Papers

In this paper, we introduce two new classes of cyclic-type mappings, namely, co-cyclic mappings and sub-cyclic mappings, defined on the union of two subsets of a metric space. Under suitable Jungck-type contractive conditions and compatibility assumptions, we establish unique common fixed point theorems for these mappings in complete metric spaces. The proofs are based on the construction of appropriate sequences, the associated contractive conditions, and the completeness of the underlying metric space. Examples are provided to illustrate the proposed concepts and to demonstrate the applicability of the main results. Finally, some open questions and possible directions for further research concerning co-cyclic and sub-cyclic mappings are presented.

In this paper we investigate the exact controllability of $n \times n$ first order one-dimensional quasilinear hyperbolic systems by $m<n$ internal controls that are localized in space in some part of the domain. We distinguish two situations. The first one is when the equations of the system have the same speed. In this case, we can use the method of characteristics and obtain a simple and complete characterization for linear systems. Thanks to a linear test this also provides some sufficient conditions for the local exact controllability around the trajectories of semilinear systems. However, when the speed of the equations are not anymore the same, we see that we encounter the problem of loss of derivatives if we try to control quasilinear systems with a reduced number of controls. To solve this problem, as in a prior article by J.-M. Coron and P. Lissy on a Navier-Stokes control system, we first use the notion of algebraic solvability due to M. Gromov. However, in contrast with this prior article where a standard fixed point argument could be used to treat the nonlinearities, we use here a fixed point theorem of Nash-Moser type due to M. Gromov in order to handle the problem of loss of derivatives.

The two-dimensional one-component plasma forms a droplet in the case that the one-body potential is proportional to the number of charges. Our interest is in studying the large $N$ form of the corresponding partition function, with the plasma confined to the surface of the cylinder. The one-body potential is (up to technical restrictions) allowed to be arbitrary in the direction of the axis of the cylinder, whereas the coupling is restricted to the exactly solvable value $\beta = 2$. It is demonstrated that a change of variables can be made to an annular droplet plasma model, up to a certain Jacobian factor. The latter can be interpreted as the characteristic function of a certain linear statistic, which generalises the original aim of our study to also analysing the large $N$ form of the characteristic function for the annular droplet plasma model. While this is well known in the case of soft wall boundary conditions, our analysis (based on Laplace's method applied to certain integrals, and the Euler-Maclaurin summation formula) covers the case of one or two hard walls at the boundary, or inside, of the droplet for which the soft edge formulas are shown to no longer hold. Use of this to study the large $N$ expansion of the cylinder partition function requires knowledge of the same expansion in the annular case. This is not available in the case of two hard walls inside the droplet, which we treat instead via a direct analysis. In distinction to the case of the annular partition function, the large $N$ expansion of the logarithm of the cylinder partition function is found to always have no term proportional to $\log N$, independent of the presence of hard walls.

We consider an agent acting against a general non-Markovian environment. The agent maintains its agent states, but is free to choose a transition kernel across those states and optimize its state-feedback control policies. We study the bi-level agent design problem that optimizes the transition kernel and the policy it induces, given said kernel with offline data of observations and actions obtained via a behavioral policy. For general environments, we show that a soft $Q$-learning algorithm converges almost surely to the fixed point of a soft Bellman equation defined by the stationary averages that the behavioral policy and the chosen kernel induce, and we delineate what separates the resulting policy from an optimal one. In partially observed Markov decision problems, we analyze convergence properties of parametrized transition kernel design via zero-th order and Bayesian optimization techniques.

This article establishes sharp strong error estimates for the fully discrete approximation of semilinear parabolic stochastic partial differential equations (SPDEs) driven by multiplicative noise and subject to nonsmooth initial data. The spatial discretization is based on a standard finite element method, coupled with the linearly implicit Euler scheme in time. By mapping the diffusion operator into negative fractional spaces, our framework accommodates both trace-class and space-time white noise. For nonsmooth initial data, by decoupling the noise regularity parameter $\beta \in (0,2)$ from the initial data regularity parameter $\mu \in (0,2]$, we derive sharp regularity estimates that isolate the exact loss of initial regularity into an integrable temporal singularity. Furthermore, we establish sharp strong convergence rates of $O(h^{\beta-\varepsilon} + k^{\frac{1}{2}\min\{\beta-\varepsilon, 1\}})$ for $\varepsilon>0$ away from $t = 0$. Finally, we consider physically relevant stochastic models, such as the modified Langmuir fractional surface coverage model and the parabolic Anderson model, in our numerical experiments to confirm the theoretical convergence rates.

