Papers

This paper is devoted to introduce new probabilistic tree-based models complementing already known Matsumoto-Yor and Hamza-Vallois tree models with two tree beta-type models.

Let $K$ be a number field. We give new practical algorithms that determine whether $K$ is $p$-rational. For real cyclotomic fields $K=\mathbb{Q}(\zeta_{n})^{+}$, our method avoids computing either the class number or the full unit group. We also use the notion of $p$-rationality to explain the practical efficiency of an algorithm for determining whether a subgroup of the unit group $\mathcal{O}_{K}^{\times}$ is $p$-saturated. This in turn yields a new algorithm for the unconditional verification of unit groups of number fields that substantially outperforms existing unconditional algorithms in practice. Finally, by combining our algorithms for determining $p$-rationality with work of Greenberg, we construct certain Galois representations $\rho : \mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) \rightarrow \mathrm{GL}_{n}(\mathbb{Z}_{p})$ with open image and further prescribed properties.

We propose a hybridized staggered discontinuous Galerkin–mixed finite element method for the strain gradient elasticity model, a fourth-order singularly perturbed problem governed by a material length scale parameter $\iota$ and the Lamé constants $\lambda$ and $\mu$. Introducing the total stress together with the scaled Cauchy and hyper stresses as auxiliary unknowns, we recast the model as a first-order system and discretize the displacement and the total stress by staggered discontinuous Galerkin spaces, the two scaled stresses by Raviart–Thomas pairs, with symmetry imposed strongly on all three stresses. The higher-order Dirichlet boundary condition enters the variational formulation naturally, so that the scheme avoids the numerical boundary layer that limits displacement-based methods. By establishing an inf-sup condition on the symmetric subspace and a discrete Korn inequality, we prove algebraic stability and optimal convergence, both uniform in $\iota$ and $\lambda$. We further develop a hybridized scheme in which local static condensation leaves only two multipliers on the mesh skeleton as globally coupled unknowns. Numerical experiments confirm the predicted convergence rates and the parameter robustness.

We prove a local stability criterion for the no focal points property on compact Riemannian surfaces. More precisely, if a smooth metric $g$ has non-positive Gaussian curvature and its zero-curvature set is finite, then $g$ belongs to the $C^2$-interior of the set of metrics without focal points. Thus, every sufficiently small $C^2$-perturbation of $g$ still has no focal points, although arbitrarily small perturbations may create regions of positive Gaussian curvature. By Ruggiero's characterization of the $C^2$-interior of the set of metrics without conjugate points, all metrics in the resulting neighborhood are Anosov. We also show that, starting from any hyperbolic metric on a compact surface, one can prescribe an arbitrary finite set as the zero set of the Gaussian curvature of a smooth conformal non-positively curved metric. These metrics can be approximated smoothly by negatively curved metrics, so they lie on the boundary of the negatively curved regime while remaining interior points of the no-focal-points regime. The proof of the stability theorem combines uniform local convexity, Gulliver's bound on the length of geodesic segments contained in small balls, and an inductive argument on the Riccati equation that controls successive passages through the regions where positive curvature may appear.

In this paper, we establish the bilinear Strichartz estimates for the following linear Schrödinger equation with repulsive inverse-square potential $i\partial_tu+\La u=0$ where $\La=-\Delta+a|x|^{-2}$ with $a\geq0$ for non-radial data. Our proof relies on the physical space method (interaction Morawetz estimate). Such type of method was first introduced in Planchon-Vega [Ann. Sci. Éc. Norm. Supér. (4) 42 (2009), 261–290] in flat case, and Planchon-Tzvetkov-Visciglia [Rev. Mat. Iberoam, 39(2023), no. 4, 1405-1436] for harmonic oscillator. Our bilinear estimate extends results obtained in Camps [Ann. Inst. H. Poincaré Anal. Non Linéaire 39 (2022), 1–58] and Pusateri-Soffer [Mem. Amer. Math. Soc. 299 (2024), v+107 pp] to the critical case. As an application, we establish the global well-posedness for cubic NLS with inverse-square potential \begin{equation*} iu_{t} +\La u = -|u|^2u,\ \ (t,x)\in \mathbb{R}\times\mathbb{R}^{3}. \end{equation*} for radial initial data $u_0\in H_a^s$ with $s>\frac{11}{13}$ and $a>0$ where $H_a^s$ is the Sobolev space adapted to the Schrödinger operator with inverse-square potential. The proof relies on Bourgain's high-low frequency decomposition argument and our bilinear estimates. This is the first result on the low regularity well-posedness for NLS with scaling-critical potential.

