Papers

We prove a criterion for a finitely presented surjection of algebraic spaces to be descendable in a topological weave. We apply this to show that étale motivic spectra satisfy $h$-descent on noetherian finite-dimensional schemes with residue fields of uniformly bounded étale cohomological dimension. We also show that rational motivic cohomology satisfies arc-descent in weights $\le 1$, and we construct the “forgetting supports” isomorphism $f_! \simeq f_*$ for a proper DM-type morphism of Artin stacks, in rational motivic sheaves.

In this work, we investigate the formation of naked singularities for the $3+1$-dimensional Einstein-scalar field system without symmetry assumptions. We generalize the spherically symmetric and self-similar naked-singularity solution constructed by Christodoulou in [4] by prescribing non-spherically symmetric initial data along both incoming and outgoing initial null hypersurfaces. We then establish global existence for the resulting solutions and analyze the singular structure of the inner Cauchy horizon. Our construction is based on employing a notion of four-type differences and designing a system of scale-invariant weighted norms to control the corresponding geometry. We show that the constructed spacetimes retain a global naked-singularity structure, characterized by an incomplete future null infinity and a singular inner Cauchy horizon. Moreover, we derive detailed asymptotics near the inner Cauchy horizon and prove the desired $C^{1, \frac{\kappa}{1-\kappa}+}$ inextendibility of these solutions, where $\kappa\in (0,1/3)$ is the self-similar parameter. This indicates a connection between weak and strong cosmic censorship: for the class of non-spherically symmetric solutions constructed here, the failure of weak cosmic censorship in its strict formulation is accompanied by a quantitative inextendibility mechanism at the inner Cauchy horizon.

The quasilinear Keller-Segel system with flux limitation and indirect signal production u_t=\nabla\cdot\left(D(u)\nabla u\right) -\nabla\cdot\left(u(1+\left|\nabla v\right|^2)^\sigma\nabla v\right), &x\in\Omega, t>0, \\ 0=\Delta v-v+w,x\in\Omega, t>0, \\ w_t=-w+u,x\in\Omega, t>0, under homogeneous Neumann boundary conditions in a smooth bounded domain \Omega\subsetR^N is considered, where D(u)\simeq u^{m-1} as u\simeq\infty. We conclude that For N=1 and any \sigma\in\mathbb{R}, if m\geq0, the classical solution exists globally, and it is moreover bounded if m>0. However, if m<0 and \Omega is a ball, there exist radially symmetric initial data such that the classical solution exhibits finite-time blow-up. For any N\geq2 and m>1-\frac{1}{N}, if \sigma\leq \frac{mN+2-2N}{2N-2}, the classical solution is global. Furthermore, if \sigma<\frac{mN+2-2N}{2N-2}, the corresponding solution is uniformly bounded. For any N\geq2 and m<2-\frac{2}{N}, if \sigma>\max\left\{\frac{N}{2-2N},\frac{mN+2-2N}{2N-2}\right\} and \Omega is a ball, there exist radially symmetric initial data such that the classical solution blows up in finite time.

Let $G=(V,E)$ be a finite simple graph with Laplacian eigenvalues $\lambda_1(L(G))\ge\cdots\ge\lambda_{|V|}(L(G))$, and define \[ \eps_k(G)= \sum_{j=1}^{\min\{k,|V|\}}\lambda_j(L(G))-|E|. \] Let $\nu(G)$ be the matching number of $G$, and let $n(G)$ be the number of non-isolated vertices of $G$. Lew proved that \(\eps_k(G)\le k\nu(G)+\lfloor k/2\rfloor\), and conjectured that the additive term can be removed in the non-endpoint range. We prove this conjecture: \[ \eps_k(G)\le k\nu(G) \qquad (1\le k\le n(G)-2). \] We also characterize all equality cases. Up to isolated vertices, equality holds precisely for stars, for \(K_1\vee(K_k\cup\overline{K_{n-k-1}})\) with \(k\) odd, and for \(K_n-E(K_{1,t})\) with \(n\) odd, \(k=n-2\), and \(1\le t\le n-2\). We also analyze the endpoint range \(k\ge n(G)-1\), where \(\eps_k(G)=|E|\), and determine the specific cases where the inequality \(\varepsilon_k(G)\le k\nu(G)\) fails or holds with equality.

