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Ognyan Kounchev

Publications and source records attributed to Ognyan Kounchev.

At least 19 recordsLinked to original sources
arXiv2025

On the critical length conjecture for spherical Bessel functions in CAGD

A conjecture of J.M. Carnicer, E. Mainar and J.M. Peña states that the critical length of the space $P_{n}\odot C_{1}$ generated by the functions $x^{k}\sin x$ and $x^{k}\cos x$ for $k=0,...n$ is equal to the first positive zero $j_{n+\frac{1}{2},1}$ of the Bessel function $J_{n+\frac{1}{2}}$ of the first kind. It is known that the conjecture implies the following statement (D3): the determinant of the Hankel matrix \begin{equation} \left( \begin{array} [c]{ccc} f & f^{\prime} & f^{\prime\prime}\\ f^{\prime} & f^{\prime\prime} & f^{\left( 3\right) }\\ f^{\prime\prime} & f^{\prime\prime\prime} & f^{\left( 4\right) } \end{array} \right) \label{eqabstract} \end{equation} does not have a zero in the interval $(0,j_{n+\frac{1}{2},1})$ whenever $f=f_{n}$ is given by $f_{n}\left( x\right) =\sqrt{\fracπ{2}} x^{n+\frac{1}{2}}J_{n+\frac{1}{2}}\left( x\right) .$ In this paper we shall prove (D3) and various generalizations.

math.CA↗
arXiv2024

Inequalities for exponential polynomials with applications to moment sequences

Let $Φ_{Λ_{n}}$ be the unique solution of the differential operator $L=\prod_{j=0}^{n}\left( \frac{d}{dx}-λ_{j}\right) $ such that $Φ_{Λ_{n}}^{\left( j\right) }\left( 0\right) =0$ for $j=0,...,n-1,$ and $Φ_{Λ_{n}}^{\left( n\right) }\left( 0\right) =1.$ Assume that $Φ_{Λ_{n}}$ is real-valued and $Φ_{Λ_{n} }^{\left( n+1\right) }\left( x\right) \geq0$ for all $x\in\left[ 0,B\right] .$ Then, if a polynomial $R\left( x\right) = {\displaystyle\sum_{k=0}^{n}} a_{k}x^{k}$ is non-negative on the interval $\left[ 0,B\right] ,$ the inequality \[ {\displaystyle\sum_{k=0}^{n}} a_{k}k!Φ_{Λ_{n}}^{\left( n-k\right) }\left( x\right) \geq R\left( x\right) \] holds for $x\in\left[ 0,B\right] $. From this we derive several interesting inequalities for exponential polynomials. An important consequence is that for a non-negative measure $μ$ over the interval $\left[ a,b\right] $ with $b-a<B$ the sequence defined by \[ s_{k}:=\int_{a}^{b}k!Φ_{Λ_{n}}^{\left( n-k\right) }\left( x-a\right) dμ\left( x\right) \] for $k=0,...,n$ is a moment sequence, i.e. there exists a non-negative measure $ν$ with support in $\left[ a,b\right] $ such that $s_{k}=\int_{a} ^{b}\left( t-a\right) ^{k}dν\left( t\right) $ for $k=0,....,n.$

math.CA↗
arXiv2023

Spectral properties of the Laplacian of temporal networks following a constant block Jacobi model

We study the behavior of the eigenvectors associated with the smallest eigenvalues of the Laplacian matrix of temporal networks. We consider the multilayer representation of temporal networks, i.e. a set of networks linked through ordinal interconnected layers. We analyze the Laplacian matrix, known as supra-Laplacian, constructed through the supra-adjacency matrix associated with the multilayer formulation of temporal networks, using a constant block Jacobi model which has closed-form solution. To do this, we assume that the inter-layer weights are perturbations of the Kronecker sum of the separate adjacency matrices forming the temporal network. Thus we investigate the properties of the eigenvectors associated with the smallest eigenvalues (close to zero) of the supra-Laplacian matrix. Using arguments of perturbation theory, we show that these eigenvectors can be approximated by linear combinations of the zero eigenvectors of the individual time layers. This finding is crucial in reconsidering and generalizing the role of the Fielder vector in supra-Laplacian matrices.

math.NA↗
arXiv2022

Error estimates for harmonic and biharmonic interpolation splines with annular geometry