We prove the Bárány–Kalai–Tverberg conjecture extending the affine version of Tverberg's theorem to simplicial balls and polytopes other than the simplex: For any affine map $\phi : P\to \mathbb R^d$ from a convex polytope $P$ of dimension $N=(d+1)(r-1)$ with $r\geq 2$, there exist $r$ pairwise disjoint faces $F_1,\ldots,F_r\subset\partial P$ such that $\phi(F_1)\cap\dots\cap\phi(F_r)\neq\varnothing$. A similar statement holds for a simplicial ball $P$ with $\phi$ assumed affine on all of its faces.

For a positive integer $h$, a subset $A = \{a_1, \dots, a_k\}$ of an additive abelian group $G$ is called an $S_h$-set of size $k$ if all sums of $h$ distinct elements in $S$ are distinct. For fixed positive integers $h$ and $k$, let $v_h(k)$ denote the order of the smallest abelian group containing an $S_h$-set of size $k$. A lower bound for $v_2(k)$ is known. In this paper, we establish a lower bound for $v_3(k)$. Using our argument for $h=2$, we recover the known bound for $v_2(k)$.

We establish the abundance theorem for log surfaces without assuming $\mathbb{Q}$-factoriality or log canonicity.

We formulate and analyze the linear exponential quadratic Gaussian (LEQG) covariance steering problem in continuous time over a given deadline (finite time horizon). The solution for this problem can be seen as a risk-sensitive Schrödinger bridge between Gaussian endpoints in the linear quadratic setting. Unlike the risk-neutral case, the LEQG covariance steering controller–still a linear state feedback–can no longer be written in closed form. We show that the optimal controller is parameterized by a symmetric matrix solving an algebraic equation that encodes the implicit dependence on the risk-sensitivity parameter. We explain how the structure of this optimal controller significantly generalizes the existing results for the risk-neutral case. Building on these results, for the matched noise and input channel case, we prove the existence-uniqueness of solution for the LEQG covariance steering problem in the neighborhood of the known risk-neutral optimal solution. We give an illustrative numerical example.

In this paper, we study the structural relationship between immersed finite element (IFE) spaces for scalar elliptic and Stokes interface problems on unfitted meshes. Although IFE methods have been widely developed for individual partial differential equation models, the algebraic connection between scalar and vector IFE spaces has not been systematically explored. We establish a precise unisolvence relationship between the immersed Crouzeix-Raviart (CR) element for elliptic interface problems and the immersed CR-$P_0$ element for Stokes interface problems in both gradient and stress formulations. By analyzing the block structure of the local IFE matrices, we show that the determinant of the Stokes IFE matrix factorizes in terms of the determinant of the corresponding elliptic IFE matrix. This factorization reveals an intrinsic algebraic link between the two classes of immersed elements and explains the unisolvence of the Stokes IFE spaces through the scalar elliptic case. The same result extends naturally to three dimensions and significantly simplifies the corresponding analysis. Numerical experiments confirm optimal convergence rates for both velocity and pressure in three-dimensional Stokes interface problems.

Community detection in bipartite networks is a fundamental problem in modern data analysis, with applications in recommendation systems, biological networks, and social network analysis. Unlike conventional unipartite graphs, bipartite networks consist of two distinct types of nodes with edges only connecting across types, so recovering latent communities requires estimating labels on the two node types. The stochastic co-blockmodel is a classical probabilistic framework for such networks, yet theoretical guarantees for exact community recovery in this setting remain limited, especially when the number of communities grows, the community sizes are unbalanced, or the degrees are heterogeneous. In this work, we prove that a simple spectral clustering algorithm based on the diagonal-deleted Gram matrix achieves exact recovery with high probability under mild conditions on sparsity, community balance, and the number of clusters. We further extend the result to the degree-corrected stochastic co-blockmodel, where each node carries its own degree heterogeneity parameter, and show that a row-normalized version of the same algorithm maintains the exact recovery guarantee. Extensive experiments validate our theoretical findings.

