Papers by Andriy Bondarenko
Bulletin of the London Mathematical Society, 2015
holds for arbitrary integers 1 ≤ n 1 < · · · < n N . This bound is essentially better than that f... more holds for arbitrary integers 1 ≤ n 1 < · · · < n N . This bound is essentially better than that found in a recent paper of Aistleitner, Berkes, and Seip and can not be improved by more than possibly a power of 1/ log log log N . The proof relies on ideas from classical work of Gál, the method of Aistleitner, Berkes, and Seip, and a certain completeness property of extremal sets of squarefree numbers.
Mathematika, 2015
We consider the random functions S N (z) := N n=1 z(n), where z(n) is the completely multiplicati... more We consider the random functions S N (z) := N n=1 z(n), where z(n) is the completely multiplicative random function generated by independent Steinhaus variables z(p). It is shown that E|S N | N (log N ) −0.05616 and that (E|S N | q ) 1/q q N (log N ) −0.07672 for all q > 0.
Proceedings of the American Mathematical Society
In terms of the minimal N-point diameter D d (N) for ℝ d , we determine, for a class of continuou... more In terms of the minimal N-point diameter D d (N) for ℝ d , we determine, for a class of continuous real-valued functions f on [0,+∞], the N-point f-best-packing constant min{f(∥x-y∥):x,y∈ℝ d }, where the minimum is taken over point sets of cardinality N. We also show that N 1/d Δ d -1/d -2≤D d (N)≤N 1/d Δ d -1/d ,N≥2, where Δ d is the maximal sphere packing density in ℝ d . Further, we provide asymptotic estimates for the f-best-packing constants as N→∞.
We consider the random functions $S_N(z):=\sum_{n=1}^N z(n) $, where $z(n)$ is the completely mul... more We consider the random functions $S_N(z):=\sum_{n=1}^N z(n) $, where $z(n)$ is the completely multiplicative random function generated by independent Steinhaus variables $z(p)$. It is shown that $({\Bbb E} |S_N|^q)^{1/q}\gg_{q} \sqrt{N}(\log N)^{-0.02152}$ for all $q>0$.
We answer Totik's question on weighted Bernstein's inequalities showing that $$ \|T_n'... more We answer Totik's question on weighted Bernstein's inequalities showing that $$ \|T_n'\|_{L_p(\omega)} \le C(p,\omega)\, {n}\,\|T_n\|_{L_p(\omega)},\qquad 0<p\le \infty, $$ holds for all trigonometric polynomials $T_n$ and certain nondoubling weights $\omega$. Moreover, we find necessary conditions on $\omega$ for Bernstein's inequality to hold. We also prove weighted Bernstein-Markov, Remez, and Nikolskii inequalities for trigonometric and algebraic polynomials.
Journal of Number Theory, 2015
The L q norm of a Dirichlet polynomial F (s) = N n=1 a n n −s is defined as
holds for arbitrary integers 1 ≤ n 1 < · · · < n N . This bound is essentially better than that f... more holds for arbitrary integers 1 ≤ n 1 < · · · < n N . This bound is essentially better than that found in a recent paper of Aistleitner, Berkes, and Seip and can not be improved by more than possibly a power of 1/ log log log N . The proof relies on ideas from classical work of Gál, the method of Aistleitner, Berkes, and Seip, and a certain completeness property of extremal sets of squarefree numbers.
Constructive Approximation, 2014
Ukrainian Mathematical Journal, 2009
ABSTRACT Let Δ3 be the set of functions three times continuously differentiable on [−1, 1] and su... more ABSTRACT Let Δ3 be the set of functions three times continuously differentiable on [−1, 1] and such that f″′(x) ≥ 0, x ∈ [−1, 1]. We prove that, for any n ∈ ℕ and r ≥ 5, there exists a function f ∈ C r [−1, 1] ⋂ Δ3 [−1, 1] such that ∥f (r)∥ C[−1, 1] ≤ 1 and, for an arbitrary algebraic polynomial P ∈ Δ3 [−1, 1], there exists x such that $$ \left| {f(x) - P(x)} \right| \geq C\sqrt n {{\uprho}}_n^r(x), $$ where C &gt; 0 is a constant that depends only on r, and $$ {{{\uprho }}_n}(x): = \frac{1}{{{n^2}}} + \frac{1}{n}\sqrt {1 - {x^2}} . $$
SIAM Journal on Discrete Mathematics, 2010
For each N&amp;amp;gt;=c_d*n^{2d*(d+1)/(d+2)} we prove the existence of a spherical n-design ... more For each N&amp;amp;gt;=c_d*n^{2d*(d+1)/(d+2)} we prove the existence of a spherical n-design on S^d consisting of N points, where c_d is a constant depending only on $d$.
