Papers

We identify Gaussian curvature as a local mechanism for cubic cancellation of Laplace eigenfunctions on Riemannian surfaces, without arithmetic or dynamical assumptions. For $L^2$-normalized local solutions at frequency $\lambda$, cubic integrals against smooth compactly supported weights tend to zero whenever the curvature is nonzero almost everywhere where the weight is nonzero. Finite-order nondegeneracy yields sharp polynomial decay rates. In particular, on closed surfaces with nowhere-vanishing curvature, we obtain the rate $O(\lambda^{-1/6})$, attained by spherical harmonics on the round sphere. This proves the cubic case of the Gaussian moment conjecture for all normalized real eigenfunctions on closed negatively curved surfaces. The local proof rests on an exact identity relating the reduced phase at a Fermat point to Gaussian curvature. On complete hyperbolic surfaces with positive systole, we also prove global $O((\log\lambda)^{-3/2})$ decay for $L^2$-normalized eigenfunctions and smooth weights with bounded covariant derivatives through order two, without assuming finite area.

We determine the Poisson algebras generated by partial Casimirs of undeformed and non-standard $q$-deformed $\mathfrak{sl}(2)$ coalgebras, respectively. The derivation is performed on the abstract algebraic level and does not depend on any particular symplectic realization. For the 3- and 4-site cases the constructions associated with the undeformed $\mathfrak{sl}(2)$ produce the rank-one and rank-two Racah Poisson algebras, respectively, while those with the non-standard deformed $\mathfrak{sl}$ yield finite quadratic Poisson algebras generated by connected partial Casimirs. At both the three and four sites, the final closure coefficients do not explicitly depend on the deformation parameter. The results hold at the Poisson coalgebra level and hence apply to any realizations which preserve the defining brackets and coproducts. Canonical symplectic realizations give rise to Hamiltonians in spaces of non-constant curvature. For the Stäckel $q$-deformed Smorodinsky-Winternitz model we show explicitly how the universal coalgebra sector is completed to maximal superintegrability. An exact two-block identity, proved directly in the localized Poisson algebra without appealing to a symplectic realization, controls the additional Stäckel integrals. In three dimensions, it generates a quadratic Poisson block with its own Casimir relation; in four dimensions, its coupling to the full left/right coalgebra family is organized into three nested quadratic blocks, and the same identity extends this nested coupling to arbitrary $N$. Explicit higher-site partial Casimirs and factorized bracket generators are collected in the appendices.

A ghost-cell immersed boundary method is proposed for the two-dimensional shallow water equations with a variable bed (topography), solved with an explicit Godunov-type finite volume scheme (unsplit second-order MUSCL, HLLC flux, hydrostatic reconstruction) on Cartesian grids. Ghost-cell methods leak mass through the wall. To find a robust and conservative treatment, eight constructions of the ghost values and a cut-cell form of the continuity equation were applied to seven test problems, which include wetting and drying on sloping beds and a laboratory dam break. In the resulting scheme, the ghost values are interpolated bilinearly at the image point; solid and dry nodes are excluded, and the weights of the remaining nodes are renormalized. Mass is conserved by solving the continuity equation in cut-cell form, while the momentum equations are left unchanged. The proposed scheme is about as accurate as the ghost-cell method without this modification and preserves a lake at rest whose shoreline crosses an immersed wall. Because only the continuity equation is modified, it avoids the treatment of the momentum equations on cut cells (wall pressure, cell merging) that a full cut-cell method requires.

This paper develops a unified framework for distributional regression with arbitrary response variables. Rather than relying on scalar thresholds or Euclidean structure, we model conditional event probabilities indexed by measurable sets, which extends distributional regression to general response spaces while retaining a direct connection to classical event-based formulations. Within this framework, we study two complementary estimators: a preliminary estimator based on separate regression of event indicators, and a constrained estimator that enforces inclusion-consistent predictions through a directed acyclic graph (DAG) over a finite set family. We establish general error bounds for both estimators and further specialize the results to dense ReLU networks, yielding explicit convergence rates overcoming the curse of dimensionality. Simulation studies and real-data experiments further demonstrate the practical effectiveness of the proposed framework.

