Papers

We characterize all algebraic numbers $\alpha$ of degree $8$ for which there exist four distinct algebraic conjugates $\alpha_1$, $\alpha_2$, $\alpha_3$, $\alpha_4$ of $\alpha$ satisfying the linear relation $\alpha_{1}=\alpha_{2}+\alpha_{3}+\alpha_{4}$. Analogous characterization is obtained for the linear relation $\alpha_{1}+\alpha_{2}+\alpha_{3}+\alpha_{4}=0$. In particular, when an algebraic number $\alpha$ of degree $8$ has a non-even minimal polynomial and possesses exactly six distinct linear relations of the form $\alpha_{i_1}+\alpha_{i_2}+\alpha_{i_3}+\alpha_{i_4}=0$, we prove that $\alpha$ is a sum of a quadratic and a quartic algebraic number.

Stability of robust heteroclinic cycles and networks is typically studied by constructing return maps and analysing their associated transition matrices. This analysis can be simplified with the network's projected map, derived by projecting the linear action of the transition matrix onto a simplex. This projection produces a piecewise-smooth map, which is one-dimensional for heteroclinic networks in $\mathbb{R}^{4}$. We consider two such networks, the Kirk–Silber network and the $\Delta$-clique network. For the Kirk–Silber network, the projected map is discontinuous on its switching manifold, while it is continuous for the $\Delta$-clique network. In this paper, we address the dynamical phenomena that produce a discontinuity in the case of the Kirk–Silber network, and explain the value of the projected map at the switching manifold. We construct a completed return map near both networks, which captures the behaviour of all trajectories that begin near the network but may move away from it temporarily. We show that the discontinuity in the projected map of the Kirk–Silber network emerges due to two phenomena: first, there exists a discontinuity in a component of the completed return map in the limit as trajectories approach the Kirk–Silber network, as a result of the presence of a separatrix near the network, and, second, the procedure that defines the projected map. For the $\Delta$-clique network, there is no such separatrix, and so no such discontinuity emerges. This analysis is a necessary step towards understanding the more complicated dynamics observed near heteroclinic networks of five or more equilibria.

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 $\Gamma$ 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 $\Gamma$-covers of $X$, étale over $X\setminus S_X$, dominating $Y$; and also the minimum of the genera of $\Gamma$-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} -\Delta_p u = u^{\alpha}v^{\beta} \, \ \ \ \ \ \text{in}\,\ \ \R^N, \\ -\Delta_p v = u^{\beta}v^{\alpha} \,\ \ \ \ \ \text{in}\,\ \ \R^N, \end{cases}\end{equation*} where $1<p<N$, $N\geq2$, $0\leq \alpha \leq \beta,$ 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 $\alpha$ is called fence-decreasing if $x\alpha \preceq x$ for all $x$ in the domain of $\alpha$, and fence-preserving if $x\prec y$ implies $x\alpha \preceq y\alpha$ for all $x$ and $y$ in the domain of $\alpha$. 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 $(\Gamma(l), \Gamma_{00}(p^\infty))$-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.

Let $G$ be a finite group and $p$ a prime. It is interesting to determine when two ordinary irreducible representations of $G$ have the same $p$-modular reduction; this is the same as saying that the corresponding rows of the decomposition matrix are equal, or that the characters of the two representations agree on $p$-regular conjugacy classes. In fact we consider the more general problem of asking when two rows of the decomposition matrix are proportional. In the case where $G$ is a double cover of the alternating or symmetric group, this problem has been solved except when $p=3$. Here we resolve the missing case for spin characters (i.e. characters which are not lifted from the covered group), which completely solves the problem for the double cover of the symmetric group. There are surprising parallels to our solution to the corresponding problem for $p=2$.

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

We classify when a spin and a non-spin irreducible character of the alternating group have proportional 2-modular reductions. Equivalently, we classify when such a pair of characters are proportional on elements of odd order.

We prove an isoperimetric inequality in terms of the mean shadow area.