We study the one-dimensional Gross-Pitaevskii equation with a traveling delta potential and non-zero conditions at infinity. This model describes the effect of a moving impurity in a quantum fluid. Firstly, we show that the associated Cauchy problem is globally well-posed in the energy space. This requires the definition of a conserved energy, which involves the notion of renormalized momentum. Secondly, we study the existence and stability of stationary states in a co-moving reference frame. It is known that there exists an impurity-dependent critical velocity above which no stationary state exists. For velocities below the critical one, two different stationary states appear. We show the orbital stability of the one with higher minimal density.

Classical fractal geometry describes the metric and topological structure of attractors generated by contractive Iterated Function Systems (IFSs). Much less is known about their global qualitative dynamics once the contractive hypothesis is abandoned. In this article we focus on IFSs with "compact dynamics", namely those IFSs for which there exists a non-empty compact set invariant under the Hutchinson map that attracts every compact subset of the phase space. We call such a set the "global attractor" of the IFS. We introduce the "chain graph" of a general IFS as a directed graph encoding the qualitative dynamics features of the IFS. For an IFS with compact dynamics, the chain graph contains at least one node. Our main results are that the chain graph of an IFS with compact dynamics coincides with the chain graph of its restriction to its global attractor and that the chain graph of an IFS has at most as many nodes as the graph of its Hutchinson map.

We prove a coarse analogue of the classic fact that every graph can be decomposed along its cut-vertices into $2$-connected components. Precisely, we prove that for every graph $G$ and a positive integer $d$, $G$ admits a tree decomposition whose adhesion sets have weak diameter at most $3d+2$ so that no two vertices $u,v$ lying in the same bag can be separated by a set of weak diameter at most $d$ whose distance from $u$ and $v$ is more than $d$. By the Coarse Menger's Theorem for two paths, this condition admits also a dual formulation, phrased in terms of the existence of two paths that are far from each other and connect the vicinity of $u$ with the vicinity of $v$.

A set of vertices $S$ of a directed graph $G$ is geodetic if every vertex of $G$ lies on a shortest path from a vertex of $S$ to a vertex of $S$. A directed graph is geodetic if there is at most one shortest path from every vertex of $G$ to every vertex of $G$. We prove the NP-completeness of the following decision problem. Given a directed acyclic planar geodetic graph $G$ and an integer $k$, does $G$ have a geodetic set with at most $k$ vertices? This implies that the question of whether $G$ has a strong or a monitoring geodetic set with at most $k$ vertices is also NP-complete for directed acyclic planar geodetic graphs. Furthermore, we prove that the number of vertices in a minimum geodetic set and the number of vertices in a minimum edge geodetic set can be computed in linear time for directed acyclic series-parallel graphs.

This paper is concerned with the zero-mass $(p,q)$-Laplacian equation $$ -\Delta_p u-\Delta_q u = |u|^{r-2}u+\lambda |u|^{s-2}u \quad \text{in } \mathbb{R}^N, $$ where $1<q<p<N$. The exponents $r$ and $s$ may belong to either the subcritical or the supercritical range with respect to the critical Sobolev exponents $p^*$ and $q^*$. We establish existence and nonexistence results and show that the sign of the parameter $\lambda$ determines the solvability regimes of the equation. The existence proofs rely on variational methods, truncation arguments, and regularity theory, while the nonexistence results are derived from a suitable Pohozaev-type identity for the $(p,q)$-Laplacian operator.

In this paper we are interested in isoperimetric inequalities for plane n-gons in relation with a old conjecture proposed by P. Levy.

