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 a closed $d$-rectifiable set $A$ embedded in Euclidean space, we investigate minimal weighted Riesz energy points on $A$; that is, $N$ points constrained to $A$ and interacting via the …
Given a closed $d$-rectifiable set $A$ embedded in Euclidean space, we investigate minimal weighted Riesz energy points on $A$; that is, $N$ points constrained to $A$ and interacting via the weighted power law potential $V=w(x,y)\left |x-y\right |^{-s}$, where $s>0$ is a fixed parameter and $w$ is an admissible weight. (In the unweighted case ($w\equiv 1$) such points for $N$ fixed tend to the solution of the best-packing problem on $A$ as the parameter $s\to \infty$.) Our main results concern the asymptotic behavior as $N\to \infty$ of the minimal energies as well as the corresponding equilibrium configurations. Given a distribution $\rho (x)$ with respect to $d$-dimensional Hausdorff measure on $A$, our results provide a method for generating $N$-point configurations on $A$ that are "well-separated" and have asymptotic distribution $\rho (x)$ as $N\to \infty$.
In this paper, we show that certain sequences of polynomials $\{ p_k (z)\} _{k = 0}^n $, generated from three-term recurrence relations, have no zeros in parabolic regions in the …
In this paper, we show that certain sequences of polynomials $\{ p_k (z)\} _{k = 0}^n $, generated from three-term recurrence relations, have no zeros in parabolic regions in the complex plane of the form $y^2 \leqq 4\alpha (x + \alpha )$, $x > - \alpha $. As a special case of this, no partial $s_n (z) = {{\sum _{k = 0}^n z^k } / {k!}}$ of $e^z $ has a zero in $ y^2 \leqq 4(x + 1)$, $x > - 1$, for any $n \geqq 1$. Such zero-free parabolic regions are obtained for Padé approximants of certain meromorphic functions, as well as for the partial sums of certain hypergeometric functions.
Abstract We consider the s -energy for point sets 𝒵 = {𝒵 k,n : k = 0, …, n } on certain compact sets Γ in ℝ d having finite …
Abstract We consider the s -energy for point sets 𝒵 = {𝒵 k,n : k = 0, …, n } on certain compact sets Γ in ℝ d having finite one-dimensional Hausdorff measure, is the Riesz kernel. Asymptotics for the minimum s -energy and the distribution of minimizing sequences of points is studied. In particular, we prove that, for s ≥ 1, the minimizing nodes for a rectifiable Jordan curve Γ distribute asymptotically uniformly with respect to arclength as n → ∞.
We derive the complete asymptotic expansion in terms of powers of $N$ for the Riesz $s$-energy of $N$ equally spaced points on the unit circle as $N\to \infty$. For $s\ge …
We derive the complete asymptotic expansion in terms of powers of $N$ for the Riesz $s$-energy of $N$ equally spaced points on the unit circle as $N\to \infty$. For $s\ge -2$, such points form optimal energy $N$-point configurations with respect to the Riesz potential $1/r^{s}$, $s\neq0$, where $r$ is the Euclidean distance between points. By analytic continuation we deduce the expansion for all complex values of $s$. The Riemann zeta function plays an essential role in this asymptotic expansion.
Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f left-parenthesis z right-parenthesis equals z plus normal upper Sigma 2 Superscript normal infinity Baseline a Subscript k Baseline z Superscript k"> <mml:semantics> <mml:mrow> <mml:mi>f</mml:mi> …
Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f left-parenthesis z right-parenthesis equals z plus normal upper Sigma 2 Superscript normal infinity Baseline a Subscript k Baseline z Superscript k"> <mml:semantics> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>z</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:mi>z</mml:mi> <mml:mo>+</mml:mo> <mml:msubsup> <mml:mi mathvariant="normal">Σ</mml:mi> <mml:mn>2</mml:mn> <mml:mi mathvariant="normal">∞</mml:mi> </mml:msubsup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>a</mml:mi> <mml:mi>k</mml:mi> </mml:msub> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>z</mml:mi> <mml:mi>k</mml:mi> </mml:msup> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">f(z) = z + \Sigma _2^\infty {a_k}{z^k}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be analytic and univalent in the unit disk <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper E colon StartAbsoluteValue z EndAbsoluteValue greater-than 1"> <mml:semantics> <mml:mrow> <mml:mi>E</mml:mi> <mml:mo>:</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mi>z</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mo>></mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">E:|z| > 1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and map the disk onto a domain which is convex in the direction of the imaginary axis. We show by example that for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartRoot 2 EndRoot minus 1 greater-than r greater-than 1"> <mml:semantics> <mml:mrow> <mml:msqrt> <mml:mn>2</mml:mn> </mml:msqrt> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> <mml:mo>></mml:mo> <mml:mi>r</mml:mi> <mml:mo>></mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">\sqrt 2 - 1 > r > 1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, the function <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f left-parenthesis z right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>z</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">f(z)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> need not map the disk <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue z EndAbsoluteValue greater-than r"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mi>z</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mo>></mml:mo> <mml:mi>r</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">|z| > r</mml:annotation> </mml:semantics> </mml:math> </inline-formula> onto a domain convex in the direction of the imaginary axis. We also find the largest domain contained in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f left-parenthesis upper E right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>E</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">f(E)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for every normalized <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f left-parenthesis z right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>z</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">f(z)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> that maps <italic>E</italic> onto a domain convex in the direction of the imaginary axis.
