Papers

To test the potential of so-called quantum physics-informed neural network (QPINN) technology for solving evolutionary partial differential equations, we consider the problem of multiple wave propagation in a weakly nonlinear medium described by the Korteweg-de Vries equation with periodic boundary conditions. Although this problem is solvable analytically, high-genus solutions could represent a rather challenging testbed for numerical methods. Therefore, our idea is to test whether a QPINN is sufficiently mature for application in this context by providing a set of indicators of merit and performing several numerical experiments.

We give some existence results of Reeb chords using the $\textit{generation-criteria}$ of the Fukaya category. In particular, for $n\geq3$, we construct a Legendrian $\ell'\subset S^*\mathbb T^n$ such that every closed spin Legendrian $\ell\subset S^*\mathbb T^n$ either admits a self-Reeb chord or a Reeb chord joining it to $\ell'$.

Dickman and Tomé (1991) introduced models on $Z$ in which $k$ consecutive occupied sites give birth at rate $\lambda$, individual particles die at rate 1, and the configuration is subject to nearest neighbor stirring. They claimed that the pair creation model ($k=2$) has a continuous phase transition and was in the universality class of directed percolation, but the triplet creation model ($k=3$) has a discontinuous (or first-order) phase transition when the stirring rate is large. Hinrichsen (2000a) disputed the second claim and argued that first-order phase transitions in 1+1-dimensional nonequilibrium systems with fluctuating ordered phases are impossible. In this paper we prove rigorously that the phase transitions are discontinuous in both the pair and triplet models when the stirring rate is large. At the end of Section 2 we state some open problems and suggest reasons that the results in the physics literature differ from those presented here,

We construct high-order Runge–Kutta extensions of Barrett–Garcke–Nürnberg (BGN) parametric finite element methods for curve-shortening flow and curve diffusion of planar curves and for mean curvature flow and surface diffusion of closed genus-$0$ surfaces. On each time slab $I_m=(t_m,t_{m+1}]$, the continuous equations are posed on the left-endpoint surface $\Gamma^m=\Gamma^{t_m}$ through the map $\X^{m,t}:\Gamma^m\to\Gamma^t$, whose target is the evolving surface at time $t$. For curves, this formulation follows by harmonic pullback from $\Gamma^0$ to $\Gamma^m$. For surfaces, an orientation-preserving harmonic diffeomorphism is posed separately on each time slab, and conformality yields the weight $\frac12|\nabla_{\Gamma^m}\X^{m,t}|^2$. Evaluating the time-slab equations at the Runge–Kutta internal times and applying mass-lumped parametric finite elements yields systems on the common domain $\Gamma^m$, with $\Gamma^{t_m+c_i\tau_m}$ as the intermediate target geometry. This internal-time discretization constructs high-order BGN-structured extensions for the four flows. For an algebraically stable tableau with nonnegative weights, every exact stage solution with nondegenerate intermediate configurations satisfies monotone decay of the discrete curve length or surface area for every positive time step for which that solution exists. Radau IIA experiments exhibit energy decay for all four flows and BGN-type mesh redistribution in the curve tests. The Hausdorff self-convergence results exhibit high-order behavior consistent with the corresponding design orders.

In the whole space of low spatial dimensions, namely $d \leq 2$, we study the minimizers of an energy functional with an attractive cubic nonlinearity and a repulsive quintic nonlinearity, which describes a quantum Bose gas with a two-body attraction and a three-body repulsion. We prove the existence of minimizers at fixed effective statistics parameter. In the limit of a large effective statistics parameter of the mass-critical nonlinearity (with respect to the kinetic energy), we derive an effective Thomas-Fermi-like model for the homogeneous Bose gas. We also consider other limit regimes depending on the mass-critical nonlinearity.

We consider a homogeneous Bose gas composed of $N$ identical bosons occupying an infinite two-dimensional space. This gas exhibits both an attractive two-body interaction and a repulsive three-body interaction. Via the intermediate Hartree theory, we rigorously derive the homogeneous cubic-quintic nonlinear Schroedinger functional as the mean-field limit of the model. Our investigation focuses on the system's behavior in relation to the two-body interaction.

