Relationship between Transport Phenomena and Characteristics of the Cluster Structure

Polyanskaya, A. V.; Polyanskiǐ, A. M.; Polyanskii, V. A.

doi:10.1134/s106378421906015x

Cited by 4 publications

(3 citation statements)

References 17 publications

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“…The following conjecture, already proven for d = 2 in [19, Thm. 1.3], is from [19] and [26]. Notice that c(d, n, n) was considered in [19] as an application to understand the behavior of the circumradius with respect to the Minkowski addition of n centrally symmetric sets in R d .…”

Section: History and Resultsmentioning

confidence: 99%

Large signed subset sums

Ambrus,

Merino

2020

Preprint

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We study the following question: for given d 2, n d and k n, what is the largest value c(d, n, k) such that from any set of n unit vectors in R d , we may select k vectors with corresponding signs ±1 so that their signed sum has norm at least c(d, n, k)?The problem is dual to classical vector sum minimization and balancing questions, which have been studied for over a century. We give asymptotically sharp estimates for c(d, n, k) in the general case. In several special cases, we provide stronger estimates: the quantity c(d, n, n) corresponds to the ℓp-polarization problem, while determining c(d, n, 2) is equivalent to estimating the coherence of a vector system, which is a special case of p-frame energies. Two new proofs are presented for the classical Welch bound when n = d + 1. For large values of n, volumetric estimates are applied for obtaining fine estimates on c(d, n, 2). Studying the planar case, sharp bounds on c(2, n, k) are given. Finally, we determine the exact value of c(d, d + 1, d + 1) under some extra assumptions.

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Section: History and Resultsmentioning

confidence: 99%

Large signed subset sums

Ambrus,

Merino

2020

Preprint

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“…For n = d + 1, natural intuition and numerical experiments suggest that each extremal configuration is, up to sign changes, the union of the vertex set of an even dimensional regular simplex and an orthonormal basis of the orthogonal complement of its subspace. The following conjecture was stated in a slightly incorrect form in [11] and has been corrected by [23].…”

Section: Sign Sequences: P =mentioning

confidence: 99%

Polarization, sign sequences and isotropic vector systems

Ambrus

Nietert

2019

Pacific J. Math.

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We determine the order of magnitude of the nth ℓp-polarization constant of the unit sphere S d−1 for every n, d 1 and p > 0. For p = 2, we prove that extremizers are isotropic vector sets, whereas for p = 1, we show that the polarization problem is equivalent to that of maximizing the norm of signed vector sums. Finally, for d = 2, we discuss the optimality of equally spaced configurations on the unit circle.2010 Mathematics Subject Classification. 52A40(primary), and 31C20(secondary).

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“…It does not seem easy to guess the optimal bounds in extending Theorem 1.3 to arbitrary n ∈ N and i ≥ n + 1. Nevertheless, we were told by Alexandr Polyanskii [18] (see also [3]) that the following conjecture for i = n + 1 seems reasonable. Then max l1,...,ln+1…”

Section: Generalized Minkowski Spacesmentioning

confidence: 99%