In this paper, we establish a proof for the first variation formula of the total mass within the $L_p$ framework. Our main result removes an extra restrictive determinant condition imposed in a theorem originally proved by Fang,Xing and Ye

Associated to any smooth map $F\colon M\to N$ equipped with a closed, nondegenerate relative $(n+1)$-form $\varpi$ – a \emph{relative $n$-plectic structure} – is an $L_\infty$-algebra of relative observables $L_{\infty}(F,\varpi)$, constructed by the author in earlier work. In this article we develop the corresponding theory of moment maps: for a Lie group $G$ acting compatibly on $M$ and $N$ and preserving $\varpi$, we define a relative homotopy moment map as an $L_\infty$-morphism from $\mathfrak{g}$ into $L_{\infty}(F,\varpi)$ lifting the infinitesimal action, thereby providing a full relative generalization of the homotopy moment maps of Callies, Frégier, Rogers and Zambon. We characterize such morphisms by explicit component equations, show that a relative homotopy moment map is equivalent to a homotopy moment map on the target $N$ together with a coherent trivialization of its pullback to $M$, and relate relative moment maps to a relative Cartan model computing relative equivariant de Rham cohomology. Every cocycle in the relative Cartan model extending $\varpi$ induces a relative homotopy moment map via explicit formulas, and we prove the one-step case in full detail. In the existence theory a new phenomenon appears: under a mild connectivity hypothesis the Lie-algebra-cohomology obstruction present in the absolute theory vanishes identically in the relative setting. Finally, we show that quasi-Hamiltonian $G$-spaces with group-valued moment map $\mu\colon M\to G$ fit into this framework: the pair $(\eta,\omega)$ built from the Cartan $3$-form is a relative $2$-plectic structure whose Alekseev–Malkin–Meinrenken axioms amount precisely to a canonical one-step cocycle in the relative Cartan model, and hence every quasi-Hamiltonian $G$-space carries a canonical relative homotopy moment map, which we compute explicitly.

Data-Enabled Predictive Control (DeePC) has recently emerged as a framework for controlling unknown systems from data. However, its performance relies on the relevance of the collected data, and as such, disturbances lead to inevitable errors. This paper addresses this problem by proposing an augmentation of DeePC using Model-Free Disturbance Observer with Online Modification (MFDOOM). The method corrects output predictions based on previous prediction errors using a dedicated continuously updated Hankel matrix. We compare our method, both theoretically and through simulation, to other recent algorithms designed for time-varying systems in the DeePC framework. It is shown that for disturbances that can be modeled as the output of an autonomous linear time-invariant system, this approach can reduce tracking error and online-update burden compared with existing online DeePC variants.

Our objective is to solve large systems of ordinary differential equations (ODEs) commonly used to model biological processes. These equations are typically nonlinear, complex, and high-dimensional. In computational biology, such ODEs are generally solved using numerical methods. In this work, we focus on explicit numerical methods because of their flexibility. However, their limited stability regions may result in high computational costs. To mitigate this issue, we investigate mixed precision algorithms designed to reduce computational effort by performing selected parts of the numerical method in lower arithmetic precision. We develop several mixed precision explicit methods and assess their performance on two large scale biological benchmark ODE models. Our theoretical analysis highlights the effectiveness of partially reducing arithmetic precision within explicit methods. Numerical experiments demonstrate that our mixed methods-implemented in both sequential and parallel versions using MPI-combining single (float) and double precision arithmetic can achieve up to twice the speed of a fully double precision implementation while preserving the same level of accuracy. Furthermore, the results indicate that decreasing the timestep improves the performance and robustness of our mixed methods, while the single precision method fails to converge.