We exhibit new links between approximation theory in the complex domain and a family of inverse problems for the 2D Laplacian related to non-destructive testing.
We exhibit new links between approximation theory in the complex domain and a family of inverse problems for the 2D Laplacian related to non-destructive testing.
We provide an introduction to logarithmic potential theory in the complex plane that particularly emphasizes its usefulness in the theory of polynomial and rational approximation. The reader is invited to …
We provide an introduction to logarithmic potential theory in the complex plane that particularly emphasizes its usefulness in the theory of polynomial and rational approximation. The reader is invited to explore the notions of Fekete points, logarithmic capacity, and Chebyshev constant through a variety of examples and exercises. Many of the fundamental theorems of potential theory, such as Frostman's theorem, the Riesz Decomposition Theorem, the Principle of Domination, etc., are given along with essential ideas for their proofs. Equilibrium measures and potentials and their connections with Green functions and conformal mappings are presented. Moreover, we discuss extensions of the classical potential theoretic results to the case when an external field is present.
Journal Article Asymptotics of orthogonal polynomials with respect to an analytic weight with algebraic singularities on the circle Get access A. Martínez-Finkelshtein, A. Martínez-Finkelshtein Department of Statistics and Applied Mathematics, …
Journal Article Asymptotics of orthogonal polynomials with respect to an analytic weight with algebraic singularities on the circle Get access A. Martínez-Finkelshtein, A. Martínez-Finkelshtein Department of Statistics and Applied Mathematics, University of AlmeríaAlmería, Spain E-mail address: [email protected] Search for other works by this author on: Oxford Academic Google Scholar K. T.-R. McLaughlin, K. T.-R. McLaughlin Department of Mathematics, The University of ArizonaTucson, AZ 85721, USA E-mail address: [email protected] Search for other works by this author on: Oxford Academic Google Scholar E. B. Saff E. B. Saff Department of Mathematics, Vanderbilt UniversityNashville, TN 37240, USA E-mail address: [email protected] Search for other works by this author on: Oxford Academic Google Scholar International Mathematics Research Notices, Volume 2006, 2006, O91426, https://doi.org/10.1155/IMRN/2006/91426 Published: 01 January 2006 Article history Published: 01 January 2006 Received: 20 June 2006 Accepted: 28 June 2006
Consider the integro-differential equation (*) U(x) = x' + A(t, x) + ίV(ί, s, x(s))ds = T(t), t e [a, b] subject to the initial condition (**) x(a) = h …
Consider the integro-differential equation (*) U(x) = x' + A(t, x) + ίV(ί, s, x(s))ds = T(t), t e [a, b] subject to the initial condition (**) x(a) = h .Then a problem in approximation theory is whether a solution x(t) of ((*), (**)) can be approximated, uniformly on [a, b], by a sequence of polynomials P n , which satisfy (**) and minimize the expression ||JΓ( )-U(P n )\\ 9 where || || is a certain norm.It is shown here that such a sequence of minimizing: polynomials, or, more generally, hyperpolynomials, exists with respect to the Lp-norm (1 < p ^ oo) and converges to x(t), uniformly on [a, b], under the mere assumption of existence and uniqueness of x(t).
We derive fundamental asymptotic results for the expected covering radius |$\rho (X_N)$| for |$N$| points that are randomly and independently distributed with respect to surface measure on a sphere as …
We derive fundamental asymptotic results for the expected covering radius |$\rho (X_N)$| for |$N$| points that are randomly and independently distributed with respect to surface measure on a sphere as well as on a class of smooth manifolds. For the unit sphere |$\mathbb {S}^d \subset \mathbb {R}^{d+1}$|, we obtain the precise asymptotic that |$\mathbb {E}\rho (X_N)[N/\log N]^{1/d}$| has limit |$[(d+1)\upsilon _{d+1}/\upsilon _d]^{1/d}$| as |$N \to \infty $|, where |$\upsilon _d$| is the volume of the |$d$|-dimensional unit ball. This proves a recent conjecture of Brauchart et al. as well as extends a result previously known only for the circle. Likewise, we obtain precise asymptotics for the expected covering radius of |$N$| points randomly distributed on a |$d$|-dimensional ball, a |$d$|-dimensional cube, as well as on a three-dimensional polyhedron (where the points are independently distributed with respect to volume measure). More generally, we deduce upper and lower bounds for the expected covering radius of |$N$| points that are randomly and independently distributed on a compact metric measure space, provided the measure satisfies certain regularity assumptions.
