A problem of practical importance is that of generating a large number of random elements in a group. Of particular interest are the orthogonal groups SO(n + 1). A closely …
A problem of practical importance is that of generating a large number of random elements in a group. Of particular interest are the orthogonal groups SO(n + 1). A closely related problem is the one of placing a large number of points on the sphere S n in a uniform way. For a survey and many references see Sloane [Sl] and Berger [Beg].
Abstract We study four different methods for distributing points on the sphere and numerically analyze their relative merits with respect to certain metrics. Keywords: Distributing points on a sphereFekete pointslogarithmic …
Abstract We study four different methods for distributing points on the sphere and numerically analyze their relative merits with respect to certain metrics. Keywords: Distributing points on a sphereFekete pointslogarithmic potentialColumb potential
In this work, we present uniformly distributed sequences on theunit sphere, and we show that this property is equivalent torequiring the sequences to have a low discrepancy. Numericalintegration over the …
In this work, we present uniformly distributed sequences on theunit sphere, and we show that this property is equivalent torequiring the sequences to have a low discrepancy. Numericalintegration over the sphere is taken as a direct application, andthe corresponding errors are estimated. Special care is taken inrelating these concepts and properties to those for the euclideancase. Several examples of uniformly distributed sequences of nodes(ensembles) are presented.
How to distribute a set of points uniformly on a spherical surface is a longstanding problem that still lacks a definite answer. In this work, we introduce a physical measure …
How to distribute a set of points uniformly on a spherical surface is a longstanding problem that still lacks a definite answer. In this work, we introduce a physical measure of uniformity based on the distribution of distances between points, as an alternative to commonly adopted measures based on interaction potentials. We then use this new measure of uniformity to characterize several algorithms available in the literature. We also study the effect of optimizing the position of the points through the minimization of different interaction potentials via a gradient descent procedure. In this way, we can classify different algorithms and interaction potentials to find the one that generates the most uniform distribution of points on the sphere.
How to distribute a set of points uniformly on a spherical surface is a very old problem that still lacks a definite answer. In this work, we introduce a physical …
How to distribute a set of points uniformly on a spherical surface is a very old problem that still lacks a definite answer. In this work, we introduce a physical measure of uniformity based on the distribution of distances between points, as an alternative to commonly adopted measures based on interaction potentials. We then use this new measure of uniformity to characterize several algorithms available in the literature. We also study the effect of optimizing the position of the points through the minimization of different interaction potentials via a gradient descent procedure. In this way, we can classify different algorithms and interaction potentials to find the one that generates the most uniform distribution of points on the sphere.
Points on a Sphere 10855 [2001, 171]. Proposed by Ernesto Bruno Cossi, Universidade Federal do Rio Grande do Sul, Porto Alegre, Brazil. Given n + 2 points P1, P2, ..., …
Points on a Sphere 10855 [2001, 171]. Proposed by Ernesto Bruno Cossi, Universidade Federal do Rio Grande do Sul, Porto Alegre, Brazil. Given n + 2 points P1, P2, ..., Pn+2 in R, let f(j) be the volume of the n-dimensional simplex whose vertex set is {P1, P2, ..., Pn+2} {Pj), and let g(j) = +f(j) according to whether the vertices P1, P2,... , Pj-1, Pj+, ... Pn+2 give the positive orientation on the simplex. Prove that the Pj lie on a in R if and only if Ej+2(-l)ys(Pj)g(j) = 0, where s(P) is the square of the distance from the origin to the point P. Solution by Robin Chapman, University of Exeter, Exeter, U. K. For the problem statement to be correct, we must replace the phrase common sphere with common or hyperplane. Denote the coordinates of a point Q in IRn by (x (Q), ... , X (Q)). If Q1, ... , Qn+ are n + 1 points in Rn, then the determinant 1 xl(QI) ... xn(Qi) 1 xl(Q2) ... Xn(Q2) D(Ql,..., Qn+l) =
We consider the error term P 3 ( R ) = # { n ∈ Z 3 : | n | ⩽ R } − 4 3 π R 3 …
We consider the error term P 3 ( R ) = # { n ∈ Z 3 : | n | ⩽ R } − 4 3 π R 3 which occurs in the counting of lattice points in a sphere of radius R. By considering second and third power moments, we prove that P 3 ( R ) = Ω ± ( R l o g R ) . An upper bound for the gap between the sign changes of P3(R) is also proved. 1991 Mathematics Subject Classification 11P21.