We investigate the behavior of a homogeneous one-dimensional Bose gas composed of $N$ identical bosons experiencing attractive two-body and repulsive three-body interactions. Utilizing intermediate Hartree theory, we rigorously derive the homogeneous cubic-quintic nonlinear Schrödinger functional as the mean-field limit of the many-body Hamiltonian. Particular attention is given to the impact of the repulsive three-body term on ground state energy estimates and mass concentration properties

We prove local laws for the resolvents of separable covariance matrices of the form $\mathcal Q=A^{1/2}XBX^*A^{1/2}$, where $X=(x_{ij})$ is a $p\times n$ random matrix whose entries $x_{ij}$ are i.i.d.~random variables with mean 0 and variance $n^{-1}$, and $A,B$ are deterministic non-negative definite symmetric (or Hermitian) matrices. Following the method developed in arXiv:1611.05364, we first establish a self-consistent equation for the resolvent of $\mathcal Q$ and use it to prove optimal local laws without the technical assumption $\mathbb{E}[x_{ij}^{3}]=0$, which was essential in the previous derivation of the local laws in arXiv:1809.04572. As an application of our local law, we compute the asymptotic distribution of the outlier eigenvalues for spiked separable covariance matrices, extending the corresponding result in arXiv:2008.11903.

We introduce new notions of controllability and observability for networked linear time-invariant (LTI) systems based on $\sigma$-semistability of quiver representations. Utilizing King's criterion for $\sigma$-semistability, we define a network generalization of the Kalman decomposition for networked LTI systems, which systematically decomposes the local and interconnection dynamics while respecting the underlying network structure. Furthermore, we present efficient algorithms for deciding the proposed controllability and observability of a given networked LTI system and for finding the Kalman-type decomposition. We also show efficient algorithms for deciding the $\sigma$-semistability of representations of acyclic quivers with self-loops if the weight $\sigma$ has the same sign for all vertices with self-loops. Such quiver representations and weights arise from networked LTI systems.

The Virasoro conjecture predicts a set of universal relations among all genera Gromov–Witten invariants of any smooth projective variety. The conjecture is well understood for semisimple theories, but remains largely open in the non-semisimple setting. We prove the genus-one Virasoro conjecture on the ambient state space of smooth Fano complete intersections in projective space. For most of these complete intersections, the big quantum cohomology is nowhere semisimple. We also generalize the wall-crossing formula for quasimap invariants with weighted markings to the equivariant twisted setting, allowing descendant insertions at light markings. Together with genus-one quantum Lefschetz for quasimaps with light markings, this wall-crossing formula provides the key bridge from the Gromov–Witten theory of the complete intersection to the semisimple equivariant twisted theory of the projective space.

We propose a differential step-down thresholding algorithm with randomized grid dithering for robust model order selection. It detects the value clustering of noise extrema. Our algorithm asymptotically achieves exact order recovery and prevents false alarms even under lowered detection thresholds.

Let $M$ be a finite-volume hyperbolic $d$-manifold with $d\ge3$. We prove that the number of primitive nonsimple closed geodesics has exponential growth rate strictly smaller than $d-1$. Consequently, asymptotically almost every primitive closed geodesic in $M$ is simple. In contrast, we show that on an arithmetic hyperbolic manifold of type~I, the unit vectors tangent to nonsimple closed geodesics are dense in the unit tangent bundle.

Let $X$ be a time-homogeneous one-dimensional diffusion whose coefficients satisfy the Engelbert–Schmidt conditions, and let $q$ be a bounded potential. We show that the associated Feynman–Kac semigroup is smooth in time and $C^1$ in space, with a locally absolutely continuous first spatial derivative; the Kolmogorov equation holds almost everywhere. If the drift, second-order coefficient, and potential are continuous, the Feynman–Kac value is $C^{1,2}$ in the interior. Thus, in the time-homogeneous one-dimensional setting, continuity suffices for interior $C^{1,2}$ regularity. We identify the corresponding semigroups with the killed and reflected Feynman–Kac functionals, treat nonzero, time-independent Dirichlet data, and give examples showing the sharpness of the continuity assumptions and the failure of the corresponding statement in two dimensions.