Motivated by the Morse–Seelinger conjecture on type C $k$-Schur functions, we develop a permutation model for type C strong marked tableaux. Using type C transpositions, we give an explicit correspondence with the symmetric-core model that preserves the marking data. We also describe the induced affine Weyl group action on bounded partitions.

The discrete Calderón problem aims at recovering the conductivity on the edges of a graph from boundary measurements, encoded by the discrete Dirichlet-to-Neumann (DtN) map. The problem has been intensively studied in the linear case on square lattices since the seminal works of Curtis and Morrow. In this work, we investigate a nonlinear analogue of the discrete Calderón problem for a semilinear second-order elliptic equation on square lattices. We establish three uniqueness results for the conductivity recovery. First, we show that conductivity-dependent corner excitations allow a layer-by-layer reconstruction of the conductivity. Second, we study the linearization of the nonlinear DtN map at an arbitrary background boundary datum and prove that the conductivity and the background potential are uniquely determined by one pair of nonlinear Cauchy data and the linearized DtN map. Third, we show that the linearized data can be replaced by finitely many nonlinear measurements, which uniquely determine the conductivity.

We study the horizontal projection of the upper-half-plane Hardy–Szegő zero process after independent height-dependent thinning, with height retained as an unobserved mark. We identify the reduced Palm law of the projected process by disintegrating over the missing height coordinate, describe the conditional law of the two heights associated with a pair of projected points and its near- and far-separation limits, and obtain the small-spacing asymptotic for the right nearest neighbour. We then study recovery of the height profile from the projected second-order structure. The resulting covariance transform uniquely determines every compactly supported finite positive height measure and, for separated finite unions of intervals, yields uniform Lipschitz recovery of all endpoints, even when the number of intervals is unknown but bounded.

For an integer $n > 1$, let $P^+(n)$ be the largest prime factor of $n$, and let $P_y^+(n)$ denote the largest prime factor of $n$ not exceeding $y$. One of Erdős and Turán's conjectures asserts that the asymptotic density of integers $n$ satisfying $P^+(n)<P^+(n+1)$ is $1/2$. Furthermore, Wang conjectures that for $x\rightarrow\infty$, one has $\#\{p\leq x:P^+(p-1)<P^+(p+1)\}\sim \pi(x)/2$. In this paper, we show the following results. For any $3\leq y\ll x^{o(1)}$, for $x\rightarrow\infty$, we have \begin{align*} &\#\{n\leq x:P^+_y(n)<P^+_y(n+1)\}\sim \frac{1}{2}x, \&\#\{p\leq x:P^+_y(p-1)<P^+_y(p+1)\}\sim \frac{1}{2}\pi(x). \end{align*} For any $0<\alpha<17/32$, let $y=x^\alpha$. Then there exists $h(\alpha)>0$ such that \begin{align*} \#\{p\leq x:P_y^+(p-1)<P_y^+(p+1)\}\geq(h(\alpha)+o(1))\pi(x). \end{align*} In particular, the function $h$ satisfies $\lim_{\alpha\rightarrow 0^+}h(\alpha)=1/2$. Similar result also holds for $\#\{n\leq x:P_y^+(n)<P_y^+(n+1)\}$. These improve Rivat's result (2001) and Wang's result (2019).

We prove that $B_kA_6$ is stably rational over every field $k$ in which $2$ and $-3$ are nonzero squares. Combining this with a theorem of Plans, we deduce that $B_kA_7$ is stably rational over every field $k$ of characteristic zero in which $2$ and $-3$ are squares.