The concept of a Point Cloud has played an increasingly important role in many areas of Engineering, Science, and Mathematics. Examples are: LIDAR, 3D-Printing, Data Analysis, Computer Graphics, Machine Learning, …
The concept of a Point Cloud has played an increasingly important role in many areas of Engineering, Science, and Mathematics. Examples are: LIDAR, 3D-Printing, Data Analysis, Computer Graphics, Machine Learning, Mathematical Visualization, Numerical Analysis, and Monte Carlo Methods. Entering point cloud into Google returns nearly 3.5 million results! A point cloud for a finite volume manifold M is a finite subset or a sequence in M, with the essential feature that it is a representative sample of M. The definition of a point cloud varies with its use, particularly what constitutes being representative. Point clouds arise in many different ways: in LIDAR they are just 3D data captured by a scanning device, while in Monte Carlo applications they are constructed using highly complex algorithms developed over many years. In this article we outline a rigorous mathematical theory of point clouds, based on the classic Cauchy Crofton formula of Integral Geometry and its generalizations. We begin with point clouds on surfaces in R^3, which simplifies the exposition and makes our constructions easily visualizable. We proceed to hyper-surfaces and then sub-manifolds of arbitrary co-dimension in R^n, and finally, using an elegant result of Jurgen Moser to arbitrary smooth manifolds with a volume element.
The concept of a Point Cloud has played an increasingly important role in many areas of Engineering, Science, and Mathematics. Examples are: LIDAR, 3D-Printing, Data Analysis, Computer Graphics, Machine Learning, …
The concept of a Point Cloud has played an increasingly important role in many areas of Engineering, Science, and Mathematics. Examples are: LIDAR, 3D-Printing, Data Analysis, Computer Graphics, Machine Learning, Mathematical Visualization, Numerical Analysis, and Monte Carlo Methods. Entering point cloud into Google returns nearly 3.5 million results! A point cloud for a finite volume manifold M is a finite subset or a sequence in M, with the essential feature that it is a representative sample of M. The definition of a point cloud varies with its use, particularly what constitutes being representative. Point clouds arise in many different ways: in LIDAR they are just 3D data captured by a scanning device, while in Monte Carlo applications they are constructed using highly complex algorithms developed over many years. In this article we outline a rigorous mathematical theory of point clouds, based on the classic Cauchy Crofton formula of Integral Geometry and its generalizations. We begin with point clouds on surfaces in R^3, which simplifies the exposition and makes our constructions easily visualizable. We proceed to hyper-surfaces and then sub-manifolds of arbitrary co-dimension in R^n, and finally, using an elegant result of Jurgen Moser to arbitrary smooth manifolds with a volume element.
Uniformly inserting points on the sphere has been found useful in many scientific and engineering fields. Different from the offline version where the number of points is known in advance, …
Uniformly inserting points on the sphere has been found useful in many scientific and engineering fields. Different from the offline version where the number of points is known in advance, we consider the online version of this problem. The requests for point insertion arrive one by one and the target is to insert points as uniformly as possible. To measure the uniformity we use gap ratio which is defined as the ratio of the maximal gap to the minimal gap of two arbitrary inserted points. We propose a two-phase online insertion strategy with gap ratio of at most 3.69 . Moreover, the lower bound of the gap ratio is proved to be at least 1.78 .
Abstract We consider the assignment problem between two sets of N random points on a smooth, two-dimensional manifold $$\Omega $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>Ω</mml:mi></mml:math> of unit area. It is known that the …
Abstract We consider the assignment problem between two sets of N random points on a smooth, two-dimensional manifold $$\Omega $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>Ω</mml:mi></mml:math> of unit area. It is known that the average cost scales as $$E_{\Omega }(N)\sim {1}/{2\pi }\ln N$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>E</mml:mi><mml:mi>Ω</mml:mi></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mi>N</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>∼</mml:mo><mml:mn>1</mml:mn><mml:mo>/</mml:mo><mml:mrow><mml:mn>2</mml:mn><mml:mi>π</mml:mi></mml:mrow><mml:mo>ln</mml:mo><mml:mi>N</mml:mi></mml:mrow></mml:math> with a correction that is at most of order $$\sqrt{\ln N\ln \ln N}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msqrt><mml:mrow><mml:mo>ln</mml:mo><mml:mi>N</mml:mi><mml:mo>ln</mml:mo><mml:mo>ln</mml:mo><mml:mi>N</mml:mi></mml:mrow></mml:msqrt></mml:math> . In this paper, we show that, within the linearization approximation of the field-theoretical formulation of the problem, the first $$\Omega $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>Ω</mml:mi></mml:math> -dependent correction is on the constant term, and can be exactly computed from the spectrum of the Laplace–Beltrami operator on $$\Omega $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>Ω</mml:mi></mml:math> . We perform the explicit calculation of this constant for various families of surfaces, and compare our predictions with extensive numerics.