We prove two sharp monotonicity formulae for the natural Green's function of a closed three-manifold with positive scalar curvature, analogous to the Munteanu-Wang theorem for non-negative scalar curvature. We also show that both results may be interpreted as the monotonicity of a suitably defined "enclosed mass" of the sublevel sets of Green's function in the corresponding "conformal" blow-up of the manifold.

In this second chapter of the $\textit{Arithmetic of critical values}$ series (ACV), we study certain double covers $E_f\to\mathbb{P}^1$ whose branch locus coincides with that of a quartic polynomial $f$. We give a direct proof of the fact, already shown non-constructively in ACV I, that the elliptic curves $E_f$ admit a $3$-isogeny. Our methods are Galois-theoretic, and lead us to a thorough analysis of the Galois closure of $f:\mathbb{P}^1\to\mathbb{P}^1$. We exploit its rich geometry to prove a Selmer companionship theorem for the family $E_f$, allowing us to exhibit elements in certain Tate-Shafarevich groups which are visible in an abelian surface. We also give some dynamical and Diophantine applications of our constructions, as well as new examples of Jacobians isogenous to a power of an elliptic curve.

In this paper, we introduce the controlling $L_{\infty}[1]$-algebra of extended Rota-Baxter Lie algebras. As applications, we obtain the cohomology of extended Rota-Baxter Lie algebras. We also study abelian extensions and formal deformations by the lower cohomology group. Lastly, we get the notion of homotopy extended Rota-Baxter Lie algebras. Specifically, we present the controlling $L_{\infty}[1]$-algebra of modified Rota-Baxter Lie algebras and of Rota-Baxter Lie algebras with weight.

We establish an analogue of André Weil's classical theorem for irreducible nodal curves. Let \(X_0\) be an irreducible projective nodal curve. We prove that an indecomposable vector bundle or torsion-free coherent sheaf \(E\) on \(X_0\) admits a holomorphic logarithmic connection \(\nabla\colon E\to E\otimes\omega_{X_0}\) with respect to the dualizing sheaf if and only if \(\deg E=0\). Moreover, when \(\deg E=0\), such a connection can be chosen so that the induced logarithmic connection on the normalization has scalar residues \(\lambda\cdot I\) at one preimage of the node and \(-\lambda\cdot I\) at the other, for some \(\lambda\in\mathbb{C}\). Explicit one-parameter families of flat connections are constructed on the irreducible rational nodal cubic curve as an illustration.

We extend the Wigner-Eckart factorization of the spectral Boltzmann collision operator to polyatomic gases with continuous internal energy. Because internal energies are invariant under spatial rotations, the SO(3) reduction survives the Borgnakke-Larsen energy exchange, and the twelve-dimensional collision integral collapses onto a nine-dimensional kinematic core. The core splits into a sparse geometric tensor, evaluated exactly, and a dense physical tensor, integrated by singularity-resolving Gauss rules with an auxiliary Laplace representation of the fractional energy couplings. The quadrature attains near machine precision at the fractional exponents of real gases. The collision invariants are embedded exactly, preserving the translational-internal energy exchange. The factorization compresses the operator by three to nearly four orders of magnitude and accelerates its evaluation 40-fold over dense formulations. The method is validated against the exact monatomic limit, Landau-Teller relaxation, and an analytic frozen-channel Prandtl number, and it matches a published calibration of the same kernel for N2, CO, and H2.

In this paper, we study the long-time behavior of the viscous conservation law equation on the whole real line, subject to a white-in-time force and arbitrarily small damping and viscosity constants. We establish polynomial mixing rates in two settings: (a) a general super-quadratic flux with hyperviscous dissipation, where the growth of the nonlinearity is controlled by the order of hyperviscosity, and the force is sufficiently non-degenerate on a generic L2-basis; and (b) a general quadratic flux with standard second-order dissipation, and the force is non-degenerate on some unconditional basis. Both settings include the classical Burgers-type nonlinearity. The proof is based on an abstract polynomial coupling criterion developed in arXiv:2408.00592 and a Foiaş–Prodi estimate.

