In this article, we derive several significant upper bounds for the numerical radius and $a$-numerical radius of an element in a ${C}^*$-algebra by improving inequalities for positive linear functionals. Our findings refine and generalize the existing inequalities. Furthermore, we introduce a new notion to derive improved upper bounds of the numerical radius for an element in a ${C}^*$-algebra using the Moore-Penrose inverse.

There are up to isomorphism exactly three strongly regular graphs with parameters $(120,56,28,24)$ whose automorphism group acts primitively on the vertices. Two of these graphs belong to classical families: one is the non-orthogonality graph on anisotropic points of the hyperbolic quadric $\mathcal Q^+(7,2)$, and the other one belongs to the Johnson scheme. The third one is not well understood. In this paper, we give a description of this graph in terms of ovoids and spreads of $\mathcal Q^+(7,2)$, or equivalently in terms of overlarge sets of Steiner systems with parameters $(3,4,8)$.

Let $Y\longrightarrow X$ be a $G$-Galois connected cover of smooth projective curves over an algebraically closed field $k$ of positive characteristic $p$ with the branch locus contained in a finite subset of closed points $S_X$ in $X$, where $G$ is a finite cyclic $p$-group and $l$ is a prime number other than $p$. Let $Γ$ be an extension of an elementary abelian $l$-group $H$ by $G$. We find $G$-stable submodules of $l$-torsion of the Picard group of $Y$ and its generalisation. This is used to describe a method for finding the minimum genus of $Γ$-covers of $X$, étale over $X\setminus S_X$, dominating $Y$; and also the minimum of the genera of $Γ$-covers of $\mathbb{P}^1$ étale over $\mathbb{A}^1$.

In this paper, we develop a generalized framework for constructing reversible linear, reversible self dual and reversible DNA codes using a matrix-theoretic approach based on involutory matrices. The proposed concatenation scheme gives a large class of generator matrices and yields codes with good parameters. The construction is carried out at the level of linear codes and then extended to DNA codes. Using a matrix product approach, we provide a unified method for analysis and proof. Further, we resolve an open problem raised by Oztas et.al. and also we correct and improve some results of them.

In this paper, we mainly consider positive solution to the $D^{1,p}(\R^{N})$-critical quasi-linear Schrödinger system with $p$-Laplacian: \begin{equation*}\begin{cases} -Δ_p u = u^αv^β \, \ \ \ \ \ \text{in}\,\ \ \R^N, \\ -Δ_p v = u^βv^α \,\ \ \ \ \ \text{in}\,\ \ \R^N, \end{cases}\end{equation*} where $1<p<N$, $N\geq2$, $0\leq α\leq β,$ and $u,v\in D^{1,p}(\R^N)$. We establish regularity and the sharp estimates on asymptotic behaviors for any positive solution $(u,v)$. Then, we prove that all positive solutions are radially symmetric and strictly decreasing about some point. Furthermore, we obtain the uniqueness and complete classification of positive solutions. Our results extend the uniqueness results in \cite{LM,QS} for $p=2$ to general cases $1<p<N$.

For a natural number $n$, a fence $[n]=\{1\prec 2\succ 3\prec 4\succ 5\prec \cdots n\}$ is a partial ordered set. A partial transformation $α$ is called fence-decreasing if $xα\preceq x$ for all $x$ in the domain of $α$, and fence-preserving if $x\prec y$ implies $xα\preceq yα$ for all $x$ and $y$ in the domain of $α$. In this paper, we consider the monoids $\mathcal{DF}_{n}$ ($\mathcal{PDF}_{n})$ of all fence-decreasing full (partial) transformations as well as the monoid $\mathcal{PCF}_{n}$ of all fence-preserving transformations of $\mathcal{PDF}_{n}$. For these three monoids and some of their ideals, we determine the unique minimal generating set. Moreover, we calculate the rank of $\mathcal{DF}_{n}$, $\mathcal{PDF}_{n}$, and $\mathcal{PCF}_{n}$. Additional, we provide several combinatorial results concerning these three monoids.

In this work, we extend the construction of the Eisenstein-Kronecker classes of Kings-Sprang to the universal abelian schemes lying over Hilbert moduli spaces parametrizing objects with $(Γ(l), Γ_{00}(p^\infty))$-level. We then compare the associated ($p$-adic) Hilbert modular forms to the Eisenstein series constructed by Katz. Similarly to the procedure of Kings-Sprang, we produce a $p$-adic measure which interpolates the Eisenstein-Kronecker classes, and use the mentioned comparison results to show that this recovers essentially the $p$-adic Eisenstein measure constructed by Katz.

