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

Some remarks on a generalization of the superintegrable chiral Potts model

The spontaneous magnetization of a two-dimensional lattice model can be expressed in terms of the partition function $W$ of a system with fixed boundary spins and an extra weight dependent on the value of a particular central spin. For the superintegrable case of the chiral Potts model with cylindrical boundary conditions, W can be expressed in terms of reduced hamiltonians H and a central spin operator S. We conjectured in a previous paper that W can be written as a determinant, similar to that of the Ising model. Here we generalize this conjecture to any Hamiltonians that satisfy a more general Onsager algebra, and give a conjecture for the elements of S.Comment: 18 pages, one figur

Ferromagnetic models for cooperative behavior: Revisiting Universality in complex phenomena

Ferromagnetic models are harmonic oscillators in statistical mechanics. Beyond their original scope in tackling phase transition and symmetry breaking in theoretical physics, they are nowadays experiencing a renewal applicative interest as they capture the main features of disparate complex phenomena, whose quantitative investigation in the past were forbidden due to data lacking. After a streamlined introduction to these models, suitably embedded on random graphs, aim of the present paper is to show their importance in a plethora of widespread research fields, so to highlight the unifying framework reached by using statistical mechanics as a tool for their investigation. Specifically we will deal with examples stemmed from sociology, chemistry, cybernetics (electronics) and biology (immunology).Comment: Contributing to the proceedings of the Conference "Mathematical models and methods for Planet Heart", INdAM, Rome 201

Ratios of BB and DD Meson Decay Constants in Relativistic Quark Model

We calculate the ratios of $B$ and $D$ meson decay constants by applying the variational method to the relativistic hamiltonian of the heavy meson. We adopt the Gaussian and hydrogen-type trial wave functions, and use six different potentials of the potential model. We obtain reliable results for the ratios, which are similar for different trial wave functions and different potentials. The obtained ratios show the deviation from the nonrelativistic scaling law, and they are in a pretty good agreement with the results of the Lattice calculations.Comment: 13 pages, 1 Postscript figur

Adjustment to colostomy: stoma acceptance, stoma care self-efficacy and interpersonal relationships

‘The definitive version is available at www.blackwell-synergy.com.’ Copyright Blackwell Publishing. DOI: 10.1111/j.1365-2648.2007.04446.xThis paper is a report of a study to examine adjustment and its relationship with stoma acceptance and social interaction, and the link between stoma care self-efficacy and adjustment in the presence of acceptance and social interactions.Peer reviewe

The critical Ising model via Kac-Ward matrices

The Kac-Ward formula allows to compute the Ising partition function on any finite graph G from the determinant of 2^{2g} matrices, where g is the genus of a surface in which G embeds. We show that in the case of isoradially embedded graphs with critical weights, these determinants have quite remarkable properties. First of all, they satisfy some generalized Kramers-Wannier duality: there is an explicit equality relating the determinants associated to a graph and to its dual graph. Also, they are proportional to the determinants of the discrete critical Laplacians on the graph G, exactly when the genus g is zero or one. Finally, they share several formal properties with the Ray-Singer \bar\partial-torsions of the Riemann surface in which G embeds.Comment: 30 pages, 10 figures; added section 4.4 in version

On the asymmetric simple exclusion process with multiple species

In the asymmetric simple exclusion process on the integers each particle waits exponential time, then with probability p it moves one step to the right if the site is unoccupied, otherwise it stays put; and with probability q=1-p it moves one step to the left if the site is unoccupied, otherwise it stays put. In previous work the authors, using the Bethe Ansatz, found for N-particle ASEP a formula --- a sum of multiple integrals --- for the probability that a system is in a particular configuration at time t given an initial configuration. The present work extends this to the case where particles are of different species, with particles of a higher species having priority over those of a lower species. Here the integrands in the multiple integrals are defined by a system of relations whose consistency requires verifying that the Yang-Baxter equations are satisfied.Comment: 17 pages. Version 3 corrects misprints, clarifies some points, and adds reference

Glassiness and constrained dynamics of a short-range non-disordered spin model

We study the low temperature dynamics of a two dimensional short-range spin system with uniform ferromagnetic interactions, which displays glassiness at low temperatures despite the absence of disorder or frustration. The model has a dual description in terms of free defects subject to dynamical constraints, and is an explicit realization of the ``hierarchically constrained dynamics'' scenario for glassy systems. We give a number of exact results for the statics of the model, and study in detail the dynamical behaviour of one-time and two-time quantities. We also consider the role played by the configurational entropy, which can be computed exactly, in the relation between fluctuations and response.Comment: 10 pages, 9 figures; minor changes, references adde

A leap forward in geographic scale for forest ectomycorrhizal fungi

Published versio

Leptonic and Semileptonic Decays of Charm and Bottom Hadrons

We review the experimental measurements and theoretical descriptions of leptonic and semileptonic decays of particles containing a single heavy quark, either charm or bottom. Measurements of bottom semileptonic decays are used to determine the magnitudes of two fundamental parameters of the standard model, the Cabibbo-Kobayashi-Maskawa matrix elements $V_{cb}$ and $V_{ub}$. These parameters are connected with the physics of quark flavor and mass, and they have important implications for the breakdown of CP symmetry. To extract precise values of $|V_{cb}|$ and $|V_{ub}|$ from measurements, however, requires a good understanding of the decay dynamics. Measurements of both charm and bottom decay distributions provide information on the interactions governing these processes. The underlying weak transition in each case is relatively simple, but the strong interactions that bind the quarks into hadrons introduce complications. We also discuss new theoretical approaches, especially heavy-quark effective theory and lattice QCD, which are providing insights and predictions now being tested by experiment. An international effort at many laboratories will rapidly advance knowledge of this physics during the next decade.Comment: This review article will be published in Reviews of Modern Physics in the fall, 1995. This file contains only the abstract and the table of contents. The full 168-page document including 47 figures is available at http://charm.physics.ucsb.edu/papers/slrevtex.p