Let $A \in \mathbb{C}^{n \times n}$ have spectrum in the closed unit disk. Its maximal power growth $\operatorname{Power}(A) := \sup_{k \ge 0} \Vert{}A^k\Vert{}$ measures transient amplification, whereas the Kreiss constant $\operatorname{Kreiss}(A) := \sup_{\vert{}z\vert{}>1} (\vert{}z\vert{}-1) \Vert{}(zI-A)^{-1}\Vert{}$ measures the corresponding resolvent growth outside the disk. The classical finite-dimensional Kreiss theorem gives $\operatorname{Power}(A) \le en \operatorname{Kreiss}(A)$, and the linear dependence on $n$ is unavoidable for general matrices. We show that, for matrices selfadjoint with respect to an indefinite metric, the ambient dimension $n$ can be replaced by an effective dimension determined by the minimal polynomial and the inertia of the metric. Specifically, if $A^*J = JA$, where $J$ is a fundamental symmetry with inertia $(n-q,q)$, then $\operatorname{Power}(A) \le e \min\{d(A), 2q+1, 2(n-q)+1\} \operatorname{Kreiss}(A)$, where $d(A)$ is the degree of the minimal polynomial. Our proof requires no assumption on diagonalizability or reality on the spectrum. Instead, we associate each cyclic orbit with a finite-rank selfadjoint Hankel operator and transfer its rank and inertia to a coefficient estimate. Examples based on scaled nilpotent shifts show that the linear dependence on the smaller inertia index is asymptotically sharp, even when this index is negligible relative to the matrix size and $d(A)=n$. We also obtain scaled-disk decay estimates and weighted-norm extensions to arbitrary nonsingular Hermitian metrics.

We prove uniform-in-time quantitative relative entropy estimates for Kac's particle approximation of the three-dimensional spatially homogeneous Landau equation for Maxwellian molecules. Using the relative entropy method and logarithmic derivative estimates, we obtain a finite-time normalized entropy bound of order $N^{-1/2}$. The proof relies on explicit covariance identities and a new law-of-large-numbers estimate for the particle system using a new duality method. The uniform-in-time propagation of chaos is derived from an interpolation with an algebraic relaxation estimate toward equilibrium.

We establish boundary Harnack inequalities for non-negative solutions of the Kolmogorov equation \[ \mathcal Ku=\Delta_xu+x\cdot\nabla_yu-\partial_tu=0 \] in non-characteristic intrinsic Lipschitz graph domains whose defining functions may depend on the higher-order variable \(y_m\). Earlier theory required independence of \(y_m\). We replace this symmetry by quantitative asymptotic cylindricality, measured by a scale-invariant \(C^{0,1/3}_{y_m}\) defect. The perturbative comparison theorem applies whenever this defect is sufficiently small, and hence at all sufficiently small scales for \(C^{0,\alpha}_{y_m}\) graphs with \(\alpha>1/3\), as well as under a vanishing critical cylindrical modulus. We obtain local comparability and intrinsic Hölder continuity of quotients of positive solutions, together with a Hölder estimate for their logarithmic quotient. The proof combines local boundary estimates, valid without \(y_m\)-independence, with a global rigidity argument for cylindrical blow-up limits. The central issue is uniform balance of forward and backward reference values. If this balance fails at collapsing scales, maximal-scale selection and a second rescaling produce a non-negative polynomial-growth Dirichlet solution in an unbounded cylindrical domain that vanishes at the backward reference point but not at the forward one. We exclude this configuration using boundary-energy and mean-value estimates, finite-dimensionality of spaces of polynomial-growth solutions, and translation invariance in \(y_m\). The resulting finite-dimensional translation representation yields real-analytic dependence on \(y_m\). Propagation of zeros, analytic continuation along invariant fibres, and a Tikhonov uniqueness argument then force the limiting solution to vanish identically. The exponent \(1/3\) is critical because \(y_m\) has homogeneous degree three.