The $r$ largest observations of a block form a dependent random vector whose support moves with all three parameters of the $r$-largest generalized extreme-value ($r$GEV) model. For a fixed number $r$ of order statistics, we show that the model is differentiable in quadratic mean if and only if $k_0<1/2$, where $k=-\gamma$ is the shape parameter; the threshold is thus the same as for block maxima, because only the largest or the smallest retained order statistic can reach the moving boundary. In this range the Fisher information is positive definite and decomposes into the block-maximum information plus contributions of the lower order statistics, and it equals the negative expected Hessian, which we prove without using any estimator. The model is then locally asymptotically normal, whereas for $k_0\ge1/2$ root-$n$ local asymptotic normality fails. Numerically, using the five largest order statistics instead of the block maximum alone lowers the efficient asymptotic standard error by about $45\%$ for the shape parameter and by $20$–$24\%$ for the 100-year return level in heavy-tailed cases ($k_0\le-0.2$), but by only about $2\%$ for the return level when $k_0=0.4$.

Let $0\leq\alpha<n$, and let $\sigma$ and $\omega$ be locally finite positive Borel measures on $\R^n$, possibly with common point masses. Suppose that one measure is supported on a compact regular $C^{1,\mathrm{Dini}}$ curve. We prove that the $\alpha$-fractional Riesz vector maps $L^2(\sigma)$ to $L^2(\omega)$ if and only if the two $\cA_2^\alpha$ conditions with holes, the two punctured $A_2^\alpha$ conditions, and the forward and dual testing conditions over rotated cubes hold. The theorem includes curves whose tangent is not Hölder continuous of any positive order. Straightening such a curve produces a kernel whose first derivatives need not satisfy the Hölder estimates required by the energy-side-condition $T1$ theorem. We prove both strong energy conditions in the straightened coordinates using only a sufficiently small graph slope, and show that these conditions are preserved under biLipschitz changes of variables. The Dini condition makes the errors in the kernel and testing comparisons summable. We then transfer the energy conditions back and apply the $T1$ theorem to the original smooth Riesz kernel. We also obtain a two-weight criterion for Green gradients with curve-supported sources. For arclength measure on the curve and observation measure $\dist(x,\Gamma)^\beta\,dx$, the range is $\beta>n-3$ in a bounded tube and $n-3<\beta<n-2$ in $\R^n$. The endpoints fail logarithmically whenever the measure is locally finite. In dimension three, the corresponding whole-space range for the Biot–Savart field with arbitrary vector densities is $0<\beta<1$.

Linear complementary dual (LCD) codes are linear codes that intersect their dual trivially. We study double circulant (Euclidean) LCD codes over the finite chain ring $R=\mathbb F_2+u\mathbb F_2$, where $u^2=0$. Using the Chinese Remainder Theorem and the factorization of $x^n-1$, we derive an explicit enumeration formula, with a separate treatment of the constituent corresponding to $x-1$, which is not obtained by a formal specialization of the previously known counting formula over $\mathbb F_q[u]/(u^k)$. Assuming Artin's primitive root conjecture for $2$, we show that the binary Gray images contain an asymptotically good family of rate $1/2$ whose relative distance can approach $H_2^{-1}(1/4)$ from below. More importantly from the constructive viewpoint, by combining Gray images with a known extension construction, we obtain a binary LCD code with parameters $[29,15,7]$ and construct two explicit permutation-inequivalent binary LCD codes with parameters $[45,23,9]$.