Abstract Anticoherent spin states are introduced as quantum states that exhibit maximally nonclassical behaviors which are important in the study of quantum entanglement. Using Majorana representation of spin states, Crann–Pereira–Kribs …
Abstract Anticoherent spin states are introduced as quantum states that exhibit maximally nonclassical behaviors which are important in the study of quantum entanglement. Using Majorana representation of spin states, Crann–Pereira–Kribs studied the relation between anticoherent spin states and spherical designs. They proposed a conjecture that a spin- s state is anticoherent to order t if and only if its Majorana representation is a spherical t -design on S 2 . In this paper, we prove that this conjecture is true for s = 2.
We review what is known, unknown and expected about the mathematical properties of Coulomb and Riesz gases. Those describe infinite configurations of points in $\mathbb{R}^d$ interacting with the Riesz potential …
We review what is known, unknown and expected about the mathematical properties of Coulomb and Riesz gases. Those describe infinite configurations of points in $\mathbb{R}^d$ interacting with the Riesz potential $\pm |x|^{-s}$ (resp. $-\log|x|$ for $s=0$). Our presentation follows the standard point of view of statistical mechanics, but we also mention how these systems arise in other important situations (e.g. in random matrix theory). The main question addressed in the article is how to properly define the associated infinite point process and characterize it using some (renormalized) equilibrium equation. This is largely open in the long range case $s<d$. For the convenience of the reader we give the detail of what is known in the short range case $s>d$. In the last part we discuss phase transitions and mention what is expected on physical grounds.
Abstract The spherical ensemble is a well-known ensemble of N repulsive points on the two-dimensional sphere, which can realized in various ways (as a random matrix ensemble, a determinantal point …
Abstract The spherical ensemble is a well-known ensemble of N repulsive points on the two-dimensional sphere, which can realized in various ways (as a random matrix ensemble, a determinantal point process, a Coulomb gas, a Quantum Hall state...). Here we show that the spherical ensemble enjoys remarkable convergence properties from the point of view of numerical integration. More precisely, it is shown that the numerical integration rule corresponding to N nodes on the two-dimensional sphere sampled in the spherical ensemble is, with overwhelming probability, nearly a quasi-Monte-Carlo design in the sense of Brauchart-Saff-Sloan-Womersley for any smoothness parameter $$s\le 2.$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>s</mml:mi> <mml:mo>≤</mml:mo> <mml:mn>2</mml:mn> <mml:mo>.</mml:mo> </mml:mrow> </mml:math> The key ingredient is a new explicit sub-Gaussian concentration of measure inequality for the spherical ensemble.
Understanding how particles are arranged on the sphere is not only central to numerous physical, biological, and materials systems but also finds applications in mathematics and in analysis of geophysical …
Understanding how particles are arranged on the sphere is not only central to numerous physical, biological, and materials systems but also finds applications in mathematics and in analysis of geophysical and meteorological measurements. In contrast to particle distributions in Euclidean space, restriction that the particles should lie on the sphere brings about several important constraints. These require a careful extension of quantities used for particle distributions in Euclidean space to those confined to the sphere. We introduce a framework designed to analyze and classify structural (dis)order in particle distributions constrained to the sphere. The classification is based on the concept of hyperuniformity, which was introduced 15 years ago and since then studied extensively in Euclidean space, yet has only very recently been considered for the sphere. We build our framework on a generalization of the structure factor on the sphere, which we relate to the power spectrum of the corresponding multipole expansion. The spherical structure factor is then shown to couple with cap number variance, a measure of local density fluctuations, allowing us to derive different forms of the variance. In this way, we construct a classification of hyperuniformity for scale-free particle distributions on the sphere and show how it can be extended to other distributions as well. We demonstrate that hyperuniformity on the sphere can be defined either through a vanishing spherical structure factor at low multipole numbers or through a scaling of the cap number variance, in both cases extending the Euclidean definition while pointing out crucial differences. Our work provides a comprehensive tool for detecting long-range order on spheres and the analysis of spherical computational meshes, biological and synthetic spherical assemblies, and ordering phase transitions in spherically-distributed particles.
