Papers

We study the spectrality of measures with positive integrable densities supported on two line segments in $\mathbb R^d$. We prove that, when the two segments are non-overlapping, spectrality forces the density on each segment to be constant almost everywhere. When the two segments are overlapping, spectrality forces the total density to be constant almost everywhere on their union. We then study the resulting measures with positive constant densities according to the geometric of the segments. For two non-coplanar segments, every such measure admits a spectrum contained in a straight line. For two segments lying on distinct parallel lines, the measure is spectral if and only if the two densities are equal. For non-parallel coplanar segments, a suitable invertible linear transformation converts the measure into an unweighted arc-length measure. When these segments are viewed in their affine plane, every spectrum of a spectral measure is contained in a straight line. Finally, we give examples showing how the densities determine the directions of line spectra and construct explicit spectra for weighted measures.

If $G$ is a connected reductive algebraic group over a field $k$ of characteristic $\ell$, we show that both the Lie-theoretic and group-theoretic commuting schemes associated to $G$ are normal and Cohen–Macaulay when $\ell = 0$, or $\ell$ is larger than the Coxeter numbers of all simple factors of $G$, and that $\ell \neq 19$ (resp. $\ell \neq 31$) when $G$ has a simple factor of type $E_7$ (resp. $E_8$). The main new ingredients are provided by studying the unipotent categorical Langlands functor of Zhu. Along the way we prove a $t$-exactness result for a finite unipotent categorical Langlands functor which may be of independent interest.

Andrews and Dastidar introduced the statistic $\SOME(n)$, defined as the difference between the sum of all odd parts and the sum of all even parts occurring in the partitions of $n$, and investigated its arithmetic properties. Motivated by their work, Baruah and Gogoi defined two analogous statistics for overpartitions, namely $\OSOMEo(n)$ and $\OSOMEe(n)$, which record the sums of all odd and all even overlined parts, respectively. They established several congruences for these statistics and their difference, and proposed a conjecture on congruences modulo $7$. In this paper, we prove their conjecture by elementary $q$-series methods. Our proof combines two specializations of Watson's quintuple product identity, a $(p,k)$-parameterization of divisor-sum series, and termwise logarithmic differentiation

We study trust-region minimization of a smooth, possibly nonconvex functional plus a convex functional composed with a bounded linear operator. A relative primal–dual gap condition controls both the error in an approximate proximal-gradient step and its linear-model decrease. Together with a computable absolute stationarity test, it yields a finite Cauchy search, convergence of the proximal stationarity measure to zero, and an $O(\varepsilon^{-2})$ bound on outer trials. The outer analysis allows the linear operator to take values in a Banach space and does not require dual attainment. When the operator takes values in a Hilbert space and the regularizer is finite and Lipschitz, the dual proximal-gradient method produces finite gaps tending to zero, provided the required proximal maps and functional values can be evaluated. We prove $O(j^{-1})$ gap bounds for both recovered and averaged primal candidates and give a sharper bound on the primal error for exactly recovered points. A semilinear elliptic control problem with unsmoothed total-variation regularization and an $L^2$ control cost illustrates the method in the full $H^1$ metric. Across five meshes, outer and state Newton counts remain constant, while interior-point iteration counts vary mildly.

In this paper, we study sampled-data stabilization of general nonlinear stochastic systems under a local exponential-type Lyapunov condition for the continuous-time closed-loop system. The coefficients are assumed only locally Lipschitz, with neither linear-growth nor Khasminskii-type conditions. Two counterexamples show that almost-sure stabilization is generally unattainable. Instead, given an initial state, we establish exponential stability with arbitrarily high probability under sufficiently fast sampling. Separate methods are developed for \(p\ge 2\) and \(0<p<2\), where \(p\) is the growth order in the Lyapunov condition. Numerical examples illustrate the results.

