The paper gives a constructive method, based on greedy algorithms, that provides for the classes of functions with small mixed smoothness the best possible in the sense of order approximation error for the -term approximation with respect to the trigonometric system.
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48 results for “trigonometric system”
Paper extends trigonometric summation formula with weights.
problem Trigonometric summation formula by Grigor'yan, Lin and Yau.
method Weighted trigonometric summation formula derivation.
result Extension of trigonometric summation formula.
Survey on greedy algorithms for sparse approximation.
problem Sparse approximation using greedy algorithms.
method Greedy algorithms, Lebesgue-type inequalities, trigonometric system, tensor product structure.
result Constructive sparse approximation methods provided for various cases.
Computed distortion coefficients for the α-Grushin plane.
problem Analyzing the distortion coefficients of the α-Grushin plane.
method Using generalised trigonometric functions and synthetic curvature conditions.
result Estimates for distortion coefficients and a curvature condition conjecture.
Turbiner's conjecture posits that a Lie-algebraic Hamiltonian operator whose domain is a subset of the Euclidean plane admits a separation of variables. A proof of this conjecture is given in those cases where the generating Lie-algebra acts imprimitively. The general form of the conjecture is false. A counter-example …
The study connects circle actions, trigonometric polynomials, and thickenings of metric spaces.
problem Understanding the topology and connectivity of thickenings of metric spaces.
method Combining circle actions, trigonometric polynomials, and geometric proofs.
result Optimal bounds on thickenings of the sphere and circle, proving zero points of trigonometric polynomials.
Deep Gaussian Processes are reinterpreted as deep trigonometric networks for tractable inference.
problem Challenging inference in DGPs due to intractable marginalization in latent function space.
method Viewing DGPs as deep trigonometric networks with Bochner's theorem, and using the wide limit with a bottleneck to translate DGPs into deep trigonometric networks.
result The weight space view yields the same effective covariance functions as obtained in function space, and varying prior distributions over network parameters is equivalent to employing different kernels.
Deformed σ-models linked to Ricci flow and Toda theories.
problem Understanding the relationship between deformed σ-models and geometric flows.
method Exploring trigonometric deformations of CP^n-1 models and their duals, linking to Ricci flow and Toda field theories.
result Trigonometric deformations of CP^n-1 models solve the Ricci flow equation and relate to Toda field theories.
Sigma models linked to Gross-Neveu models via quiver varieties.
problem Understanding the relationship between sigma models and Gross-Neveu models.
method Exploring the mathematical correspondence between sigma models and Gross-Neveu models, including their geometric and trigonometric/elliptic deformations.
result Sigma models are mathematically equivalent to Gross-Neveu models under certain conditions.
New approach for classification using trigonometric polynomial kernels from signal processing.
problem Classifying objects in arbitrary compact metric spaces.
method Localized trigonometric polynomial kernels for separating different probability measures.
result Minimal number of queried labels for perfect classification.
We study sparse approximation by greedy algorithms. We prove the Lebesgue-type inequalities for the Weak Chebyshev Greedy Algorithm (WCGA), a generalization of the Weak Orthogonal Matching Pursuit to the case of a Banach space. The main novelty of these results is a Banach space setting instead of a Hilbert space setti…
New trigonometric method for convex sets aids control problems.
problem Optimal control problems with two-dimensional control.
method Proposes a new method to describe convex sets.
result Investigates sub-Finsler problems in various groups.
Quantum flag manifold σ-models are integrable and satisfy Ricci flow equations.
problem Integrating quantum flag manifold σ-models with fermions.
method Gauging bosonic Thirring/Gross-Neveu-type systems, adding fermions to cancel anomalies, and checking Ricci flow equations.
result Trigonometrically deformed geometries of flag manifold σ-models satisfy generalized Ricci flow equations.
Geodesics on Grushin spacesmath.DG
Study geodesics on Grushin spaces, proving upper bounds on conjugate times.
problem Classify geodesics on higher-dimensional Grushin spaces.
method Solve Hamilton's equations using calculus of generalized trigonometric functions, analyze symmetries, and use density arguments.
result Prove a conjectured cut time provides an upper bound on conjugate times.
In this paper we investigate the construction of state models for link invariants using representations of the braid group obtained from various gauge choices for a solution of the trigonometric Yang-Baxter equation. Our results show that it is possible to obtain invariants of regular isotopy (as defined by Kauffman) w…
Study shows overparameterization helps in generalizing from smooth interpolants.
problem Understanding generalization in overparameterized linear models.
method Analysis of random Fourier series model with weighted trigonometric interpolation.
result Weighted trigonometric interpolation leads to lower generalization error in overparameterized scenarios.