We study the convexification of sets defined by box constraints and bilinear inequality constraints in which all bilinear terms share a common variable. For this structured class of nonconvex sets, we provide a complete characterization of the extreme points of the convex hull and derive an explicit description of all facet-defining inequalities, thereby obtaining a full polyhedral description of the convex hull. In addition, we develop a polynomial-time separation algorithm for the facet-defining inequalities and illustrate its effectiveness through examples. These results offer both structural insight into the geometry of the convex hull and algorithmic tools for identifying violated inequalities.

This paper develops a sector-constrained guidance (SCG) law for interception of a maneuvering target while maintaining a prescribed approach geometry. The line of sight is required to remain within a specified angular sector behind the target throughout the engagement, resulting in a state-constrained guidance problem. To enforce this constraint, the directional angle between the target heading and the line of sight is mapped from the prescribed sector to an unconstrained coordinate using a sigmoid diffeomorphism. A guidance law is then designed in the lifted coordinate to generate a desired transverse velocity and corresponding pursuer lead angle. The desired lead angle is tracked through an inner-loop turn-rate controller, which is implemented through the normal-acceleration command of a nonlinear point-mass model. This construction embeds the sector constraint directly into the guidance law and does not require online constrained optimization. Numerical simulations against a maneuvering target show that SCG achieves interception while maintaining the prescribed rear-sector constraint, including when the pursuer normal-acceleration command is subject to actuator saturation. Comparisons with proportional guidance under the same engagement conditions demonstrate that proportional guidance may achieve interception but does not maintain the prescribed directional constraint.

Let \(A,B\subset\R^2\) be nonempty Borel sets. We prove that \[ \dimH(gA-B)\ge\min\{\dimH A+\dimH B,2\} \] for almost all \(g\in\mathrm O(2)\). This bound is sharp in general, and the proof gives a corresponding statement for the lower Hausdorff dimension of convolutions of Frostman measures. We also prove that \(\LL^2(gA-B)>0\) for almost all \(g\) if both dimensions exceed \((\sqrt{13}-1)/2\). The second result follows from an angular decomposition, refined decoupling and a two-ends Furstenberg estimate. Some applications on intersection patterns are also addressed.

We prove boundary Hölder regularity for fractional porous medium-type equations in bounded smooth domains with zero exterior data. For $0<\sigma<1$, $n>2\sigma$, and $m>1$, we show that nonnegative weak dual solutions $v$ of $\partial_t v+(-\Delta)^\sigma(v^m)=0$ satisfy \[ \frac{v}{d^{\sigma/m}}\in C_{x,t}^{\alpha,\frac{\alpha m}{\sigma(m+1)}} \] up to the boundary and away from the initial time, for some $\alpha\in(0,\sigma)$, where $d$ denotes the distance to the boundary. More generally, we establish boundary Hölder estimates for $u/d^\sigma$ for nonnegative viscosity solutions of $\partial_t u=u^\gamma Lu$, with $0<\gamma<1$ and initial data comparable to $d^\sigma$. Here $L$ is a symmetric stable operator of order $2\sigma$ whose measurable angular density is bounded above and below by positive constants. Our proof combines comparison principles and boundary barriers with a blow-up and compactness argument based on a Liouville theorem for ancient solutions in a half-space.

We prove a Cohen-Lenstra type result commenting on the distribution of class group structures amongst function fields of superelliptic curves over finite fields. We also prove a result commenting on the sparsity of such superelliptic curves whose zeta functions vanish at a fixed complex number. These results are proven via point-counting techniques on certain Hurwitz schemes generalizing techniques of Ellenberg-Venkatesh-Westerland and Ellenberg-Li-Shusterman. To obtain these point counts, We prove a unitary big monodromy theorem generalizing big monodromy theorems of Yu and Achter-Pries by appealing to an arithmeticity theorem of Venkataramana.