We discuss a problem posed by A.~Agrachev asking how many times a usual circle in $\mathbb R^n$ should be traversed to admit a deformation by curves with nowhere degenerating Frenet frame. It turns out that the answer depends on a specific topology which we consider. For the literal $C^n$ curve topology, the least number of turns of a plane circle admitting arbitrarily small nondegenerate perturbations is \[ k(2)=1,\qquad k(3)=2,\qquad k(n)=1\quad(n\ge4). \] This jet-level problem is different from the original Frenet-control problem by Agrachev. We show that in the literal interpretation of Agrachev's problem one has a simple spherical Fenchel obstruction in all dimensions $n\ge4$. To retain a nontrivial turn-counting problem, we introduce decorated turn data. In $\R^4$ the datum is a pair $(p,q)$ recording tangent-plane and normal-plane turns; we prove that every nonresonant pair $(p,q)$ with $p,q>0$, $p\ne q$, is accessible by small positive constant Frenet controls. In even dimension $2r$ the analogous datum is a vector $(p_1,\ldots,p_r)$, and every vector with pairwise distinct positive entries is accessible by constant controls. Odd dimensions require genuinely time-dependent openings since constant controls cannot close the base curve.

Generating high-quality meshes for arbitrary geometries remains a fundamental bottleneck in computational engineering, often demanding heuristic tuning and semi-manual workflows. In this paper, we introduce Dmsh, a first fully automated reinforcement learning pipeline that unifies geometric decomposition and quadrilateral mesh generation within a single learning-based framework. Dmsh decomposes the problem through three coordinated agents handling topology simplification, geometric regularization, and mesh generation. The meshing process is formulated as a Markov Decision Process and solved using a parametric Soft Actor-Critic architecture with decoupled critics, enabling efficient exploration of a hybrid discrete-continuous action space. A curriculum learning strategy ensures scalability from simple domains to highly complex geometries, suppressing seed variance. By design, the recursive decomposition enables parallel meshing of subregions, yielding globally conforming all-quadrilateral meshes without post hoc correction. Across a wide range of benchmarks, Dmsh consistently outperforms existing methods in automation, robustness, and mesh quality, establishing a new paradigm for learning-based mesh generation.

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 a priori Hölder continuous weak solutions to the parabolic double phase equation \[ \partial_t u - \operatorname{div} \left(|Du|^{p-2}Du + a(z)|Du|^{q-2}Du\right) = 0 \qquad \text{in } \Omega_T. \] We work under a relaxed gap condition on the exponents $p$ and $q$. The coefficient $a$ is assumed to belong to the class $\mathcal{Z}^{\kappa}(\Omega_T)$ for some $\kappa \in (0,\infty)$. 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 $u \in C^{0,\gamma,\gamma/q}_{\mathrm{loc}}(\Omega_T)$ with $\gamma \in [0,1)$, our main result holds under the gap bound \begin{equation}\tag{G}\label{eq:G} 2 \le p \le q, \,\,\,\,\,\,\,\,\, (q-p)(q-2\gamma)\leq q\kappa \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 $\phi\colon X \to X_0$ over a number field $k$, and an integer $d$, when is the set \[\{p \in X_0(\overline{k})|\ \mathrm{deg}\ p = d, \text{ and the fiber } \phi^{-1}(p) \text{ is reducible over } k(p)\}\] 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 $\phi:X \to \mathbb{P}^1$ 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.

We present NOVA, an autonomous symbolic regression framework that identifies interpretable car-following and lane-change structures from raw trajectory data with minimal behavioral priors. Applied to 4,765,788 active driving observations from the NGSIM I-80 and US-101 datasets, NOVA's deterministic Rust-powered search engine evaluates over 10,000 candidate algebraic structures and identifies a compact two-term acceleration model under a forward-shifted rolling-mean prediction target. Evaluated under two complementary preprocessing pipelines, NOVA achieves $RMSE = 1.376 m/s^2$ ($R^2 = 15.57\%$) on the intent-forecasting benchmark, outperforming the best recalibrated symbolic-regression baseline (SR-LLM, PNAS~2025) by 0.135 m/s$^2$ in RMSE under an identical evaluation protocol. Across eight independent experiments, a single dominant nonlinear term emerges as a robust backbone of human car-following; a residual-guided extension further links the selected structure to an established psychophysical theory of collision avoidance. The discovered feature operators transfer zero-shot between freeway sites with under 3 pp $R^2$ loss. Extended to lane-change modelling within a multinomial logit framework, NOVA achieves 67.4% balanced accuracy under strict vehicle-ID holdout on 502 unseen drivers, surpassing existing lane-changing baselines by +29.8 percentage points on a three-class problem.