We study supersingular isogeny graphs of rank-two Drinfeld modules over $A=\mathbb{F}_q[T]$. For distinct finite primes $\mathfrak p$ and $\mathfrak q$ of $A$, we prove that the graph in characteristic $\mathfrak p$, with edges given by cyclic $\mathfrak q$-isogenies, is connected. The proof combines Gekeler's ideal-class correspondence with strong approximation to realize the graph as a quotient of the Bruhat–Tits tree of the homothety classes of $A_\mathfrak q$-lattices in $(\mathbb{F}_q(T))_\mathfrak q^2$. We also introduce the completeness number $E(\mathfrak p)$, the least integer such that the graph is complete for every prime $\mathfrak q\neq\mathfrak p$ with $\operatorname{deg}\mathfrak q\geq E(\mathfrak p)$, and derive explicit upper bounds using the Ramanujan–Petersson bound for the eigenvalues of operators associated to Brandt matrices over function fields. In particular, if $d=\operatorname{deg}\mathfrak p$, then $E(\mathfrak p)\leq 2d+2$, with sharper bounds depending on $q$ and the parity of $d$. This confirms a conjecture of Micheli and Papikian that the graph becomes complete once $\operatorname{deg}\mathfrak q$ is sufficiently large relative to $\operatorname{deg}\mathfrak p$. We further obtain parity-dependent lower bounds and prove that $E(\mathfrak p)\geq d+\log_q d-K$ for an absolute constant $K>0$ and sufficiently large odd $d$, thereby refuting the previously suggested bound $E(\mathfrak p)\leq d+1$. Finally, we prove a criterion yielding an algorithm to compute $E(\mathfrak p)$.

Quantum resonances and black-hole quasinormal modes (QNMs) are described by closely related mathematics: radiative boundary conditions, analytic continuation of Green functions or resolvents, poles on nonphysical sheets, and non-self-adjoint spectral representations. This article does not introduce that equivalence but reconstructs its genealogy. Nuclear resonance theory grew from radioactive decay, reaction cross sections, metastable compound states, and complex energies, whereas black-hole perturbation theory grew from spacetime stability and causal response. We place the two developments on a common chronology, from Sommerfeld's radiation condition and early nuclear-decay theory through Regge–Wheeler perturbations, Vishveshwara's 1968 dissertation and 1970 papers, Press's 1971 quasi-normal terminology, and the independent Aguilar–Balslev–Combes theory of analytic dilation, then follow their later mathematical reunification in scattering-resonance theory. Alongside the chronology, we give a working dictionary relating no-incoming and horizon-ingoing/infinity-outgoing conditions, pole energy and complex frequency, width and damping rate, residues and excitation factors, and nonresonant background with branch-cut and prompt contributions. The aim is to make the cross-disciplinary equivalence operational without flattening its history, and to distinguish mathematical equivalence from historical genealogy.

This study introduces and analyzes a three-step sequential decoupling algorithm designed to address nonlinear, fully coupled quasi-static thermo-poroelasticity systems incorporating convective transport. The finite element method is employed for spatial discretization and the backward Euler method for temporal discretization. The proposed sequential method has a higher computing efficiency than the fully implicit nonlinear numerical scheme, since it does not require any internal iterations. The well-posedness of the numerical solution is discussed by introducing a cut-off operator and the stability analysis of the algorithm is performed. Rigorous analysis yields optimal convergence order estimates for both spatial and temporal discretizations. In order to confirm the theoretical results and the effectiveness of the suggested approach, numerical experiments are finally carried out.

We study branching SDEs with law-dependent immigration and their mean-field particle approximations. Under a dissipativity condition and sufficiently weak interaction, a uniform propagation-of-chaos bound in time of order $N^{-1/2}$ is established. On every fixed finite time horizon, the same order of propagation of chaos holds for arbitrary finite interaction strength. A two-stage scaling limit connects continuous-time discrete-state mean-field birth–death processes to interacting branching diffusions and then to the nonlinear equation. For a logistic mean-field diffusion we prove a sharp criterion for extinction/non-extinction, and further show that weak enough interaction strength is necessary for a uniform-in-time approximation.

