In this paper we discuss how to associate a suitable non-transitive version of a Cartan connection to sub-Riemannian manifolds of corank 1 (including contact and quasi-contact sub-Riemannian manifolds) with non-necessarily constant sub-Riemannian symbols. In particular, we recast the variation of the sub-Riemannian symbols into a suitable "type" map, which is constant if and only if the symbols are constant. We then consider the (non-transitive) groupoid of sub-Riemannian symmetries and investigate its smoothness, properness, regularity, and other properties in relation with the type map. Last, we describe how to build a "non-transitive" analogue of a Cartan connection on top of such (Lie) groupoid, obtained as the sum of a tautological form with a multiplicative Ehresmann connection. We conclude by illustrating our results on concrete examples in dimension 5.

The Grothendieck constant $K_{G}$ is a fundamental quantity in functional analysis, with important connections to quantum information, combinatorial optimization, and the geometry of Banach spaces. Despite decades of study, the value of $K_{G}$ is unknown. The best known lower bound on $K_{G}$ was obtained independently by Davie and Reeds in the 1980s. In this paper we show that their bound is not optimal. We prove that $K_{G} \ge K_{DR} + 10^{-12}$, where $K_{DR}$ denotes the Davie-Reeds lower bound. Our argument is based on a perturbative analysis of the Davie-Reeds operator. We show that every near-extremizer for the Davie-Reeds problem has $Ω(1)$ weight on its degree-3 Hermite coefficients, and therefore introducing a small cubic perturbation increases the integrality gap of the operator.

We study contact 3-manifolds $Y$ with a special global frame inspired by Cartan's structure equations. This frame is dual to a generalized Finsler structure defined by Bryant. We present some examples and rigidity results on the class of manifolds whose frame satisfies certain natural conditions on a scalar function $K\colon Y\to \mathbb{R}$, related to the frame. This function realizes the curvature when $Y$ is the unit tangent bundle with respect to a metric on a surface. As applications, we obtain sharp estimates for the action of a Reeb orbit in terms of this scalar function, under the assumption that the frame satisfies specific conditions. In particular, we recover a classical upper bound on the systole of positively curved metrics on $S^2$ due to Toponogov.

We present a theoretical framework for recovering the amplitude and carrier phase of a single received RF field with a Rydberg-atom receiver, without injecting an RF local oscillator (LO) into the atoms. The key enabling mechanism is a static DC bias applied to the vapor cell: by Stark-mixing a near-degenerate Rydberg pair, the bias activates an otherwise absent upper optical pathway and closes a phase-sensitive loop within a receiver driven only by the standard probe/coupling pair and the received RF field. For a spatially uniform bias, we derive an effective four-level rotating-frame Hamiltonian of Floquet form and show that the periodic steady state obeys an exact harmonic phase law, so that the $n$th probe harmonic carries the factor $e^{inΦ_S}$. This yields direct estimators for the signal phase and amplitude from a demodulated probe harmonic, with amplitude recovery obtained by inverting an injective harmonic response map. In the high-SNR regime, we derive explicit RMSE laws and use them to identify distinct phase-optimal and amplitude-optimal bias-controlled mixing angles, together with a weighted joint-design criterion and a balanced compromise angle that equalizes the fractional phase and amplitude penalties. We then extend the analysis to nonuniform DC bias through quasistatic spatial averaging and show that bias inhomogeneity reduces coherent gain for phase readout while also reshaping the amplitude-response slope. Numerical examples validate the phase law, illustrate response-map inversion and mixing-angle trade-offs, and quantify the penalties induced by bias nonuniformity. The results establish a minimal route to coherent Rydberg reception of a single RF signal without an auxiliary RF LO in the atoms.

This note outlines a mean-field approach to dynamic optimal transport problems based on the recently proposed McKean-Pontryagin maximum principle. Key aspects of the proposed methodology include i) avoidance of sampling over stochastic paths, ii) a fully variational approach leading to constrained Hamiltonian equations of motion, and iii) a unified treatment of deterministic and stochastic optimal transport problems. We also discuss connections to well-known dynamic formulations in terms of forward-backward stochastic differential equations and extensions beyond classical entropic-regularized transport problems.