The main result in this paper is an error estimate for interpolation biharmonic polysplines in an annulus $A\left( r_{1},r_{N}\right) $, with respect to a partition by concentric annular domains $A\left( r_{1} ,r_{2}\right) ,$ ...., $A\left( r_{N-1},r_{N}\right) ,$ for radii $0<r_{1}<....<r_{N}.$ The biharmonic polysplines interpolate a smooth function on the spheres $\left\vert x\right\vert =r_{j}$ for $j=1,...,N$ and satisfy natural boundary conditions for $\left\vert x\right\vert =r_{1}$ and $\left\vert x\right\vert =r_{N}.$ By analogy with a technique in one-dimensional spline theory established by C. de Boor, we base our proof on error estimates for harmonic interpolation splines with respect to the partition by the annuli $A\left( r_{j-1},r_{j}\right) $. For these estimates it is important to determine the smallest constant $c\left( Ω\right) ,$ where $Ω=A\left( r_{j-1},r_{j}\right) ,$ among all constants $c$ satisfying \[ \sup_{x\inΩ}\left\vert f\left( x\right) \right\vert \leq c\sup _{x\inΩ}\left\vert Δf\left( x\right) \right\vert \] for all $f\in C^{2}\left( Ω\right) \cap C\left( \overline{Ω}\right) $ vanishing on the boundary of the bounded domain $Ω$ . In this paper we describe $c\left( Ω\right) $ for an annulus $Ω=A\left( r,R\right) $ and we will give the estimate \[ \min\{\frac{1}{2d},\frac{1}{8}\}\left( R-r\right) ^{2}\leq c\left( A\left( r,R\right) \right) \leq\max\{\frac{1}{2d},\frac{1}{8}\}\left( R-r\right) ^{2}% \] where $d$ is the dimension of the underlying space.

math.NA↗
arXiv2021

Fast algorithms for interpolation with L-splines for differential operators L of order 4 with constant coefficients

In the classical theory of cubic interpolation splines there exists an algorithm which works with only $O\left( n\right)$ arithmetic operations. Also, the smoothing cubic splines may be computed via the algorithm of Reinsch which reduces their computation to interpolation cubic splines and also performs with $O\left( n\right)$ arithmetic operations. In this paper it is shown that many features of the polynomial cubic spline setting carry over to the larger class of $L$-splines where $L$ is a linear differential operator of order $4$ with constant coefficients. Criteria are given such that the associated matrix $R$ is strictly diagonally dominant which implies the existence of a fast algorithm for interpolation.

math.NA↗
arXiv2020

Error bounds for interpolation with piecewise exponential splines of order two and four

Explicit pointwise error bounds for the interpolation of a smooth function by piecewise exponential splines of order four are given. Estimates known for cubic splines are extended to a natural class of piecewise exponential splines which are appearing in the construction of multivariate polysplines. The error estimates are derived in an inductive way using error estimates for the interpolation of a smooth function by exponential splines of order two.

math.NA↗
arXiv2020

The TVBG-SEIR spline model for analysis of COVID-19 spread, and a Tool for prediction scenarios

Mathematical models are traditionally used to analyze the long-term global evolution of epidemics, to determine the potential and severity of an outbreak, and to provide critical information for identifying the type of disease interventions and intensity. One of the widely used mathematical models of long-term spreading of epidemics are the so-called deterministic compartmental models (SIR/SEIR type models). One of the main purposes of applying such models is to assess how the expensive restriction measures imposed by the authorities (home and social isolation/quarantine, travel restrictions, etc.) can effectively reduce the control reproduction number of the disease and its transmission risk. However the classical SIR/SEIR models have been primarily studied in what may be called stationary case, where the main parameters, the Transmission rate Beta (reflecting the virus spread by infected individuals) and the Removed rate Gamma (reflecting the hospitalization/isolation measures) remain constant during the whole period of interest. Hence, it is important to extend the classical SIR/SEIR models by creating new ansatzes for the dynamics of the transmission rates Beta(t) (which we will call further just Beta) and removed rates Gamma(t) (which we will call further just Gamma). The main purpose of the present research is to introduce a spline-based SEIR model with Time-varying Beta and Gamma parameters, or abbreviated TVBG-SEIR model, which is used to estimate the practical implications of the public health interventions and measures. We have designed a Tool based on the TVBG-SEIR model, which may be used as a Decision Support Tool to assist the health decision- and policy-makers in creating predictive scenarios.

math.NA↗
arXiv2019

Wavelet Analysis of Big Data in the Global Investigation of Magnetic Field Variations in Solar-Terrestrial Physics

We provide a Wavelet analysis of Big Data in Solar Terrestrial Physics. In order to explain and predict the dynamics of the geomagnetic phenomena we analyze high frequency time series data from different sources: 1. The Interplanetary Magnetic Field (from the ACE satellite). 2. The Ionospheric parameters - TEC (from ionospheric sounding stations). 3. The ground Geomagnetic data (from ground geomagnetic observatories, located in middle geographic latitudes). We seek for correlations in the wavelet coefficients which explain the dynamics of different magnetic phenomena in the Solar Terrestrial Physics. The large variety of data used in our research from both Solar Astronomy and Earth Observations makes it a contribution to the newly developing area of AstroGeoInformatics.

physics.space-ph↗
arXiv2015

A symmetry property for polyharmonic functions vanishing on equidistant hyperplanes