We investigate the Riesz energy minimization problem on a $d$-dimensional ball in the presence of an external field created by a point charge above the ball in $\R^{d+1}$, $d\geq1$. Both …
We investigate the Riesz energy minimization problem on a $d$-dimensional ball in the presence of an external field created by a point charge above the ball in $\R^{d+1}$, $d\geq1$. Both cases of an attractive charge and a repulsive charge are considered. The notion of a signed equilibrium measure is one of the main tools in the present study. For the case of a positive (repulsive) charge, the determination of the support of the equilibrium measure is a nontrivial question. We solve it in the one-dimensional case by making use of iterated balayage, a method already applied in logarithmic potential theory. Here we use a modified version of it, in order to handle the phenomenon of mass loss, characteristic of the Riesz balayage of positive measures. Moreover, we also consider minimization of Coulomb energy on the ball in dimension $d\geq2$, and of logarithmic energy on the segment in dimension 1. Different techniques are used for these two cases.
This article is devoted to the study of discrete potentials on the sphere in \mathbb{R}^{n} for sharp codes. We show that the potentials of most of the known sharp codes …
This article is devoted to the study of discrete potentials on the sphere in \mathbb{R}^{n} for sharp codes. We show that the potentials of most of the known sharp codes attain the universal lower bounds for polarization for spherical \tau -designs previously derived by the authors, where “universal” is meant in the sense of applying to a large class of potentials that includes absolutely monotone functions of inner products. We also extend our universal bounds to T -designs and the associated polynomial subspaces determined by the vanishing moments of spherical configurations, and thus obtain the minima for the icosahedron, the dodecahedron, and sharp codes coming from E_{8} and the Leech lattice. For this purpose, we investigate quadrature formulas for certain subspaces of Gegenbauer polynomials P^{(n)}_{j} which we call PULB subspaces, particularly those having basis \{P_{j}^{(n)}\}_{j=0}^{2k+2}\setminus \{P_{2k}^{(n)}\}. Furthermore, for potentials with h^{(\tau+1)}<0 , we prove that the strong sharp codes and the antipodal sharp codes attain the universal bounds and their minima occur at points of the codes. The same phenomenon is established for the 600 -cell when the potential h satisfies h^{(i)}\geq 0 , i=1,\dots,15 , and h^{(16)}\leq 0 .
We consider Riesz energy problems with radial external fields. We study the question of whether or not the equilibrium is the uniform distribution on a sphere. We develop general necessary …
We consider Riesz energy problems with radial external fields. We study the question of whether or not the equilibrium is the uniform distribution on a sphere. We develop general necessary as well as general sufficient conditions on the external field that apply to powers of the Euclidean norm as well as certain Lennard--Jones type fields. Additionally, in the former case, we completely characterize the values of the power for which dimension reduction occurs in the sense that the support of the equilibrium measure becomes a sphere. We also briefly discuss the relation between these problems and certain constrained optimization problems. Our approach involves the Frostman characterization, the Funk--Hecke formula, and the calculus of hypergeometric functions.
Universal bounds for the potential energy of weighted spherical codes are obtained by linear programming. The universality is in the sense of Cohn-Kumar -- every attaining code is optimal with …
Universal bounds for the potential energy of weighted spherical codes are obtained by linear programming. The universality is in the sense of Cohn-Kumar -- every attaining code is optimal with respect to a large class of potential functions (absolutely monotone), in the sense of Levenshtein -- there is a bound for every weighted code, and in the sense of parameters (nodes and weights) -- they are independent of the potential function. We derive a necessary condition for optimality (in the linear programming framework) of our lower bounds which is also shown to be sufficient when the potential is strictly absolutely monotone. Bounds are also obtained for the weighted energy of weighted spherical designs. We explore our bounds for several previously studied weighted spherical codes.
We study the problem originally communicated by E. Meckes on the asymptotics for the eigenvalues of the kernel of the unitary eigenvalue process of a random $n \times n$ matrix. …
We study the problem originally communicated by E. Meckes on the asymptotics for the eigenvalues of the kernel of the unitary eigenvalue process of a random $n \times n$ matrix. The eigenvalues $p_{j}$ of the kernel are, in turn, associated with the discrete prolate spheroidal wave functions. We consider the eigenvalue counting function $|G(x,n)|:=\#\{j:p_j>Ce^{-x n}\}$, ($C>0$ here is a fixed constant) and establish the asymptotic behavior of its average over the interval $x \in (\lambda-\varepsilon, \lambda+\varepsilon)$ by relating the function $|G(x,n)|$ to the solution $J(y)$ of the following energy problem on the unit circle $S^{1}$, which is of independent interest. Namely, for given $\theta$, $0<\theta< 2 \pi$, and given $q$, $0<q<1$, we determine the function $J(y) =\inf \{I(\mu): \mu \in \mathcal{P}(S^{1}), \mu(A_{\theta}) = y\}$, where $I(\mu):= \iint \log\frac{1}{|z - \zeta|} d\mu(z) d\mu(\zeta)$ is the logarithmic energy of a probability measure $\mu$ supported on the unit circle and $A_{\theta}$ is the arc from $e^{-i \theta/2}$ to $e^{i \theta/2}$.