This paper presents a constraint-enforcing control framework for a class of discrete-time strict- feedback nonlinear systems. The objective is to guarantee closed-loop stability while ensuring forward invariance of a prescribed safe set defined by state constraints. The pro- posed approach transforms the constrained control problem into an equivalent unconstrained one through smooth constraint-lifting mappings constructed using strictly increasing sigmoid functions. Controller synthesis is then performed in the lifted coordinates, enabling recursive backstepping design while preserving the admissibility of the constrained states. Conditions on the controller gains are derived to guarantee both asymptotic stability of the closed-loop system and forward invariance of the admissible domain of the lifting functions. The analysis also establishes a conditional deadbeat property for the second backstepping step once the system trajectory enters a region in which the lifting-domain admissibility conditions are satisfied. Numerical simulations of a constrained Roup oscillator demonstrate the effectiveness of the proposed method in enforcing state constraints while tracking a reference command.

We prove that for every choice of five positive masses the planar five-body problem has only finitely many central configurations up to rotation, translation and scaling. This answers Smale's sixth problem for five bodies. Albouy and Kaloshin proved the statement for all masses outside a closed algebraic subset of codimension two of the mass space. We keep their classification of singular sequences and replace the construction of the exceptional set by a direct exclusion: we use finite boundary charts for seven of the eight diagrams that can carry an unbounded four-product, and an exhaustive order census for Diagram 5. We derive the necessary mass conditions where needed and show that no germ with an unbounded four-product lies on a curve of normalized central configurations through a positive configuration. Five diagrams are excluded for all complex germs, two on curves through real configurations, and one on curves through positive configurations. Parts of the argument are computer-assisted: exact algebra, interval arithmetic and exact linear programming.

This article develops matrix-free Q-less QR algorithms for orthogonalizing tall linear operators. Our main algorithm, CQRRTO, uses randomized preconditioning to make a single Cholesky QR pass numerically safe for any numerically full-rank operator, overcoming the stability issues of Cholesky QR and related methods that operate on the Gram matrix. Compared with deterministic stabilized variants such as Q-less Cholesky QR2 (reorthogonalized Cholesky QR), it remains numerically stable under substantially weaker conditioning assumptions while requiring fewer operator applications. Matrix-free Q-less QR can be useful for improving the numerical robustness of least-squares solves via the normal equations, spectral approximations in a low-dimensional basis, and inversion of operators with low-rank updates via the Woodbury identity. We present a componentwise stability analysis of preconditioned Q-less Cholesky QR that applies to matrix-free operators and is modular with respect to the preconditioner, yielding guarantees for Q-less Cholesky QR, Cholesky QR2, shifted Cholesky QR3, and CQRRTO as special cases. All methods are implemented in the open-source RandLAPACK library.

We use centered differences to construct Chernoff families and derive even asymptotic expansions for their iterated products. Richardson extrapolation is then used to cancel polynomial part and thereby raise the convergence order. As illustrations, we apply the construction to uniformly parabolic, Schrödinger-type second-order operators and to a third-order dispersive operator, obtaining, at extrapolation depth $m$, the rates $ O(n^{-2m}) $ on the Sobolev scales $H^{2m+\delta}$, $H^{4m}$, and $H^{5m}$, respectively, with arbitrary $\delta>0$. In the analytic category, quantitative all-order estimates allow the extrapolation depth to grow with the base resolution and yield the super-fast rates $e^{-cn^2}$ for the two second-order models and $e^{-cn} $ for the third-order model. The constructions are explicit in terms of the coefficients of the underlying differential operators and do not use the target semigroup in their definition. This provides a constructive positive answer to the challenge of super-fast Chernoff approximation formulated by Remizov in Problem 30 of arXiv:2410.00416.

We introduce a multiplicity one defect that quantitatively measures the failure of multiplicity one for the Jacobian of the Shimura curve \(X^D(M)\) attached to an indefinite rational quaternion algebra of discriminant \(D\) and level \(M\), localized at a maximal ideal of the Hecke algebra. By constructing a certain lattice inside the local Tate module, we obtain an explicit upper bound for the defect in terms of congruences between eigenforms and the lattice-Tate module discrepancies associated with the individual newforms. We then show that the local ratio between the congruence exponent and the modular exponent of modular abelian varieties attached to newforms is bounded above by the multiplicity one defect. Finally, we examine the relation with the Gorenstein property and give an explicit example at level \(126\) where the local Hecke algebra is Gorenstein even though multiplicity one fails. In this example, we compute the multiplicity one defect to be \(3\), which is also the local congruence–modular exponent ratio.