We determine the exact $L^p(\mathbb{R})$ norm of the centered Hardy–Littlewood maximal operator for every $1<p<\infty$, proving that it equals $p/(p-1)$. This settles the sharp strong-type problem for the centered maximal operator on the real line. For every $p$ in this range, the norm is strictly larger than the constant associated with the critical power $|x|^{-1/p}$, thereby disproving the conjecture of Dror, Ganguli, and Strichartz that this profile determines the unrestricted norm.

We study the compressible Euler-alignment system with pressure under a singular pressure scaling, where the hypersingular communication weight induces a fractional alignment operator of order $2\alpha$, $0<\alpha<1$. The scaling corresponds to a large-time and small-velocity regime and leads to a low Mach number problem in which the density is forced to remain close to a constant state. Our main result is a uniform regularity theory for this scaled pressure system. We establish uniform estimates with respect to the scaling parameter and construct global strong solutions near the constant state. A key feature of the analysis is that, in the low-order fractional regime $0<\alpha\le\frac12$, the estimates close under the lower Sobolev condition $s>\frac d2+1-2\alpha$, gaining $\alpha$ derivatives over the threshold $s>\frac d2 + 1 - \alpha$ arising from a direct use of the fractional alignment dissipation. This is achieved by combining refined commutator estimates for the singular alignment operator with the density dissipation induced by the pressure scaling. As an application of the uniform estimates, we justify the low Mach number limit toward the incompressible Navier–Stokes system with fractional dissipation. For general small, possibly ill-prepared initial data, a Helmholtz decomposition combined with dispersive estimates for the acoustic component yields subsequential strong convergence locally in space-time to a distributional solution of the limiting system. For well-prepared initial data, a relative-energy argument further identifies the limit with a prescribed sufficiently regular solution and yields global strong convergence in the fractional dissipation norm.

Quasi-exactly solvable (QES) deformations of harmonic and singular oscillators have been widely studied via a range of approaches. In this paper, we obtain families of new QES deformations of solvable quantum systems associated with exceptional orthogonal polynomials (EOPs). The construction builds upon the theory of Darboux-Crum transformations and the classification of exactly solvable quantum systems associated with EOPs of Hermite type. It is shown that the deformations break the exact solvability of the undeformed systems and introduce model parameters into the deformed systems that permit the existence of a finite number of polynomial solutions whose roots satisfy systems of algebraic equations. The new families presented and studied consist of deformations of quantum systems related to Hermite EOPs of type III with arbitrary codimensions. We present polynomial and rational deformations and analyze, in each case, the conditions for quasi-exact solvability in terms of Bethe ansatz equations and parameter constraints. In general, the structures of the underlying polynomial solutions are no longer associated with well-known classical orthogonal polynomials. So for the general cases, we mainly focus on presenting the new approach as well as the Bethe ansatz equations and the constraints for model parameters. As applications, we construct particular families of QES deformations related to solvable models allowing one, two, and up to four gaps, and obtain the closed form expressions for their wavefunctions and spectra. We analyze the existence of QES solutions in the spaces of model parameters, providing information on the number of solutions for given parameters.

We prove that the Bohnenblust–Hille constants for Walsh polynomials on the Hamming cube grow at most polynomially in the degree. More precisely, there is an absolute constant $K$ such that every $f :\{-1,1\}^{n} \to \mathbb{C}$ of degree at most $m$ satisfies $$ \left(\sum_{|S|\le m}|\widehat f(S)|^{2m/(m+1)}\right)^{(m+1)/(2m)} \le Km^{27}\|f\|_\infty. $$ The estimate is uniform in the dimension. The exponent $27$ is not optimized.