We study the uniqueness and boundary behavior of nonzero convex Aleksandrov solutions to $\det D^2 u=M|u|^p\nu$ with zero boundary values on bounded convex domains in $\mathbb{R}^n (n \geq 2)$. For $0<p<n$, we prove the uniqueness of nonzero convex solutions in the finite-energy class when $\nu$ is a locally finite Borel measure with positive mass and $\int_\Omega \text{dist }(\cdot,\partial\Omega)\,d\nu<\infty$. For $p>n$, we construct an explicit two-shell measure on the unit ball for which the problem has at least three radial solutions that are globally Lipschitz and have finite energy. In the case of $\nu=\text{dist }(\cdot,\partial\Omega)^{-\alpha}\,dL^n$, $0\leq\alpha<2$, we prove global Lipschitz continuity when $p-\alpha>n-2$ and obtain sharp upper and lower estimates on domains with a flat boundary part when $n(\alpha-1)-2<p-\alpha\leq n-2$. When $\alpha=0$ and $p=n-2$, our log-Lipschitz lower estimate has the same exponent as the known upper estimate. This answers the question raised by Le (Global Lipschitz and Sobolev estimates for the Monge-Ampère eigenfunctions of general bounded convex domains. Ann. Fac. Sci. Toulouse Math. (6) 35 (2026)). We also give a sufficient condition for finite Monge-Ampère energy on every bounded convex domain, prove its necessity when the boundary contains a flat part, and apply it to prove the uniqueness of the Monge-Ampère eigenvalue among all nonzero convex solutions.

Let $P(t)$ be the transition function of an absorbing continuous-time Markov chain on a countable state space. We study asymptotic relations of the form $p_{ij}(t)\sim a_{ij} L(t)$, $t\to\infty$, where $L$ is independent of $i$ and $j$. We obtain general conditions under which the coefficient matrix $A=(a_{ij})$ has rank one and describe the resulting consequences for survival probabilities and conditional distributions. The analysis applies to both reducible and irreducible chains. We also construct an irreducible counterexample, based on a killed random walk on a homogeneous tree, for which a common asymptotic scale exists but the coefficient matrix has rank greater than one. This shows that irreducibility alone does not imply rank-one asymptotics. The results identify conditions under which fixed-state transition asymptotics determine the asymptotic behaviour of the entire transition function and clarify the limitations of such conclusions in the absence of additional structure.

We develop a sample-based framework for constructing information-driven hierarchical multi-resolution representations of probabilistic occupancy grids. Recent methods compute information-optimal abstractions via dynamic-programming-based exhaustive recursions, which become computationally prohibitive for large-scale grids and are ill-suited to robotics applications. To address this limitation, we introduce a sample-based strategy inspired by Monte Carlo Tree Search (MCTS) that incrementally constructs hierarchical abstractions through statistical estimation rather than exhaustive enumeration. The proposed method is anytime in nature, allowing computation to be terminated at any stage to produce a valid compressed representation. We compare our approach with the information-optimal Q-tree search algorithm and demonstrate its effectiveness in rapidly generating abstractions of large real-world probabilistic occupancy grids.

In this paper, we propose and analyze an augmented Lagrangian framework for solving stochastic nonconvex optimization problems with expectation-based equality constraints over a closed and convex constraint set. The framework generates a sequence of nonconvex primal subproblems, which are solved inexactly using stochastic second-order methods. We establish iteration complexity results for obtaining approximate second-order stationary points, both in expectation and with prescribed probability, under corresponding conditions on the accuracy of the primal subproblem solutions. We further establish sample complexity results for the framework with general stochastic second-order subproblem solvers under reasonable assumptions on their theoretical guarantees. Moreover, we incorporate three existing stochastic second-order solvers and derive the corresponding deterministic and probabilistic sample complexity results for obtaining approximate second-order stationary points. To the best of our knowledge, such sample complexity guarantees for obtaining approximate second-order stationary points in nonconvex expectation-constrained optimization have not been established previously. Finally, we demonstrate the empirical performance of the proposed framework on two nonconvex machine learning problems.

In this work, we propose a hybrid iterative deep Ritz method (H-IDRM) for a class of interface problems for second-order elliptic operators. It is based on a new mixed formulation of the problem and involves solving a sequence of convex minimization problems. We employ a level-set neural network architecture, featuring a level-set representation of the interface, to accommodate the piecewise smoothness of the solution and the flux. The approach involves only volumetric representations instead of duality pairing on the interface and avoids explicit interface sampling that is inconvenient for complex interface geometries. Further, we present an analysis of the method, including the errors arising from the neural network approximation, Monte Carlo approximation, iterative scheme, and penalty parameters. Numerical experiments indicate that the H-IDRM outperforms existing neural solvers on problems with high-dimensional domains, intricate interface geometries, and lower subdomain regularity.