We develop a unified multi-scale framework for sparse domination of multilinear operators beyond the multilinear Calderón–Zygmund theory. More precisely, under mild hypotheses on a support condition, weak type and restricted strong type uniform bounds for finite partial sums, and single-scale improving and regularity estimates, we establish $(r_1, \ldots, r_m, s')$-sparse form bound for a family of localized $m$-linear operators uniformly over all finite partial sums. This sparse bound yields weighted norm inequalities for the corresponding multilinear Muckenhoupt classes $A_{\vec{p}, (\vec{r}, s)}$, including the quasi-Banach range. As applications, we determine the optimal open sparse range for multilinear rough singular integrals, and characterize sparse domination for bilinear pseudo-differential operators. Further applications include sparse bounds for bilinear oscillatory Fourier multipliers, bilinear spherical maximal operators, and bilinear Bochner–Riesz means.

We study nematic liquid crystal phases confined to bounded, two-dimensional (2D) domains, in the reduced Landau-de Gennes (rLdG) theoretical framework. The rLdG framework is equivalent to the celebrated Ginzburg-Landau (GL) theoretical setting, under certain assumptions. In the asymptotic limit of large domains (or low temperatures), we recall the concept of the finite-dimensional renormalized energy in the GL theory [Bethuel et al., GL Vortices, Prog. Nonlinear Differ. Equ. Appl. 13, Birkhäuser, 1994.] and generalize it to polygons with sharp corners. The renormalized energy is fully defined by the triplet, $(\mathbf{a},\mathbf{c}, \mathbf{d}$), of topological degrees of the interior defects ($\mathbf{d}$) and the winding number contribution of polygon vertices ($\mathbf{c}$) and interior defect locations ($\mathbf{a}$), subject to sum rules dictated by the imposed Dirichlet boundary condition. We use the critical points of the renormalized energy to construct initial conditions for the rLdG solver, recover known energy-minimizing and non-energy minimizing saddle points of the rLdG energy on a disc, regular hexagon and rectangle along with the discovery of previously unreported rLdG solution branches. The renormalized energy approach is computationally substantially more efficient than a direct PDE-solver in the rLdG setting but misses solution branches with line defects and finite-size effects.

We consider the three-dimensional Fröhlich polaron at fixed coupling strength $\alpha>0$ and total momentum $P \in \bbR^3$. We show that the bottom of the spectrum reaches the essential spectrum at a finite value of $|P|$, implying that the fiber Hamiltonian has no ground state at sufficiently large total momentum. This extends the known weak-coupling result to arbitrary coupling. Our approach describes states near the spectral threshold by a fully interacting polaron with small residual momentum and one additional phonon carrying almost all of the total momentum.

We find a new presentation of the orbifold fundamental group of the moduli space of cubic surfaces, based on the Artin group of the Petersen graph. This group has two well-known quotients, which appear in the simplest possible way: killing the squares resp. sixth powers of our generators yields the Coxeter group $W(E_6)$ resp. a certain arithmetic group in $\operatorname{PU}(1,4)$. Along the way we present a certain central extension by $\mathbb{Z}$, of the genus zero mapping class group with any number of punctures. The $6$-puncture case is essential in this paper, and the general case may have independent interest.

