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35
Local Error Estimates for Radial Basis Function Interpolation of Scattered Data
 IMA J. Numer. Anal
, 1992
"... Introducing a suitable variational formulation for the local error of scattered data interpolation by radial basis functions OE(r), the error can be bounded by a term depending on the Fourier transform of the interpolated function f and a certain "Kriging function", which allows a formulation as an ..."
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Cited by 93 (20 self)
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Introducing a suitable variational formulation for the local error of scattered data interpolation by radial basis functions OE(r), the error can be bounded by a term depending on the Fourier transform of the interpolated function f and a certain "Kriging function", which allows a formulation as an integral involving the Fourier transform of OE. The explicit construction of locally wellbehaving admissible coefficient vectors makes the Kriging function bounded by some power of the local density h of data points. This leads to error estimates for interpolation of functions f whose Fourier transform f is "dominated" by the nonnegative Fourier transform / of /(x) = OE(kxk) in the sense R j f j 2 / \Gamma1 dt ! 1. Approximation orders are arbitrarily high for interpolation with Hardy multiquadrics, inverse multiquadrics and Gaussian kernels. This was also proven in recent papers by Madych and Nelson, using a reproducing kernel Hilbert space approach and requiring the same h...
Domains for Computation in Mathematics, Physics and Exact Real Arithmetic
 Bulletin of Symbolic Logic
, 1997
"... We present a survey of the recent applications of continuous domains for providing simple computational models for classical spaces in mathematics including the real line, countably based locally compact spaces, complete separable metric spaces, separable Banach spaces and spaces of probability dist ..."
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Cited by 48 (10 self)
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We present a survey of the recent applications of continuous domains for providing simple computational models for classical spaces in mathematics including the real line, countably based locally compact spaces, complete separable metric spaces, separable Banach spaces and spaces of probability distributions. It is shown how these models have a logical and effective presentation and how they are used to give a computational framework in several areas in mathematics and physics. These include fractal geometry, where new results on existence and uniqueness of attractors and invariant distributions have been obtained, measure and integration theory, where a generalization of the Riemann theory of integration has been developed, and real arithmetic, where a feasible setting for exact computer arithmetic has been formulated. We give a number of algorithms for computation in the theory of iterated function systems with applications in statistical physics and in period doubling route to chao...
A New Representation for Exact Real Numbers
, 1997
"... We develop the theoretical foundation of a new representation of real numbers based on the infinite composition of linear fractional transformations (lft), equivalently the infiite product of matrices, with nonnegative coefficients. Any rational interval in the one point compactification of the rea ..."
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Cited by 42 (8 self)
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We develop the theoretical foundation of a new representation of real numbers based on the infinite composition of linear fractional transformations (lft), equivalently the infiite product of matrices, with nonnegative coefficients. Any rational interval in the one point compactification of the real line, represented by the unit circle S¹, is expressed as the image of the base interval [0�1] under an lft. A sequence of shrinking nested intervals is then represented by an infinite product of matrices with integer coefficients such that the first socalled sign matrix determines an interval on which the real number lies. The subsequent socalled digit matrices have nonnegative integer coe cients and successively re ne that interval. Based on the classi cation of lft's according to their conjugacy classes and their geometric dynamics, we show that there is a canonical choice of four sign matrices which are generated by rotation of S¹ by =4. Furthermore, the ordinary signed digit representation of real numbers in a given base induces a canonical choice of digit matrices.
On the Asymptotic Cardinal Function of the Multiquadric ...
"... this paper, we prove that Ø c enjoys the following property: lim ..."
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Cited by 7 (1 self)
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this paper, we prove that Ø c enjoys the following property: lim
On the functions counting walks with small steps in the quarter plane
 Publ. Math. Inst. Hautes Études Sci
"... Abstract. Models of spatially homogeneous walks in the quarter plane Z 2 + with steps taken from a subset S of the set of jumps to the eight nearest neighbors are considered. The generating function (x,y,z) ↦ → Q(x,y;z) of the numbers q(i,j;n) of such walks starting at the origin and ending at (i,j) ..."
