For a finite reflection subgroup $G\leq O(n+1,1,\mR)$ of the conformal group of the sphere with standard conformal structure , we geometrically derive differential-difference Dunkl version of the series of conformally invariant differential operators with symbols given by powers of Laplace operator. The co…
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48 results for “finite difference operator”
Generalizes neural networks for infinite-dimensional mappings, including PDE solutions.
problem Learning mappings between infinite-dimensional spaces and finite-dimensional approximations.
method Graph kernel network architecture with message passing for kernel integration.
result Competitive performance compared to state-of-the-art solvers for PDEs.
Proposes a new model for traffic flow on directed graphs.
problem Modeling advection on directed graphs for traffic flow.
method Reformulates graph advection operator as finite difference scheme; proposes DGAMGP model.
result Effective modeling of traffic flow and uncertainty as an advective process.
The paper analyzes off-policy TD-learning using generalized Bellman operators and provides finite-sample bounds.
problem High variance in off-policy TD-learning due to importance sampling.
method Derives finite-sample bounds for off-policy TD-like algorithms using generalized Bellman operators.
result First-known finite-sample guarantees for several off-policy TD algorithms.
Generative classifiers' properties are linked to linear constraints.
problem Understanding the Markov property in generative classifiers.
method Characterization of discrimination functions using linear constraints and a second order finite difference operator.
result Discrimination functions of undirected Markov network classifiers are characterized by sets of linear constraints.
Asymptotic and Finite Sample Analysis of Nonexpansive Stochastic Approximations with Markovian Noisecs.LG
This work analyzes nonexpansive stochastic approximations with Markovian noise, proving convergence in reinforcement learning.
problem Applying stochastic approximation to reinforcement learning settings with nonexpansive operators.
method Investigates nonexpansive stochastic approximations with Markovian noise, providing asymptotic and finite sample analysis.
result First-time proof of convergence for classical tabular average reward temporal difference learning.
Unified framework for Arnold-type invariants via dual complexes and finite-difference structures.
problem Study of Arnold-type invariants of immersed curves and surfaces.
method Framework on dual complexes, locally normalized maps, finite-difference structures, and Shumakovitch-type identities.
result Unified evaluation of Arnold-type invariants and on dual skeleta.
Most of the 50-year history of the study of the set of knot concordance classes, C, has focused on its structure as an abelian group. Here we take a different approach, namely we study C as a metric space admitting many natural geometric operators, especially satellite operators. We consider several knot concordance sp…
Study on CR Paneitz operator on non-embeddable CR manifolds.
problem Understanding the CR Paneitz operator on non-embeddable CR manifolds.
method Analysis of the CR Paneitz operator's properties on non-embeddable CR manifolds.
result CR Paneitz operator on the Rossi sphere has infinitely many negative eigenvalues.
Pricing Derivatives under Self-Exciting Dynamics: A Finite-Difference and Transform Approachq-fin.MF
A new method for pricing derivatives using self-exciting dynamics and finite-difference transforms.
problem Pricing derivatives with accumulated marks using a self-exciting marked point process.
method Derive discounted pricing equation as a PIDE, transform to one-dimensional PIDEs, use Laplace/Fourier transform, approximate jump term, solve using finite difference scheme.
result Efficiently price derivatives with accumulated marks using a novel finite-difference and transform approach.
Study essential spectrum of differential operators on geometrically finite orbifolds.
problem Analyzing the essential spectrum of differential operators over specific geometric structures.
method Investigates first order and Laplace type elliptic differential operators on Riemannian vector bundles over geometrically finite orbifolds.
result Discovers properties of essential spectra for these operators.
Study of unitary and groupoid orbits of normal operators, focusing on manifold structures and spectral conditions.
problem Understanding the manifold structures of orbits of normal operators under different norm topologies.
method Unified treatment of unitary and groupoid orbits, using moment maps and conditional expectations.
result Differentiable structures for orbits and necessary spectral conditions for norm closure and submanifold properties.
A discrete (finite-difference) analogue of differential forms is considered, defined on simplicial complexes, including triangulations of continuous manifolds. Various operations are explicitly defined on these forms, including exterior derivative and exterior product. The latter one is non-associative. Instead, as ant…
We investigate the Dolbeault operator on a pair of pants, i.e., an elementary cobordism between a circle and the disjoint union of two circles. This operator induces a canonical selfadjoint Dirac operator on each regular level set of a fixed Morse function defining this cobordism. We show that as we approac…
In this article, a compact finite difference method is proposed for pricing European and American options under jump-diffusion models. Partial integro-differential equation and linear complementary problem governing European and American options respectively are discretized using Crank-Nicolson Leap-Frog scheme. In pro…
Estimates small eigenvalues for geometrically finite manifolds.
problem Estimating small eigenvalues of Schrödinger operators.
method Geometrically finite manifolds, Riemannian vector bundles.
result Estimates the number of small eigenvalues.