We explicitly determine the generalized superderivations of the Neveu-Schwarz and Ramond algebras.

The Favard length of a Borel set $E$ on the Euclidean plane is the average length of its orthogonal projections (mean shadow). Here we solve the following problem: Minimize the Favard length in the class of planar Borel sets with unit area (2-dimensional Lebesgue measure). It is shown that a circle with unit area is a solution, as expected.

The formation of public opinion in modern information environments is shaped by the interplay between social conformity and information exposure. While social interactions promote opinion alignment, heterogeneous visibility and selective exposure may reinforce local agreement, a mechanism commonly associated with the echo chamber effects. To describe how such reinforcement influences spatially heterogeneous opinion activity and non-consensus patterns, we propose a spatial opinion dynamics model with attention-mediated feedback. The model couples nonlocal alignment with an evolving attention field and captures a self-reinforcing mechanism in which regions of high opinion activity attract greater visibility. Starting from agent-based jump mechanism inspired by bounded confidence interactions and biased random walks induced by environments, we formally derive a nonlocal advection-cross-diffusion system, where opinion transport is driven by nonlocal conformity and modulated by attention-dependent redistribution. We characterize the transition from spatially homogeneous consensus states to non-consensus clustered regimes through linear stability analysis of the homogeneous equilibrium. The results show that attention-mediated feedback has an explicit correction on the instability threshold and enlarges the parameter regime in which clustering occurs, thereby promoting persistent spatial heterogeneity and non-consensus patterns. Numerical simulations based on a structure-preserving IMEX spectral method support the theoretical predictions and quantify the resulting aggregation phenomena. These findings provide a macroscopic description of how nonlocal alignment and environmental feedback jointly shape spatial signatures of non-consensus patterns.

We prove a local higher integrability result for the gradient of Hölder continuous weak solutions to the parabolic double phase equation \[ \partial_t u - \operatorname{div} \left(|Du|^{p-2}Du + a(z)|Du|^{q-2}Du\right) = 0 \qquad \text{in } Ω_T. \] We work under a relaxed gap condition on the exponents $p$ and $q$. The coefficient $a$ is assumed to belong to the class $\mathcal{Z}^κ(Ω_T)$ for some $κ\in (0,\infty)$. The functions in this class satisfy a one-sided pointwise bound that controls how fast $a$ can grow away from its zero set, and the class contains the Hölder continuous functions. We also impose a mild almost increasing condition on $a$, which motivates the introduction of a new mollification, which we call the slanted Steklov average. For $u \in C^{0,γ,γ/q}_{\mathrm{loc}}(Ω_T)$ with $γ\in [0,1)$, our main result holds under the gap bound \begin{equation}\tag{G}\label{eq:G} 2 \le p \le q \le p + \frac{qκ}{q - 2γ}. \end{equation} The new gap condition \eqref{eq:G} is purely parabolic in nature and is stricter than the optimal gap relation associated with the Lavrentiev phenomenon for the elliptic double phase functional.

We consider polyconvex integrands that are conformally invariant and frame indifferent. In two dimensions, we prove that the corresponding stationary points are smooth outside a discrete set; this result is new even for minimizers. We further show that every orientation-preserving stationary point is $C^1$. Since such solutions are closely related to Teichmüller-type variational problems, our result also confirms, in the case of integrands with linear growth in the distortion, a conjecture of Astala, Iwaniec, Martin, and Onninen from 2005.

We study the following problem: given a covering of curves $ϕ\colon X \to X_0$ over a number field $k$, and an integer $d$, when is the set \[\{p \in X_0(\overline{k})|\ \mathrm{deg}\ p = d, \text{ and the fiber } ϕ^{-1}(p) \text{ is reducible over } k(p)\}\] finite? In case $X$ itself admits infinitely many degree $d$ points, we consider the modified problem where the images of degree $d$ points on $X$ are removed from the set. We prove a number of theorems ensuring a positive answer. As a consequence we show that for a fixed curve $X$ and all sufficiently high-degree indecomposable rational functions $ϕ:X \to \mathbb{P}^1$ with $b$ branch points, the set of reducible fibers above degree $d<b/7-2$ points, not containing a degree $d$ point from $X$, is finite.