The ability to construct uniform deterministic samples of rotation groups is useful in many contexts, but there are inherent mathematical difficulties that prevent an exact solution. Here, we present successive …
The ability to construct uniform deterministic samples of rotation groups is useful in many contexts, but there are inherent mathematical difficulties that prevent an exact solution. Here, we present successive orthogonal images, an effective means for uniform deterministic sampling of orthogonal groups. The method is valid in any dimension, and analytical bounds are provided on the sampling uniformity. Numerical comparisons with other sampling methods are given for the special case of $SO(3)$. We make use of non-Riemannian distance metrics that are group-invariant and locally compatible with the Haar measure. In addition, our results provide a semi-unique decomposition of any orthogonal matrix into the product of planar rotations.
We formulate the problem of finding equilibrium configurations of N -point vortices in the plane in terms of a gradient flow on the smallest singular value of a skew-symmetric matrix …
We formulate the problem of finding equilibrium configurations of N -point vortices in the plane in terms of a gradient flow on the smallest singular value of a skew-symmetric matrix M whose nullspace structure determines the (real) strengths, rotational frequency and translational velocity of the configuration. A generic configuration gives rise to a matrix with empty nullspace, and hence is not a relative equilibrium for any choice of vortex strengths. We formulate the problem as a gradient flow in the space of square covariance matrices M T M . The evolution equation for drives the configuration to one with a real nullspace, establishing the existence of an equilibrium for vortex strengths that are elements of the nullspace of the matrix. We formulate both the unconstrained gradient flow problem where the point vortex strengths are determined a posteriori by the nullspace of M and the constrained problem where the point vortex strengths are chosen a priori and one seeks configurations for which those strengths are elements of the nullspace.
Sylvia Serfaty is interested in developing analysis tools to understand problems from physics. She has extensively studied the Ginzburg–Landau model of super- conductivity and this has led her to questions …
Sylvia Serfaty is interested in developing analysis tools to understand problems from physics. She has extensively studied the Ginzburg–Landau model of super- conductivity and this has led her to questions on the statistical mechanics of Coulomb-type systems. She has been splitting her career between Paris and New York.
For a compact set A in Euclidean space we consider the asymptotic behavior of optimal (and near optimal) N-point configurations that minimize the Riesz s-energy (corresponding to the potential 1/t^s) …
For a compact set A in Euclidean space we consider the asymptotic behavior of optimal (and near optimal) N-point configurations that minimize the Riesz s-energy (corresponding to the potential 1/t^s) over all N-point subsets of A, where s>0. For a large class of manifolds A having finite, positive d-dimensional Hausdorff measure, we show that such minimizing configurations have asymptotic limit distribution (as N tends to infinity with s fixed) equal to d-dimensional Hausdorff measure whenever s>d or s=d. In the latter case we obtain an explicit formula for the dominant term in the minimum energy. Our results are new even for the case of the d-dimensional sphere.
The statistical literature and practitioners have long advocated the use of confirmation experiments as the final stage of a sequence of designed experiments to verify that the optimal operating conditions …
The statistical literature and practitioners have long advocated the use of confirmation experiments as the final stage of a sequence of designed experiments to verify that the optimal operating conditions identified as part of a response surface methodology strategy are attainable and able to achieve the value of the response desired. However, until recently there has been a gap between this recommendation and details about how to perform an analysis to quantitatively assess whether the confirmation runs are adequate. Similarly, there has been little in the way of specific recommendations for the number and nature of the confirmation runs that should be performed. In this article, we propose analysis methods to assess agreement between the mean response from previous experiments and the confirmation experiment, and suggest a strategy for the design of confirmation experiments that more fully explores the region around the optimum.
For a symmetric kernel $k:X\times X \to \mathbb{R}\cup\{+\infty\}$ on a locally compact Hausdorff space $X$, we investigate the asymptotic behavior of greedy $k$-energy points $\{a_{i}\}_{1}^{\infty}$ for a compact subset $A\subset …
For a symmetric kernel $k:X\times X \to \mathbb{R}\cup\{+\infty\}$ on a locally compact Hausdorff space $X$, we investigate the asymptotic behavior of greedy $k$-energy points $\{a_{i}\}_{1}^{\infty}$ for a compact subset $A\subset X$ that are defined inductively by selecting $a_{1}\in A$ arbitrarily and $a_{n+1}$ so that $\sum_{i=1}^{n}k(a_{n+1},a_{i})=\inf_{x\in A}\sum_{i=1}^{n}k(x,a_{i})$. We give sufficient conditions under which these points (also known as Leja points) are asymptotically energy minimizing (i.e. have energy $\sum_{i\neq j}^{N}k(a_{i},a_{j})$ as $N\to\infty$ that is asymptotically the same as $\mathcal{E}(A,N):=\min\{\sum_{i\neq j}k(x_{i},x_{j}):x_{1},...,x_{N}\in A\}$), and have asymptotic distribution equal to the equilibrium measure for $A$. For the case of Riesz kernels $k_{s}(x,y):=|x-y|^{-s}$, $s>0$, we show that if $A$ is a rectifiable Jordan arc or closed curve in $\mathbb{R}^{p}$ and $s>1$, then greedy $k_{s}$-energy points are not asymptotically energy minimizing, in contrast to the case $s<1$. (In fact we show that no sequence of points can be asymptotically energy minimizing for $s>1$.) Additional results are obtained for greedy $k_{s}$-energy points on a sphere, for greedy best-packing points, and for weighted Riesz kernels.