We study small solutions to the one-dimensional nonlinear Schrödinger family \[ \mathrm{i}\partial_t u_\delta=-\partial_x^2u_\delta+\kappa|u_\delta|^{2+\delta}u_\delta, \qquad 0\leq\delta\leq\delta_0, \] uniformly as $\delta\to0^+$ and $t\to\infty$, across the threshold between cubic modified scattering and nearby-power ordinary scattering. For initial data small in $H^{0,1}(\mathbb{R})$, with one spatial weight and no physical derivative assumed, we prove a joint-limit asymptotic formula governed by an explicit nonlinear clock. The variable $ \delta\log t$ gives three regimes according as it tends to zero, a positive finite limit, or infinity. Within the regime $\delta\log t\to0$, successive Taylor terms of the clock become visible at an infinite hierarchy of time scales, each requiring finer absolute phase accuracy. These scales accumulate at the transition scale, where the exact clock describes them together. In the transition and saturated regimes, the clock is of order $\delta^{-1}$, so the first-order variation of its coefficient contributes a finite phase correction.

We study an online sketched Newton method that approximates the Newton direction at each step via a state-of-the-art sketching solver, called the generalized accelerated sketch-and-project solver (GAS), thereby mitigating the computational bottleneck of classical second-order methods. The GAS solver improves upon vanilla, unaccelerated sketch-and-project solvers by achieving accelerated convergence through Nesterov momentum updates, and accommodates a flexible projection metric whose proper choice further reduces computational cost. Building on this design, we establish asymptotic normality of the averaged sketched Newton iterates and characterize their limiting covariance matrix. The resulting covariance recovers that of the unaccelerated sketched Newton method under a specific choice of acceleration parameters, converges more rapidly (in the number of sketching steps) to the minimax-optimal covariance in general, and is smaller than that of the last iterate produced by the accelerated method. Finally, we strengthen these results by establishing a functional central limit theorem for the Newton iterates, which allows us to bypass explicit covariance estimation and develop an online inference procedure based on random scaling. Specifically, we construct a pivotal test statistic by appropriately rescaling the averaged iterates, so that its limiting distribution is free of any unknown parameters, enabling asymptotically valid online inference. Numerical experiments demonstrate superior performance of the proposed inference procedure.

Let $f:X\to S$ be a smooth projective family, and fix a semisimple flat bundle on a fiber $X_0$. Its isomonodromic deformation determines a holomorphic section $\sigma_{\mathrm{dR}}:S\to M_{\mathrm{dR}}(X/S)$ of the relative de Rham moduli space. Applying the relative non-abelian Hodge correspondence fiberwise gives a section $\sigma_{\mathrm{Dol}}:S\to M_{\mathrm{Dol}}(X/S)$, whose value at each $s\in S$ is the Higgs bundle corresponding to the flat bundle $\sigma_{\mathrm{dR}}(s)$. Unlike $\sigma_{\mathrm{dR}}$, the section $\sigma_{\mathrm{Dol}}$ is in general only real analytic. We study the geometric consequences of its holomorphicity along a complex analytic subvariety. First, we prove an absolute lifting theorem: the relative isomonodromic Higgs bundle admits an absolute Higgs lift, and the fiberwise harmonic metrics can be modified to solve the Hitchin–Simpson equation on the total space. Second, we prove that the spectral one-form on every resolved irreducible component of the relative spectral scheme is Gauss–Manin flat. As an application, we prove the nilpotent rigidity conjecture of Hu–Sun–Yang–Zuo: if the initial Higgs field is nilpotent, then the Higgs field remains nilpotent along the entire holomorphic isomonodromic locus.