Let $(M,g)$ be a compact oriented Einstein four-manifold with Einstein constant $E$ and let $\widehat{R}^+$ denote the action of the Riemann curvature tensor on self-dual two-forms. We show that if $\widehat{R}^+ < 0$, then $g$ is strictly linearly stable for the Einstein-Hilbert functional, thus giving a chiral criterion for stability. Our result is stronger than the one by Fine-Krasnov-Singer, who conclude local rigidity from $\widehat{R}^+ < 0$ by proving stability for a different action functional. Our proof proceeds by showing that $(M,g)$ admits a natural spin$^h$ structure carrying a non-zero parallel spin$^h$-spinor. We then apply a lower bound on the Lichnerowicz Laplacian on traceless symmetric two-tensors in the presence of such a spinor.

We prove lower bounds on the relaxation time $1/\gamma_{N,\beta}$ of Langevin dynamics for mixed spherical spin glasses with even mixture $\xi(x)=\sum_{p\ge4}\gamma_p^2x^p$, under a one-step-replica-symmetry-breaking-type standing assumption of strict threshold separation $E_0(\xi)>E_1(\xi)>E_2(\xi)$; the pure spherical $p$-spin glass with even $p\ge4$, for which the assumption is a theorem, is recovered as a corollary. The main result is an aggregate Eyring-Kramers bound whose exponent sums conductances over the exponentially many index-one saddles above the lowest saddle level $-NE_1(\xi)$, combining the saddle complexity $\Theta_{1,\xi}$ with a half-determinant Hessian statistic – an entropic contribution that the classical single-saddle picture misses. The single new random-matrix ingredient of the mixed model is that the conditional Hessian at a critical point is a randomly shifted GOE matrix: the radial derivative is no longer determined by the energy (Euler's identity degenerates exactly in the pure case), and every landscape rate becomes a one-dimensional supremum over the scalar shift with a Gaussian penalty. At low temperature the aggregate exponent exceeds the unconditional free-energy bound by $\frac12\log(\beta e)-\Xi^{\mathrm{EK}}_{1,\xi}(-E_1(\xi))-C_{\xi,b}\beta^{-1/2}$ with $-\Xi^{\mathrm{EK}}_{1,\xi}(-E_1(\xi))\ge\frac14\log\xi''(1)+\frac14>0$, so the refinement is a strict improvement for all sufficiently large fixed $\beta$; the onset temperature and constants $b_*(\xi)$ and $C_{\xi,b}$ produced by the proof are not numerically explicit. Two companion results – a sequential Arrhenius bound with the explicit constant $E_0(\xi)-E_1(\xi)$ and a fixed-temperature free-energy bound – come with complete, self-contained proofs. All bounds are one-sided; identifying the mechanism the dynamics actually realizes would require a matching upper bound, which remains open.

We present an example that shows that the double Hilbert transform $\H$ can have the characteristic function of an open set of finite measure in $\mathbb{R} \times \mathbb{R}$ as an approximate eigenvector for the eigenvalue $1$. The same holds for the dyadic model of $\H$, the double dyadic shift. We further construct non-diagonal sets whose indicators are fixed by $\H$ – strips bent along the Boole curve $x=y-1/y$ and, more generally, along level curves of real Herglotz functions with purely singular measures – and study the class of invariant sets which they generate. At positive density we construct an example built on three families of parallel lines, related to the carry cocycle $\{u\}+\{v\}-\{u+v\}$, and the fans – alternate sectors of an odd number of concurrent lines – which are the simplest members of the invariant cones, a class that we describe completely. We then study the classification of the invariant sets whose boundary lies on countably many lines. Its combinatorial half is settled completely: a locally finite arrangement of lines in which no point lies on exactly two lines is a family of parallel lines, a pencil of concurrent lines, or an affine image of the triangular lattice. The classification is further connected, via projective duality, with configurations of Sylvester–Gallai type, and, via a change of variables, with two-valued superpositions of traveling waves.

Motivated by the effective form of Shafarevich's conjecture, we prove that the number of classes of singular fibers of genus $g$ has the exact asymptotic order $g^{\Theta(g)}$.

In this paper, we introduce some type vector fields with respect to a semi-symmetric non-metric (SSNM) connection. We investigate several geometric properties of a K-contact manifold equipped with an SSNM connection and provide a concrete example to justify the relation between the scalar curvature of the SSNM connection and Levi-Civita connection that we have obtained in this paper. Furthermore, we have found the nature of Riemann solitons, conformal Ricci solitons and conformal $\eta$-Ricci-Yamabe solitons on K-contact manifolds admitting a SSNM connection.\\ Finally, we determine the necessary and sufficient conditions for such a manifold to be $\Tilde{\tau}$-semi-symmetric, quasi-conformal-semi-symmetric and pseudo-projective-semi-symmetric.