Compressive Sensing (CS) theory states that real-world signals can often be recovered from much fewer measurements than those suggested by the Shannon sampling theorem. Nevertheless, recoverability does not only depend …
Compressive Sensing (CS) theory states that real-world signals can often be recovered from much fewer measurements than those suggested by the Shannon sampling theorem. Nevertheless, recoverability does not only depend on the signal, but also on the measurement scheme. The measurement matrix should behave as close as possible to an isometry for the signals of interest. Therefore the search for optimal CS measurement matrices of size $m\times n$ translates into the search for a set of $n$ $m$-dimensional vectors with minimal coherence. Best Complex Antipodal Spherical Codes (BCASCs) are known to be optimal in terms of coherence. An iterative algorithm for BCASC generation has been recently proposed that tightly approaches the theoretical lower bound on coherence. Unfortunately, the complexity of each iteration depends quadratically on $m$ and $n$. In this work we propose a modification of the algorithm that allows reducing the quadratic complexity to linear on both $m$ and $n$. Numerical evaluation showed that the proposed approach does not worsen the coherence of the resulting BCASCs. On the contrary, an improvement was observed for large $n$. The reduction of the computational complexity paves the way for using the BCASCs as CS measurement matrices in problems with large $n$. We evaluate the CS performance of the BCASCs for recovering sparse signals. The BCASCs are shown to outperform both complex random matrices and Fourier ensembles as CS measurement matrices, both in terms of coherence and sparse recovery performance, especially for low $m/n$, which is the case of interest in CS.
We study the statistical mechanics of classical two-dimensional Coulomb gases with general potential and arbitrary \beta, the inverse of the temperature. Such ensembles also correspond to random matrix models in …
We study the statistical mechanics of classical two-dimensional Coulomb gases with general potential and arbitrary \beta, the inverse of the temperature. Such ensembles also correspond to random matrix models in some particular cases. The formal limit case \beta=\infty corresponds to Fekete and also falls within our analysis.
It is known that in such a system points should be asymptotically distributed according to a macroscopic measure, and that a large deviations principle holds for this, as proven by Ben Arous and Zeitouni.
By a suitable splitting of the Hamiltonian, we connect the problem to the renormalized energy W, a Coulombian interaction for points in the plane introduced in our prior work, which is expected to be a good way of measuring the disorder of an infinite configuration of points in the plane. By so doing, we are able to examine the situation at the microscopic scale, and obtain several new results: a next order asymptotic expansion of the partition function, estimates on the probability of fluctuation from the equilibrium measure at microscale, and a large deviations type result, which states that configurations above a certain threshhold of W have exponentially small probability. When \beta\to \infty, the estimate becomes sharp, showing that the system has to crystallize to a minimizer of W. In the case of weighted Fekete sets, this corresponds to saying that these sets should microscopically look almost everywhere like minimizers of W, which are conjectured to be Abrikosov triangular lattices.
Abstract In this paper, we show the uniqueness and local stability of an inverse source problem for the quasi-static Maxwell equation in a layered domain, where the source consists of …
Abstract In this paper, we show the uniqueness and local stability of an inverse source problem for the quasi-static Maxwell equation in a layered domain, where the source consists of multiple point dipoles. Also, an algebraic algorithm is proposed to identify the number, locations, and moments of the dipoles from boundary measurements of tangential components of the electric and magnetic fields. The proposed algorithm is numerically verified.