We give a precise asymptotic formula for the number of $n\times 4t$ partial Hadamard matrices in the regimes $t/n^3\to\infty$ and $t/n^3\toΘ$ for sufficiently large fixed $Θ$. This strengthens earlier results of de~Launey and Levin, who obtained the asymptotic for $t/n^{12}\to\infty$, and of Canfield, who extended this to $t/n^4\to\infty$.

We investigate stability of a new class of heteroclinic cycles that we call heteroclinic cycles of type Y. The cycles can be regarded as a generalisation of heteroclinic cycles of type Z introduced in [Podvigina, Nonlinearity 25, 2012]. The type Y cycles differ from the cycles of type Z in the following: The trajectories comprising a cycle of type Y belong to flow-invariant subspaces that can be of different dimensions. Unlike in the most studies of the stability of heteroclinic cycles, we do not require that the eigenvalues of the linearisations of the dynamical system near the equilibria are distinct. Instead of the common assumption that the cycles are robust, we prescribe flow-invariance of certain subspaces. Similarly to type Z cycles, asymptotic stability and fragmentary asymptotic stability of type Y cycles is determined by the eigenvalues and eigenvectors of transition matrices. The matrices are products of basic transition matrices that depend on the eigenvalues of linearisations and the dimensions of the contracting subspaces.

We study the geometric and dynamical structure induced by the return map associated with domains in the class \(\mathcal{O}_{C}\). This map, defined through a geometric round-trip between the convex core and the outer boundary, generates a discrete dynamical system on the boundary \(\partial C\). Building on previous results establishing global convergence of the return dynamics, we show that equilibria of the return map coincide with the critical points of the thickness function. This identification allows us to apply Morse-theoretic tools to derive global constraints on the dynamics. In particular, we obtain lower bounds on the number of equilibria in terms of the Betti numbers of \(\partial C\), as well as a global balance relation governed by the Euler characteristic. We further analyze the local behavior of the return map near equilibria. Using the differentiability of the return map inherited from the radial and reciprocal constructions, we derive a first-order expansion in which the linearization is governed by the Hessian of the thickness function and an operator arising from the geometry of the return map. This leads to an operator-valued generalization of the previously observed scalar structure, revealing that the dynamics behaves as an anisotropic gradient-like iteration rather than a purely isotropic descent. Near nondegenerate minima, we prove a quantitative descent estimate and local linear convergence under a spectral condition. Under aligned nonlocal geometry, the sign of the curvature gap between the convex core and the outer boundary determines whether the induced dynamics is contracting, neutral, or expanding in each principal direction. Finally, we discuss extensions beyond the Morse setting, including the Morse-Bott case, and highlight connections between the geometry of the domain, the topology of \(\partial C\), and the structure of the induced dynamics.

In this paper, we construct additively graceful signed graphs S from a given graph G that may be additively graceful or not be additively graceful. We also show the construction of additively graceful signed graphs from additively graceful signed graphs. We find the values of m, n in non-divisible sum graph, denoted as G(m, n), that admit additively graceful labeling.

We prove that minimal Dirac operators on the half-line are self-modeling, which means that such an operator is determined by its arbitrary unitary copy uniquely up to a transformation (shape equivalence) which changes its potential by a constant factor of modulus one. This result is obtained using the wave functional model of the minimal matrix Schrödinger operator on the half-line.

We compute the homotopy groups at each unital abelian C*-algebra $C(T)$ in the Morita $3$-category of abelian C*-algebras, C*-algebras with central maps, C*-correspondences, and adjointable bimodule maps. We describe these groups in terms of the topological data of the underlying compact Hausdorff space $T$. We also compute the actions of the first homotopy group on the second and third homotopy groups in terms of these topological invariants of $T$.

In this article, we investigate vacuum leakage detection problems in composite manufacturing. Our approach uses Voronoi diagrams, a well-known structure in discrete geometry. The Voronoi diagram of the vacuum connection positions partitions the component surface. We use this partition to narrow down potential leak locations to a small area, making an efficient manual search feasible. To further reduce the search area, we propose refined Voronoi diagrams. We evaluate both variants using a novel dataset consisting of several hundred one- and two-leak positions along with their corresponding flow values. Our experimental results demonstrate that Voronoi-based predictive models are highly accurate and have the potential to resolve the leakage detection bottleneck in composite manufacturing.