Inspired by Simpson's de Rham stack and Drinfeld's crystalline stack, we develop a stacky approach to convergent cohomology and convergent isocrystals in positive characteristic. To any scheme $X$ over $\mathbb{F}_p$ we attach a convergent stack $X_{\mathrm{conv}}$. When $X$ is of finite type over a perfect field, its finitely generated quasi-coherent $\mathcal{O}[\tfrac1p]$-modules are equivalent to convergent isocrystals over $X$, compatibly with cohomology. When $X$ embeds into a smooth $p$-adic formal scheme, we describe $X_{\mathrm{conv}}$ explicitly as the quotient of an open tube by a $p$-adic formal groupoid. For $f$-semiperfect schemes, by contrast, the convergent stack is representable by a preperfectoid adic space over $\mathbb{Q}_p$.

We develop a variational theory of structured deformations for energies whose surface densities satisfy general growth conditions. This requires a formulation in the generalised space ${\rm GBV}_\star$, introduced by Dal Maso and Toader, which is the natural setting for surface energies that are linear near the origin and bounded at infinity. In this framework, we prove three main results: an approximation theorem for structured deformations, an integral representation theorem for abstract lower semicontinuous functionals, and an explicit representation formula for relaxed energies. The proofs rely on new density results for functions of bounded variation and on Poincaré-type inequalities tailored to ${\rm GBV}_\star$. Our results extend the applicability of structured deformations to cohesive models in fracture mechanics.

We prove two representability results for rigidly-compactly generated $\infty$-categories and functors between them. The first one represents contravariant linear functionals out of a category of perfect objects with values in a category of (pseudo)-coherent objects in terms of (pseudo)-coherent objects. The second one represents covariant functionals out of coherent objects with values in a category of coherent objects in terms of perfect objects. The techniques used belong to the realm of "functional analysis" of presentable stable categories and ultimately depend on the interaction between three notion of finiteness, namely compactness, dualizability and coherence. These results apply to $\mathbb{E}_\infty$-ring spectra, quasi-proper maps of quasi-compact quasi-separated schemes and certain spectral algebraic spaces. We also reformulate Grothendieck duality in terms of internal left adjoints.

We study Innovative Air Mobility (IAM) as a dedicated hub feeder and dispersal service for Bologna Guglielmo Marconi Airport (BLQ), addressing short-range airport-to-airport connectivity where scheduled fixed-wing services are inefficient and surface access is time-consuming. Demand is generated directly from BLQ's flight timetable via a staggered rolling-window model: arriving passengers become IAM-eligible after a 20-minute buffer (a baseline 70% ''staying'' share is considered and distributed across nearby airports using a gravity model), while feeder demand for departures is timestamped backward by IAM travel time plus a 2-hour pre-flight buffer. We build an IAM-only time-expanded network (airports as vertiports) and solve two sequential subproblems: passenger routing with A* to reveal edge-level demand pressure, followed by fleet dispatch with dynamic programming to schedule a finite IAM fleet. Using OAG schedules for 5 May 2024, we simulate 09:00-17:00 with a 50-vehicle fleet (10 seats, 30 m/s) and 5-minute windows. Results show large travel-time advantages versus rail for many links (e.g., BLQ-Il Ciocco: 46 min IAM vs. 237-294 min by train), while a few pairs remain competitive for rail (e.g., BLQ-Florence). Sensitivity to IAM speed and staying share quantifies how technology and adoption shape feasible coverage, timing feasibility, and hub accessibility, demonstrating the potential of IAM to strengthen BLQ's regional reach without altering airline schedules.

According to R. Bricard there exist three types of flexible octahedra. The octahedra of Type 3 are unsymmetric and admit two flat poses. With regard to higher-dimensional analogues of octahedra called cross-polytopes, A.A. Gaifullin presented in 2014 a complete classification of flexible types in n-dimensional Euclidean, hyperbolic and spherical spaces for n>3. The goal of this presentation is a synthetic approach to a particular family in the Euclidean n-space, the flexible cross-polytopes that admit two poses within hyperplanes. We provide a construction of their flat poses and prove several properties of these higher-dimensional analogues to Bricard's type-3 octahedra. According to Gaifullin, they are the simplest from the algebraic point of view.