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Cited by 7 (6 self)
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Abstract. Models of spatially homogeneous walks in the quarter plane Z 2 + with steps taken from a subset S of the set of jumps to the eight nearest neighbors are considered. The generating function (x,y,z) ↦ → Q(x,y;z) of the numbers q(i,j;n) of such walks starting at the origin and ending at (i,j) ∈ Z 2 + after n steps is studied. For all nonsingular models of walks, the functions x ↦ → Q(x,0;z) and y ↦ → Q(0,y;z) are continued as multivalued functions on C having infinitely many meromorphic branches, of which the set of poles is identified. The nature of these functions is derived from this result: namely, for all the 51 walks which admit a certain infinite group of birational transformations of C 2, the interval]0,1/S [ of variation of z splits into two dense subsets such that the functions x ↦ → Q(x,0;z) and y ↦ → Q(0,y;z) are shown to be holonomic for any z from the one of them and nonholonomic for any z from the other. This entails the nonholonomy of (x,y,z) ↦ → Q(x,y;z), and therefore proves a conjecture of BousquetMélou and Mishna in [5].
2000), ‘The Equivalence Postulate of Quantum Mechanics
 International Journal of Modern Physics A15
"... The removal of the peculiar degeneration arising in the classical concepts of rest frame and time parameterization is at the heart of the recently formulated Equivalence Principle (EP). The latter, stating that all physical systems can be connected by a coordinate transformation to the free one with ..."
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Cited by 4 (1 self)
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The removal of the peculiar degeneration arising in the classical concepts of rest frame and time parameterization is at the heart of the recently formulated Equivalence Principle (EP). The latter, stating that all physical systems can be connected by a coordinate transformation to the free one with vanishing energy, univocally leads to the Quantum Stationary HJ Equation (QSHJE). This is a third–order non–linear differential equation which provides a trajectory representation of Quantum Mechanics (QM). The trajectories depend on the Planck length through hidden variables which arise as initial conditions. The formulation has manifest p–q duality, a consequence of the involutive nature of the Legendre transformation and of its recently observed relation with second–order linear differential equations. This reflects in an intrinsic ψ D –ψ duality between linearly independent solutions of the Schrödinger equation. Unlike Bohm’s theory, there is a non–trivial action even for bound states and no pilot–wave guide is present. A basic property of the formulation is that no use of any axiomatic interpretation of the wave–function is made. For example, tunnelling is a direct consequence of the quantum potential which differs from the Bohmian one and plays the role of particle’s self–energy. Furthermore, the QSHJE is defined only if the ratio ψ D /ψ is a local homeomorphism of the extended
Reproduction of Polynomials by Radial Basis Functions
, 1994
"... For radial basis function interpolation of scattered data in IR d , the approximative reproduction of highdegree polynomials is studied. Results include uniform error bounds and convergence orders on compact sets. x1. Introduction We consider interpolation of realvalued functions f defined on a ..."
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Cited by 3 (2 self)
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For radial basis function interpolation of scattered data in IR d , the approximative reproduction of highdegree polynomials is studied. Results include uniform error bounds and convergence orders on compact sets. x1. Introduction We consider interpolation of realvalued functions f defined on a set \Omega ` IR d ; d 1. These functions are interpolated on a set X := fx 1 ; : : : ; xNX g of NX 1 pairwise distinct points x 1 ; : : : ; xNX in \Omega\Gamma Interpolation is done by linear combinations of translates \Phi(x \Gamma x j ) of a single continuous realvalued function \Phi defined on IR d . For various reasons it is sometimes necessary to add the space IP d m of dvariate polynomials of order not exceeding m to the interpolating functions. Interpolation is uniquely possible under the requirement If p 2 P d m satisfies p(x i ) = 0 for all x i 2 X then p = 0; (1) and if \Phi is conditionally positive definite of order m (see e.g. [8]): Definition 1. A function \Phi...