The van der Waerden simplicial complex, denoted ${\tt vdw}(n,k)$, is the simpicial complex whose facets correspond to the arithmetic progressions of length $k$ in the set $\{1,\ldots,n\}$. We study the Lefschetz properties of the Artinian ring $A(n,k) = K[x_1,\ldots,x_n]/(I_{{\tt vdw}(n,k)} + \langle x_1^2,\ldots,x_n^2\rangle)$ where $I_{{\tt vdw}(n,k)}$ is the associated Stanley--Reisner ideal. If $k=1,2$ or $n-1$, the ring $A(n,k)$ will have the Weak Lefschetz Property for all $n > k$. When $k=3$, we classify the rings $A(n,3)$ that have the Weak Lefschetz Property. We conjecture that $A(n,k)$ fails to have the Weak Lefschetz Property if $n \gg k \geq 3$ and $k$ odd. We also classify when ${\tt vdw}(n,k)$ is a pseudo-manifold, which allows us to show that $A(n,k)$ satisfies the Weak Lefschetz Property in some degrees by using a result of Dao and Nair.

The extension problem asks whether positive semi-definite functions on a symmetric unital subset of a discrete group can be extended to positive semi-definite functions on the whole group. It has been known at least since the work of Rudin in the 1960s that this is closely related to the problem of finding sums of squares factorisations of positive elements in the group C*-algebra. We give an operator system perspective at these two problems explaining their equivalence: the extension property is characterised by a certain quotient map on the Fourier--Stieltjes algebra, and the factorisation property by a certain complete order embedding into the group C*-algebra. These properties are linked to the duality of the operator systems which have recently emerged from spectral and Fourier truncations in noncommutative geometry. We exemplify how one can relate certain extension problems to operator system techniques such as nuclearity and the C*-envelope.

The Hadamard rank of a point with respect to a projective variety is, if it exists, the minimum number of points of the variety whose coordinate-wise product is the given point. We classify the projective varieties for which the Hadamard rank is finite for any point. As a by-product we obtain the finiteness of the Hadamard rank with respect to varieties of tensors, such as Grassmannians, Chow varieties, varieties of reducible forms and their secant varieties, complementing previous known results on secant varieties of Segre-Veronese varieties. We prove sharp upper bounds on the maximum Hadamard rank for certain families of algebraic varieties: this is a consequence of a result on the lower semi-continuity of the Hadamard rank for curves that do not contain points with at least two zero coordinates.

We present a new general method for performing basic arithmetic in the finite field~$\mathbb{F}_p$ for any prime $p>2$ by using traditional binary operations over~$\mathbb{F}_2$. Our new approach is efficient and competitive with current state-of-art methods. We apply our new arithmetic method to the computation of the minimum Hamming distance of random linear codes for the fields $\mathbb{F}_3$ and $\mathbb{F}_7$. Our new arithmetic method allows to apply new techniques such as the isometric addition that accelerate the computation of the Hamming distance. We have developed implementations in the C programming language for computing the Hamming distance that clearly outperform both state-of-art licensed software and open-source software such as \textsc{Magma} and \textsc{GAP}/\textsc{Guava} on single-core processors, multicore processors, and shared-memory multiprocessors.

The sparse analogue of Szemerédi's regularity method has played a central role in the development of extremal results for random graphs. While the sparse embedding lemma (the KLR conjecture) has been resolved, the corresponding sparse counting lemma remains widely open. The conjecture, formulated by Gerke, Marciniszyn, and Steger, states that for every fixed graph $H$ and any $β>0$, there exists $\varepsilon>0$ such that the following holds. Consider a balanced blow-up of $H$ with vertex classes of size $n$, where each pair corresponding to an edge of $H$ forms an $(\varepsilon)$-regular bipartite graph with exactly $m$ edges. Assume that $m$ is above the natural threshold $m \gg n^{2-1/m_2(H)}$, then all but a $β^m$ proportion of such graphs contain at least $(1-δ)$ times the expected number of copies of $H$. At present, among the complete graphs, the conjecture is known only for $H=K_3$. In this paper, we establish the $H=K_4$ case of the conjecture.