We compute the homological Euler characteristics of $\mathrm{SL}_4(\mathbb{Z})$ and $\mathrm{GL}_4(\mathbb{Z})$ with coefficients in arbitrary irreducible rational highest-weight representations. Applying Wall's formula, we combine the orbifold Euler characteristics of centralizers of torsion elements with traces computed using the Jacobi-Trudi identity to derive explicit formulas and rational generating functions. Consequently, these Euler characteristics are quasi-polynomial functions of the highest-weight parameters, of total degree at most two. We further derive degreewise vanishing results and parity-sensitive lower bounds for the dimensions of cohomology groups. The two extensions of an $\mathrm{SL}_4(\mathbb{Z})$-coefficient system to $\mathrm{GL}_4(\mathbb{Z})$ yield sharper bounds, which grow linearly or quadratically in explicit infinite families. For symmetric powers, combining our formulas with Horozov's calculation of the determinant-twisted summand yields exact identities and lower bounds for the untwisted summand, together with a conjectural degreewise description of its cohomology.

In this paper, we consider the Cauchy problem for the incompressible magnetohydrodynamic equations with the Coriolis force in the three-dimensional whole space. For large initial data, it is known that the Cauchy problem admits a unique global solution in critical Sobolev space $\dot{H}^{1/2}(\mathbb{R}^3)$ provided that the speed of rotation is fast enough. By using delicate dispersive estimates, we show refined decay estimates of the velocity field and magnetic field. These decay rates significantly improve the ones obtained by Kim [J. Differential Equations., 2022] in the subcritical Sobolev framework $H^s(\mathbb{R}^3)$ with $1/2<s<3/2$.

Uncertainty quantification from finite data is central to machine learning, optimization, and automation systems, where decisions must remain reliable under limited samples and test-time distribution shift. Conformal prediction (CP) and distributionally robust optimization (DRO) offer two complementary approaches: CP constructs data-dependent prediction sets with distribution-free finite-sample validity under exchangeability, while DRO optimizes worst-case performance over an ambiguity set around an empirical distribution. We develop a unified probabilistic perspective on CP and DRO by viewing both as ways to turn finite calibration data into a data-dependent quantile estimator that a test score falls below with high probability. From this perspective, CP and DRO correct the empirical quantile along two coordinates of the same family of estimators: CP inflates the quantile level, whereas DRO shifts the quantile value through an ambiguity radius. Both methods provide the same calibration-conditional guarantee for the true distribution, requiring the target coverage to hold with high probability over the calibration sample. Their constructions differ, however: CP uses a closed-form, distribution-free level correction, while DRO uses a value-space correction whose certified radius depends on properties of the unknown distribution and additionally guarantees coverage uniformly over the ambiguity set. This distinction emerges in the tails of the score distribution. Because CP relies on sparse upper-tail order statistics of the calibration samples, its level inflation barely moves the estimator when those samples are dense near the target quantile but overshoots when they are sparse, whereas a well-chosen DRO radius corrects in value space and may avoid this overshoot.

Adapting the analytic method of Gross and Zagier, Roskam proved a prime-factorization formula for the norm of the difference of two level-two Weber singular moduli. Independently, Yang and Yin obtained an equivalent formula using Borcherds lifts. More precisely, the formula concerns the norm of \[ \omega_{2}\left(\frac{-1+\sqrt{d_{1}}}{2}\right) - \omega_{2}\left(\frac{-1+\sqrt{d_{2}}}{2}\right) \] for coprime negative fundamental quadratic discriminants $d_{1},d_{2}\equiv1\pmod 8$, where \[ \omega_{2}(\tau) = 2^{12}\frac{\eta(2\tau)^{24}}{\eta(\tau)^{24}}, \] and $\eta(\tau)$ denotes the Dedekind eta function. In this work, we revisit this formula from the perspective of arithmetic intersection theory and give a new proof.