We obtain new asymptotic results about systems of $ N $ particles governed by Riesz interactions involving $ k $-nearest neighbors of each particle as $N\to\infty$. These results include a …
We obtain new asymptotic results about systems of $ N $ particles governed by Riesz interactions involving $ k $-nearest neighbors of each particle as $N\to\infty$. These results include a generalization to weighted Riesz potentials with external field. Such interactions offer an appealing alternative to other approaches for reducing the computational complexity of an $ N $-body interaction. We find the first-order term of the large $ N $ asymptotics and characterize the limiting distribution of the minimizers. We also obtain results about the $ Γ$-convergence of such interactions, and describe minimizers on the 1-dimensional flat torus in the absence of external field, for all $ N $.
We explore the connection between supports of equilibrium measures and quadrature identities, especially in the case of point sources added to the external field $Q(z)=|z|^{2p}$ with $p \in \mathbb{N}$. Along …
We explore the connection between supports of equilibrium measures and quadrature identities, especially in the case of point sources added to the external field $Q(z)=|z|^{2p}$ with $p \in \mathbb{N}$. Along the way, we describe some quadrature domains with respect to weighted area measure $|z|^{2p}dA_z$ and complex boundary measure $|z|^{-2p}dz$.
We compute the equilibrium measure in dimension d=s+4 associated to a Riesz s-kernel interaction with an external field given by a power of the Euclidean norm. Our study reveals that …
We compute the equilibrium measure in dimension d=s+4 associated to a Riesz s-kernel interaction with an external field given by a power of the Euclidean norm. Our study reveals that the equilibrium measure can be a mixture of a continuous part and a singular part. Depending on the value of the power, a threshold phenomenon occurs and consists of a dimension reduction or condensation on the singular part. In particular, in the logarithmic case s=0 (d=4), there is condensation on a sphere of special radius when the power of the external field becomes quadratic. This contrasts with the case d=s+3 studied previously, which showed that the equilibrium measure is fully dimensional and supported on a ball. Our approach makes use, among other tools, of the Frostman or Euler-Lagrange variational characterization, the Funk-Hecke formula, the Gegenbauer orthogonal polynomials, and hypergeometric special functions.
In this article we investigate the $N$-point min-max and the max-min polarization problems on the sphere for a large class of potentials in $\mathbb{R}^n$. We derive universal lower and upper …
In this article we investigate the $N$-point min-max and the max-min polarization problems on the sphere for a large class of potentials in $\mathbb{R}^n$. We derive universal lower and upper bounds on the polarization of spherical designs of fixed dimension, strength, and cardinality. The bounds are universal in the sense that they are a convex combination of potential function evaluations with nodes and weights independent of the class of potentials. As a consequence of our lower bounds, we obtain the Fazekas-Levenshtein bounds on the covering radius of spherical designs. Utilizing the existence of spherical designs, our polarization bounds are extended to general configurations. As examples we completely solve the min-max polarization problem for $120$ points on $\mathbb{S}^3$ and show that the $600$-cell is universally optimal for that problem. We also provide alternative methods for solving the max-min polarization problem when the number of points $N$ does not exceed the dimension $n$ and when $N=n+1$. We further show that the cross-polytope has the best max-min polarization constant among all spherical $2$-designs of $N=2n$ points for $n=2,3,4$; for $n\geq 5$, this statement is conditional on a well-known conjecture that the cross-polytope has the best covering radius. This max-min optimality is also established for all so-called centered codes.
This article is devoted to the study of discrete potentials on the sphere in $\mathbb{R}^n$ for sharp codes. We show that the potentials of most of the known sharp codes …
This article is devoted to the study of discrete potentials on the sphere in $\mathbb{R}^n$ for sharp codes. We show that the potentials of most of the known sharp codes attain the universal lower bounds for polarization for spherical $\tau$-designs previously derived by the authors, where ``universal'' is meant in the sense of applying to a large class of potentials that includes absolutely monotone functions of inner products. We also extend our universal bounds to $T$-designs and the associated polynomial subspaces determined by the vanishing moments of spherical configurations and thus obtain the minima for the icosahedron, dodecahedron, and sharp codes coming from $E_8$ and the Leech lattice. For this purpose, we investigate quadrature formulas for certain subspaces of Gegenbauer polynomials $P^{(n)}_j$ which we call PULB subspaces, particularly those having basis $\{P_j^{(n)}\}_{j=0}^{2k+2}\setminus \{P_{2k}^{(n)}\}.$ Furthermore, for potentials with $h^{(\tau+1)}<0$ we prove that the strong sharp codes and the antipodal sharp codes attain the universal bounds and their minima occur at points of the codes. The same phenomenon is established for the $600$-cell when the potential $h$ satisfies $h^{(i)}\geq 0$, $i=1,\dots,15$, and $h^{(16)}\leq 0.$
We employ signed measures that are positive definite up to certain degrees to establish Levenshtein-type upper bounds on the cardinality of codes with given minimum and maximum distances, and universal …
We employ signed measures that are positive definite up to certain degrees to establish Levenshtein-type upper bounds on the cardinality of codes with given minimum and maximum distances, and universal lower bounds on the potential energy (for absolutely monotone interactions) for codes with given maximum distance and cardinality. The distance distributions of codes that attain the bounds are found in terms of the parameters of Levenshtein-type quadrature formulas. Necessary and sufficient conditions for the optimality of our bounds are derived. Further, we obtain upper bounds on the energy of codes of fixed minimum and maximum distances and cardinality.