An exactly solvable lattice with periodic inverse-square singularities exhibits two physically distinct mechanisms of spectral-gap closure. The model arises from the generalized trigonometric Pöschl-Teller potential in the coupling range where both local solutions at each singularity are square-integrable. A transmitting, current-preserving self-adjoint matching condition connects neighboring cells and generates a nontrivial Bloch spectrum. For a generic lattice with two inequivalent singular sublattices, the spectrum contains two alternating families of gaps, and the band edges reproduce the scale-invariant spectra of the associated finite singular system. One gap family closes when the two sublattices become equivalent. In this symmetric limit, however, the primitive period halves, so the apparent band crossing results from Brillouin-zone folding. A different family closes at a critical but genuinely asymmetric configuration. Because the primitive period is preserved, adjacent bands then meet at the center of the original Brillouin zone in a genuine massless Dirac-like cone. Near either critical limit, a small finite gap acts as an effective Dirac mass: the dispersion remains quadratic near a band edge but crosses over to a massive Dirac-like form beyond that narrow region. The band curvatures define effective electron and hole masses. Exact Bloch eigenfunctions further reveal that isospectral lattices can display different single-particle probability densities. This model therefore provides an analytically controlled one-dimensional setting in which conventional quadratic carriers, massive Dirac-like band dispersions, and a tunable massless contact coexist within the same singular lattice.

We prove the Fourier entropy–influence inequality for every linear threshold function, with a constant below $8.937$ and no restrictions on its weights or threshold. The argument applies to regular Boolean functions and gives logarithmic bounds for their Boolean combinations and for unions with bounded variable overlap. Character multiplication preserves the estimates and yields bounds by influence minus variance. Under arbitrary product measures, symmetric thresholds satisfy the strengthened inequality needed for read-once composition, uniformly in depth and fan-in. A joint description of each Fourier set refines the overlap estimate and recovers constant bounds for addressing. Exact selector identities extend the bound to orthogonal multiplexers with shared data. Rare activation preserves the optimal constant of unrestricted FEI. The proofs combine pivotal antichains, spectral reference measures, and product-measure composition.

The elephant random walk (ERW) is a discrete-time random walk with complete memory. At every step the walker chooses one of the previous increments at random and then either repeats with a constant probability $p\in [0,1]$ or reverses it. Since its introduction by Schütz and Trimper, the model has become a useful probabilistic framework for studying long-range dependence, anomalous diffusion, reinforcement, and phase transitions. This paper proposes a generalized ERW model with time-dependent memory. In this generalized model, the usual constant memory parameter $p$ in the standard ERW is allowed to change with time. The cental limit theorems, and the law of the iterated logarithm and some other strong laws are obtained.

We revisit the online submodular welfare maximization problem (OSW), a vast generalization of numerous well-studied online allocation problems, including online bipartite matching, display ads and AdWords. A natural greedy OSW algorithm achieves a competitive ratio of $1/2$ [LLN06], and $1-1/e$ under i.i.d. arrivals [KPV13]. Both bounds are tight for polynomial-time algorithms unless NP=RP [KPV13]. The information-theoretic limits for OSW, in contrast, remained open. This prompted researchers to ask for tighter upper bounds [Mehta13] or conjecture the existence of better-competitive (slow) algorithms [Huang23, LTX+23, CHS24, Huang2024, HTW24}. We resolve Mehta's question and refute the subsequent conjectures, proving that the greedy algorithm is in fact tight for both adversarial and i.i.d. arrivals, regardless of computation time. Seemingly contradicting recent $(1-1/e)$-competitive fractional algorithms for OSW [HJP+24,Patton26], we show that this impossibility even holds for fractional algorithms, provided their outputs correspond to marginals that are roundable in hindsight. Conceptually, our proofs can be seen as identifying that a bottleneck for OSW, even under (deterministic) adversarial arrivals, is the inherently random notion of correlation gap [ADSY12].

We show that if an automorphism $\pi$ of the truth-table degrees is induced by a Baire measurable function $F\colon 2^{\omega}\to 2^{\omega}$, or merely induced on some comeager set by a continuous function, then $\pi$ is the identity. The same results hold for the weak truth-table degrees. The argument is an adaptation to truth-table reducibility of OpenAI's recent proof that the Turing degrees are rigid.

