Recent Advances in Orthogonal Polynomials, Special Functions, and Their Applications

Arvesú, J.; Lagomasino, G. López

doi:10.1090/conm/578

Cited by 13 publications

(12 citation statements)

References 21 publications

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“…Given its broad relevance, the global minimum of the Thomson problem has been studied extensively [24][25][26][27][28][29][30][31][32][33][34][35][36][37]. Analytical solutions are known for N ¼ 2-6 and 12 [38,39].…”

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confidence: 99%

Kinetic Transition Networks for the Thomson Problem and Smale’s Seventh Problem

Mehta

Chen

et al. 2016

Phys. Rev. Lett.

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Readers may view, browse, and/or download material for temporary copying purposes only, provided these uses are for noncommercial personal purposes. Except as provided by law, this material may not be further reproduced, distributed, transmitted, modied, adapted, performed, displayed, published, or sold in whole or part, without prior written permission from the American Physical Society.Additional information: Use policyThe full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-prot purposes provided that:• a full bibliographic reference is made to the original source • a link is made to the metadata record in DRO • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders.Please consult the full DRO policy for further details. The Thomson problem, arrangement of identical charges on the surface of a sphere, has found many applications in physics, chemistry and biology. Here, we show that the energy landscape of the Thomson problem for N particles with N ¼ 132, 135, 138, 141, 144, 147, and 150 is single funneled, characteristic of a structure-seeking organization where the global minimum is easily accessible. Algorithmically, constructing starting points close to the global minimum of such a potential with spherical constraints is one of Smale's 18 unsolved problems in mathematics for the 21st century because it is important in the solution of univariate and bivariate random polynomial equations. By analyzing the kinetic transition networks, we show that a randomly chosen minimum is, in fact, always "close" to the global minimum in terms of the number of transition states that separate them, a characteristic of small world networks.

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confidence: 99%

Kinetic Transition Networks for the Thomson Problem and Smale’s Seventh Problem

Mehta

Chen

et al. 2016

Phys. Rev. Lett.

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“…, содержат некоторую фиксированную (12). С другой стороны, доказательство будет опираться на доказанную Е. А. Рах-мановым [13] полунепрерывность сверху функционала энергии для последова-тельности компактов, состоящих из конечного числа (одного и того же для всех компактов) континуумов.…”

Section: § 1 формулировка результатаunclassified

“…В имеющейся ситуации (непрерывность внешнего поля и равномерная ограниченность числа континуумов, образующих компакты K * n → F , n ⊂ Λ) Е. А. Рахманов [13] доказал полунепрерывность сверху функционала энергии (в действительности Е. А. Рахманов доказал полунепрерывность сверху функ-ционала энергии для внешних полей более общего вида). Следовательно, с учетом равенства (9)…”

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О сходимости $m$-точечных аппроксимаций Паде набора многозначных аналитических функций

Buslaev¹

2015

Математический сборник

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УДК 517.53 В. И. БуслаевО сходимости m-точечных аппроксимаций Паде набора многозначных аналитических функций В работе доказана сходимость m-точечных аппроксимаций Паде набо-ра m голоморфных ростков функций, имеющих аналитические продолже-ния вдоль любых путей в расширенной комплексной плоскости, не прохо-дящих через конечное число точек. Полученный результат распространя-ет теорему Шталя со случая m = 1 на случай произвольного m ∈ N.Библиография: 15 названий. Ключевые слова: рациональные приближения, ортогональные мно-гочлены, аппроксимации Паде, сходимость по емкости, предельное рас-пределение полюсов.

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“…The proof of Theorem 1 is based on the theory of the condenser capacities and dissymmetrization. These ideas go back to the book [4]. By taking the limit η → 0 in (1), we obtain an inequality for the Green energy of a charge concentrated in an unrestricted number of points (not necessary even).…”

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confidence: 99%

“…Introduction and statement of results. There are many studies related to the extremal problems for different kinds of energy of a discrete charge (see, e. g., the papers [2][3][4], [9], [10], [12], and the references therein). In contrast to the previous research, we consider the Green energy (see also the recent articles [1] and [5]).…”

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confidence: 99%

Green energy and extremal decompositions

Дубинин¹

2019

Issues of Analysis

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We give two precise estimates for the Green energy of a discrete charge, concentrated in an even number of points on the circle, with respect to the concentric ring. The lower estimate for the Green energy is attained for the points with a nonstandard symmetry. The well-known Pólya-Schur inequality for the logarithmic energy is a special case of this estimate. The proof is based on the application of dissymmetrization and an asymptotic formula for the conformal capacity of a generalized condenser in the case when some of its plates contract to given points. The upper bound is established for a charge that takes values of opposite signs. Its proof reduces to solving a problem on the so-called extremal decomposition of a circular ring with free poles on a circle.

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Recent Advances in Orthogonal Polynomials, Special Functions, and Their Applications

Cited by 13 publications

References 21 publications

Kinetic Transition Networks for the Thomson Problem and Smale’s Seventh Problem

Kinetic Transition Networks for the Thomson Problem and Smale’s Seventh Problem

О сходимости $m$-точечных аппроксимаций Паде набора многозначных аналитических функций

Green energy and extremal decompositions

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