Proceedings of the American Mathematical Society, 2013
In terms of the minimal N -point diameter D d (N ) for R d , we determine, for a class of continu... more In terms of the minimal N -point diameter D d (N ) for R d , we determine, for a class of continuous real-valued functions f on [0, +∞], the N -point
Journal of Combinatorial Theory, Series B, 2013
In this paper, we give a complete description of strongly regular graphs with parameters ((n 2 + ... more In this paper, we give a complete description of strongly regular graphs with parameters ((n 2 + 3n − 1) 2 , n 2 (n + 3), 1, n(n + 1)).
Journal of Approximation Theory, 2012
In recent years there has been much interest and there have been quite a few achievements in ques... more In recent years there has been much interest and there have been quite a few achievements in questions concerning the degree of approximation of a contin-uous function f, on a finite interval, which has a certain shape, by algebraic polynomials and by piecewise ...
Journal of Approximation Theory, 2008
Let N(n, t) be the minimal number of points in a spherical t-design on the unit sphere S^n in R^{... more Let N(n, t) be the minimal number of points in a spherical t-design on the unit sphere S^n in R^{n+1}. For each n &gt;= 3, we prove a new asymptotic upper bound N(n, t) &lt;= C(n)t^{a_n}, where C(n) is a constant depending only on n, a_3 &lt;= 4, a_4 &lt;= 7, a_5 &lt;= 9, a_6 &lt;= 11, a_7 &lt;= 12,
Discrete and Continuous Dynamical Systems, 2005
Abstract In the homogenization of second order elliptic equations with periodic coefficients, it ... more Abstract In the homogenization of second order elliptic equations with periodic coefficients, it is well known that the rate of convergence of the corrector un − uhom in the L2 norm is 1/n, the same as the scale of periodicity (see Jikov et al [7]). It is possible to have the same rate of ...
Annals of Mathematics, 2013
In this paper we prove the conjecture of Korevaar and Meyers: for each N ≥ c d t d there exists a... more In this paper we prove the conjecture of Korevaar and Meyers: for each N ≥ c d t d there exists a spherical t-design in the sphere S d consisting of N points, where c d is a constant depending only on d.
Arxiv preprint arXiv:1011.3478, 2010
For X ⊂ C(K), denote by X the closure of X with respect to this norm. Recall, a continuous functi... more For X ⊂ C(K), denote by X the closure of X with respect to this norm. Recall, a continuous function f on a convex set K∗ ⊂ IRd is called a convex function on K∗, if for all x1, x2 ∈ K∗ the inequality ... Theorem S[3]. Let K ⊂ IRd be a convex compact set. If f is a convex function on K, then ...
Acta Mathematica Hungarica, 2014
For N -point best-packing configurations ω N on a compact metric space (A, ρ), we obtain estimate... more For N -point best-packing configurations ω N on a compact metric space (A, ρ), we obtain estimates for the mesh-separation ratio γ(ω N , A), which is the quotient of the covering radius of ω N relative to A and the minimum pairwise distance between points in ω N . For best-packing configurations ω N that arise as limits of minimal Riesz s-energy configurations as s → ∞, we prove that γ(ω N , A) ≤ 1 and this bound can be attained even for the sphere. In the particular case when N = 5 on S 2 with ρ the Euclidean metric, we prove our main result that among the infinitely many 5-point best-packing configurations there is a unique configuration, namely a square-base pyramid ω * 5 , that is the limit (as s → ∞) of 5-point s-energy minimizing configurations. Moreover, γ(ω * 5 , S 2 ) = 1.
atlas-conferences.com
Atlas home || Conferences | Abstracts | about Atlas Optimal Configurations on the Sphere and Othe... more Atlas home || Conferences | Abstracts | about Atlas Optimal Configurations on the Sphere and Other Manifolds May 17-20, 2010 Vanderbilt University, Department of Mathematics Nashville, Tennessee, USA. Organizers Henry Cohn, Microsoft Research; Doug Hardin, Vanderbilt University; Edward Saff, Vanderbilt University; Salvatore Torquato, Princeton University. Conference Homepage. Abstracts. Congpei An Numerical Verification Method for ...
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Papers by Andriy Bondarenko