Let A be an n × n generic and irreducible matrix. We consider power series expansions for eigenvectors corresponding to the real part of e^-iθA and use them to analyze and approximate the components of the Kippenhahn algebraic curve associated with the numerical range of A. We obtain information about the local behavior of the curve. As a result we obtain new equations for the series expansions, new conditions under which these components contain no line segments as well as conditions under which a component degenerates to a point. Applying these results to work of Mirman on compressions of unitary matrices, we show that in Mirman’s setting the components are locally asymptotically quadratic at all but finitely many points.
The measure of the positivity set { x ∈ [0; 2π ] | f(x) > 0 } of a trigonometric polynomial f is bounded from below by the Motzkin density. We generalize the bound to polynomials in several variables and almost periodic functions. We then use these generalizations to extend known results on Taikov’s problem.
QMC designs were introduced in a 2014 paper by Brauchart, Saff and the present authors (Math. Comp. 83:2821–2851). They represent a novel approach to cubature on the sphere 𝕊^d , in which instead of requiring cubature rules to be exact for polynomials up to a certain degree, a sequence of cubature rules is a QMC design sequence if the worst-case error for functions in a Sobolev space ℍ^s(𝕊^d) is of order 𝒪(N^-s/d) , where N is the number of cubature points. The original paper considered only equal cubature weights, but the present paper allows positive weights that sum to 1. After reviewing the known results, the present paper presents necessary and sufficient conditions for QMC design sequences, one of which requires only a finite sum for each tested value of N. The paper also gives experimental results that aim to give insight into the design of QMC designs, and extends the known QMC designs to a 3-dimensional example.
In this paper, we will prove the distinct partition function satisfies the Laguerre inequality of order 2 and the determinantal inequality of order 3 conjectured by Dong and Ji. We also prove the Laguerre inequality of order 3 holds for the distinct partition function. Moreover, for 4≤ m≤ 11 , we conjecture the thresholds for the Laguerre inequalities of order m and the positivity of m-order determinants for the distinct partition function.
Two types of Bernstein inequalities are established on the unit ball in ℝ^d , which are stronger than those known in the literature. The first type consists of inequalities in L^p norm for a fully symmetric doubling weight on the unit ball. The second type consists of sharp inequalities in L^2 norm for the Jacobi weight, which are established via a new self-adjoint form of the spectral operator that has orthogonal polynomials as eigenfunctions.
We study point configurations on the torus 𝕋^d that minimize interaction energies with tensor product structure. Such interactions arise naturally in the context of discrepancy theory and quasi-Monte Carlo integration. Permutation sets on 𝕋^2 and Latin hypercube sets in higher dimensions (i.e. sets whose projections onto coordinate axes are equispaced points) are natural candidates to be energy minimizers. We show that such point configurations that have only one distance in the vector sense minimize the energy for a wide range of potentials, i.e. such sets satisfy a tensor product version of universal optimality. This specifically applies to three- and five-point Fibonacci lattices. We also characterize all lattices with this property and exhibit some non-lattice sets of this type. In addition, we obtain several further structural results about global and local minimizers of tensor product energies.
The complete solution of the bispectral problem for the Schrödinger operator L=-d^2dx^2+V(x) in [19] is obtained by the application of the Darboux process to the cases of V=0 and V(x)=-14x^2 . Both of these cases are trivially bispectral and after repeated applications of the Darboux process one gets either a pair of rank one bundles of bispectral situations (when starting from V=0 ) or a rank two bispectral bundle (when starting from V(x)=-14x^2 ). In the first case all operators have “trivial monodromy” as defined in [19]. In the second case the monodromy group of all operators is given by the integers. In this paper we start from V(x)=x^2 , use the Darboux process and explore the connection between the rank of certain non-polynomial bispectral families and trivial monodromy by means of examples. The main conclusion is that the results in [19] do not apply verbatim in this case.
We describe several randomized collections of $3\times 3$ rotation matrices and analyze their associated logarithmic energy. The best one (i.e. the one attaining the lowest expected logarithmic energy) is constructed by choosing $r$ spherical points, which come from the zeros of a randomly chosen degree $r$ polynomial, and considering at each of these points a set of $s$ evenly distributed rotation matrices. This construction yields a new upper bound on the minimal logarithmic energy of $n=rs$ rotation matrices.
We describe several randomized collections of 3× 3 rotation matrices and analyze their associated logarithmic energy. The best one (that is, the one attaining the lowest expected logarithmic energy) is constructed by choosing r points on the sphere, which come from the zeros of a randomly chosen degree r polynomial, and considering at each of these points a set of s evenly distributed rotation matrices. This construction yields a new upper bound on the minimal logarithmic energy of n=rs rotation matrices.
In this paper we develop the following general approach. We study asymptotic behavior of the entropy numbers not for an individual smoothness class, how it is usually done, but for the collection of classes, which are defined by integral operators with kernels coming from a given class of functions. Earlier, such approach was realized for the Kolmogorov widths.
We show that deep Heaviside networks (DHNs) have limited expressiveness but that this can be overcome by including either skip connections or neurons with linear activation. We provide lower and upper bounds for the Vapnik-Chervonenkis (VC) dimensions and approximation rates of these network classes. As an application, we derive statistical convergence rates for DHN fits in the nonparametric regression model.
