arXiv research

Study shows overparameterization helps in generalizing from smooth interpolants.

Paper extends trigonometric summation formula with weights.

New trigonometric method for convex sets aids control problems.

Study on random knots and their projections with varying decay rates of coefficients.

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 $m$-term approximation with respect to the trigonometric system.

Computed distortion coefficients for the α-Grushin plane.

Quantile-Frequency Analysis detects nonlinear dynamics in financial time series.

Book reviews Riemann's work's impact on math, philosophy, physics.

The study connects circle actions, trigonometric polynomials, and thickenings of metric spaces.

Deep Gaussian Processes are reinterpreted as deep trigonometric networks for tractable inference.

Deformed σ-models linked to Ricci flow and Toda theories.

Sigma models linked to Gross-Neveu models via quiver varieties.

New approach for classification using trigonometric polynomial kernels from signal processing.

Quantum modularity proved for SU(2) TQFT signature on genus 2 surfaces.

Study uses regression and ML for COVID-19 mortality forecasting.

Survey on greedy algorithms for sparse approximation.

Study geodesics on Grushin spaces, proving upper bounds 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 finds the minimum number of finite Gaussian mixtures for best approximation.

This paper analyzes various forms of concentrated liquidity in decentralized finance.

Study Fourier estimator for spot volatility with unbounded coefficients and jumps.

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 $π$ in the middle points of a certain pair of its skew edges.

Euler explored spherical geometry using trigonometric formulae and solid geometry methods.

Generalizes crystallographic properties to all dimensions.

Improved Gaussian Process regression using TQFF over RFF and Gaussian QFF.

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…

Quantum neural networks approximate periodic functions more efficiently.

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 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…

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.

Algorithm finds real-analytic Legendrian representatives for every link type.

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…

Recently, the explicit volume formulae for hyperbolic cone-manifolds, whose underlying space is the 3-sphere and the singular set is the knot $4_1$ and the links $5^2_1$ and $6^2_2$, 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…

Quantum flag manifold σ-models are integrable and satisfy Ricci flow equations.

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 …

Crochet patterns for minimal surfaces created using trigonometry.

We introduce SARR for symmetric object pose estimation, improving CNN performance.

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…

The authors compute distances between arbitrary elements of Lie groups SU(2) and SO(3) for special left-invariant sub-Riemannian metrics $ρ$ and $d$. To compute distances for the second metric, we essentially use the fact that canonical two-sheeted covering epimorphism $Ω$ 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.

We prove a Chern-Lashof type formula computing the expected number of critical points of smooth function on a smooth manifold $M$ randomly chosen from a finite dimensional subspace $V\subset C^\infty(M)$ equipped with a Gaussian probability measure. We then use this formula this formula to find the asymptotics of the e…

This paper is based on the introduction to the monograph ``Double affine Hecke algebras'' to be published by Cambridge University Press. The connections with Knizhnik-Zamolodchikov equations, Kac-Moody algebras, tau-function, harmonic analysis on symmetric spaces, and special functions are discussed. The rank one case …

New contact structures on spheres and tori with weakly compatible metrics.