The palindromic length $PL(u)$ of a nonempty finite word $u$ is the least number of nonempty palindromes whose concatenation is $u$. For an infinite word $w$, let $P_w(n)$ be the number of nonempty palindromic prefixes of $w[1,n]$, and let $T_w(n)$ consist of those prefixes $w[1,j]$, $1\leq j\leq n$, whose palindromic length equals the maximum attained among the nonempty prefixes of $w[1,j]$. We prove the finite inequality $|T_w(n)|\geq P_w(n)$ for every $n\geq1$. In particular, if $w$ has infinitely many palindromic prefixes, then \[ \liminf_{n\to\infty}\frac{|T_w(n)|}{P_w(n)}\geq1. \] The coefficient $1$ is optimal, already for a nonconstant periodic word. The proof uses chains of occurrences connected by palindromic factors, together with trimming and reflection arguments that preserve lower bounds on palindromic length.

Given $\rho>0$, we consider the problem \[ \text{find $(\lambda,u) \in \mathbb{R} \times H_0^1(B)$ such that } \begin{cases} -\Delta u+\lambda u = |u|^{p-1}u & \text{in } B \\ \int_B u^2\,dx = \rho, \end{cases} \] where $B$ is a ball in $\mathbb{R}^N$, $N \ge 1$, and $1<p<2^*-1$. Without any further restriction on $N$, $\rho$ and $p$, we prove the existence of infinitely many radial solutions, and, in dimension $N \ge 4$, of at least one non-radial solution.

We prove the conjecture of Xu, Li, Hua, and Pan that the resistance diameter does not increase under the line-graph operation. For every finite connected simple graph $G$ with at least one edge, we establish $D_r(L(G))\le D_r(G)$, with equality if and only if $G$ is a cycle or $K_4$. The proof combines an exact electrical representation of $L(G)$ by stars with a sharp budget inequality for the branch core. This inequality controls the combined error terms arising from the comparison of degree-two paths and is central to both the diameter bound and the equality analysis.

We establish the dynamical Mordell–Lang conjecture over \(\mathbb C\) for endomorphisms that are relatively étale over arbitrary endomorphisms of finite products of smooth projective curves. In particular, we establish the conjecture for arbitrary endomorphisms of product of smooth projective curves and relatively étale skew products. The arithmetic input is our unconditional theorem for split endomorphisms over \(\overline{\mathbb Q}\), proved by reducing to a simultaneous Hasse principle for the local periods of critical points and analyzing local inertia in joint arboreal towers. We combine this arithmetic input with relative étaleness, simultaneous algebraic specialization preserving wandering base coordinates, prescribed-reduction \(p\)-adic embeddings, and \(p\)-adic interpolation to obtain the result in the complex case.

For the Iwahori-Hecke algebras $\mathcal H_q(\mathfrak S_n)$ of the symmetric group $\mathfrak S_n$ at a primitive $e$-th root of unity, James and Mathas proved a theorem which relates $v$-decomposition numbers $d_{\lambda\mu}^{e}(v)$ for different values of $e$, by adding “empty runners” to the abacus display for the labelling partitions $\lambda,\mu$. Fayers proved a similar theorem, which involves adding “full” runners to these abacus displays. In this paper we use Uglov's map from the set of $r$-abaci for $r$-partitions to the set of $e$-abaci for partitions to extend these theorems to the cyclotomic Hecke algebras of type $G(r,1,n)$.