In this paper we study a family of non-classical Jacobi polynomials with varying parameters of the form α _n=n+1/2 and β _n=-n-1/2 . We obtain global asymptotics for these polynomials, and use this to establish results on the location of their zeros. The analysis is based on the Riemann Hilbert formulation of Jacobi polynomials derived from the non-hermitian orthogonality introduced by [Kuijlaars, A., Martinez-Finkelshtein, A., Orive, R.: Orthogonality of Jacobi polynomials with general parameters. Electron. Trans. Numer. Anal. 19, 1–17 (2005)]. This family of polynomials arises in the symbolic evaluation of integrals in the work of [Boros, G., Moll, V.: A sequence of unimodal polynomials. J. Math. Anal. Appl. 237, 272–287 (1999)], [Boros, G., Moll, V.: An integral hidden in Gradshteyn and Ryzhik. J. Comput. Appl. Math., Elsevier 106(2), 361–368 (1999), and corresponds to a limiting case, which is not considered in the works of [Kuijlaars, A., Martínez-Finkelshtein, A.: Strong asymptotics for Jacobi polynomials with varying nonstandard parameters. J. d’Analyse Math. 54, 195–234 (2004)], [Kuijlaars, A., Martinez-Finkelshtein, A., Orive, R.: Orthogonality of Jacobi polynomials with general parameters. Electron. Trans. Numer. Anal. 19, 1–17 (2005)], [Martínez-Finkelshtein, A., Martínez-González, P., Orive, R.: Zeros of Jacobi Polynomials with Varying Non-classical Parameters. Special functions, pp. 98–113. World Scientific, Singapore (2000)], [Martínez-Finkelshtein, A., Orive, R.: Riemann-Hilbert analysis for Jacobi polynomials orthogonal on a single contour. J. Approx. Theory 134(2), 137–170 (2005)]. A remarkable feature in the analysis is encountered when performing the local analysis of the RHP near the origin, where the local parametrix introduces a pole.
We give upper and lower bounds for weighted Chebyshev and residual polynomials on subsets of the real line. As an application, we prove a Szegő-type theorem in the setting of Parreau–Widom sets.
We study the non-compact Sobolev embeddings into the optimal scale of Lorentz spaces, W_0^mL^p,q(Ω ) → L^dp/d - mp,r(Ω ), where Ω⊆ℝ^d , 1 ≤ m ≤ d , 0
We consider the complex and symplectic elliptic Ginibre matrices of size (c+1)N × (c+1)N, conditioned to have a deterministic eigenvalue at p ∈ℝ with multiplicity c N. We show that their limiting spectrum is either simply connected, doubly connected, or composed of two disjoint simply connected components. Moreover, denoting by τ∈ [0,1] the non-Hermiticity parameter, we explicitly characterise the regions in the parameter space (p, c, τ) where each topological type emerges. For cases where the droplet is either simply or doubly connected, we provide an explicit description of the limiting spectrum and the corresponding electrostatic energies. As an application, we derive the asymptotic behaviour of the moments of the characteristic polynomial for elliptic Ginibre matrices in the exponentially varying regime.
In this paper, we study the problem of multivariate L_2-approximation of functions belonging to a weighted Korobov space. We propose and analyze a median lattice-based algorithm, inspired by median integration rules, which have attracted significant attention in the theory of quasi-Monte Carlo methods. Our algorithm approximates the Fourier coefficients associated with a suitably chosen frequency index set, where each coefficient is estimated by taking the median over approximations from randomly shifted rank-1 lattice rules with independently chosen generating vectors. We prove that the algorithm achieves, with high probability, a convergence rate of the L_2-approximation error that is arbitrarily close to optimal with respect to the number of function evaluations. Furthermore, we show that the error bound depends only polynomially on the dimension, or is even independent of the dimension, under certain summability conditions on the weights. Numerical experiments illustrate the performance of the proposed median lattice-based algorithm.
Given an open set T⊂ [-1,1), we introduce the concepts of T-avoiding spherical codes and designs, that is, spherical codes that have no inner products in the set T. We show that certain codes found in the minimal vectors of the Leech lattice, as well as the minimal vectors of the Barnes–Wall lattice and codes derived from strongly regular graphs, are universally optimal in the restricted class of T-avoiding codes. We also extend a result of Delsarte–Goethals–Seidel about codes with three inner products α, β, γ (in our terminology (α,β)-avoiding γ-codes). Parallel to the notion of tight spherical designs, we also derive that these codes are minimal (tight) T-avoiding spherical designs of fixed dimension and strength. In some cases, we also find that codes under consideration have maximal cardinality in their T-avoiding class for given dimension and minimum distance.
In this paper, an Askey-Wilson version of the Wronskian-Casorati determinant 𝒲(f_0, …, f_n)(x) for meromorphic functions f_0, …, f_n is introduced to establish an Askey-Wilson version of the general form of the Second Main Theorem in projective space. This improves upon the original Second Main Theorem for the Askey-Wilson operator due to Chiang and Feng. In addition, by taking into account the number of irreducible components of hypersurfaces, an Askey-Wilson version of the Truncated Second Main Theorem for holomorphic curves into projective space with hypersurfaces located in l-subgeneral position is obtained.