<p style='text-indent:20px;'>We investigate the large time behavior of <inline-formula><tex-math id="M1">\begin{document}$ N $\end{document}</tex-math></inline-formula> particles restricted to a smooth closed curve in <inline-formula><tex-math id="M2">\begin{document}$ \mathbb{R}^d $\end{document}</tex-math></inline-formula> and subject to a gradient flow …
<p style='text-indent:20px;'>We investigate the large time behavior of <inline-formula><tex-math id="M1">\begin{document}$ N $\end{document}</tex-math></inline-formula> particles restricted to a smooth closed curve in <inline-formula><tex-math id="M2">\begin{document}$ \mathbb{R}^d $\end{document}</tex-math></inline-formula> and subject to a gradient flow with respect to Euclidean hyper-singular repulsive Riesz <inline-formula><tex-math id="M3">\begin{document}$ s $\end{document}</tex-math></inline-formula>-energy with <inline-formula><tex-math id="M4">\begin{document}$ s&gt;1. $\end{document}</tex-math></inline-formula> We show that regardless of their initial positions, for all <inline-formula><tex-math id="M5">\begin{document}$ N $\end{document}</tex-math></inline-formula> and time <inline-formula><tex-math id="M6">\begin{document}$ t $\end{document}</tex-math></inline-formula> large, their normalized Riesz <inline-formula><tex-math id="M7">\begin{document}$ s $\end{document}</tex-math></inline-formula>-energy will be close to the <inline-formula><tex-math id="M8">\begin{document}$ N $\end{document}</tex-math></inline-formula>-point minimal possible energy. Furthermore, the distribution of such particles will be close to uniform with respect to arclength measure along the curve.</p>
In this paper, we investigate Riesz energy problems on unbounded conductors in $\R^d$ in the presence of general external fields $Q$, not necessarily satisfying the growth condition $Q(x)\to\infty$ as $x\to\infty$ …
In this paper, we investigate Riesz energy problems on unbounded conductors in $\R^d$ in the presence of general external fields $Q$, not necessarily satisfying the growth condition $Q(x)\to\infty$ as $x\to\infty$ assumed in several previous studies. We provide sufficient conditions on $Q$ for the existence of an equilibrium measure and the compactness of its support. Particular attention is paid to the case of the hyperplanar conductor $\R^{d}$, embedded in $\R^{d+1}$, when the external field is created by the potential of a signed measure $\nu$ outside of $\R^{d}$. Simple cases where $\nu$ is a discrete measure are analyzed in detail. New theoretic results for Riesz potentials, in particular an extension of a classical theorem by de La Vall\'ee-Poussin, are established. These results are of independent interest.
We investigate the large time behavior of $N$ particles restricted to a smooth closed curve in $\mathbb{R}^d$ and subject to a gradient flow with respect to Euclidean hyper-singular repulsive Riesz …
We investigate the large time behavior of $N$ particles restricted to a smooth closed curve in $\mathbb{R}^d$ and subject to a gradient flow with respect to Euclidean hyper-singular repulsive Riesz $s$-energy with $s>1.$ We show that regardless of their initial positions, for all $N$ and time $t$ large, their normalized Riesz $s$-energy will be close to the $N$-point minimal possible. Furthermore, the distribution of such particles will be close to uniform with respect to arclength measure along the curve.
Abstract We discuss recent results from [10] on sparse recovery for inverse potential problem with source term in divergence form. The notion of sparsity which is set forth is measure- …
Abstract We discuss recent results from [10] on sparse recovery for inverse potential problem with source term in divergence form. The notion of sparsity which is set forth is measure- theoretic, namely pure 1-unrectifiability of the support. The theory applies when a superset of the support is known to be slender, meaning it has measure zero and all connected components of its complement has infinite measure in ℝ 3 . We also discuss open issues in the non-slender case.
We introduce a projective Riesz $s$-kernel for the unit sphere $\mathbb{S}^{d-1}$ and investigate properties of $N$-point energy minimizing configurations for such a kernel. We show that these configurations, for $s$ …
We introduce a projective Riesz $s$-kernel for the unit sphere $\mathbb{S}^{d-1}$ and investigate properties of $N$-point energy minimizing configurations for such a kernel. We show that these configurations, for $s$ and $N$ sufficiently large, form frames that are well-separated (have low coherence) and are nearly tight. Our results suggest an algorithm for computing well-separated tight frames which is illustrated with numerical examples.