We show that the noncommutative second Chern number of a $(3+1)$D disordered axion pump remains quantized beyond global spectral gap closure at $W_g \approx 14.5$. Large-scale computation of the second Chern number and finite-size scaling demonstrate convergence of $\mathrm{Ch}_2$ toward $-1$ well into the mobility-gap regime ($W \lesssim 16.5$), while the Středa-Hall response $\partial_\phi \mathrm{Ch}_{0,3}$ departs rapidly after global spectral gap closure, serving as an internal control that distinguishes bulk mobility-gap protection from finite-size artifacts. Level statistics identify a 3D unitary Anderson transition at each pump slice and locate the mobility-edge bottleneck at $\tau=0$, with critical disorder $W_c \approx 18$.

We develop a geometric lifting method for inverse sumset problems in compact connected abelian groups. The first result is an analogue of Freiman's $3k-4$ theorem, that is every compact set $A\subseteq G$ of sufficiently small Haar measure satisfying $\mu_G(A+A)<3\mu_G(A)$ is contained in a one dimensional Bohr set of measure at most $\mu_G(A+A)-\mu_G(A)$. This resolves a question of Christ and Iliopoulou. The proof combines Bilu's theorem with a geometric refinement of the spillover argument. We also establish a sharp projection theorem. Under a continuous surjective homomorphism with connected kernel, a compact set of sufficiently small positive measure and doubling at most $K$, where $2\le K<3$, has image of doubling at most $2K-2$, and this factor is best possible. Further consequences include variants of the $3k-4$ theorem for popular sumsets and an inverse theorem for Tao's convolution inequality.

Let $I_h(a,b)$ be the ideal generated by the $h\times h$ permanents of a generic $a\times b$ matrix over a field of characteristic different from two. We prove that $\dim K[X]/I_h(a,b)=\max(a,b)(h-1)$ whenever $1\le h\le\min(a,b)$ and $h<\max(a,b)$. Consequently, the critical locus of the permanent of order $n$ has codimension $2n$ for every $n\ge 2$. We also prove that the maximal permanents of a generic $k\times(k+1)$ matrix generate a geometrically reduced complete intersection. The dimension argument uses a pivot expansion and cubic relations in the border variables. Two successive comparisons with highest-degree equations reduce the relevant fibers to a coordinate-support calculation. Reducedness follows from a separate regular-sequence argument and a dimension estimate excluding components over the exceptional locus.

We provide an equitable partition that gives an exact, minimal representation for the graph Cartesian product formed from quantum-like bits that preserves the relevant spectral and topological properties. We show that this result follows from the fact that the operations of taking the Cartesian product of graphs and constructing equitable partition of the graphs commute. Numerical simulations illustrate the preserved emergent eigenstates in the reduced structures. Additionally, we provide a construction of the minimal structure without passing through the Cartesian product. Finally, we frame quantum-like structures in the language of topology and fibrations.

We investigate the existence of ground states for two-component Bose–Einstein condensates with intraspecies interactions $a_1, a_2 \in (0, a^*)$ and interspecies interaction $\beta>0$. By carefully investigating an associated auxiliary minimization problem, we prove the existence of a unique, continuous, critical surface $\gamma = \gamma(a_1, a_2)$ between $\beta_*:= \sqrt{(a^* - a_1) (a^* - a_2)}$ and $\beta^*:= a^* - \frac{a_1 + a_2}{2}$. This surface provides a complete classification for the existence of minimizers: ground states exist for $\beta \in (0,\gamma)$ and do not exist for $\beta > \gamma$. Additionally, when the trapping potentials are continuous at a common zero point, there is no ground state for $\beta = \gamma$. Our results close the gap case $\beta \in [\beta_*, \beta^*]$ when $a_1\ne a_2$ in literature, significantly extending the results in \cite{Bao-Cai-1,guoBlowupSolutionsTwo2017a}.