We consider point distributions in compact connected two-point homogeneous spaces (Riemannian symmetric spaces of rank one). All such spaces are known, they are the spheres in the Euclidean spaces, the …
We consider point distributions in compact connected two-point homogeneous spaces (Riemannian symmetric spaces of rank one). All such spaces are known, they are the spheres in the Euclidean spaces, the real, complex and quaternionic projective spaces and the octonionic projective plane. Our concern is with discrepancies of distributions in metric balls and sums of pairwise distances between points of distributions in such spaces. Using the geometric features of two-point spaces, we show that Stolarsky's invariance principle, well-known for the Euclidean spheres, can be extended to all projective spaces and the octonionic projective plane (Theorem 2.1 and Corollary 2.1). We obtain the spherical function expansions for discrepancies and sums of distances (Theorem 9.1). Relying on these expansions, we prove in all such spaces the best possible bounds for quadratic discrepancies and sums of pairwise distances (Theorem 2.2). Applications to $t$-designs on such two-point homogeneous spaces are also considered. It is shown that the optimal $t$-designs meet the best possible bounds for quadratic discrepancies and sums of pairwise distances. (Corollaries 3.1 and 3.2).
In this article we consider the distribution of $N$ points on the unit sphere $\mathbb{S}^{d-1}$ in $\mathbb{R}^d$ interacting via logarithmic potential. A characterization theorem of the stationary configurations is derived …
In this article we consider the distribution of $N$ points on the unit sphere $\mathbb{S}^{d-1}$ in $\mathbb{R}^d$ interacting via logarithmic potential. A characterization theorem of the stationary configurations is derived when $N=d+2$ and two new log-optimal configurations minimizing the logarithmic energy are obtained for six points on $\mathbb{S}^3$ and seven points on $\mathbb{S}^4$. A conjecture on the log-optimal configurations of $d+2$ points on $\mathbb{S}^{d-1}$ is stated and three auxiliary results supporting the conjecture are presented.
This paper presents an algebraic method for an inverse source problem for the Poisson equation where the source consists of dipoles and quadrupoles. This source model is significant in the …
This paper presents an algebraic method for an inverse source problem for the Poisson equation where the source consists of dipoles and quadrupoles. This source model is significant in the magnetoencephalography inverse problem. The proposed method identifies the source parameters directly and algebraically using data without requiring an initial parameter estimate or iterative computation of the forward solution. The obtained parameters could be used for the initial solution in an optimization-based algorithm for further refinement.
Cubature formulas and geometrical designs are described in terms of reproducing kernels for Hilbert spaces of functions on the one hand, and Markov operators associated to orthogonal group representations on …
Cubature formulas and geometrical designs are described in terms of reproducing kernels for Hilbert spaces of functions on the one hand, and Markov operators associated to orthogonal group representations on the other hand. In this way, several known results for spheres in Euclidean spaces, involving cubature formulas for polynomial functions and spherical designs, are shown to generalize to large classes of finite measure spaces (Ω, σ) and appropriate spaces of functions inside L2(Ω, σ). The last section points out how spherical designs are related to a class of reflection groups which are (in general dense) subgroups of orthogonal groups.
In the singularly perturbed limit of an asymptotically small diffusivity ratio ${\varepsilon}^2$, the existence and stability of localized quasi-equilibrium multispot patterns is analyzed for the Brusselator reaction-diffusion model on the …
In the singularly perturbed limit of an asymptotically small diffusivity ratio ${\varepsilon}^2$, the existence and stability of localized quasi-equilibrium multispot patterns is analyzed for the Brusselator reaction-diffusion model on the unit sphere. Formal asymptotic methods are used to derive a nonlinear algebraic system that characterizes quasi-equilibrium spot patterns and to formulate eigenvalue problems governing the stability of spot patterns to three types of “fast” ${\mathcal O}(1)$ time-scale instabilities: self-replication, competition, and oscillatory instabilities of the spot amplitudes. The nonlinear algebraic system and the spectral problems are then studied using simple numerical methods, with emphasis on the special case where the spots have a common amplitude. Overall, the theoretical framework provides a hybrid asymptotic-numerical characterization of the existence and stability of spot patterns that is asymptotically correct to within all logarithmic correction terms in powers of $\nu={-1/\log{\varepsilon}}$. From a leading-order-in-$\nu$ analysis, and with an asymptotically large inhibitor diffusivity, some rigorous results for competition and oscillatory instabilities are obtained from an analysis of a new class of nonlocal eigenvalue problem (NLEP). Theoretical results for the stability of spot patterns are confirmed with full numerical computations of the Brusselator PDE system on the sphere using the closest point method.
We introduce a class of variational principles on measure spaces which are causal in the sense that they generate a relation on pairs of points, giving rise to a distinction …
We introduce a class of variational principles on measure spaces which are causal in the sense that they generate a relation on pairs of points, giving rise to a distinction between spacelike and timelike separation. General existence results are proved. It is shown in examples that minimizers need not be unique. Counter examples to compactness are discussed. The existence results are applied to variational principles formulated in indefinite inner product spaces.