We describe a framework for extending the asymptotic behavior of a short-range interaction from the unit cube to general compact subsets of $ \mathbb R^d $. This framework allows us …
We describe a framework for extending the asymptotic behavior of a short-range interaction from the unit cube to general compact subsets of $ \mathbb R^d $. This framework allows us to give a unified treatment of asymptotics of hypersingular Riesz energies and optimal quantizers. We further obtain new results about the scale-invariant nearest neighbor interactions, such as the $ k $-nearest neighbor truncated Riesz energy. Our generalized approach has applications to methods for generating distributions with prescribed density: strongly-repulsive Riesz energies, centroidal Voronoi tessellations, and a popular meshing algorithm due to Persson and Strang.
We investigate the large time behavior of $N$ particles restricted to a smooth closed curve in $\mathbb{R}^d$ and subject to a gradient flow with respect to Euclidean hyper-singular repulsive Riesz …
We investigate the large time behavior of $N$ particles restricted to a smooth closed curve in $\mathbb{R}^d$ and subject to a gradient flow with respect to Euclidean hyper-singular repulsive Riesz $s$-energy with $s>1.$ We show that regardless of their initial positions, for all $N$ and time $t$ large, their normalized Riesz $s$-energy will be close to the $N$-point minimal possible. Furthermore, the distribution of such particles will be close to uniform with respect to arclength measure along the curve.
We employ signed measures that are positive definite up to certain degrees to establish Levenshtein-type upper bounds on the cardinality of codes with given minimum and maximum distance, and universal …
We employ signed measures that are positive definite up to certain degrees to establish Levenshtein-type upper bounds on the cardinality of codes with given minimum and maximum distance, and universal lower bounds on the potential energy (for absolutely monotone interactions) for codes with given maximum distance and fixed cardinality. In particular, we extend the framework of Levenshtein bounds for such codes.
We introduce and study the unconstrained polarization (or Chebyshev) problem which requires to find an $N$-point configuration that maximizes the minimum value of its potential over a set $A$ in …
We introduce and study the unconstrained polarization (or Chebyshev) problem which requires to find an $N$-point configuration that maximizes the minimum value of its potential over a set $A$ in $p$-dimensional Euclidean space. This problem is compared to the constrained problem in which the points are required to belong to the set $A$. We find that for Riesz kernels $1/|x-y|^s$ with $s>p-2$ the optimum unconstrained configurations concentrate close to the set $A$ and based on this fundamental fact we recover the same asymptotic value of the polarization as for the more classical constrained problem on a class of $d$-rectifiable sets. We also investigate the new unconstrained problem in special cases such as for spheres and balls. In the last section we formulate some natural open problems and conjectures.
Based on the Delsarte-Yudin linear programming approach, we extend Levenshtein's framework to obtain lower bounds for the minimum $h$-energy of spherical codes of prescribed dimension and cardinality, and upper bounds …
Based on the Delsarte-Yudin linear programming approach, we extend Levenshtein's framework to obtain lower bounds for the minimum $h$-energy of spherical codes of prescribed dimension and cardinality, and upper bounds on the maximal cardinality of spherical codes of prescribed dimension and minimum separation. These bounds are universal in the sense that they hold for a large class of potentials $h$ and in the sense of Levenshtein. Moreover, codes attaining the bounds are universally optimal in the sense of Cohn-Kumar. Referring to Levenshtein bounds and the energy bounds of the authors as ``first level", our results can be considered as ``next level" universal bounds as they have the same general nature and imply necessary and sufficient conditions for their local and global optimality. For this purpose, we introduce the notion of Universal Lower Bound space (ULB-space), a space that satisfies certain quadrature and interpolation properties. While there are numerous cases for which our method applies, we will emphasize the model examples of $24$ points ($24$-cell) and $120$ points ($600$-cell) on $\mathbb{S}^3$. In particular, we provide a new proof that the $600$-cell is universally optimal, and in so doing, we derive optimality of the $600$-cell on a class larger than the absolutely monotone potentials considered by Cohn-Kumar.
We introduce a linear programming framework for obtaining upper bounds for the potential energy of spherical codes of fixed cardinality and minimum distance. Using Hermite interpolation we construct polynomials to …
We introduce a linear programming framework for obtaining upper bounds for the potential energy of spherical codes of fixed cardinality and minimum distance. Using Hermite interpolation we construct polynomials to derive corresponding bounds. These bounds are universal in the sense that they are valid for all absolutely monotone potential functions and the required interpolation nodes do not depend on the potentials.