We study the problem of maximizing a linear function over the integer points of an origin-centered $\ell_p$ ball, which we call \BallIPp{p}. For every fixed integer $p\ge2$, we prove that the decision problem over an $\ell_p$-ball is NP-complete. We then focus on the Euclidean case and study how the difficulty of the problem depends on the numerical parameters of the instance. We give two complementary pseudo-polynomial exact algorithms. The first specializes a known radius-budget dynamic program; the same nonlinear-knapsack framework also applies to every fixed finite integer $p$. We then develop a complementary dynamic program over candidate objective values for the Euclidean case. The latter is polynomial in the encoding size of the radius and pseudo-polynomial in the magnitude of the cost coefficients. For a fixed objective value, feasibility can be formulated as a closest vector problem (CVP) instance. This connection gives an exact algorithm whose running time depends on the covering radius of that lattice. Conversely, we show that a rank-$k$ Euclidean closest-vector instance in ambient dimension $d$ reduces to the decision version of a Euclidean \BallIP{} instance in dimension $d+k\le2d$, transferring known bounds under the Exponential Time Hypothesis (ETH). Finally, we study dimension-dependent approaches based on proximity to the continuous optimizer and describe the fixed-level constructions that extend to ellipsoids.

Predict-then-optimize methods learn a mapping from features to unknown edge costs. They then solve the shortest-path problem on the predicted costs. We study a feature-free alternative for the case where the decision-maker observes many noisy cost vectors on a fixed network. The cost vectors are denoised by an eigenvalue decomposition (PCA) of their covariance matrix. Dijkstra's algorithm then runs on the result. We test this on the $5\times5$ grid of \citet{elmachtoub2022}, with $1000$ independent replications and a fresh ground truth in each. We compare against SPO+, against naive Dijkstra on noisy costs, and against two alternatives to a hand-picked rank $k$ (the Gavish-Donoho threshold and a Ledoit-Wolf filter). We also test a hybrid of SPO+ and PCA. Four findings emerge. First, the rank matters. Keeping $k=2$ components performs worse than no denoising. Second, when costs are close to linear in the features, the Gavish-Donoho threshold recovers the true rank without supervision. PCA-denoised Dijkstra outperforms SPO+ at $\deg=1$, but underperforms SPO+ at $\deg\geq 2$. The hybrid works best at $\deg=1$ and is slightly better than SPO+ at $\deg=2$. Third, when costs are strongly nonlinear ($\deg=4$), the signal is no longer low-rank, and only the Ledoit-Wolf filter clearly beats the naive baseline. Fourth, the benefit of denoising grows with network size on grids, but not on a layered non-grid network at high nonlinearity.

This paper addresses the existence of pullback attractors for a class of nonlinear non-autonomous strongly damped plate equations with delay. The model contains the biharmonic operator, the strong damping term, a nonlinear source term and a delay operator acting on the history of the solution. The interaction between the strong damping mechanism and the hereditary delay effect gives rise to several analytical difficulties in deriving uniform estimates and proving pullback asymptotic compactness. To overcome these difficulties, we construct a modified energy functional adapted to the strong damping structure and employ the contractive function method to handle the delay term in the history phase space. By establishing the existence and uniqueness of weak solutions, uniform pullback estimates and the pullback asymptotic compactness of the associated non-autonomous process, we prove the existence of pullback attractors for the strongly damped plate equation with delay.

For the $k$-self-similar naked singularity solutions of Einstein–scalar field equations constructed by Christodoulou, we construct a family of interior gravitational perturbations leading to trapped surface formation that remain large but finite at the threshold Hölder regularity, while converging to zero in all regularities below the threshold. This extends the interior spherically symmetric result below the threshold in \cite{Li25} to non-spherically symmetric setting. We further construct a genuinely localized family of perturbations whose angular support shrinks to a single point, with smooth convergence to zero away from that point. This exploits the additional angular freedom available beyond spherical symmetry. Moreover, when measured instead in Sobolev regularity, such genuinely localized family of perturbations still converges to zero in all regularities below the same threshold.

In this paper, we study rigidity phenomena associated with a class of overdetermined two-phase problems on the $n$-dimensional sphere $\mathbb{S}^n$. Specifically, we analyze the solutions of coupled elliptic equations defined on a domain $\Omega \subset \mathbb{S}^n$ and its complement, subject to Dirichlet boundary conditions on the interface $\partial\Omega$ and a compatibility condition on the gradient. We utilize two different methods. The method of moving planes works when $\partial\Omega$ is contained inside an open hemisphere. The other method works when $\Omega$ is simply connected in $\mathbb{S}^2$. Both lead to the same conclusion. Namely, the existence of a solution to such overdetermined problems necessarily implies that $\Omega$ is a geodesic ball. In the latter, we prove this rigidity property for three distinct scenarios: a baseline problem with piecewise constant source terms, which is the two-phase version of the torsion problem; a generalization including a class of positive and regular nonlinearities; and finally an eigenvalue problem involving the first Dirichlet eigenvalue of the respective phases.