Every cluster variety admits an action of its cluster dilation group. We prove that, in the case of braid varieties for simple Lie groups, this action always extends to a regular action on each of the brick compactifications. We explore two applications of this result. First, we show that any closed Richardson variety admits a faithful action of a torus of rank the Kazhdan-Lusztig $d$-invariant, answering affirmatively a recent question of E. Gorsky–S. Kim–M. Sherman-Bennett. The same result holds for projected Richardson varieties. Second, we show that the braid variety is a torus if and only if for each of its brick compactifications, the polar dual of the moment polytope for this action realizes the corresponding subword complex. The braid words satisfying this property turn out to be precisely the double root free words of V. Pilaud and C. Stump. This provides a novel approach to the longstanding open question of the polytopality of spherical subword complexes asked by A. Knutson and E. Miller, and in particular gives infinite families of subword complexes admitting polytopal realizations in dimension higher than the rank of the corresponding Coxeter group. As a common consequence of these two applications, we classify all Bruhat intervals in finite crystallographic Coxeter groups which are isomorphic to face lattices of convex polytopes via certain double root free words.

Microwave ablation (MWA) is a percutaneous ablation that induces thermal necrosis by generating localized high-temperature regions within tumor tissue. However, when a tumor is located close to a large vessel such as the aorta, blood flow continuously removes heat from the ablation zone. This phenomenon causes the vascular heat-sink effect and thereby increases the risk of incomplete ablation. This study constructs an ANSYS-based numerical framework to investigate the heat-sink effect during MWA of tumors located close to the aorta. The computational model consists of the tumor, surrounding tissue, vessel wall, blood flow domain, and microwave antenna. Microwave-induced heating was represented by an equivalent heat source mapped from the High-Frequency Structure Simulator (HFSS)-derived volume loss density, rather than by direct electromagnetic-thermal coupling within the thermal-fluid solver. Two parametric studies were performed to quantify the heat-sink effect. First, the tumor-aorta distance was varied at D = 0 mm, 4 mm, and 8.5 mm to assess how vessel proximity affects temperature elevation and ablation zone formation. Second, at each tumor-aorta distance, three blood-flow conditions were compared: no blood, stagnant blood V = 0 m/s, and normal aortic blood flow V=0.65 m/s. This comparison enabled the heat-sink effect of blood flow to be independently evaluated. The results highlight the clinical need for simulation-based treatment planning in patients with tumors located close to major vessels.

Han, Jouhet and Zeng established $q$-analogues of the $\gamma$-expansion formulas for Eulerian polynomials of types $A$ and $B$. Combinatorial interpretations of the corresponding coefficients$a_{n,k}(q)$ and $b_{n,k}(q)$, however, remained open. In this paper, we provide combinatorial interpretations for thesecoefficients by using the model of increasing binary trees, thereby resolving a problem posed by Han, Jouhet and Zeng. Our combinatorial approach consists of three main steps: 1. construct a Carlitz-type insertion bijection for increasing binary trees and derive a new combinatorial interpretation ofCarlitz's $q$-Eulerian polynomials of type $A$ in terms of such trees; 2. introduce a generalized Foata–Strehl action on increasing binary trees to interpret the coefficients $a_{n,k}(q)$;3. derive a new combinatorial interpretation for the $q$-Eulerianpolynomials of type $B$ introduced by Chow and Gessel in terms of increasing binary trees of type $B$, and develop a generalizedFoata–Strehl action on these trees to interpret the coefficients $b_{n,k}(q)$. We further give combinatorial interpretations for the quotients ${a_{n,k}(q)/(-q;q)_{k-1}}$ and ${b_{n,k}(q)/(1+q)^k(-q;q^2)_k}$ in terms of André trees and a certain class of increasing binary trees of type $B$, respectively. As an application of the latter interpretation, we obtain a combinatorial interpretation for a $q$-analogue of the secant number and prove the positivity conjecture of Han, Jouhet and Zeng.

We prove several necessary and sufficient conditions under which a point satisfies the Poincaré Recurrence Theorem for all computable (ergodic) measure-preserving transformations and all sets of a particular complexity. The necessary conditions are obtained by constructing specific measure-preserving transformations which violate recurrence. While some of these conditions pertain to standard notions of algorithmic randomness, others involve new notions of genericity developed in the context of a computable probability space, which we call $\Pi^0_n$-genericity and quasi-$\Pi^0_n$-genericity. We also provide conditions under which points recur at a positive frequency in sets containing them, regarding both simple and multiple recurrence.