We investigate lower bounds for the number of ideal and finite vertices of right-angled hyperbolic polyhedra of finite volume. We use a geometric method of orthogonal gluings to establish new bounds in low dimensions, specifically $v_\infty(P^5) \ge 3$ and $v_{fin}(P^7) \ge 4$. By combining these initial bounds with double counting arguments and recurrence relations, we obtain improved lower bounds for both types of vertices in all higher dimensions up to $n=12$, the maximal dimension where polyhedra of this class exist.

Monotone co-design enables compositional engineering design by modeling components through feasibility relations between required resources and provided functionalities. However, its standard boolean formulation cannot natively represent quantitative criteria such as cost, confidence, or implementation choice. In practice, these quantities are often introduced through ad hoc scalarization or by augmenting the resource space, which obscures system structure and increases computational burden. We address this limitation by developing a quantale-enriched theory of co-design. We model resources and functionalities as quantale-enriched categories and design problems as quantale-enriched profunctors, thereby lifting co-design from boolean feasibility to general quantitative evaluation. We show that the fundamental operations of series, parallel, and feedback composition remain valid over arbitrary commutative quantales. We further introduce heterogeneous composition through change-of-base maps between quantales, enabling different subsystems to be evaluated in different local semantics and then composed in a common framework. The resulting theory unifies feasibility-, cost-, confidence-, and implementation-aware co-design within one compositional formalism. Numerical examples on a target-tracking system and a UAV delivery problem demonstrate the framework and highlight how native quantitative enrichment can avoid the architectural and computational drawbacks of boolean-only formulations.

In the analysis of real-world data, extracting meaningful features from signals is a crucial task. This is particularly challenging when signals contain non-stationary frequency components. The Iterative Filtering (IF) method has proven to be an effective tool for decomposing such signals. However, such a technique cannot handle directly data that have been sampled non-uniformly. On the other hand, graph signal processing has gained increasing attention due to its versatility and wide range of applications, and it can handle data sampled both uniformly and non-uniformly. In this work, we propose two algorithms that extend the IF method to signals defined on graphs. In addition, we provide a unified convergence analysis for the different IF variants. Finally, numerical experiments on a variety of graphs, including real-world data, confirm the effectiveness of the proposed methods. In particular, we test our algorithms on seismic data and the total electron content of the ionosphere. Those data are by their nature non-uniformly sampled, and, therefore, they cannot be directly analyzed by the standard IF method.

We prove that for any smooth polarized complex $n$-dimensional manifold $(X, L_X)$ which admits an extremal Kähler metric in $c_1(L_X)$, and for any integer $k$ large enough (in terms of a bound depending on $(X, L_X)$), the $(n+k+1)$-dimensional complex cone $\mathcal{Y}:= \overline{(L_X \otimes \mathcal{O}_{\mathbb{P}^k}(1))^{\times}}$ with section $X \times \mathbb{P}^k$ admits a scalar-flat Kähler cone metric. Equivalently, the unweighted Sasaki join of a smooth compact quasi-regular extremal Sasaki manifold with a regular Sasaki sphere $\mathbb{S}^{2k+1}$ of sufficiently large dimension $(2k+1)$ admits a Sasaki metric of constant (positive) scalar curvature. This gives an affirmative answer to an asymptotic version of a question raised by Boyer--Huang--Legendre--Tønnesen-Friedman in arXiv:1906.04827.

The classical bar-cobar adjunction between dg algebras and dg coalgebras goes back to the origins of differential homological algebra as developed by Cartan, Eilenberg, Moore, and many others, and is part of the broader framework of Koszul duality. In recent years, several $\infty$-categorical analogues of this adjunction have been developed, notably by Lurie, Francis--Gaitsgory, and Heuts. However, there is no comparison in the literature between the classical chain-level constructions and their higher-categorical counterparts, and in fact the two constructions are not quite compatible. In this paper we provide a unified framework relating these different forms of Koszul duality in the differential graded setting. We construct a commutative square of adjunctions, called the inclusion-restriction square, intertwining the classical operadic bar-cobar adjunction with its completed variant due to Le Grignou--Lejay. We show that this square induces an $\infty$-categorical adjunction between algebras and their Koszul dual coalgebras, recovering in particular the differential graded case of Lurie's bar-cobar adjunction, and explain precisely how our constructions relate to those of Francis--Gaitsgory and Heuts.