This review paper provides a comprehensive overview of the Constrained Energy Minimizing Generalized Multiscale Finite Element Method (CEM-GMsFEM) for solving elliptic PDEs characterized by highly heterogeneous, high-contrast coefficients. We detail the construction of multiscale basis functions via spectral auxiliary spaces, combined with an oversampling strategy that enables localized computations and guarantees exponential error decay. Rigorous error estimates are outlined for reference to confirm the method's optimal convergence and robustness. Numerical simulations are provided to verify the exponential decay property of the multiscale basis functions. Additionally, we discuss and comment several up-to-date applications of CEM-GMsFEMs.

The Ekedahl sieve is a powerful tool for enumerating arithmetic objects, but traditional formulations relying on inductive steps often yield suboptimal bounds when applied to highly skew boxes. This limitation is particularly restrictive when introducing large modular conditions that compete with the tail-end variables of a binary form. In this paper, we develop a specialized variant of the Ekedahl sieve tailored to the singular locus of the discriminant polynomial. By exploiting the specific non-degeneracy properties of the discriminant outside its two most extreme coefficients, we bypass the standard inductive framework, reducing the sieve to a highly efficient two-step process in some cases, and a one step process in others. We establish robust generic tail-end estimates as well as squarefree, power-saving bounds that seamlessly incorporate external modular conditions. This optimization maximizes the permissible range of the sieving modulus, yielding improved error terms for the enumeration of bounded squarefree values of certain polynomials and providing the foundational geometric sieve estimates required for the weighted enumeration of number fields by discriminant.

We establish a Fundamental Theorem of Orders (FTO), which allows us to express any order (in a number field) uniquely as an intersection of \textit{irreducible orders}. Along this decomposition, the index (in the ring of integers) distributes multiplicatively, and the conductor factors into pairwise co-prime ideals. We use it to show a more general version of Furtwangler criterion about the structure of conductors of orders over $\Z$, as this answers a wide variations of such questions. In a future work, we will also give applications to weighted enumeration of number fields.

We prove two sliding-window inequalities for maximal convolutions. The first concerns the multiplicative maximal convolution. If $f$ and $g$ are nonnegative continuous functions on $[0,A]$ and $[0,B]$, respectively, define \[ h(x)=\max_{\substack{0\le u\le A\\0\le x-u\le B}} f(u)g(x-u),\qquad 0\le x\le A+B. \] Then there exists a window $[a,a+B]$ of length $B$ such that \[ \frac1B\int_a^{a+B}h(x)\,dx\ge \left(\frac1A\int_0^A f(x)\,dx\right) \left(\frac1B\int_0^B g(x)\,dx\right). \] The second concerns the additive maximal convolution. Let $f$ and $g$ be nonnegative continuous functions on $[0,C]$, and define \[ H(x)=\max_{\substack{0\le u\le C\\0\le x-u\le C}}\{f(u)+g(x-u)\},\qquad 0\le x\le 2C. \] Then, for every $p\ge1$, there exists a window $[a,a+C]$ of length $C$ such that \[ \left(\int_a^{a+C}H(x)^p\,dx\right)^{1/p} \ge \left(\int_0^C f(x)^p\,dx\right)^{1/p} + \left(\int_0^C g(x)^p\,dx\right)^{1/p}. \] We also record discrete analogues. The main point is that, in one-dimensional maximal-convolution settings, certain global Brunn--Minkowski or Prékopa--Leindler type phenomena admit natural sliding-window localizations.

We derive a generating function identity for the joint distribution of the numbers of peaks of a permutation and its inverse, via enriched $P$-partitions. The coefficients of the corresponding peak polynomial $W_n(s,t)$ satisfy a second-order recurrence. A martingale formulation of this recurrence yields a bivariate central limit theorem, showing that the two statistics are asymptotically independent. We also give an exact closed form for their covariance, which is of order $n^{-1}$.

We consider a finite-horizon, zero-sum stochastic differential game in which one player controls a path-dependent stochastic system, while the opponent is given the opportunity to terminate the game prematurely. We introduce a control randomization formulation, which allows us to establish that the upper and lower value functions coincide. Our approach also yields a representation of the common game value in terms of a nonlinear Snell envelope, where the underlying stopped process is given by the unique maximal solution of a backward stochastic differential equation (BSDE) with constrained jumps.

The fractional Laplacian has been widely used to model non-local diffusion on graphs, allowing interactions that extend beyond immediate neighbors. However, it suffers from a structural inconsistency as it breaks compatibility with the topology of the original network. To address this issue, a combination of the standard and fractional Laplacians aimed at restoring compatibility while retaining the spectral richness of the fractional operator was recently proposed. In this work, we provide a thorough analysis of the diffusion properties of the resulting regularized operator. We prove that it yields superdiffusive behavior independently of whether the underlying graph is weighted or unweighted -a property not generally satisfied by other nonlocal Laplacian variants. An efficient construction of the regularized operator which preserves the same asymptotic computational cost of the fractional Laplacian is also given. Our numerical experiments demonstrate the effectiveness and computational practicality of the regularized operator for modeling non-local diffusion on real-life networks.