Andrews and Bachraoui (Int. J. Number Theory (2026)) studied the two-color partition function $F(n)$ of a non-negative integer $n$ wherein odd parts may appear in two colors (red and blue) and even parts appear in one color (blue). For any non-negative integer $n,$ they also considered some restricted versions of $F(n)$: $F_0(n)$: the number of partitions of $n$ counted by $F(n)$ such that the number of odd parts in red color is even; $F_1(n)$: the number of partitions counted by $F(n)$ such that the number of odd parts in red color is odd; $H(n)$: the number of partitions of $n$ counted by $F(n)$ such that the parts of the same color do not repeat. The main purpose of this paper is to present the analytic proofs of the $q$-series identities connected with $F(n)$ and $H(n),$ which appeared as open problems in the original paper. We also prove some congruences of $F_0(n)$ and $F_1(n)$ modulo $2,$ $4,$ and $8$ by using $ q $-series.

This paper studies dynamic equilibrium tracking in online two-player zero-sum games with time-varying convex-concave payoff functions. Existing regret metrics for online saddle-point problems usually aggregate historical payoffs with uniform weights, and hence may fail to characterize the real-time tracking performance with respect to the current Nash equilibrium (NE). To address this issue, we introduce a zero-sum game regret function with a forgetting factor, which assigns exponentially decaying weights to past saddle gaps and emphasizes recent performance. This metric directly links regret minimization to the tracking of time-varying NEs. Within this framework, we investigate three online algorithms under different computational and information settings. For first-order feedback, we analyze projected gradient descent-ascent and design a projection-free online Frank-Wolfe method to reduce the computational cost of projections. For zeroth-order feedback, we develop a deterministic finite-difference method that only uses function-value queries. For all three algorithms, we establish forgetting-factor regret bounds that explicitly characterize the effects of NE variation, payoff variation, and gradient-estimation error. We further provide sufficient conditions under which the proposed regret converges to zero, thereby certifying asymptotic tracking of time-varying NEs. The numerical example validates the theoretical results and illustrates the tracking advantage of the proposed regret metric.

This paper introduces the weak augmented Zarankiewicz number $z_{wA}(m,n)$ and the weak limited augmented Zarankiewicz number $z_{wL}(m,n)$, which are combinatorial extensions of the classical Zarankiewicz number obtained by relaxing the original admissibility conditions for augmented bipartite graphs. We show that the resulting weak framework still guarantees irreducibility of the associated doubly simple biquadratic forms, with SOS rank equal to the total number of edges. This yields the inequality chain \[ \mathrm{BSR}(m,n) \geq z_{wA}(m,n) \geq z_{wL}(m,n) \geq z_L(m,n) \geq z(m,n). \] We provide three complementary constructions demonstrating the power of the weak framework. First, a $5\times 3$ construction using degenerate 2-edges yields $z_{wL}(5,3)\ge 10>9=z_L(5,3)$, giving $\mathrm{BSR}(5,3)\ge 10$. Second, a $15\times 6$ construction on the incidence graph of $K_6$ with 14 nondegenerate 2-edges gives $z_{wL}(15,6)\ge 44>43$, improving the previously known bound. Third, a critical $6\times 3$ construction with complementary 2-cycles gives $z_{wL}(6,3)\ge 12>11=z_L(6,3)$, yielding $\mathrm{BSR}(6,3)\ge 12$ and demonstrating that complementary 2-cycles are safe.

Transformers have demonstrated a remarkable ability to learn algorithmic reasoning, yet mechanistic analyses have mostly focused on globally invertible operations such as cyclic addition and group composition. In this work, we investigate how small transformers learn modular integer multiplication over composite moduli, a fundamentally non-invertible operation due to the presence of zero-divisors. We propose the monoid extension: a localized generalization of Group Composition via Representation (GCR) that suggests the learned computation does not rely on a single global representation space. Instead, the model partitions the input space into local hierarchical algebraic regions, where group-like structure survives and Fourier mechanisms can be applied. In transformers trained on square-free modular multiplication, we find that embeddings organize around these regions, attention exhibits class-sensitive routing and low-rank write directions, and local character features explain a large fraction of the model's output logits. Our results suggest that representation-theoretic mechanisms previously identified for group operations can extend beyond groups to more general structures.