We develop a method for constructing minimal subshifts with a prescribed bound on the factor complexity within any strong orbit equivalence (SOE) class. This implies realizations of topological entropies within any SOE class, strengthening a series of classical results by Sugisaki. While prior work established complexity controls for zero-entropy systems, the approach of the present work extends to positive-entropy regimes. Our construction introduces the class of pseudo-Toeplitz subshifts, and we show that they exist in any SOE class. More precisely, for any $\alpha \geq 1$ and sequence $g_n$ growing exponentially at rate $\log(\alpha)$, we construct a system in this class whose topological entropy is $\log(\alpha)$ and whose complexity grows strictly faster, or, under certain conditions, slower than $g_n$. As a consequence, any Choquet simplex can be realized as the set of invariant measures of a Toeplitz subshift with precisely controlled complexity, in either a zero or positive-entropy regime.

We study the locally analytic vectors of completed cohomology of Hilbert modular varieties below middle degree. Our main result says that the universal enveloping algebra action has to factor through certain quotients. Application includes bounds on the Gelfand-Kirillov dimension and a density result. The method is a continuation of our joint work with Kai-Wen Lan.

We prove that the amplitude of the unique limit cycle of the van der Pol equation $\ddot{x}+\mu(x^2-1)\dot{x}+x=0$ is strictly greater than $2$ for every $\mu>0$. The proof combines orbit comparison in the relaxation oscillation regime, trigonometric comparison curves in the weakly nonlinear regime, and polynomial energy estimates in the intermediate regime. The polynomial sign conditions arising in the latter two regimes are certified using exact rational arithmetic and Bernstein coefficients. The orbit comparison method also applies to a class of symmetric Liénard systems. Under suitable monotonicity assumptions, we establish a sufficient criterion for the limit cycle amplitude to exceed a critical value determined by the damping function.

We revisit the Hodge conjecture for Fermat (and weighted Fermat) varieties. We review the reduction of the Hodge conjecture in terms of Hodge characters introduced by Shioda and its further reduction in terms of the formal module of tuples introduced by Aoki. Following Aoki, we introduce several lengths on this formal module, and by means of Kang's theorem on the Hodge conjecture for Fermat fourfolds we reduce the conjecture to bound the lengths of exceptional tuples by 6. By applying length reduction algorithms we prove the Hodge conjecture for all Fermat varieties (of any dimension) of degree less than 65 and different from 44, 51 and 52. The main novelty of our length reduction method is the introduction of a lift operation on the degree of the tuple.

We provide a generators and relations presentation of pivotal pointed fusion categories, $Vec(G,\omega,\pi)$. Unlike the well-known skeletal model, our presentation is strict and allows multiple isomorphic objects. Our main tool is skein theory, which allows us to apply topological tools to understand the relations of morphisms in the category.

Short-horizon forecasts of atmospheric temperature are needed to support climate-aware digital-twin systems, but such forecasts must be produced where thermal observations are incomplete. This study evaluates a physics-informed neural network for potential-temperature forecasting, constrained by a pressure-coordinate thermodynamic advection-source equation and a diabatic-source closure fit from the preceding 12-hour period and frozen before future-time training. Using hourly ERA5 reanalysis at three pressure levels, the model is evaluated as a conditional hindcast at lead times of one, two and three hours against persistence, local-trend, and two matched neural-network baselines, one of which receives the same future meteorological forcing as the PINN, helping distinguish the physical constraint from access to future forcing. In an Oklahoma development case, mean RMSE improvement over the strongest baseline grew from 8.1% at one hour to 23.8% at three hours; under an observation-density sweep down to 5% of candidate locations, this 3-hour advantage remained 14.6–16.9%, with no evidence that lower density improves performance. Under a fixed protocol transferred to an Alabama heat event with three virtual-observation layouts, three-hour improvement ranged 19.7-24.4% with consistent origin-level wins. A parallel Montana stress test, in which fixed pressure levels intersected complex terrain, produced a three-hour degradation of roughly 17.5%, identifying a terrain-related applicability limit of the formulation. Together, these results indicate that the physics constraint's benefit grows with forecast horizon, persists under severe observation sparsity, and transfers across regions, but is bounded by the validity of a fixed vertical-coordinate representation over complex terrain, evidence relevant to physics-constrained components of climate-aware forecasting and digital-twin systems.