This paper develops a mean-field theory for a simplified single-layer causal multi-head self-attention model trained by cross-entropy minimization. Each attention head is treated as a particle in parameter space, and the empirical law of the heads is used as the large-head state variable. In the infinite-head limit, the averaged attention logits define a risk functional on probability measures, whose first variation generates a nonlinear Wasserstein gradient-flow equation. Unlike classical mean-field analyses of shallow networks that often focus on square-loss regression, the present model contains the softmax residual from the cross-entropy objective and the query-key-value structure of masked self-attention. We prove a static finite-head approximation bound for the optimal risk, characterize global minimizers through a variational support condition, and establish a quantitative finite-time propagation-of-chaos estimate comparing finite-head stochastic gradient descent with the limiting PDE. We then study the long-time behavior of the PDE: energy dissipation, convergence to the stationary set under compactness, convergence to a single stationary measure under topological or Kurdyka--Łojasiewicz assumptions, and explicit convergence rates under gradient-domination conditions. Finally, we prove local exponential stability under a Wasserstein strong-monotonicity condition and give verifiable stability and instability criteria for Dirac stationary measures. The results provide a rigorous baseline mean-field framework for attention-head training and clarify the additional compactness, landscape, and curvature assumptions needed to pass from stationarity to convergence and stability.

We develop a distributionally robust formulation of principal component analysis that minimizes worst-case reconstruction risk over distributions lying within a Wasserstein neighborhood of the empirical measure. The Wasserstein neighborhood, viewed as an ambiguity set of distributions, is adaptively calibrated through a transport matrix $G$ to capture heterogeneous uncertainty across dimensions. The homogeneous case, in which G is a scalar multiple of the identity matrix, recovers classical PCA. Under a general transport matrix G, we derive a dual characterization of the associated minimax optimization problem and introduce a tractable surrogate objective function consisting of the square-root empirical reconstruction error plus a geometry-dependent residual exposure penalty. The exact and surrogate estimators are shown to be consistent for the population PCA subspace and asymptotically equivalent at the projector level. The transport geometry is allowed to be data adaptive, while the Wasserstein radius is calibrated via robust Wasserstein profile inference, yielding a data-driven radius of order $n^{-1/2}$. Comprehensive theoretical guarantees are established, including consistency and local Grassmannian asymptotics exhibiting an explicit Wasserstein-induced drift determined by the limiting transport geometry and calibration level. Numerical experiments and a real-data application demonstrate that the proposed method can substantially improve finite-sample out-of-sample performance under structured covariance shifts, moderate contamination, and certain same-distribution regimes.

Let $\mathcal{C}$ be any stable presentably symmetric monoidal $\infty$-category. In this paper, we characterize $\mathrm{Shv}(-,\mathcal{C})$ on locally compact Hausdorff spaces as the unique six functor formalism satisfying a list of very natural properties. As a consequence, we deduce that every continuous six functor formalism $D$ in the sense of Zhu is equivalent to $\mathrm{Shv}(-, D(\mathrm{pt}))$.

We study the Cauchy problem for inhomogeneous evolution equations with time-dependent, potentially degenerate, and unbounded coefficients. A key feature of our work is allowing the principal coefficients to undergo arbitrary blow-up at both the initial and terminal times.

In 2023, Greaves, Syatriadi, and Yatsyna found a set of $57$ equiangular lines in $\mathbb{R}^{18}$, breaking the previous record. In 2025, Lin, Munemasa, Taniguchi, and Yoshino constructed a large number of sets of $57$ equiangular lines in $\mathbb{R}^{18}$ as affine equiangular sets in an integral overlattice of $A_9^2 \oplus A_1$. In this paper, we construct further sets of $57$ equiangular lines in $\mathbb{R}^{18}$ from Latin squares of order $6$ and Pasch configurations, realized as affine equiangular sets in an integral overlattice of $A_5^3 \oplus A_1^4$. Unlike the previously known examples, these sets are not strongly maximal. Moreover, some of them have only five distinct Seidel eigenvalues, fewer than any previously known examples.