In his quest to disprove a claim by Peirce that all lattices are distributive, Ernst Schröder considered 135 years ago a list of 990 equational laws on quasigroups, analogous to associativity, such as $(x // y) * z = (y // x) \backslash\backslash z$. A quasigroup is a non-associative analogue of groups, specifically a set equipped with multiplication and right/left conjugate-division operations that are compatible. Each equation of interest identifies two three-variable expressions built from these operations. I determine all $114$ equivalence classes of their conjunctions, and all implications between them. This includes as a small corner the five-element non-distributive lattice identified by Schröder.

This article provides a naturel sequel of previous works [6, 4] regarding the stability of travelling waves for a general one-dimensional Schrödinger equation (N LS) with non-zero condition at infinity. The aim of this article is twofold. First, we prove the asymptotic stability of well-prepared chains of dark solitons and secondly, we construct an asymptotic N -soliton-like solution, which is an exact solution of (N LS), the large-time dynamics of which is similar to a decoupled chain of solitons.

We propose a new frame work of $p$-adic neural network. Unlike the original $p$-adic neural network by S.\ Albeverio, A.\ Khrennikov, and B.\ Tirrozi using a family of characteristic functions indexed by hyperparameters of precision as activation functions, we use a single injective $p$-adic character on the topological Abelian group $\mathbb{Z}_p$ of $p$-adic integers as an activation function. We prove the $p$-adic universal approximation theorem for this formulation of $p$-adic neural network, and reduce it to the feasibility problem of polynomial equations over the finite ring of integers modulo a power of $p$.

We consider discrete Schrödinger operators $H_{μQ}=Δ+μQ$ with real periodic potentials $Q$ on periodic graphs, where $Δ$ is the adjacency operator and $μ\in\mathbb R$ is a coupling constant. The spectra of the operators consist of a finite number of closed intervals (bands). In the large coupling regime, we obtain an asymptotic upper bound for the measure of the spectrum of $H_{μQ}$ which depends essentially on a "degeneracy degree" of the potential $Q$. This result extends the result of Y. Last obtained for the one-dimensional lattice $\mathbb Z$ to the case of general periodic graphs. It also may serve as a certain quantitative complement to the recent criterion of J. Fillman for the measure of the spectrum of $H_{μQ}$ to go to zero as $μ\to\infty$.

We study the stabiliser formalism for qudits of arbitrary dimension $d$. In the free case, we show that the basic theorem of the stabiliser formalism remains valid: if the stabiliser subgroup $H$ is free as a $Z/dZ$-module and contains no non-trivial scalars, then the protected space $V^H$ is naturally identified with the state space of a smaller number of qudits of the same dimension, and the quotient $N(H)/H$ is identified with the Pauli group on a smaller number of qudits. We then remove the freeness assumption and describe the resulting structure in general. In this case, the protected space is identified with a tensor product of qudit spaces of possibly smaller dimensions, and the quotient $N(H)/H$ is described by a corresponding product of qudit Pauli groups, possibly of smaller dimensions, over a common center. We also characterise the shifted free case, which is exactly the situation in which $N(H)/H$ is again an ordinary qudit Pauli group. Our approach is algebraic and uniform, and applies in particular to the qudit Kitaev model and to its shifted and twisted variants.

We present a new theory of categorization based on an information-theoretic rational analysis. To evaluate this theory, we investigate how well it can account for key findings from classic categorization experiments conducted by Hayes-Roth and Hayes-Roth (1977), Medin and Schaffer (1978), and Smith and Minda (1998). We find that it explains the human categorization behavior at least as well (or better) than the independent cue and context models (Medin & Schaffer, 1978), the rational model of categorization (Anderson, 1991), and a hierarchical Dirichlet process model (Griffiths et al., 2007).

In this work, initial-boundary value problems for the time-fractional Airy equation are considered on different intervals. We study the properties of potentials for this equation and, using these properties, construct solutions to the considered problems. The uniqueness of the solution is proved using an analogue of the Gronwall-Bellman inequality and an a priori estimate.

For the fully nonlinear stationary logistic equation ${\mathcal F}(x,D^2u)+μu=k(x)u^p$ with $p>1$ and $k(x)\geq 0$, in a bounded domain with Dirichlet boundary condition, we determine, in terms of $μ$, the existence and uniqueness or the nonexistence of a positive solution. Furthermore, we study the asymptotic behavior of the solutions when $μ$ approaches the boundary points of the existence range.