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 propose a new optimization method, the Nyström-enhanced relaxed scalar auxiliary variable method (N-RSAV), which incorporates curvature information into the RSAV framework to accelerate convergence while preserving an unconditional modified energy dissipation law. Existing RSAV-based methods rely solely on first-order information and often suffer from slow convergence, particularly for ill-conditioned problems such as those arising in physics-informed neural networks (PINNs). To address this limitation, we design the linear operator in the RSAV scheme using approximate Hessian information obtained from a randomized low-rank Nyström approximation. To preserve the dissipation structure, we enforce positive semidefiniteness through eigenvalue truncation. Furthermore, we introduce an adaptive strategy that reuses the approximate Hessian based on the deviation between the original and modified energies, significantly reducing computational cost. We also provide a convergence analysis of the RSAV scheme with a general positive semidefinite operator under the Polyak-Lojasiewicz (PL) condition and establish corresponding convergence guarantees for N-RSAV under the PL condition and an additional convexity assumption. Numerical experiments on ill-conditioned problems with effectively low-rank structure, including convex quadratic problems and training of PINNs, demonstrate that the proposed methods achieve substantially faster convergence than conventional RSAV-based approaches.

If the denominator in a tamed stochastic gradient Langevin update uses the current stochastic-gradient draw, the conditional mean can be biased even when the stochastic-gradient oracle is unbiased. A state-dependent denominator fixed before that draw removes this coupling. We build practical deterministic denominators from a short pilot run. A log-scale proxy is fitted to the growth score $G_\star(x)=\|b(x)\|/(1+\|x\|)$, and empirical pilot quantiles set the activation thresholds of a local proxy-quantile envelope. The reported proxy-quantile experiments use this local denominator. Separately, we describe a final denominator with a norm-polynomial tail-floor correction that can be used when one wants to certify the global effective-linearity input required by the companion deterministic-envelope Lyapunov theory. We show how proxy and threshold errors enter denominator errors and the resulting stationary observable errors. In the reported experiments, the local proxy-quantile denominator improves over random-denominator tamed SGLD at comparable production cost. It also gives observable behavior close to the $G_\star$-envelope benchmark, without the full-gradient growth-score evaluations required by that benchmark.

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 paper addresses the backstepping boundary stabilization of coupled multidimensional first-order hyperbolic systems. We consider systems whose transport velocity fields are collinear, meaning that each velocity field is a scalar multiple of a common base velocity field. Building upon a recent framework developed for scalar multidimensional first-order hyperbolic equations, we introduce a change of variables, based on characteristic curves defined entirely in the spatial domain, that converts the original multidimensional system into a continuum of coupled one-dimensional first-order hyperbolic systems. By designing a backstepping controller for each system in the continuum representation, and assuming that the transit times of the characteristic curves are uniformly bounded, we achieve finite-time stabilization of the multidimensional system.

In the implicit version of a propositional proof system Q, we work with Q-proofs that are not written down directly, but are succinctly encoded by circuits. Thus implicit Q-proofs are potentially exponentially shorter than usual Q-proofs. We study narrow implicit proofs, a restricted version of this notion, in which lines in the encoded proof can only have polynomial size. We use a cut-elimination construction to show that G_{i+1} is equivalent to narrow implicit G_i, for i >= 1, where G_i is the extension of Frege allowing reasoning with Sigma^q_i quantified propositional formulas. We show that G_1 is equivalent to implicit resolution.

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.