We employ signed measures that are positive definite up to certain degrees to establish Levenshtein-type upper bounds on the cardinality of codes with given minimum and maximum distances, and universal …
We employ signed measures that are positive definite up to certain degrees to establish Levenshtein-type upper bounds on the cardinality of codes with given minimum and maximum distances, and universal lower bounds on the potential energy (for absolutely monotone interactions) for codes with given maximum distance and cardinality. The distance distributions of codes that attain the bounds are found in terms of the parameters of Levenshtein-type quadrature formulas. Necessary and sufficient conditions for the optimality of our bounds are derived. Further, we obtain upper bounds on the energy of codes of fixed minimum and maximum distances and cardinality.
We study configurations of points on the unit sphere that minimize potential energy for a broad class of potential functions (viewed as functions of the squared Euclidean distance between points). …
We study configurations of points on the unit sphere that minimize potential energy for a broad class of potential functions (viewed as functions of the squared Euclidean distance between points). Call a configuration sharp if there are <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> distances between distinct points in it and it is a spherical <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis 2 m minus 1 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mn>2</mml:mn> <mml:mi>m</mml:mi> <mml:mo>−<!-- − --></mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(2m-1)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-design. We prove that every sharp configuration minimizes potential energy for all completely monotonic potential functions. Examples include the minimal vectors of the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper E 8"> <mml:semantics> <mml:msub> <mml:mi>E</mml:mi> <mml:mn>8</mml:mn> </mml:msub> <mml:annotation encoding="application/x-tex">E_8</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and Leech lattices. We also prove the same result for the vertices of the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="600"> <mml:semantics> <mml:mn>600</mml:mn> <mml:annotation encoding="application/x-tex">600</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-cell, which do not form a sharp configuration. For most known cases, we prove that they are the unique global minima for energy, as long as the potential function is strictly completely monotonic. For certain potential functions, some of these configurations were previously analyzed by Yudin, Kolushov, and Andreev; we build on their techniques. We also generalize our results to other compact two-point homogeneous spaces, and we conclude with an extension to Euclidean space.
Given a closed $d$-rectifiable set $A$ embedded in Euclidean space, we investigate minimal weighted Riesz energy points on $A$; that is, $N$ points constrained to $A$ and interacting via the …
Given a closed $d$-rectifiable set $A$ embedded in Euclidean space, we investigate minimal weighted Riesz energy points on $A$; that is, $N$ points constrained to $A$ and interacting via the weighted power law potential $V=w(x,y)\left |x-y\right |^{-s}$, where $s>0$ is a fixed parameter and $w$ is an admissible weight. (In the unweighted case ($w\equiv 1$) such points for $N$ fixed tend to the solution of the best-packing problem on $A$ as the parameter $s\to \infty$.) Our main results concern the asymptotic behavior as $N\to \infty$ of the minimal energies as well as the corresponding equilibrium configurations. Given a distribution $\rho (x)$ with respect to $d$-dimensional Hausdorff measure on $A$, our results provide a method for generating $N$-point configurations on $A$ that are "well-separated" and have asymptotic distribution $\rho (x)$ as $N\to \infty$.
Abstract We consider the s -energy for point sets 𝒵 = {𝒵 k,n : k = 0, …, n } on certain compact sets Γ in ℝ d having finite …
Abstract We consider the s -energy for point sets 𝒵 = {𝒵 k,n : k = 0, …, n } on certain compact sets Γ in ℝ d having finite one-dimensional Hausdorff measure, is the Riesz kernel. Asymptotics for the minimum s -energy and the distribution of minimizing sequences of points is studied. In particular, we prove that, for s ≥ 1, the minimizing nodes for a rectifiable Jordan curve Γ distribute asymptotically uniformly with respect to arclength as n → ∞.
Let E be a compact set in the complex plane C having connected and regular complement. For a nonentire function f analytic in the interior of E and continuous on …
Let E be a compact set in the complex plane C having connected and regular complement. For a nonentire function f analytic in the interior of E and continuous on E, we investigate the limiting distribution of the zeros of the sequence of polynomials { p n ∗ } 1 ∞ of best uniform approximation to f on E. The zeros of best Lp polynomial approximants and best uniform rational approximants having a bounded number of free poles are also considered.
The theory of General Relativity, after its invention by Albert Einstein, remained for many years a monument of mathemati cal speculation, striking in its ambition and its formal beauty, but …
The theory of General Relativity, after its invention by Albert Einstein, remained for many years a monument of mathemati cal speculation, striking in its ambition and its formal beauty, but quite sep
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.
We survey known results and present estimates and conjectures for the next-order term in the asymptotics of the optimal logarithmic energy and Riesz s-energy of N points on the unit …
We survey known results and present estimates and conjectures for the next-order term in the asymptotics of the optimal logarithmic energy and Riesz s-energy of N points on the unit sphere in Rd+1, d ≥ 1. The conjectures are based on analytic continuation assumptions (with respect to s) for the coefficients in the asymptotic expansion (as N →∞) of the optimal s-energy.