We investigate the low Mach number limit of the three-dimensional full compressible Navier-Stokes-Fourier (NSF) equations on \(\mathbb{R}\times\mathbb{T}^2\) for two different classes of initial data, corresponding respectively to a well-prepared regime and an ill-prepared regime. The density and temperature are allowed to approach different asymptotic states at infinity. For the well-prepared regime, the solutions of compressible NSF equations converge to a planar diffusion wave solution globally in time as the Mach number tends to zero, where the difference between the states at the far fields is small independently of the Mach number and the non-zero modes of the initial perturbations are exponentially small. Moreover, the optimal time-decay rate can be obtained. It can be viewed as the first global-in-time result on the low Mach number limit of three-dimensional NSF equations with large temperature variations. The corresponding local-in-time result is proved while the difference between the states at the far fields can be arbitrarily large for the ill-prepared data.

Multicomponent reacting-flow discretizations must conserve total species mass, preserve positive mass fractions, and dissipate a convex mixing potential. These properties are controlled at different stages of a solver, however, and a certificate established at one stage need not survive the next. We study this compatibility problem for mixture-averaged diffusion, non-orthogonal finite-volume reconstruction, and positivity-preserving Patankar time integration. The analysis is isothermal, uses prescribed density, and uses synthetic detailed-balance reaction networks for the chemistry tests. At the face level, the entropy-variable jump is shown to equal a log-mean Gibbs Hessian applied exactly to the mass-fraction jump. Requiring conservation and entropy neutrality of the mass-correction term on the simplex tangent space then yields a unique componentwise-positive face composition and an exact two-point entropy certificate. The arithmetic-weight face criterion is also satisfied by production neutral-air/carbon data and a quasi-neutral ambipolar-air control, but a face-level certificate does not automatically extend to subsequent discretization layers. A weighted least-squares non-orthogonal correction can, in the tested prototype, convert a dissipative two-point operator into an anti-dissipative multipoint operator. Likewise, Patankar rescaling can preserve positivity and conservation while increasing the free energy; an explicit finite-step rational counterexample establishes this failure independently of ensemble statistics. A fully discrete two-cell example further shows that a positive face entropy margin does not guarantee positivity of the fully discrete update. Thus conservation, positivity, and free-energy dissipation are layer-specific properties and require distinct discrete certificates.

We address two questions of Huybrechts concerning rigidity and singular deformation spaces of tangent bundles on Calabi–Yau threefolds, without assuming simple connectedness. Our first example is a free $(\mathbb Z/2)^3$ quotient of a smooth intersection of four quadrics in $\mathbb P^7$. Its tangent bundle is stable and infinitesimally rigid, and it also provides a three-dimensional counterexample to the proposed classification in Peternell's Question~1.6. For a classical Igusa quotient of a product of three elliptic curves, we determine the analytic semiuniversal deformation germ of its polystable tangent bundle, keeping the underlying threefold fixed. This germ is isomorphic to the product of two copies of the union of the three coordinate axes in $\mathbb C^3$. It is therefore reduced and singular, with nine two-dimensional irreducible components.

Self-regulation together with mutual promoter competition provides a simple mechanism for bistability in gene-regulatory models. We study a two-gene system with regulatory terms of Hill exponent one, allowing distinct basal production rates and distinct degradation rates. Each gene product, when bound to its own promoter, may enhance or reduce production relative to the basal rate, while the two products compete through promoter occupancy. We show that the system has at least one and at most three equilibria in the positive quadrant. Exactly two positive equilibria can occur only if one nullcline intersection is degenerate; consequently, a configuration in which all positive nullcline intersections are transverse has either one or three positive equilibria. If there are exactly three distinct positive equilibria, then all three are automatically hyperbolic: the two outer equilibria are asymptotically stable nodes and the middle equilibrium is a saddle. Moreover, every positive solution converges to an equilibrium. Hence the positive quadrant is the disjoint union of the basins of attraction of the two stable nodes and the one-dimensional stable manifold of the saddle, yielding global bistability.