The knot quandle, introduced independently by Joyce and Matveev, admits a geometric description in terms of nooses. Motivated by this description, we introduce the loop fundamental quandle $LQ(X,A,\ast)$ of a pair $(X,A)$ with a fixed base point $\ast\in X$. Its elements are represented by pairs consisting of a loop in $A$ and a path joining the base point of the loop to the fixed base point $\ast$. More generally, for a loop condition $P$ on $A$, invariant under free homotopy, we define the loop fundamental quandle $LQ(X,A,\ast;P)$. We establish functoriality and homotopy invariance of the loop fundamental quandle and show that it admits a natural augmented quandle structure over $\pi_1(X,\ast)$. For a knot $K$ in $S^3$, applying the construction to its exterior gives the knot loop quandle $LQ(K,\ast;P)$. Appropriate choices of the loop condition recover the knot quandle, its $n$-th power quandles, and the full knot quandle. The construction is inspired by Joyce's noose-theoretic approach, but allows loop data that need not arise as the boundary of a disk satisfying the noose condition.

A quickest change detection problem is studied where the decision-maker has access to multiple experiments, but can choose only one experiment at a time to detect the change. The true post-change distribution under each experiment is unknown, but known to belong to a family of distributions. It is assumed that each of these families of distributions has an element that is least favorable in a well-defined sense. An algorithm is proposed that is designed using the least favorable distributions. It is shown that the algorithm is first-order asymptotically optimal, as the rate of false alarms goes to zero. This algorithm minimizes the worst-case average detection delay over all post-change distributions within the assumed families, subject to a constraint on the average run length to false alarm and constraints on the average fraction of time each experiment is performed before change.

Nonlinear coupled dispersive models arise in numerous physical settings, including quantum electrodynamics, nonlinear optics, and plasma dynamics. In the nonrelativistic limit, such systems exhibit rapid temporal oscillations of wavelength $\mathcal{O}(\varepsilon^{2})$, controlled by a dimensionless parameter $0<\varepsilon\ll 1$. Moreover, owing to the Yukawa-coupled interaction terms, numerical schemes that simultaneously achieve high-order uniform accuracy and approximately preserve the mass and the energy are essential for reliable simulations. In this work, we propose a class of fully discrete schemes for nonlinear coupled dispersive models in the nonrelativistic regime that achieve higher-order uniform accuracy and approximately preserve both invariants over long times. Spatial discretization is based on the Fourier pseudospectral method, and time advancement employs two-scale exponential Runge-Kutta methods. Numerical experiments on the highly oscillatory Klein-Gordon-Dirac and Klein-Gordon-Schrödinger equations in one, two, and three dimensions confirm that the resulting schemes attain the theoretically predicted uniform accuracy and the long-time near-conservation of both invariants.

We propose a conservative local neural operator (CLNO) for accurate and generalizable autoregressive prediction of compressible flows. Building upon the locality and translation invariance of the local neural operator (LNO), CLNO introduces a conservative evolution framework based on local subdomains of fixed physical size, separating local mean-state evolution from fine-scale reconstruction. The flow is encoded into subdomain-integrated conservative states and fine-scale features, while a shared local network predicts interface fluxes for finite-volume updates that enforce discrete conservation. A convolutional reconstruction network recovers fine-scale structures using zero-mean basis functions, preserving the updated local means and providing information for subsequent evolution. To accommodate different physical parameters, Reynolds–Mach number conditioning modulates the encoding, flux-prediction, and reconstruction networks, enabling a single model to learn parameter-dependent dynamics and generalize to unseen parameter combinations without retraining. CLNO supports prediction across spatial resolutions, computational domain sizes, and boundary conditions. Numerical experiments demonstrate improved generalization and competitive or higher accuracy compared with LNO, including order-of-magnitude error reductions in representative cases, at approximately three to five times the inference cost on periodic benchmarks. These results demonstrate accurate, generalizable, and physically consistent flow prediction through conservative local evolution and fine-scale reconstruction.