Stochastic variance reduced gradient (SVRG) is a variant of stochastic gradient descent and is a promising iterative method for solving large-scale inverse problems. Nevertheless, the development of theoretically grounded a posteriori stopping rules for SVRG remains an open challenge. In this work, we provide a convergence analysis of SVRG equipped with the discrepancy principle, the most well-known a posteriori stopping rule, for solving a class of linear inverse problems in Hilbert spaces. We establish the regularizing property of SVRG, and moreover, under suitable source conditions, we derive convergence rates of SVRG iterates. To the best of our knowledge, these are the first convergence rate results of any stochastic iterative method for inverse problems under the a posteriori stopping rule. The theoretical findings are supported by numerical experiments.

The study of $t$-intersecting families in symmetric groups received lots of attention in the last two decades. In this paper, we study two different kinds of stability results for $t$-intersecting families in symmetric groups. Let $\Sigma_n$ denote the symmetric group on $\{1,2,\ldots,n\}$. The $t$-diversity $\gamma_t(\mathcal{F})$ of $\mathcal{F}\subseteq\Sigma_n$ is defined as the minimum number of members of $\mathcal{F}$ whose deletion results in a family with transversal number $t$. The star $t$-diversity $\gamma_t^{\star}(\mathcal{F})$ is defined as the minimum number of members of $\mathcal{F}$ whose deletion results in a $t$-star. For $n$ relatively large with respect to $t$, we determine the best possible bounds for both $\gamma_t(\mathcal{F})$ and $\gamma_t^{\star}(\mathcal{F})$ over all $t$-intersecting families $\mathcal{F}\subseteq\Sigma_n$. The equality holding conditions are characterized. For $\gamma_t(\mathcal{F})$, the extremal case is generated by all $2t$-subsets of a $3t$-partial permutation or, when $t=2$, the family complements of the lines of the Fano plane. For $\gamma_t^{\star}(\mathcal{F})$, the extremal family is generated by all $(t+1)$-subsets of a $(t+2)$-partial permutation.

Probabilistic bits (p-bits) provide a physical and algorithmic primitive for stochastic inference, but highly parallel updates can alter their collective dynamics. We study the decoding of random-regular (3, 6) low-density parity-check (LDPC) codes using Jacobi-type p-bit annealing with stochastic partial activation. An additive response-path rule stores each bit's saturated response and reuses it before stochastic readout. High-statistics simulations with independent parameter optimization for each method show pooled bit-error-rate reductions of 33.5%, 74.8%, and 81.8% relative to memoryless probabilistic simulated annealing (pSA) for representative codes of block lengths 96, 192, and 288, respectively. Static gain, normalized averaging, response shuffling, and same-bit binary-state feedback with only its coefficient tuned under the same nonmemory parameters do not reproduce the full saturated-response benefit. Trajectory analysis links the improvement to acquisition of the channel-consistent valid-codeword basin and enhanced post-acquisition stability; the acquisition advantage persists from random and channel-hard-decision starts under the tested conditions. Across all 30 independent code realizations, fixed additive parameter-and-readout packages achieve lower bit- and frame-error rates than separately optimized pSA-specific packages, with neither package retuned for individual codes. These results show that the computational effect of temporal state depends on the retained quantity and its reinjection into stochastic dynamics.

Let $H$ be the generic Hecke algebra over $\mathbb{Z}[\mathbf{q}^{\pm 1}]$ associated to a split Kac–Moody group $G$, arising as the deformation of the group algebra of its Weyl group $W$. The cocenter $\overline{H} = H/[H, H]$ encodes the trace and character theory of $H$, playing a fundamental role in representation theory and harmonic analysis. Through deep combinatorial results on cyclic reductions in Coxeter groups, each conjugacy class $\mathcal{O}$ of $W$ determines a canonical element $T_{\mathcal{O}}$ in $\overline{H}$, and these elements are known to span the cocenter. However, establishing their linear independence has remained an open problem outside of finite and affine types. In this paper, we solve this problem: the canonical elements form a $\mathbb{Z}[\mathbf{q}^{\pm 1}]$-basis of the cocenter $\overline{H}$. Our approach differs from earlier representation-theoretic methods in finite and affine types. To construct explicit functionals that separate all conjugacy classes, we develop a new framework based on re-normalized orbital integrals. This framework synthesizes parabolic induction, traces of infinite-dimensional bimodules, and Kac–Moody harmonic analysis into an almost-dual basis for the cocenter. As a key local ingredient, we establish a generic duality theorem for finite groups of Lie type relating the cocenter to regular semisimple conjugacy classes, and determine precisely when this pairing is non-degenerate. Finally, we deduce the existence and uniqueness of generic class polynomials for $W$, and prove a uniform “dimension=degree” theorem for basic Deligne–Lusztig varieties of the split Kac–Moody group $G$.

