorphism for the $n$-fold metaplectic cover of the reductive group $\mathrm{SL}_2$ over two sufficiently close non-archimedean local fields $F$ and $F'$, both of which have residue characteristic coprime to $n$ and contain all distinct $n$th roots of unity.

Persistence diagrams are common representations in topological data analysis, but they do not naturally live in a vector space, and the statistical tools developed for comparing them have largely evolved separately from those used for downstream prediction. We introduce STRAND (Survival Topological Representation ANalysis of Diagrams), which treats (collections of) PDs as survival data: each topological feature with persistence value $p = d - b$ is a fully observed time-to-event, and the persistence survival function $S(t) = \mathbb{P}(p > t)$ is the central object for comparing diagrams. From this single representation we derive (i) a non-parametric two-sample test with calibrated Type I error and high power from a small number of diagrams; (ii) interpretable effect sizes; and (iii) a 1-Wasserstein-stable feature vector for downstream machine learning. We validate calibration and power on synthetic manifolds with controlled topology, demonstrate competitive vectorisation across 14 graph and 3D point cloud benchmarks, and apply the method to study functional brain connectivity in fMRI/neuroscience data. To our knowledge, STRAND is the first method to provide hypothesis testing and vectorisation for persistence diagrams from a single coherent and interpretable representation.

Aumann defined common knowledge mathematically and established his now famous Agreement Theorem. We present a novel approach to quantifying how close individuals are to commonly knowing events, $(\delta,\epsilon)$-common knowledge, which is defined for any (and not just countable) probability spaces, and provide quantitative versions of the key results in this field. Specifically, we do this for Aumann's Agreement Theorem and Nielsen's extension thereof to random variables, as well as for the setting in which posteriors are communicated back and forth between individuals. Our results apply in particular to noisy communication settings.

To a full $m\times n$ Mealy automaton $A$ we associate a bijection $\theta_A$, a one-vertex rank-two graph $F_{\theta_A}$, and a one-vertex $VH$-square complex $Y_A$ tiled by $mn$ Wang tiles. We prove that $Y_A$ contains an anti-torus if and only if $A$ is bi-reversible and $F_{\theta_A}$ is aperiodic. The two hypotheses are independent and play disjoint roles: bi-reversibility is exactly what makes $Y_A$ a complete square complex, so that its universal cover splits as a product of two trees and anti-tori can be discussed at all; and, within that setting, an anti-torus is precisely a period-free configuration in the two-sided path space of $F_{\theta_A}$, whose existence is the aperiodicity condition. Working at the level of configurations removes any appeal to the geometry of products of trees from the main equivalence; the geometric (loop-spanned) form of Wise is shown to be strictly stronger, the lamplighter being aperiodic with no loop-spanned anti-torus.

We extend the dendroidal Rezk nerve to the setting of relative $\infty$-operads. Our main theorem relates it to localization of $\infty$-operads, generalizing a theorem of Mazel-Gee. By exploiting the relation, we obtain a surprisingly effective tool to prove localization results in operadic contexts. As applications, we obtain a number of new results on operadic localizations, including a generalization of Willwacher's recent result on cyclic operads and operadic modules, and a description of locally constant factorization algebras on spheres in terms of discrete geometry.

The emergence of cooperation in structured populations is fundamental to the success of human societies. Physical and online networks can drive behavioural change by altering who people interact with, thereby modifying social pressures. In this paper, we study imitation dynamics in a population of self-interested agents playing a continuous strategy Prisoner's Dilemma on a dynamically evolving weighted network. In the fast-network regime, we incorporate the edge weights into the strategy evolution before deriving and analysing the large population mean-field limit. Without noise, we establish well-posedness and show the solution collapses to a single Dirac mass. For initially separated clusters, we identify a payoff threshold and sufficient conditions for the overall level of cooperation to increase. We then introduce stochastic strategy updates, and obtain a non-local Fokker-Planck equation in the mean-field limit. We rigorously prove existence and uniqueness of stationary distributions, and show linear stability under sufficient noise. Numerics illustrate that noise can transform the deterministic consensus into stable cooperative stationary behaviour. These findings show that the fast adaptive interactions and stochastic exploration can jointly support the emergence of stable cooperation at a population level.