We extend the average case analysis of numerical integration in the sense of Traub and Wozniakowski to homogeneous spaces. Various exam- ples are described and the connections to minimal energy …
We extend the average case analysis of numerical integration in the sense of Traub and Wozniakowski to homogeneous spaces. Various exam- ples are described and the connections to minimal energy point sets are outlined.
We enumerate and classify all stationary logarithmic configurations of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="d plus 2"> <mml:semantics> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>+</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">d+2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> points on the …
We enumerate and classify all stationary logarithmic configurations of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="d plus 2"> <mml:semantics> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>+</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">d+2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> points on the unit sphere in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="d"> <mml:semantics> <mml:mi>d</mml:mi> <mml:annotation encoding="application/x-tex">d</mml:annotation> </mml:semantics> </mml:math> </inline-formula>–dimensions. In particular, we show that the logarithmic energy attains its local minima at configurations that consist of two orthogonal to each other regular simplexes of cardinality <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="m"> <mml:semantics> <mml:mi>m</mml:mi> <mml:annotation encoding="application/x-tex">m</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding="application/x-tex">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. The global minimum occurs when <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="m equals n"> <mml:semantics> <mml:mrow> <mml:mi>m</mml:mi> <mml:mo>=</mml:mo> <mml:mi>n</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">m=n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="d"> <mml:semantics> <mml:mi>d</mml:mi> <mml:annotation encoding="application/x-tex">d</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is even and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="m equals n plus 1"> <mml:semantics> <mml:mrow> <mml:mi>m</mml:mi> <mml:mo>=</mml:mo> <mml:mi>n</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">m=n+1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> otherwise. This characterizes a new class of configurations that minimize the logarithmic energy on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper S Superscript d minus 1"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">S</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>d</mml:mi> <mml:mo>−<!-- − --></mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:annotation encoding="application/x-tex">\mathbb {S}^{d-1}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for all <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="d"> <mml:semantics> <mml:mi>d</mml:mi> <mml:annotation encoding="application/x-tex">d</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. The other two classes known in the literature, the regular simplex (<inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="d plus 1"> <mml:semantics> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">d+1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> points on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper S Superscript d minus 1"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">S</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>d</mml:mi> <mml:mo>−<!-- − --></mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:annotation encoding="application/x-tex">\mathbb {S}^{d-1}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>) and the cross-polytope (<inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="2 d"> <mml:semantics> <mml:mrow> <mml:mn>2</mml:mn> <mml:mi>d</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">2d</mml:annotation> </mml:semantics> </mml:math> </inline-formula> points on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper S Superscript d minus 1"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">S</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>d</mml:mi> <mml:mo>−<!-- − --></mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:annotation encoding="application/x-tex">\mathbb {S}^{d-1}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>), are both universally optimal configurations.
It is shown that configuratin of N point charges 1/N on the unist sphere S,which generate a small electrostatic field inside the unit ball, correspond to good N-tuples of nodes …
It is shown that configuratin of N point charges 1/N on the unist sphere S,which generate a small electrostatic field inside the unit ball, correspond to good N-tuples of nodes for Chebyshev-type quadrature on S.Quadraure results for S are obtained with the aid of spherical harmonics and an existence theorem of S.N. Bernstein for Chebyshev-type quadrature on an interval. It follows in particular that there exist so-called shperical pdesigns consisting of O(p3 )points. A more physically motivated approach to good nodes on S is briefly indicated. Finally, the above correspondence is used to give lower and upper bounds for the smallest possible electrostaic field on ballsB(0,γ) with γ < 1, due to N point charges 1/N on S
An upper bound for the sum of the Ath powers of all distances determined by N points on a unit sphere is given for Let plt ■ • •, pN …
An upper bound for the sum of the Ath powers of all distances determined by N points on a unit sphere is given for Let plt ■ • •, pN be points on the unit sphere UmofE'", the w-dimensional Euclidean space.Let
A concept of generalized discrepancy, which involves pseudodifferential operators to give a criterion of equidistributed pointsets, is developed on the sphere. A simply structured formula in terms of elementary functions …
A concept of generalized discrepancy, which involves pseudodifferential operators to give a criterion of equidistributed pointsets, is developed on the sphere. A simply structured formula in terms of elementary functions is established for the computation of the generalized discrepancy. With the help of this formula five kinds of point systems on the sphere, namely lattices in polar coordinates, transformed two-dimensional sequences, rotations on the sphere, triangulations, and "sum of three squares sequence," are investigated. Quantitative tests are done, and the results are compared with one another. Our calculations exhibit different orders of convergence of the generalized discrepancy for different types of point systems.