Unified analysis of stochastic iterative algorithms using Lyapunov functions.
problem Analyzing convergence of stochastic iterative algorithms for fixed-point equations.
method Lyapunov-based techniques for finite-time analysis of stochastic approximation algorithms.
result Unified mean-square convergence guarantees for various algorithms.
The paper proposes a method to sample quantum field configurations using neural operators and flows.
problem Sampling lattice field configurations from Boltzmann distributions in quantum field theories.
method Approximating a time-dependent neural operator to map between free and target theories, discretizing to a normalizing flow, and training to diffeomorphism.
result The method can generalize to larger lattice sizes when pre-trained on smaller ones, improving efficiency.
Improved bounds for Carleson-Sjölin operators on manifolds with specific curvature conditions.
problem Bounding Carleson-Sjölin operators on manifolds with special curvature conditions.
method Two different methods: one using distance function conditions and the other using contact orders of oscillatory integral operators.
result Improved bounds for Carleson-Sjölin operators on manifolds with constant sectional curvature and those satisfying Sogge's chaotic curvature condition.
Paper optimizes aquaculture feeding and harvesting strategies for profit maximization.
problem Maximizing farm profit through optimal feeding and harvesting decisions under stochastic price dynamics.
method Developed a simplified aquaculture model and two numerical solution approaches: finite difference scheme and PINN-based method combined with DeepOS algorithm.
result PINN-based method achieves comparable accuracy to finite differences but is more scalable.
Physics-informed neural operator learns from coarse to fine discretized data.
problem Lack of high-fidelity training data and uneven grid resolution.
method Physics-informed multi-resolution neural operator framework.
result Learn from arbitrarily discretized input functions using latent embedding and finite difference solver.
Operator Analysis of MACDq-fin.MF
The paper analyzes MACD using operator theory.
problem Understanding the mathematical foundation of MACD.
method Developed a functional-analytic framework interpreting MACD as a phase-corrected, smoothed derivative operator.
result MACD is structurally equivalent to a band-pass filter and can be expressed as a finite difference of delayed and doubly averaged signals.
We give an explicit formula for the analytic torsion of the finite metric cone over an oriented compact connected Riemannian manifold. We provide an interpretation of the different factors appearing in this formula. We prove that the analytic torsion of the cone is the finite part of the limit obtained collapsing…
Deep learning approximates PDE evolution operators from solution data.
problem Recovering unknown time-dependent PDEs from solution data.
method Approximate evolution operator in modal space, train deep neural network.
result Deep learning method accurately approximates PDE solutions.
We analyze the Hessian spectra of large models up to 100B parameters.
problem Accurate Hessian spectra of large foundation models are difficult to obtain.
method We use shard-local finite-difference Hessian vector products and stochastic Lanczos quadrature.
result We produce the first large-scale spectral density estimates of foundation models.
Optimal reinsurance strategies for multi-line insurance companies.
problem Choosing the best dynamic reinsurance policies for multi-line insurance companies.
method Characterized the optimal survival function as the unique nondecreasing viscosity solution of the HJB equation, solved numerically using the finite difference method.
result Provided proof of convergence of numerical solution to the survival probability function.
Study connects curvature to graph theory and reveals differences.
problem Exploring differences between Quadratic Orthogonal Bisectional Curvature and Real Bisectional Curvature.
method Real (1,1)--forms and Weitzenböck curvature operator used to represent graph Dirichlet energy.
result Curvature differences illuminated between Quadratic Orthogonal Bisectional Curvature and Real Bisectional Curvature.
Optimal proof of finite small eigenvalues for specific geometric manifolds.
problem Proving finiteness of small eigenvalues for geometrically finite manifolds.
method Analyzing the spectrum of the Laplace operator on geometrically finite rank one locally symmetric manifolds.
result Optimal proof of finite small eigenvalues in a specific interval.
The paper studies strong topologies for complex Monge-Ampère equations on Kähler manifolds.
problem Analyzing strong topologies for complex Monge-Ampère equations on Kähler manifolds.
method Proving the Monge-Ampère operator is a homeomorphism between finite energy potentials and energy measures with their strong topologies.
result The Monge-Ampère operator produces an homeomorphism between sets of finite energy potentials and measures on Kähler manifolds.
Study non-perturbative quantum geometry of string theories using finite difference equations and resurgence analysis.
problem Non-perturbative quantum geometry of open and closed topological string on the resolved conifold.
method Finite difference equations, resurgence analysis, exact WKB techniques.
result Identify 5d BPS states and relate spectral problems to quantum integrable systems.
On contact manifolds we describe a notion of (contact) finite-type for linear partial differential operators satisfying a natural condition on their leading terms. A large class of linear differential operators are of finite-type in this sense, and for any such operator we construct a partial connection on a (finite ra…
The paper explores how polynomial roots and operator eigenvalues change with parameters.
problem How do roots of polynomials and eigenvalues of operators vary with parameter changes?
method Analyzes parameter dependence of polynomials and linear operators, covering real analytic to differentiable of finite order.
result Definitive optimal results for perturbation theory of polynomials and linear operators, including hyperbolic polynomials.