We develop a biorthogonal time-dependent variational principle for real-time dynamics of non-Hermitian quantum many-body systems. Independent left and right matrix-product states obey coupled bivariational tangent-space equations whose cross-Gram matrix defines an oblique projection. A matrix-free scaled Taylor action propagates the resulting non-normal local generators without assembling dense matrices or storing a Krylov basis. We distinguish the fully coupled algorithm, which solves the cross-pairing problem and truncates the two bond bases jointly, from an efficient independently propagated approximation used for large systems. Independent truncation can make the retained left-right pairing nearly singular; overlap drift and the smallest singular value of the bond cross matrix expose this failure, while coupled truncation substantially delays it. Exact benchmarks and convergence tests validate the method. Applied to an interacting long-range non-Hermitian Ising chain, it resolves a biorthogonal dynamical quantum phase transition and shows that a weak imaginary field shifts the leading critical time from $t^{\ast}|J|=1.84$ to $1.04$.

We prove the existence and uniqueness of global-in-time solutions to the Navier–Stokes–Maxwell (NSM) system for small initial data in the critical Fujita–Kato space $\dot H^{\frac{d}{2}-1}(\mathbb R^d)$, in any dimension $d\geq 3$. Under the additional assumption that the initial data belong to the Besov space $\dot B^{-\frac{d}{2}}_{2,\infty}(\mathbb R^d)$, we also establish the optimal decay rate $t^{-\frac{s}{2}-\frac{d}{4}}$ at infinity of the solution in $\dot H^s(\mathbb R^d)$, for any $s\in\left(-\frac{d}{2},\frac{d}{2}-1\right]$. This is achieved by constructing a Lyapunov functional that is equivalent to the pointwise energy on the Fourier side, and by combining it with an adaptation of the Fourier splitting method to establish its optimal decay rate. All the analysis carried out here - from the global existence theory to the study of the large-time behavior - is performed within a framework that is uniform with respect to the speed of light $c\in(0,\infty)$. In particular, in the non-relativistic limit $c\to\infty$, this allows us to recover the same results for the corresponding limiting magnetohydrodynamic system.

In this work, we construct analytical kink-antikink $(K\bar{K})$ configurations in the non-integrable $\varphi^4$ model by mapping exact solutions of the integrable sine-Gordon system via a field deformation. This procedure yields two distinct classes of configurations, parametrized by the initial half-separation and velocity. We compare these profiles with the standard additive superposition Ansatz in terms of vacuum structure and equation-of-motion residuals, deriving explicit closed-form expressions for the corresponding integrated squared residuals. For small separations, the mapped soliton-antisoliton configurations exhibit appreciably smaller residuals than the naive superposition Ansatz, which in turn performs better than the mapped two-soliton configurations. Although the mapped fields are not exact solutions of the $\varphi^4$ equation of motion, they provide mathematically consistent, topologically sound, and physically motivated initial data for numerical studies of kink-antikink scattering and resonance phenomena.

We find and classify every stationary configuration of the system of four identical point vortices in the plane. Up to the natural symmetries of the system, there are exactly three configurations, which are relative equilibria: a square, a collinear configuration, and an equilateral triangle with a central vortex. Among them, the square configuration is the global minimum of the Hamiltonian, whereas the other two configurations are saddle points. In particular, the square is the unique stable configuration among the stationary configurations. Using the special algebraic structure of four points, we introduce a change of variables based on the Hadamard matrix of order four, which transforms the original four-vortex problem into an equivalent three-point problem.

In this paper, we prove an asymptotic expansion of the $L^2$-extension index of a smooth Hermitian metric on a holomorphic vector bundle, in which the Chern curvature appears as the second-order coefficient. By using this expansion, we show in a unified way that there is an equivalence between how sharp the $L^2$-extension is and how positive or negative the curvature is. We also introduce a new notion of $q$-$L^2$-extension indices and investigate partial positivity and flatness in terms of these indices.