Further implications of the Bessis-Moussa-Villani conjecture

We find further implications of the BMV conjecture, which states that for hermitian matrices A and B, the function Tr exp(A - t B) is the Laplace transform of a positive measure.Comment: LaTeX, 8 page

On kernel engineering via Paley–Wiener

A radial basis function approximation takes the form $$s(x)=\sum_{k=1}^na_k\phi(x-b_k),\quad x\in {\mathbb{R}}^d,$$ where the coefficients a 1,…,a n are real numbers, the centres b 1,…,b n are distinct points in ℝ d , and the function φ:ℝ d →ℝ is radially symmetric. Such functions are highly useful in practice and enjoy many beautiful theoretical properties. In particular, much work has been devoted to the polyharmonic radial basis functions, for which φ is the fundamental solution of some iterate of the Laplacian. In this note, we consider the construction of a rotation-invariant signed (Borel) measure μ for which the convolution ψ=μ φ is a function of compact support, and when φ is polyharmonic. The novelty of this construction is its use of the Paley–Wiener theorem to identify compact support via analysis of the Fourier transform of the new kernel ψ, so providing a new form of kernel engineering

A simple proof of Hardy-Lieb-Thirring inequalities

We give a short and unified proof of Hardy-Lieb-Thirring inequalities for moments of eigenvalues of fractional Schroedinger operators. The proof covers the optimal parameter range. It is based on a recent inequality by Solovej, Soerensen, and Spitzer. Moreover, we prove that any non-magnetic Lieb-Thirring inequality implies a magnetic Lieb-Thirring inequality (with possibly a larger constant).Comment: 12 page

On spherical averages of radial basis functions

A radial basis function (RBF) has the general form $$s(x)=\sum_{k=1}^{n}a_{k}\phi(x-b_{k}),\quad x\in\mathbb{R}^{d},$$ where the coefficients a 1,…,a n are real numbers, the points, or centres, b 1,…,b n lie in ℝ d , and φ:ℝ d →ℝ is a radially symmetric function. Such approximants are highly useful and enjoy rich theoretical properties; see, for instance (Buhmann, Radial Basis Functions: Theory and Implementations, [2003]; Fasshauer, Meshfree Approximation Methods with Matlab, [2007]; Light and Cheney, A Course in Approximation Theory, [2000]; or Wendland, Scattered Data Approximation, [2004]). The important special case of polyharmonic splines results when φ is the fundamental solution of the iterated Laplacian operator, and this class includes the Euclidean norm φ(x)=‖x‖ when d is an odd positive integer, the thin plate spline φ(x)=‖x‖2log  ‖x‖ when d is an even positive integer, and univariate splines. Now B-splines generate a compactly supported basis for univariate spline spaces, but an analyticity argument implies that a nontrivial polyharmonic spline generated by (1.1) cannot be compactly supported when d>1. However, a pioneering paper of Jackson (Constr. Approx. 4:243–264, [1988]) established that the spherical average of a radial basis function generated by the Euclidean norm can be compactly supported when the centres and coefficients satisfy certain moment conditions; Jackson then used this compactly supported spherical average to construct approximate identities, with which he was then able to derive some of the earliest uniform convergence results for a class of radial basis functions. Our work extends this earlier analysis, but our technique is entirely novel, and applies to all polyharmonic splines. Furthermore, we observe that the technique provides yet another way to generate compactly supported, radially symmetric, positive definite functions. Specifically, we find that the spherical averaging operator commutes with the Fourier transform operator, and we are then able to identify Fourier transforms of compactly supported functions using the Paley–Wiener theorem. Furthermore, the use of Haar measure on compact Lie groups would not have occurred without frequent exposure to Iserles’s study of geometric integration

On the class SI of J-contractive functions intertwining solutions of linear differential equations

In the PhD thesis of the second author under the supervision of the third author was defined the class SI of J-contractive functions, depending on a parameter and arising as transfer functions of overdetermined conservative 2D systems invariant in one direction. In this paper we extend and solve in the class SI, a number of problems originally set for the class SC of functions contractive in the open right-half plane, and unitary on the imaginary line with respect to some preassigned signature matrix J. The problems we consider include the Schur algorithm, the partial realization problem and the Nevanlinna-Pick interpolation problem. The arguments rely on a correspondence between elements in a given subclass of SI and elements in SC. Another important tool in the arguments is a new result pertaining to the classical tangential Schur algorithm.Comment: 46 page

On the Two Spectra Inverse Problem for Semi-Infinite Jacobi Matrices

We present results on the unique reconstruction of a semi-infinite Jacobi operator from the spectra of the operator with two different boundary conditions. This is the discrete analogue of the Borg-Marchenko theorem for Schr{\"o}dinger operators in the half-line. Furthermore, we give necessary and sufficient conditions for two real sequences to be the spectra of a Jacobi operator with different boundary conditions.Comment: In this slightly revised version we have reworded some of the theorems, and we updated two reference

The Two-Spectra Inverse Problem for Semi-Infinite Jacobi Matrices in The Limit-Circle Case

We present a technique for reconstructing a semi-infinite Jacobi operator in the limit circle case from the spectra of two different self-adjoint extensions. Moreover, we give necessary and sufficient conditions for two real sequences to be the spectra of two different self-adjoint extensions of a Jacobi operator in the limit circle case.Comment: 26 pages. Changes in the presentation of some result

Charged lepton electric dipole moments with the localized leptons and the new Higgs doublet in the two Higgs doublet model

We study the lepton electric dipole moments in the split fermion scenario, in the two Higgs doublet model, where the new Higgs scalars are localized around the origin in the extra dimension, with the help of the localizer field. We observe that the numerical value of the electron (muon, tau) electric dipole moment is at the order of the magnitude of 10^{-31} (10^{-24}, 10^{-22}) (e-cm) and this quantity is sensitive the new Higgs localization in the extra dimension.Comment: 20 pages, 7 figure

Schur functions and their realizations in the slice hyperholomorphic setting

we start the study of Schur analysis in the quaternionic setting using the theory of slice hyperholomorphic functions. The novelty of our approach is that slice hyperholomorphic functions allows to write realizations in terms of a suitable resolvent, the so called S-resolvent operator and to extend several results that hold in the complex case to the quaternionic case. We discuss reproducing kernels, positive definite functions in this setting and we show how they can be obtained in our setting using the extension operator and the slice regular product. We define Schur multipliers, and find their co-isometric realization in terms of the associated de Branges-Rovnyak space

Hierarchical Spherical Model from a Geometric Point of View

A continuous version of the hierarchical spherical model at dimension d=4 is investigated. Two limit distribution of the block spin variable X^{\gamma}, normalized with exponents \gamma =d+2 and \gamma =d at and above the critical temperature, are established. These results are proven by solving certain evolution equations corresponding to the renormalization group (RG) transformation of the O(N) hierarchical spin model of block size L^{d} in the limit L to 1 and N to \infty . Starting far away from the stationary Gaussian fixed point the trajectories of these dynamical system pass through two different regimes with distinguishable crossover behavior. An interpretation of this trajectories is given by the geometric theory of functions which describe precisely the motion of the Lee--Yang zeroes. The large--$N$ limit of RG transformation with L^{d} fixed equal to 2, at the criticality, has recently been investigated in both weak and strong (coupling) regimes by Watanabe \cite{W}. Although our analysis deals only with N=\infty case, it complements various aspects of that work.Comment: 27 pages, 6 figures, submitted to Journ. Stat. Phy