We study how determinant recurrences of banded Toeplitz matrices behave when their Laurent symbols are multiplied. Two classical structures underlie the problem. Clean banded Toeplitz determinants have a root-product description of their recurrences, while a product of two finite Toeplitz sections differs from the finite section of the product by finite-rank corner corrections. We connect these structures by identifying multiplication-induced boundary defects with exterior-degree sectors of the determinant recurrence. To each polynomial core we associate recurrence polynomials for all exterior degrees. Multiplication of cores becomes a graded convolution via composed products, while complementary degrees are related by a scaled reciprocal duality. On the matrix side, one-sided factors produce a defect filtration whose successive pieces are exactly these sectors. For general two-sided factors, independently weighting the two corner corrections gives an interval filtration: increasing either defect order adds one adjacent sector, and the cumulative recurrences are generically minimal. The multiplication-induced same-corner minors used in these filtrations lie in the standard row-column state module, whereas cross-corner imbalance corresponds instead to Laurent recentering. Opposite Laurent recenterings realize the individual sectors directly, and for several factors the admissible recenterings form an explicit lattice polytope. The pentadiagonal case gives the basic $1+4+1$ decomposition. The resulting framework gives a factor-level description of how finite-section boundary effects generate the recurrence spectrum of structured banded Toeplitz products.

We develop a quantitative framework for the long-time behavior of McKean–Vlasov stochastic differential equations with singular coefficients and possible phase transitions. The framework separates the existence of invariant measures from their uniqueness. For existence, we introduce a generalized Lyapunov condition with a one-level trapping mechanism, which requires dissipativity at only one admissible moment level rather than global contraction. This permits distribution dependence with supercritical growth and thus is more compatible with phase-transition models. For uniqueness and convergence, we establish an anchored uniqueness and quantitative ergodicity principle. An explicit \(L^1\)-smallness condition, expressed through a convolution kernel built from derivative estimates of the anchored frozen semigroup, yields uniqueness and transfers exponential or polynomial mixing rates of the frozen dynamics to the nonlinear McKean–Vlasov system. Moreover, the principle is sensitive to the choice of topology: different distances lead to different perturbation kernels and hence different stability thresholds. A local version gives local uniqueness and quantitative attraction near a prescribed equilibrium. We apply the framework to two representative models. For non-symmetric granular media dynamics, we derive two explicit uniqueness thresholds that reveal the topology-sensitive nature of the theory. For the dynamical Curie–Weiss model, the Wasserstein-1 criterion recovers the sharp bifurcation threshold up to the critical equality. We identify all invariant measures, establish basin-dependent exponential convergence in the phase-transition regime, and show that the loss of anchored smallness at criticality leads to polynomial slowing down.

We give algorithms to explicitly compute local Arthur packets and their intersections for even orthogonal groups $G_n$ over $p$-adic fields using extended multi-segments. In addition, we provide an algorithm which determines whether a representation of $G_n$ lies in a local Arthur packet. As applications, these results are used to identify distinguished parameters, prove the closure ordering conjecture, the enhanced Shahidi conjecture, the weak local Arthur packet conjecture, and the good parity unitary dual conjecture for $G_n$.

We show that the minimum volume among normal stable surface pairs with nonzero reduced boundary containing a zero-dimensional non-klt center is $\frac{1}{42}$, and show that the surface pair attaining this minimum is unique up to isomorphism. As an application, we show that $\frac{1}{42}$ is an infinite accumulation point of the volume set of normal stable surfaces.

Consider the compressed shift $T=P_{K_\theta}M_z|_{K_\theta}$ on $K_\theta=H^2\ominus\theta H^2$ associated with the atomic singular inner function $\theta(z)=\exp(-a(\zeta+z)/(\zeta-z))$, $a>0$ and $|\zeta|=1$. Let $k_0=P_{K_\theta}1$ and define real powers $T^s$ using a holomorphic logarithm near the singleton spectrum of $T$. For any temporal multisequence $\mathsf S=(s_k)_{k\in I}\subset[0,\infty)$ with uniformly bounded numbers of samples in unit intervals, we prove that $\{T^{s_k}k_0:k\in I\}$ is a frame if and only if $\mathsf S$ is Kummer-thick: the frequencies $\omega(s_k)=\sqrt{8s_k+4}$, weighted by $1/\omega(s_k)$, have uniformly positive total mass in every sufficiently distant interval of some fixed length. Equivalently, the occupied unit cells fill a fixed positive proportion of every window $[N-C\sqrt N,N+C\sqrt N]$ for some $C>0$ and all sufficiently large $N$. The proof combines a Volterra transmutation into Bessel waves with the Fourier–Bessel Logvinenko–Sereda theorem.