In this paper, we show that certain sequences of polynomials $\{ p_k (z)\} _{k = 0}^n $, generated from three-term recurrence relations, have no zeros in parabolic regions in the …
In this paper, we show that certain sequences of polynomials $\{ p_k (z)\} _{k = 0}^n $, generated from three-term recurrence relations, have no zeros in parabolic regions in the complex plane of the form $y^2 \leqq 4\alpha (x + \alpha )$, $x > - \alpha $. As a special case of this, no partial $s_n (z) = {{\sum _{k = 0}^n z^k } / {k!}}$ of $e^z $ has a zero in $ y^2 \leqq 4(x + 1)$, $x > - 1$, for any $n \geqq 1$. Such zero-free parabolic regions are obtained for Padé approximants of certain meromorphic functions, as well as for the partial sums of certain hypergeometric functions.
The authors prove a theorem which characterizes the limit distribution of the zeros of polynomials , , defined by one (for each ) extremal relation with a variable (depending on …
The authors prove a theorem which characterizes the limit distribution of the zeros of polynomials , , defined by one (for each ) extremal relation with a variable (depending on ) weight function.Bibliography: 9 titles.
Preliminaries.- Hp Spaces.- Conjugate Functions.- Some Extremal Problems.- Some Uniform Algebra.- Bounded Mean Oscillation.- Interpolating Sequences.- The Corona Construction.- Douglas Algebras.- Interpolating Sequences and Maximal Ideals.
Preliminaries.- Hp Spaces.- Conjugate Functions.- Some Extremal Problems.- Some Uniform Algebra.- Bounded Mean Oscillation.- Interpolating Sequences.- The Corona Construction.- Douglas Algebras.- Interpolating Sequences and Maximal Ideals.
The thesis concentrates on two problems in discrete geometry, whose solutions are obtained by analytic, probabilistic and combinatoric tools. The first chapter deals with the strong polarization problem. This states …
The thesis concentrates on two problems in discrete geometry, whose solutions are obtained by analytic, probabilistic and combinatoric tools.
The first chapter deals with the strong polarization problem. This states that for
any sequence u1,...,un of norm 1 vectors in a real Hilbert space H , there exists a unit vector \vartheta \epsilon H , such that \sum 1 over [ui, v]2 \leqslant n2.
The 2-dimensional case is proved by complex analytic methods. For the higher dimensional extremal cases, we prove a tensorisation result that is similar to F. John's
theorem about characterisation of ellipsoids of maximal volume. From this, we deduce that the only full dimensional locally extremal system is the orthonormal system. We also obtain the same result for the weaker, original polarization problem.
The second chapter investigates a problem in probabilistic geometry. Take n independent, uniform random points in a triangle T. Convex chains between two fixed
vertices of T are defined naturally. Let Ln denote the maximal size of a convex chain.
We prove that the expectation of Ln is asymptotically \alpha n1/3, where \alpha is a constant between 1:5 and 3:5 - we conjecture that the correct value is 3. We also prove strong concentration results for Ln, which, in turn, imply a limit shape result for the longest convex chains.
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.
We use linear programming techniques to obtain new upper bounds on the maximal squared minimum distance of spherical codes with fixed cardinality. Functions Q/sub j/(n,s) are introduced with the property …
We use linear programming techniques to obtain new upper bounds on the maximal squared minimum distance of spherical codes with fixed cardinality. Functions Q/sub j/(n,s) are introduced with the property that Q/sub j/(n,s)<0 for some j>m if and only if the Levenshtein bound L/sub m/(n,s) on A(n,s)=max{|W|:W is an (n,|W|,s) code} can be improved by a polynomial of degree at least m+1. General conditions on the existence of new bounds are presented. We prove that for fixed dimension n/spl ges/5 there exists a constant k=k(n) such that all Levenshtein bounds L/sub m/(n, s) for m/spl ges/2k-1 can be improved. An algorithm for obtaining new bounds is proposed and discussed.
Universal bounds for the cardinality of codes in the Hamming space F/sub r//sup n/ with a given minimum distance d and/or dual distance d' are stated. A self-contained proof of …
Universal bounds for the cardinality of codes in the Hamming space F/sub r//sup n/ with a given minimum distance d and/or dual distance d' are stated. A self-contained proof of optimality of these bounds in the framework of the linear programming method is given. The necessary and sufficient conditions for attainability of the bounds are found. The parameters of codes satisfying these conditions are presented in a table. A new upper bound for the minimum distance of self-dual codes and a new lower bound for the crosscorrelation of half-linear codes are obtained.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">></ETX>
The problem of the?limiting distribution of the?zeros of the?polynomial extremal in the?-metric with respect to a?measure with finitely many points of growth is studied under the?assumption that the?degree of this …
The problem of the?limiting distribution of the?zeros of the?polynomial extremal in the?-metric with respect to a?measure with finitely many points of growth is studied under the?assumption that the?degree of this polynomial and the?number (n$ SRC=http://ej.iop.org/images/1064-5616/187/8/A04/tex_sm_153_img4.gif/>) of points of growth of the?measure approach infinity so that .