Let $u\left( t,y\right) $ be a polyharmonic function of order $N$ defined on the strip $\left( a,b\right) \times\mathbb{R}^{d}$ satisfying the growth condition $$ \sup_{t\in K}\left\vert u\left( t,y\right) \right\vert \leq o\left( \left\vert y\right\vert ^{\left( 1-d\right) /2}e^{\fracπ{c}\left\vert y\right\vert }\right) $$ for $\left\vert y\right\vert \rightarrow\infty$ and any compact subinterval $K$ of $\left( a,b\right) $, and suppose that $u\left( t,y\right) $ vanishes on $2N-1$ equidistant hyperplanes of the form $\left\{ t_{j}\right\} \times\mathbb{R}^{d}$ for $t_{j}=t_{0}+jc\in\left( a,b\right) $ and $j=-\left( N-1\right) ,...,N-1.$ Then it is shown that $u\left( t,y\right) $ is odd at $t_{0},$ i.e. that $u\left( t_{0}+t,y\right) =-u\left( t_{0}-t,y\right) $ for $y\in\mathbb{R}^{d}$. The second main result states that $u$ is identically zero provided that $u$ satisfies the growth condition and vanishes on $2N$ equidistant hyperplanes with distance $c.$

math.AP↗
arXiv2013

Klein-Dirac Quadric and Multidimensional Toda Lattice via Pseudo-positive Moment Problem

In 1974 Jürgen Moser has shown that the classical Moment Problem plays a fundamental role for the theory of completely integrable systems, by proving that the simplest case of the finite Toda lattice is described exhaustively in its terms. In particular, the Jacobi matrix defined by the Flaschka variables corresponds to the Schrödinger operator related to the Korteweg-de Vries operator. In the present paper we use a recent breakthrough in the area of the multidimensional Moment Problem in order to develop a multidimensional generalization of the finite Toda lattice. Our construction is based on the notion of pseudo-positive measure and the notion of multidimensional Markov-Stieltjes transform naturally defined on the Klein-Dirac quadric. An important consequence of the properties of the Markov-Stieltjes transform is a new method for summation of multidimensional divergent series on the Klein-Dirac quadric.

math.AP↗
arXiv2012

Polyharmonic functions of infinite order on annular regions

Polyharmonic functions f of infinite order and type τ on annular regions are systematically studied. The first main result states that the Fourier-Laplace coefficients f_{k,l}(r) of a polyharmonic function f of infinite order and type 0 can be extended to analytic functions on the complex plane cut along the negative semiaxis. The second main result gives a constructive procedure via Fourier-Laplace series for the analytic extension of a polyharmonic function on annular region A(r_{0},r_{1}) of infinite order and type less than 1/2r_{1} to the kernel of the harmonicity hull of the annular region. The methods of proof depend on an extensive investigation of Taylor series with respect to linear differential operators with constant coefficients.

math.AP↗
arXiv2012

Compressive Sensing for Polyharmonic Subdivision Wavelets With Applications to Image Analysis

We apply successfully the Compressive Sensing approach for Image Analysis using the new family of Polyharmonic Subdivision wavelets. We show that this approach provides a very efficient recovery of the images based on fewer samples than the traditional Shannon-Nyquist paradigm. We provide the results of experiments with PHSD wavelets and Daubechies wavelets, for the Lena image and astronomical images.

math.NA↗
arXiv2012

Polyharmonic Hardy Spaces on the Complexified Annulus and Error Estimates of Cubature Formulas

The present paper has a twofold contribution: first, we introduce a new concept of Hardy spaces on a multidimensional complexified annular domain which is closely related to the annulus of the Klein-Dirac quadric important in Conformal Quantum Field Theory. Secondly, for functions in these Hardy spaces, we provide error estimate for the polyharmonic Gauß-Jacobi cubature formulas, which have been introduced in previous papers.

math.NA↗
arXiv2011

Polyharmonicity and algebraic support of measures

We introduce a multivariate Markov transform which generalizes the well-known one-dimensional Stieltjes transform from the Moment problem and Spectral theory. Our main result states that two measures μ and ν with bounded support contained in the zero set of a polynomial P(x) are equal if they coincide on the subspace of all polynomials of polyharmonic degree N_{P} where the natural number N_{P} is explictly computed by the properties of the polynomial P(x). The method of proof depends on a definition of a multivariate Markov transform which another major objective of the present paper. The classical notion of orthogonal polynomial of second kind is generalized to the multivariate setting: it is a polyharmonic function which has similar features as in the one-dimensional case.

math.CV↗
arXiv2011

Padé approximation for a multivariate Markov transform

Methods of Padé approximation are used to analyse a multivariate Markov transform which has been recently introduced by the authors, and which is generalizing the well-known in Spectral theory Stieltjes transform (Markov function) of one-dimensional measure. The first main result is a characterization of the rationality of the Markov transform via Hankel determinants. The second main result is a cubature formula for a special class of measures.

math.NA↗
arXiv2011

Multidimensional Chebyshev Systems - just a definition

We provide a definition of Multidimensional Chebyshev Systems of order N which is satisfied by the solutions of a wide class of elliptic equations of order 2N. This definition generalizes a very large class of Extended Complete Chebyshev systems in the one-dimensional case. This is the first of a series of papers in this area, which solves the longstanding problem of finding a satisfactory multidimensional generalization of the classical Chebyshev systems introduced already by A. Markov more than hundered years ago, and studied later by S. Bernstein and M. Krein.

math.FA↗