Moon and Moser proved a sharp edge-extremal theorem for Hamilton cycles in balanced bipartite graphs with minimum degree at least $k$. Li and Ning obtained spectral analogues for Hamiltonicity in balanced bipartite graphs of order $2n$ and for traceability in nearly balanced bipartite graphs with part sizes $n$ and $n-1$, under the assumption $n\ge (k+1)^2$. We show that their sharp spectral thresholds remain valid in the linear ranges $n\ge 2k$ and $n\ge 2k+1$, respectively. More precisely, we determine the extremal values of the adjacency spectral radius and the signless Laplacian spectral radius for non-Hamiltonian balanced bipartite graphs with minimum degree $\delta(G)\ge k$, and for non-traceable nearly balanced bipartite graphs with $\delta(G)\ge k$. In each case, the extremal graph is unique up to isomorphism. Our proof is based on feasible graph parameters: parameters that increase under edge addition and are nondecreasing under Kelmans operations. This yields Moon–Moser type extremal theorems for a general class of parameters, from which the spectral results follow.

Let $R$ be an order in a number field, let $\overline{\mathrm{Cl}}(R)$ be its ideal class monoid, and let $\mathrm{Cl}(R)$ act on it by multiplication. The local-global product formula identifies the orbit set $\mathrm{Cl}(R)\backslash\overline{\mathrm{Cl}}(R)$ with a product of local orbit sets; in this sense, it is the genus set of fractional $R$-ideals. For a Gorenstein order $R$ in a cubic extension of number fields, we give a closed Euler product formula for the cardinality of this genus set. The local factors come from an explicit classification of local cubic overorders: for arbitrary local cubic orders, we parametrize all overorders, determine their inclusion relations, and identify the Gorenstein ones. As an application to Bhargava's parametrization of $2\times3\times3$ cubes, our formula gives the exact number of $\mathrm{Cl}(R)$-equivalence classes of integral $\mathrm{GL}_2(\mathbb Z)\times\mathrm{SL}_3(\mathbb Z)\times\mathrm{SL}_3(\mathbb Z)$-orbits whose associated cubic ring is the prescribed Gorenstein order $R$.

We used persistent homology to quantify the multiscale topological and geometric organization of porous media, including solid connectivity and the formation of loop-like and cavity-like structures across spatial scales. Through statistical analysis, we show that these topological and geometric features are closely associated with diffusion-driven release behavior in porous media. In particular, even within each target-porosity level, samples with richer topological features tend to exhibit long-tailed release, indicating that release behavior depends not only on the amount of pore space but also on the multiscale organization of the solid phase. We further show that persistent homology-based features can classify release-curve regimes using a simple classification model. Notably, feature extraction is substantially faster than finite element diffusion simulations. Together, these results suggest that persistent homology provides a lightweight, interpretable, and geometry-based descriptor for screening diffusive release behavior in porous media.

We construct a simply connected solvable Lie group $G$ admitting lattices and a left-invariant complex structure $J$ such that $(G,J)$ is not Stein. This provides a counterexample to Hasegawa's conjecture on the Stein property of simply connected unimodular solvable Lie groups with left-invariant complex structures.

We develop a geometric reduction mechanism generated by systems of particular integrals, namely, families of functions whose time derivatives close linearly on the family. Their common zero set is dynamically invariant. In the Hamiltonian case, under a weak involution condition, the restricted dynamics is presymplectic, and its characteristic quotient carries a reduced Hamiltonian flow. This yields a direct bridge between particular integrals, presymplectic reduction, and lower-dimensional Hamiltonian dynamics, and leads to a Liouville-type notion of particular integrability. We illustrate the framework through mechanical examples and lift constructions, including variants of the Eisenhart lift.