Study finds the minimum number of finite Gaussian mixtures for best approximation.
problem Finding the minimum number of finite Gaussian mixtures for best approximation.
method Local moment matching for upper bound and spectral analysis for lower bound.
result Corrects a previous lower bound in the case of Gaussian mixing distributions.
Algorithm finds real-analytic Legendrian representatives for every link type.
problem Finding explicit expressions for Legendrian representatives and Bateman fields.
method Algorithm based on trigonometric polynomials and solving linear equations.
result No compact subset of R^3 can contain an electromagnetic knot indefinitely.
This paper analyzes various forms of concentrated liquidity in decentralized finance.
problem Understanding different models of concentrated liquidity in DeFi.
method Algebraic and geometric analysis of existing models.
result An authoritative overview of concentrated liquidity models in DeFi.
A functional ansatz is developed which gives certain elliptic solutions of the Witten-Dijkgraaf-Verlinde-Verlinde (or WDVV) equation. This is based on the elliptic trilogarithm function introduced by Beilinson and Levin. For this to be a solution results in a number of purely algebraic conditions on the set of vectors …
The Fourier estimator of spot volatility: Unbounded coefficients and jumps in the price processq-fin.CP
Study Fourier estimator for spot volatility with unbounded coefficients and jumps.
problem Estimating spot volatility with unbounded coefficients and jumps in price process.
method Fourier estimator for spot volatility, convergence analysis for unbounded coefficients and jumps.
result Convergence of trigonometric polynomial to volatility's path, almost sure convergence of reconstructed volatility.
The classical Tait-Kneser theorem states that the osculating circles of a smooth plane curve, free from curvature extrema, are pairwise disjoint. We prove a number of analogs of this theorem, e.g., for ovals of osculating cubics, osculating polynomials and trigonometric polynomials; in each case, we will obtain a non-d…
The present paper considers volume formulae, as well as trigonometric identities, that hold for a tetrahedron in 3-dimensional spherical space of constant sectional curvature +1. The tetrahedron possesses a certain symmetry: namely rotation through angle in the middle points of a certain pair of its skew edges.
Euler explored spherical geometry using trigonometric formulae and solid geometry methods.
problem Developing trigonometric formulae for spherical geometry.
method Used calculus of variations and classical solid geometry methods.
result Established trigonometric formulae and computed spherical triangle areas.
Generalizes crystallographic properties to all dimensions.
problem Analytic eigenfunctions in crystallographic groups.
method Algebraic, geometric, and analytic proofs.
result Equivalent conditions for real analytic eigenfunctions in crystallographic polytopes.
Improved Gaussian Process regression using TQFF over RFF and Gaussian QFF.
problem Limited performance of Quadrature Fourier Features (QFF) in approximating highly oscillatory functions.
method Developed Trigonometric Quadrature Fourier Features (TQFF) using a novel non-Gaussian quadrature rule.
result TQFF provides better approximation accuracy and fewer features compared to RFF and Gaussian QFF.
Transitional geometrymath.GT
We develop a transitional geometry, that is, a family of geometries of constant curvatures which makes a continuous connec-tion between the hyperbolic, Euclidean and spherical geometries. In this transitional setting, several geometric entities like points, lines, dis-tances, triangles, angles, area, curvature, etc. as…
Approximation rates of quantum neural networks for periodic functions via Jackson's inequalityquant-ph
Quantum neural networks approximate periodic functions more efficiently.
problem Approximating periodic functions with quantum neural networks.
method Using Jackson's inequality to construct a QNN that approximates a trigonometric polynomial of the function.
result Quantum neural networks can achieve better approximation results with fewer parameters for smoother functions.
Study on random knots and their projections with varying decay rates of coefficients.
problem Understanding the probability of knot types with a given number of crossings.
method Defined random knots as random periodic functions with Gaussian coefficients, analyzed decay rates of coefficients, and examined projections.
result For certain decay rates of coefficients, the probability of forming a knot type with a given number of crossings decays at least as fast as \(1/N\).
The BC(n) Sutherland Hamiltonian with coupling constants parametrized by three arbitrary integers is derived by reductions of the Laplace operator of the group U(N). The reductions are obtained by applying the Laplace operator on spaces of certain vector valued functions equivariant under suitable symmetric subgroups o…
Proves functional equation for twisted Ruelle zeta function on hyperbolic surfaces.
problem Determines the functional equation for twisted Ruelle zeta functions.
method Analyzes scattering matrix and uses topological data of hyperbolic surfaces.
result Determines the order of the divisor of R(s,χ) at s=0 and computes its Laurent expansion.