In this article, we study some classes of solutions for a stationary Navier-Stokes equation where we consider a rough drift given by a singular integral operator which does not belong to the classical Calder{ó}n-Zygmund family of singular integral operators. Given a small external force, we will construct solutions to this system in the framework of weighted Morrey-Sobolev spaces. The use of Morrey-based Sobolev spaces provides a more general setting than the usual Lebesgue-based Sobolev spaces, and the presence of Muckenhoupt weights will allow us to present some existence and uniqueness results from several points of view.

In this paper, we introduce a class of preference robust risk measures-\emph{robust optimized certainty equivalents} (ROCE)-which encompasses several widely used measures, including Conditional Value-at-Risk and expectiles, as special cases. Motivated by recent developments in distributionally robust optimal reinsurance (DROR), we investigate DROR problems under the ROCE risk measure and consider two prominent uncertainty sets: the mean-variance uncertainty set and the Wasserstein uncertainty set. For the mean-variance uncertainty set, we reformulate the infinite-dimensional optimization problem into a finite-dimensional one by showing that it suffices to consider three-point distributions. This leads to a unified and explicit formulation for a broad class of ROCE risk measures and offers a simplified framework that also recovers earlier results for Conditional Value-at-Risk and expectiles. For the Wasserstein uncertainty set, we also derive a tractable finite-dimensional formulation. The resulting data-driven models enable efficient computation and facilitate a systematic comparison between moment-based and Wasserstein-based uncertainty sets in the optimal deductible design. Numerical experiments are exhibited to illustrate the performance of our reformulated programs.

For a set $X$ of integer points, the relaxation complexity $\operatorname{rc}(X)$ is the smallest number of facets of any polyhedron $P$ such that $P \cap \mathbb{Z}^d = X$. In this paper, we focus on the case where $X$ is the discrete standard simplex $\Delta_d = \{\mathbf{0}, \mathbf{e}_1, \dots, \mathbf{e}_d\}$. We show that $\operatorname{rc}(\Delta_d) = O(\log d)$ by an explicit, elementary construction. This improves upon the previously best-known upper bound $\operatorname{rc}(\Delta_d) = O(d / \sqrt{\log d})$ due to Aprile, Averkov, Di Summa, and Hojny (2024) and matches an asymptotic lower bound by Averkov and Schymura (2022).

In this work we consider tensor train varieties. These are varieties of tensors arising in a range of fields, including quantum many-body physics and machine learning. Using methods from integral geometry, we obtain a combinatorial expression for their degrees. We provide the ready-to-use julia package this http URL.

We revisit the large population limit of our epidemic model with infection age dependent infectivity and progressive immunity waning, under the assumption that the supremum in $t$ of the random infectivity function has a finite expectation, while the previous proofs assumed that this supremum admits a deterministic upper bound.

We study Brauer groups of the smooth loci in linear systems on simply connected smooth projective varieties. Under a suitable ampleness condition, we prove that the Brauer group is at most $\mathbb Z/2 \mathbb Z$. This applies when the underlying variety is the projective plane, a very general K3 surface, or a general cubic fourfold. As an application, we compute the Tate--Shafarevich group parametrizing torsors over the relative Jacobians of universal smooth plane curves. Our approach is via a study of the $2$-nodal locus in the linear system.