In the spirit of the recent work by Demmel and Wiedemann \cite{[DW]}, we investigate the validity of the energy balance law for weak solutions to the three-dimensional Navier-Stokes equations with only horizontal dissipation. We prove that famous Lions-Shinbrot type energy conservation criteria, originally established for the classical isotropic Navier-Stokes equations, continue to hold in this anisotropic setting. Furthermore, we present two distinct approaches that handle the torus case and the whole space case, respectively.

In this paper, we study the ergodicity of geodesic flows on closed rank-one manifolds of nonpositive sectional curvature. We introduce the infinite-order vanishing set of the Gaussian curvature in dimension two and of the fiberwise second moment of the reduced Jacobi determinant in higher dimensions. We bound the Liouville measure of the singular set in terms of the volume of this subset and, for surfaces, obtain a corresponding Hausdorff dimension bound. In particular, if this subset has zero volume, then the singular set has zero Liouville measure, and the geodesic flow is ergodic. We also give a geometric criterion for finitely many exceptional regions whose fundamental groups have virtually Abelian images of rank smaller than the dimension of the manifold. Under strict negativity of sectional curvature outside these regions, we obtain a Hausdorff dimension bound for the singular set and prove that the geodesic flow is ergodic with respect to Liouville measure. By extending this method to real-analytic metrics, we characterize the singular set as the zero set of a nontrivial real-analytic determinant constructed from curvature operators, thereby establishing ergodicity in this setting without further assumptions.

A multipartite tournament is an orientation of a complete multipartite graph. We prove that every $r$-regular $c$-partite tournament with common partite-set cardinality $\alpha$ is $4$-arc-pancyclic whenever $c\ge93$; that is, every arc belongs to a cycle of each length from $4$ to $c\alpha$. This confirms the conjecture of Zhou and Zhang for all sufficiently large $c$ and provides a multipartite analog of Alspach's arc-pancyclicity theorem. Moreover, we also give a construction to show that 4-arc-pancyclic is the best possible. Next, we prove that every arc belongs to at least $c\alpha-\alpha-1$ cycles of pairwise distinct lengths when $c\ge7$ and $\alpha\ge2$. For regular $3$-partite tournaments with common partite-set cardinality $\alpha\ge2$, we obtain the sharp lower bound $\alpha$, settling the remaining case of a conjecture of Xia, Cai, Guo, and Wang.

We propose spectral and spectral-element methods for accurately computing many eigenvalues of the Schrödinger operator with Robin boundary condition on simple and complex geometries, respectively. Due to their spectral-type accuracy in approximating those eigenfunctions corresponding to high-index eigenvalues, which are usually highly oscillatory, the proposed approaches have excellent resolution in computing thousands of eigenvalues accurately and efficiently with the resolution property that the number of eigenvalues with reasonable accuracy is proportional to the number of degrees of freedom. Using a standard HPC node with 128 GB of memory, we can obtain numerically more than 5,000 reliable eigenvalues with relative errors below $10^{-8}$ for different complex domains in two dimensions (2D). Based on these computed eigenvalues, we first confirm some theoretical results on the Robin-to-Neumann (RtN) gaps of the Laplacian operator in 2D, which were recently studied by Rudnick et al. [Comm. Math. Phys. 388 (2021)]. Then we systematically study the RtN gaps of the Schrödinger operator and their convergence rates. Based on our extensive numerical results, we formulate a unified conjecture on the cumulative averages of the RtN gaps of the Schrödinger operator.