An innovative extension of Geometric Brownian Motion model is developed by incorporating a weighting factor and a stochastic function modelled as a mixture of power and trigonometric functions. Simulations based on this Modified Brownian Motion Model with optimal weighting factors selected by goodness of fit tests, sub…
Wavelets improve accuracy in solving backward SDEs.
problem Solving backward stochastic differential equations (SDEs) with high accuracy and simplicity.
method Time discretization combined with trigonometric wavelets, enhanced by antireflective boundary technique.
result Improved numerical algorithm for SDEs with enhanced accuracy and ease of implementation.
Recently, the explicit volume formulae for hyperbolic cone-manifolds, whose underlying space is the 3-sphere and the singular set is the knot and the links and , have been obtained by the second named author and his collaborators. In this paper we explicitly find the hyperbolic volume for cone-mani…
The notion of a generalized harmonic inverse mean curvature surface in the Euclidean four-space is introduced. A backward Bäcklund transform of a generalized harmonic inverse mean curvature surface is defined. A Darboux transform of a generalized harmonic inverse mean curvature surface is constructed by a backward Bäck…
New theory shows EDMD works well in chaotic systems.
problem Uncertainty in EDMD's properties in chaos.
method Developed rigorous theory of EDMD on chaotic maps using OPUC and transfer operator methods.
result EDMD converges to correct limits in chaotic systems with small polynomial dictionaries.
The paper shows the computation of the noncommutative generalization of the A-polynomial of the trefoil knot. The classical A-polynomial was introduced by Cooper, Culler, Gillet, Long and Shalen, and was generalized to the context of Kauffman bracket skein modules by the author in joint work with Frohman and Lofaro. A …
Rotation minimizing frames and spherical curves in simply isotropic and pseudo-isotropic 3-spacesmath.DG
The paper develops RM frames for isotropic and pseudo-isotropic spaces and characterizes spherical curves.
problem Differential geometry of curves in isotropic and pseudo-isotropic 3-spaces.
method Developing rotation minimizing frames and applying them to spherical curves in isotropic and pseudo-isotropic spaces.
result Characterization of spherical curves via linear equations involving curvatures and RM frames.
Crochet patterns for minimal surfaces created using trigonometry.
problem Creating crochet patterns for minimal surfaces.
method Using trigonometric identities to calculate arc lengths.
result Crochet instructions for Enneper's surface.
Towards Symmetry-sensitive Pose Estimation: A Rotation Representation for Symmetric Object Classescs.CV
We introduce SARR for symmetric object pose estimation, improving CNN performance.
problem Ambiguities in symmetric object orientations hinder deep learning pose estimation.
method Numeric rotation representation using symmetry-derived trigonometric identities.
result SARR enables standard CNNs to achieve state-of-the-art performance.
Book reviews Riemann's work's impact on math, philosophy, physics.
problem None explicitly stated, focuses on Riemann's contributions.
method Review of Riemann's work and its impact.
result Riemann's work has broad impacts on math, philosophy, physics.
Painleve Field Theorymath-ph
We propose multidimensional versions of the Painlevé VI equation and its degenerations. These field theories are related to the isomonodromy problems of flat holomorphic infinite rank bundles over elliptic curves and take the form of non-autonomous Hamiltonian equations. The modular parameter of curves plays the role o…
We calculate the volumes of the hyperbolic twist knot cone-manifolds using the Schläfli formula. Even though general ideas for calculating the volumes of cone-manifolds are around, since there is no concrete calculation written, we present here the concrete calculations. We express the length of the singular locus in t…
Quantum modularity proved for SU(2) TQFT signature on genus 2 surfaces.
problem Proving quantum modularity of SU(2) TQFT signature for genus 2 surfaces.
method Using quantum modularity of generalized Dedekind sums associated with modular forms and trigonometric sum expressions.
result Quantum modularity of SU(2) TQFT signature on genus 2 surfaces proved.
The authors compute distances between arbitrary elements of Lie groups SU(2) and SO(3) for special left-invariant sub-Riemannian metrics and . To compute distances for the second metric, we essentially use the fact that canonical two-sheeted covering epimorphism of the Lie group SU(2) onto the Lie group SO(3…
Given a knot K in an Euclidean space E and a finite dimensional space V of smooth functions on K, we express the expected number of critical points of a random function in V in terms of an integral-geometric invariant of K and V. When V consists of the restrictions to K of homogeneous polynomials of degree d on E, this…
This paper is focused on geometric aspects of two particular types of finite-variable reductions in the dispersionless Toda hierarchy. The reductions are formulated in terms of "Landau-Ginzburg potentials" that play the role of reduced Lax functions. One of them is a generalization of Dubrovin and Zhang's trigonometric…
Deep neural networks improve mean function estimation for functional data.
problem Estimating mean functions of functional data.
method Deep neural networks with ReLU activation, sparsely connected.
result Achieves optimal nonparametric convergence rate in empirical norm.