The spectral flow theorem is applied to operators on finite intervals.
problem Operators on finite intervals without boundary conditions are not Fredholm.
method Interpolation theory is used to define boundary conditions making the operators Fredholm. The spectral flow theorem is applied to find the Fredholm index.
result The Fredholm index is given by the spectral flow of the operator path.
Controlling Statistical, Discretization, and Truncation Errors in Learning Fourier Linear Operatorsstat.ML
Study identifies and analyzes three types of errors in learning Fourier operators.
problem Statistical, discretization, and truncation errors in learning Fourier operators.
method Analysis of a Discrete Fourier Transform (DFT) based least squares estimator.
result Established upper and lower bounds on statistical, discretization, and truncation errors.
Ancient flows by curvature powers in 2D have finite entropy.
problem Existence of non-homothetic ancient flows by powers of curvature in .
method Determined Morse indices and kernels of the linearized operator of shrinkers. Constructed flows using unstable eigenfunctions.
result Existence of ancient flows with finite entropy.
This study approximates distances between Gaussian processes and covariance operators using RKHS.
problem Approximating distances between Gaussian processes and covariance operators from finite samples.
method Using reproducing kernel Hilbert space (RKHS) covariance and cross-covariance operators, the study shows how to consistently and efficiently estimate Sinkhorn divergence from finite samples.
result Convergence rates are dimension-independent and of the same order as Hilbert-Schmidt distance.
Folding sequencesmath.GT
Bestvina and Feighn showed that a morphism S --> T between two simplicial trees that commutes with the action of a group G can be written as a product of elementary folding operations. Here a more general morphism between simplicial trees is considered, which allow different groups to act on S and T. It is shown that t…
A Covariate-Adjusted Homogeneity Test with Application to Facial Recognition Accuracy Assessmentstat.AP
The paper proposes a test to assess rater accuracy while accounting for rater covariates.
problem Assessing the accuracy of raters in medical imaging and forensic studies.
method Covariate-adjusted homogeneity test to determine differences in accuracy among multiple rater groups.
result The proposed test identifies statistically significant differences among five participant groups in a face recognition study.
Infinite-dimensional SBDMs improve image generation across multiple resolutions.
problem Efficient image generation at high resolutions and across different levels.
method Developed SBDMs in infinite-dimensional setting, using trace class operators and operator networks.
result Improved efficiency and generalization across resolution levels.
Hilbert bundles with endsmath.KT
New Hilbert bundles with ends defined from indexed bases.
problem Defining new structures in Hilbert bundles.
method Indexed bases and unitary operators of finite propagation.
result Characteristic classes of Hilbert bundles with ends.
The purpose of this note is to extend the results of V. Guillemin on elliptic self-adjoint pseudodifferential operators of order one, from operators defined on smooth functions on a closed manifold to operators defined on smooth sections in a vector bundle of Hilbert modules of finite type over a finite von Neumann alg…
Paper introduces FNM framework for learning finite-dimensional parametrized models.
problem Efficiently learning finite-dimensional parametrized models from limited data.
method Fourier Neural Mappings (FNMs) framework for operator learning.
result End-to-end learning of PtO maps can be less data-efficient than learning the solution operator first.
In this paper, we make some remarks on José Espinar's paper "Finite index operators on surfaces" [\texttt{arXiv:0911.3767}, to appear in Journal of Geometric Analysis (2011)].
We describe two-dimensional potential Schrodinger and Dirac operators which are finite-gap at one energy level and have singular spectral curves. It appears that the singularities can be rather complicated. Such Dirac operators appear as the spectral curves of tori immersed into the three-space.
We prove a lower bound for the -th Steklov eigenvalues in terms of an isoperimetric constant called the -th Cheeger-Steklov constant in three different situations: finite spaces, measurable spaces, and Riemannian manifolds. These lower bounds can be considered as higher order Cheeger type inequalities for the Ste…
This note presents an operational measure of fat-tailedness for univariate probability distributions, in where 0 is maximally thin-tailed (Gaussian) and 1 is maximally fat-tailed. Among others,1) it helps assess the sample size needed to establish a comparative needed for statistical significance, 2) allows…
Improved method for estimating derivatives of discontinuous functions using stochastic algorithmic differentiation and regression.
problem High Monte-Carlo error in finite difference approximation of discontinuous functions.
method Combining stochastic algorithmic differentiation and regression to estimate derivative of expectations of discontinuous functions.
result Reduction in Monte-Carlo error through decoupling integration of Dirac delta and conditional expectation.
New TD method stabilizes average-reward learning.
problem Stability issues in average-reward TD learning.
method Implicit fixed point update for average-reward TD().
result Improved numerical stability and broader step-size range.