We review the formulation and solutions of a number of extremal problems associated with points and unit charges on the surface of a sphere in E 3 . For one …
We review the formulation and solutions of a number of extremal problems associated with points and unit charges on the surface of a sphere in E 3 . For one of these problems, namely[Formula: see text]where d pq is the Euclidean distance between points P and Q and m is the number of points, we discuss the results for m ≤ 16 and 1 ≤ n ≤ ∞. For the cases m = 5, 11, 13–16 we find hitherto undiscovered solutions. Our solutions for m = 5 and 11 correct earlier results in the literature. We also sharpen the existing literature results for m = 7 and 10.
GivenTV points X\, x 2 , ... , XN on a unit sphere S in Euclidean d space (d > 3), we investigate the α-sum ]ζ \x -x } \ …
GivenTV points X\, x 2 , ... , XN on a unit sphere S in Euclidean d space (d > 3), we investigate the α-sum ]ζ \x -x } \ a , a > 1d, of their distances from a variable point x on S. We obtain an essentially best possible lower bound for the Z^-norm of its deviation from the mean value.As an application, we prove similar bounds for the α-sums Σ\ X J ~ χ k\ a of mutual distances.
We investigate the energy of arrangements of N points on the surface of a sphere in R 3 , interacting through a power law potential V = r α , …
We investigate the energy of arrangements of N points on the surface of a sphere in R 3 , interacting through a power law potential V = r α , -2 < α < 2, where r is Euclidean distance.For α = 0, we take V = log(1/r).An area-regular partitioning scheme of the sphere is devised for the purpose of obtaining bounds for the extremal (equilibrium) energy for such points.For α = 0, finer estimates are obtained for the dominant terms in the minimal energy by considering stereographical projections on the plane and analyzing certain logarithmic potentials.A general conjecture on the asymptotic form (as N → ∞) of the extremal energy, along with its supporting numerical evidence, is presented.Also we introduce explicit sets of points, called "generalized spiral points", that yield good estimates for the extremal energy.At least for N ≤ 12, 000 these points provide a reasonable solution to a problem of M. Shub and S. Smale arising in complexity theory.
Given <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper N"> <mml:semantics> <mml:mi>N</mml:mi> <mml:annotation encoding="application/x-tex">N</mml:annotation> </mml:semantics> </mml:math> </inline-formula> points on a unit sphere in Euclidean <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="m"> <mml:semantics> <mml:mi>m</mml:mi> <mml:annotation encoding="application/x-tex">m</mml:annotation> …
Given <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper N"> <mml:semantics> <mml:mi>N</mml:mi> <mml:annotation encoding="application/x-tex">N</mml:annotation> </mml:semantics> </mml:math> </inline-formula> points on a unit sphere in Euclidean <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="m"> <mml:semantics> <mml:mi>m</mml:mi> <mml:annotation encoding="application/x-tex">m</mml:annotation> </mml:semantics> </mml:math> </inline-formula> space, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="m greater-than-over-equals 2"> <mml:semantics> <mml:mrow> <mml:mi>m</mml:mi> <mml:mo>≧</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">m \geqq 2</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, we show that the sum of all distances which they determine plus their discrepancy is a constant. As applications we obtain (i) an upper bound for the sum of the distances which for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="m greater-than-over-equals 5"> <mml:semantics> <mml:mrow> <mml:mi>m</mml:mi> <mml:mo>≧</mml:mo> <mml:mn>5</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">m \geqq 5</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is smaller than any previously known and (ii) the existence of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper N"> <mml:semantics> <mml:mi>N</mml:mi> <mml:annotation encoding="application/x-tex">N</mml:annotation> </mml:semantics> </mml:math> </inline-formula> point distributions with small discrepancy. We make use of W. M. Schmidt’s work on the discrepancy of spherical caps.
GivenTV points x\, ... , x N on the unit sphere S in Euclidean d space (d > 3), lower bounds for the deviation of the sum ]Γ \x-Xj\ a …
GivenTV points x\, ... , x N on the unit sphere S in Euclidean d space (d > 3), lower bounds for the deviation of the sum ]Γ \x-Xj\ a , a > 1d x £ S, from its mean value were established in terms of L λ -norms in the first part of this paper.In the present part it is shown that these bounds are best possible.Our main tool is a multidimensional quadrature formula with equal weights.For 0 < a < 2 and N > 2, the sum E a (ω^) is known to be negative (see Theorem 2 in [4]).An application of Theorem A immediately yields