Identifying Linear Combinations of Ridge Functions
 Adv. Appl. Math
, 1999
"... . This paper is about an inverse problem. We assume we are given a function f(x) which is some sum of ridge functions of the form P m i=1 g i (a i \Delta x) and we just know an upper bound on m. We seek to identify the functions g i and also the directions a i from such limited information. S ..."
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Cited by 3 (1 self)
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. This paper is about an inverse problem. We assume we are given a function f(x) which is some sum of ridge functions of the form P m i=1 g i (a i \Delta x) and we just know an upper bound on m. We seek to identify the functions g i and also the directions a i from such limited information. Several ways to solve this nonlinear problem are discussed in this work. x1. Introduction A ridge function is a multivariate function h : IR n ! IR of the simple form h(x 1 ; : : : ; xn ) = g(a 1 x 1 + \Delta \Delta \Delta + anxn ) = g(a \Delta x); where g : IR ! IR and a = (a 1 ; : : : ; an ) 2 IR n nf0g. In other words, it is a multivariate function constant on the parallel hyperplanes a \Delta x = c, c 2 IR. The vector a 2 IR n nf0g is generally called the direction. Ridge functions appear in various areas and under various guises. We find them in the area of partial differential equations (where they have been known for many, many years under the name of plane waves [7]). We al...
”New” Veneziano amplitudes from ”old” Fermat (hyper)surfaces
, 2003
"... The history of the discovery of bosonic string theory is well documented. This theory evolved as an attempt to find a multidimensional analogue of Euler’s beta function to describe the multiparticle Veneziano amplitudes. Such an analogue had in fact been known in mathematics at least in 1922. Its ma ..."
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Cited by 3 (2 self)
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The history of the discovery of bosonic string theory is well documented. This theory evolved as an attempt to find a multidimensional analogue of Euler’s beta function to describe the multiparticle Veneziano amplitudes. Such an analogue had in fact been known in mathematics at least in 1922. Its mathematical meaning was studied subsequently from different angles by mathematicians such as Selberg, Weil and Deligne among others. The mathematical interpretation of this multidimensional beta function that was developed subsequently is markedly different from that described in physics literature. This work aims to bridge the gap between the mathematical and physical treatments. Using some results of recent publications (e.g. J.Geom.Phys.38 (2001) 81; ibid 43 (2002) 45) new topological, algebrogeometric, numbertheoretic and combinatorial treatment of the multiparticle Veneziano amplitudes is developed. As a result, an entirely new physical meaning of these amplitudes is emerging: they are periods of differential forms associated with homology cycles on Fermat (hyper)surfaces. Such (hyper)surfaces are considered as complex projective varieties of Hodge type. Although the computational formalism developed in this work resembles that used in mirror symmetry calculations, many additional results from mathematics are used along with their suitable physical interpretation. For instance, the Hodge spectrum of the Fermat (hyper)surfaces is in onetoone correspondence with the possible spectrum of particle masses. The formalism also allows us to obtain correlation functions of both conformal field theory and particle physics using the same type of the PicardFuchs equations whose solutions are being interpreted in terms of periods.
Comments on the links between su(3) modular invariants, simple factors in the Jacobian of Fermat curves, and rational triangular billiards
, 1997
"... We examine the proposal made recently that the su(3) modular invariant partition functions could be related to the geometry of the complex Fermat curves. Although a number of coincidences and similarities emerge between them and certain algebraic curves related to triangular billiards, their meaning ..."
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Cited by 2 (1 self)
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We examine the proposal made recently that the su(3) modular invariant partition functions could be related to the geometry of the complex Fermat curves. Although a number of coincidences and similarities emerge between them and certain algebraic curves related to triangular billiards, their meaning remains obscure. In an attempt to go beyond the su(3) case, we show that any rational conformal field theory determines canonically a Riemann surface. a b