We introduce numerical invariants called hyperbolic indices, which measure the hyperbolicity of compact Kähler manifolds using directed positive closed currents. We prove that if a manifold $X$ has positive hyperbolic indices, then $X$ is Kobayashi hyperbolic; and if $X$ satisfies Demailly's condition of negative jet curvature, then it has positive hyperbolic indices. In particular, by combining the method of jet differentials and density currents, we can prove that for a general hypersurface $X_d$ of degree $d$ in $\mathbb{P}^{n+1}$, the hyperbolic indices of $X_d$ grows to $\infty$ with at least linear growth in $d$. Finally, we discuss an analytic approach to the Kobayashi conjecture.

We prove that for a very general abelian variety of dimension $\geq 4$, a divisor $D\in {\rm CH}^1(A)$ that satisfies $D^2=0$ in ${\rm CH}^2(A)$ is of torsion. The same result is also established for a very general Jacobian in genus $4$. We use then the second statement in order to prove a conjecture of Pirola, which states that any rational section of the Kummer fibration $K=J/\pm {\rm Id}\rightarrow \mathcal{M}_4$, where $J\rightarrow \mathcal{M}_4 $ is the Jacobian fibration, must be a multiple of the Griffiths-Pirola section given by the difference of the two trigonal divisors.

In previous joint work, we proposed a new method to study local orbital integrals for $\mathrm{GL}_n$ (where $n=3$ or in the Bass case), by employing a smoothening method of a certain scheme defined over a henselian ring. In this paper, we geometrize this local smoothening method within the framework of the global Hitchin fibration for $\mathrm{GL}_n$ in the Bass case. Consequently, we provide a closed formula for the $\ell$-adic cohomology of the compactified Jacobian of a spectral curve over a finite field with double singularities whose local rings are integral domains, provided that the normalization of a spectral curve is isomorphic to $\mathbb{P}^1$.

We obtain a coarse version of the block-cutvertex tree-decomposition of a connected graph.

Symmetric Poisson CGL extensions form a particular class of polynomial Poisson algebras that are shown by K. Goodearl and M. Yakimov to admit compatible cluster structures. In this paper, we give explicit formulas for a family of mutation matrices in the Goodearl-Yakimov theory via matrix products as well as by entry-wise description.

In this paper, we derive a new model to simulate the incompressible resistive magnetohydrodynamic (MHD) free surface flow. A thermodynamically consistent diffuse interface method is adopted to characterize the moving interface in the modeling process. The formal convergence of the proposed MHD free surface flow model to the sharp interface model is established via a matched asymptotic argument, and the model can be solved without the need for sophisticated free surface capturing schemes. We design a fully decoupled linear finite element scheme that preserves the divergence-free constraint of the magnetic field at a discrete level. The reliability and robustness of the proposed model and algorithm are validated through numerical investigations of the magnetic damping effect on bubble dynamics. In particular, we provide a quantitative numerical comparison of the present results with those obtained from an inductionless MHD model and a sharp interface arbitrary Lagrangian–Eulerian model.

We consider the linear stability of three-dimensional sessile drops with a free contact line. The equilibrium surface is an axisymmetric solution of the Young–Laplace equation with gravity, fixed volume, and prescribed contact angle. We derive the constrained second variation of the gravity–capillary energy and formulate the associated Jacobi problem. Although variational stability gives nonnegativity, the second variation is necessarily degenerate because horizontal translations preserve the energy. Our main result identifies this degeneracy completely. Under the pressure–volume nondegeneracy condition $(dV/d\lambda\neq 0)$, we prove that the kernel of the constrained Jacobi operator is exactly the two-dimensional space generated by horizontal translations. The proof combines the geometric structure of the Jacobi operator with a Fourier-mode analysis: the axisymmetric mode is ruled out by the pressure–volume condition, the first mode gives translations, and all higher modes are excluded by comparison. This provides the precise linear nondegeneracy underlying stability of droplet dynamics modulo translations.