Hyperspectral standoff detection systems such as Focal Plane Array (FPA) Fourier Transform Infrared (FTIR) spectrometers provide high spatial resolution in detecting airborne chemical contaminants that are invisible to the human eye but potentially hazardous. When two such systems are operated simultaneously with a suitable opening angle, they enable tomographic reconstruction of contaminant plumes with improved spatial and temporal accuracy. This work presents a mathematical model of these measurement capabilities and an algorithm to identify, localize, and quantify contaminant release sources. The objective is to develop a a tool that reconstructs release locations and predict the future plume evolution from standoff measurement data, thereby supporting early warning and situational awareness in hazardous material release scenarios. The transport of contaminants is modeled by an advection-diffusion equation, and the corresponding inverse problem for source identification is formulated accordingly. Owing to the severe ill-posedness and underdetermination of the problem, a sparsity-promoting regularization approach is employed together with a high-performance optimization algorithm. To incorporate the tomographic measurement data into the discrete formulation, a level-set description of a threshold concentration is used, allowing the measurements to be represented independently of the computational mesh and avoiding costly remeshing procedures.

We prove a singularity theorem in which the classical focusing hypothesis of Hawking--Penrose theory is replaced by a condition on asymptotic volume growth. Under the strong energy condition, we introduce asymptotic volume-expansion invariants associated with a compact Cauchy hypersurface and show that a uniform positive lower bound on these invariants implies past timelike geodesic incompleteness. More precisely, we obtain an explicit upper bound on the time-separation from the hypersurface to its chronological past. The theorem extends to globally hyperbolic Lorentzian length spaces satisfying the synthetic strong energy condition $\mathsf{TCD}^e_p(0,N)$, yielding an inextendibility result valid without any smoothness or differentiability assumption. We also prove an area comparison theorem for equidistant hypersurfaces and a volume singularity theorem based on related asymptotic expansion invariants.

We consider linear programming in the oracle model: $\max\{c^\top x \,:\, x\in P\}$, where the polyhedron $P=\{x\in\mathbb{R}^n\,:\, Ax\le b\}$ is given by a separation oracle. We present an algorithm that finds exact primal and dual solutions using $O(n^2\log(n/\delta))$ oracle calls and $O(n^4\log(n/\delta)+n^5\log\log(1/\delta))$ arithmetic operations, where $\delta$ is a geometric condition number associated with the system $(A,b)$. These bounds do not depend on the cost vector $c$ and do not require a priori knowledge of $\delta$. For rational data, $\log(1/\delta)$ is polynomially bounded in the encoding size of $(A,b)$, thus providing a polynomial-time algorithm. The algorithm works in a black box manner, requiring a subroutine for approximate primal and dual solutions; the above running times are achieved when using the cutting plane method of Jiang, Lee, Song, and Wong (STOC 2020) for this subroutine. Whereas approximate solvers may return primal solutions only, we develop a general framework for extracting dual certificates based on the work of Burrell and Todd (Math. Oper. Res. 1985). Our algorithm strengthens results by Grötschel, Lovász, and Schrijver (Prog. Comb. Opt. 1984), and by Frank and Tardos (Combinatorica 1987) that rely on bit-complexity arguments. Our algorithm avoids rounding-based arguments such as simultaneous Diophantine approximation and uses geometric arguments instead.

We prove sub-Riemannian Selberg trace formulae for compact quotients of SL(2, R). Using the Fourier decomposition along the SO(2)-fibers, we reduce the heat trace computation to the Selberg trace formula for Maass Laplacians on the hyperbolic plane. The resulting formula has an identity contribution and a hyperbolic contribution, the latter involving a character-dependent theta factor over closed geodesics. We then use this trace formula to compute the zeta-regularized determinant of the sub-Laplacian. The determinant formula is remarkably compact and is expressed in terms of a determinant depending only on the base hyperbolic surface and an explicit relative Selberg product.

This paper investigates the well-posedness issue for a reduced two-dimensional model of micropolar fluids. This reduced model presents a coupling between an Euler-type equation for the velocity field of the fluid and a transport-diffusion equation for the microrotation field (which is a scalar field, in this setting). We establish the local existence and uniqueness of strong solutions in the scale of Besov space $B^s_{p,1}$ having regularity index $s\geq1+2/p$. Furthermore, in the subcritical case when $s>1+2/p$, we prove that these solutions exist globally in time. The global persistence of regularity in the critical setting $s=1+2/p$ remains open.