Rapid evaluation of radial basis functions

Over the past decade, the radial basis function method has been shown to produce high quality solutions to the multivariate scattered data interpolation problem. However, this method has been associated with very high computational cost, as compared to alternative methods such as finite element or multivariate spline interpolation. For example. the direct evaluation at M locations of a radial basis function interpolant with N centres requires O(M N) floating-point operations. In this paper we introduce a fast evaluation method based on the Fast Gauss Transform and suitable quadrature rules. This method has been applied to the Hardy multiquadric, the inverse multiquadric and the thin-plate spline to reduce the computational complexity of the interpolant evaluation to O(M + N) floating point operations. By using certain localisation properties of conditionally negative definite functions this method has several performance advantages against traditional hierarchical rapid summation methods which we discuss in detail

Radial basis functions for the sphere

In this paper we compute the ultraspherical series expansions for the more commonly used radial basis functions. In several special cases we provide asymptotic estimates for the decay rate of the coefficients involved. knowledge of the decay rate of these coefficients is useful because they enable error estimates for spherical interpolation

Phase-field approach to heterogeneous nucleation

We consider the problem of heterogeneous nucleation and growth. The system is described by a phase field model in which the temperature is included through thermal noise. We show that this phase field approach is suitable to describe homogeneous as well as heterogeneous nucleation starting from several general hypotheses. Thus we can investigate the influence of grain boundaries, localized impurities, or any general kind of imperfections in a systematic way. We also put forward the applicability of our model to study other physical situations such as island formation, amorphous crystallization, or recrystallization.Comment: 8 pages including 7 figures. Accepted for publication in Physical Review

Susceptibility of the Spin 1/2 Heisenberg Antiferromagnetic Chain

Highly accurate results are presented for the susceptibility, $\chi (T)$ of the $s=1/2$ Heisenberg antiferromagnetic chain for all temperatures, using the Bethe ansatz and field theory methods. After going through a rounded peak, $\chi (T)$ approaches its asympotic zero-temperature value with infinite slope.Comment: 8 pages and 3 postscript figures appended (uuencoded), Revtex, Report #:UBCTP-94-00

Adolescent trajectories of aerobic fitness and adiposity as markers of cardiometabolic risk in adulthood

Purpose: The aim of this study was to investigate whether adolescent growth trajectories of aerobic fitness and adiposity were associated with mid-adulthood cardiometabolic risk (CMR). Methods: Participants were drawn from the Saskatchewan Growth and Development Study (1963-1973). Adolescent growth trajectories for maximal aerobic capacity (absolute VO2 (AbsVO2)), skinfolds (SF), representing total body (Sum6SF) and central adiposity (TrunkSF), and body mass index (BMI) were determined from 7 to 17 years of age. In mid-adulthood (40 to 50 years of age), 61 individuals (23 females) returned for follow-ups. A CMR score was calculated to group participants as displaying either high or a low CMR. Multilevel hierarchical models were constructed, comparing the adolescent growth trajectories of AbsVO2, Sum6SF, TrunkSF, and BMI between CMR groupings. Results: There were no significant differences in the adolescent development of AbsVO2, Sum6SF, TrunkSF, and BMI between adult CMR groupings (p > 0.05). Individuals with high CMR accrued 62% greater adjusted total body fat percentage from adolescence to adulthood (p=0.03). Conclusions: Growth trajectories of adolescent aerobic fitness and adiposity do not appear to be associated with mid-adulthood CMR. Individuals should be encouraged to participate in behaviours that promote healthy aerobic fitness and adiposity levels throughout life to reduce lifelong CMR

Regional Conformational Flexibility Couples Substrate Specificity and Scissile Phosphate Diester Selectivity in Human Flap Endonuclease 1

Human flap endonuclease-1 (hFEN1) catalyzes the divalent metal ion-dependent removal of single-stranded DNA protrusions known as flaps during DNA replication and repair. Substrate selectivity involves passage of the 5′-terminus/flap through the arch and recognition of a single nucleotide 3′-flap by the α2–α3 loop. Using NMR spectroscopy, we show that the solution conformation of free and DNA-bound hFEN1 are consistent with crystal structures; however, parts of the arch region and α2–α3 loop are disordered without substrate. Disorder within the arch explains how 5′-flaps can pass under it. NMR and single-molecule FRET data show a shift in the conformational ensemble in the arch and loop region upon addition of DNA. Furthermore, the addition of divalent metal ions to the active site of the hFEN1–DNA substrate complex demonstrates that active site changes are propagated via DNA-mediated allostery to regions key to substrate differentiation. The hFEN1–DNA complex also shows evidence of millisecond timescale motions in the arch region that may be required for DNA to enter the active site. Thus, hFEN1 regional conformational flexibility spanning a range of dynamic timescales is crucial to reach the catalytically relevant ensemble

Algebraic Self-Similar Renormalization in Theory of Critical Phenomena

We consider the method of self-similar renormalization for calculating critical temperatures and critical indices. A new optimized variant of the method for an effective summation of asymptotic series is suggested and illustrated by several different examples. The advantage of the method is in combining simplicity with high accuracy.Comment: 1 file, 44 pages, RevTe

Determinant Representations for Correlation Functions of Spin-1/2 XXX and XXZ Heisenberg Magnets

We consider correlation functions of the spin-\half XXX and XXZ Heisenberg chains in a magnetic field. Starting from the algebraic Bethe Ansatz we derive representations for various correlation functions in terms of determinants of Fredholm integral operators.Comment: 23 pages, TeX, BONN-TH-94-14, revised version: typos correcte

Birkhoff type decompositions and the Baker-Campbell-Hausdorff recursion

We describe a unification of several apparently unrelated factorizations arisen from quantum field theory, vertex operator algebras, combinatorics and numerical methods in differential equations. The unification is given by a Birkhoff type decomposition that was obtained from the Baker-Campbell-Hausdorff formula in our study of the Hopf algebra approach of Connes and Kreimer to renormalization in perturbative quantum field theory. There we showed that the Birkhoff decomposition of Connes and Kreimer can be obtained from a certain Baker-Campbell-Hausdorff recursion formula in the presence of a Rota-Baxter operator. We will explain how the same decomposition generalizes the factorization of formal exponentials and uniformization for Lie algebras that arose in vertex operator algebra and conformal field theory, and the even-odd decomposition of combinatorial Hopf algebra characters as well as to the Lie algebra polar decomposition as used in the context of the approximation of matrix exponentials in ordinary differential equations.Comment: accepted for publication in Comm. in Math. Phy