Some results on Oppenheim's ``Factorisatio Numerorum'' function

Luca, Florian; Mukhopadhyay, Anirban; Kotyada, Srinivas

doi:10.4064/aa142-1-3

Cited by 10 publications

(8 citation statements)

References 8 publications

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“…This motivates estimating the growth of L Λn with n. In order to estimate L Λ , we introduce the univariate Lebesgue constants 64) where the supremum is taken over all non-zero real valued functions u that are everywhere defined and uniformly bounded on [−1, 1]. We define an analogous quantity for the difference operator ∆ k , namely 65) and observe that…”

Section: Stabilitymentioning

confidence: 99%

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Approximation of high-dimensional parametric PDEs

Cohen¹,

DeVore²

2015

Acta Numerica

235

266

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Parametrized families of PDEs arise in various contexts such as inverse problems, control and optimization, risk assessment, and uncertainty quantification. In most of these applications, the number of parameters is large or perhaps even infinite. Thus, the development of numerical methods for these parametric problems is faced with the possible curse of dimensionality. This article is directed at (i) identifying and understanding which properties of parametric equations allow one to avoid this curse and (ii) developing and analyzing effective numerical methodd which fully exploit these properties and, in turn, are immune to the growth in dimensionality.The first part of this article studies the smoothness and approximability of the solution map, that is, the map a → u(a) where a is the parameter value and u(a) is the corresponding solution to the PDE. It is shown that for many relevant parametric PDEs, the parametric smoothness of this map is typically holomorphic and also highly anisotropic in that the relevant parameters are of widely varying importance in describing the solution. These two properties are then exploited to establish convergence rates of n-term approximations to the solution map for which each term is separable in the parametric and physical variables. These results reveal that, at least on a theoretical level, the solution map can be well approximated by discretizations of moderate complexity, thereby showing how the curse of dimensionality is broken. This theoretical analysis is carried out through concepts of approximation theory such as best n-term approximation, sparsity, and n-widths. These notions determine a priori the best possible performance of numerical methods and thus serve as a benchmark for concrete algorithms.The second part of this article turns to the development of numerical algorithms based on the theoretically established sparse separable approximations. The numerical methods studied fall into two general categories. The first uses polynomial expansions in terms of the parameters to approximate the solution map. The second one searches for suitable low dimensional spaces for simultaneously approximating all members of the parametric family. The numerical implementation of these approaches is carried out through adaptive and greedy algorithms. An a priori analysis of the performance of these algorithms establishes how well they meet the theoretical benchmarks.

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Section: Stabilitymentioning

confidence: 99%

“…In [65], it is proved that the total number of multiplicative partitions t(A) has the asymptotic behaviour…”

Section: Summability Of Multi-indexed Sequencesmentioning

confidence: 99%

Approximation of high-dimensional parametric PDEs

Cohen¹,

DeVore²

2015

Acta Numerica

235

266

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“…We will show that additive entropy is easily measured by the additive partition function, and that multiplicative entropy is directly related to the multiplicative partition function. This project is strongly connected to previous work on the additive partition function [7,8] and on the multiplicative partition function [9,10,11,12,13]. In particular, our approach is closely related to previous work [11,12,13] on the unordered factorization of natural numbers, in the sense that the multiplicative entropy of a natural number has to be computed by such factorization.…”

Section: Introductionmentioning

confidence: 55%

“…This project is strongly connected to previous work on the additive partition function [7,8] and on the multiplicative partition function [9,10,11,12,13]. In particular, our approach is closely related to previous work [11,12,13] on the unordered factorization of natural numbers, in the sense that the multiplicative entropy of a natural number has to be computed by such factorization. In this paper we will present a computable formula for the multiplicative entropy of natural numbers with two prime factors, and introduce the comparative density of two algebraic entropies, and use this to analyze their asymptotic behavior.…”

Section: Introductionmentioning

confidence: 55%

Algebraic Entropies of Natural Numbers With One or Two Prime Factors

Jeong¹,

Kim²,

Kim³

2016

The Pure and Applied Mathematics

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Abstract. We formulate the additive entropy of a natural number in terms of the additive partition function, and show that its multiplicative entropy is directly related to the multiplicative partition function. We give a practical formula for the multiplicative entropy of natural numbers with two prime factors. We use this formula to analyze the comparative density of additive and multiplicative entropy, prove that this density converges to zero as the number tends to infinity, and empirically observe this asymptotic behavior.

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“…Our next result addresses this question. We observe that if we write um(n) for the number of unordered factorizations of n, then it was shown in [6] that the set of n such that um(n) is a divisor of n has asymptotic density zero. Observe that this set contains the primes so it is a rather thick set of asymptotic density zero.…”

Section: A Variation Of the Problemmentioning

confidence: 99%

On Prime-Perfect Numbers

Knopfmacher

Luca

2011

Int. J. Number Theory

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We prove that the Diophantine equation p r 1 1 · · · p r k k =`r 1 +···+r k r 1 ,...,r k´h as only finitely many positive integer solutions k, p 1 , . . . , p k , r 1 , . . . , r k , where p 1 , . . . , p k are distinct primes. If a positive integer n has prime factorization p r 1 1 · · · p r k k , then`r 1 +···+r k r 1 ,...,r k´r epresents the number of ordered factorizations of n into prime parts. Hence, solutions to the above Diophantine equation are designated as prime-perfect numbers.

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Some results on Oppenheim's ``Factorisatio Numerorum'' function

Cited by 10 publications

References 8 publications

Approximation of high-dimensional parametric PDEs

Approximation of high-dimensional parametric PDEs

Algebraic Entropies of Natural Numbers With One or Two Prime Factors

On Prime-Perfect Numbers

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