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In this note we study the approximation of the fractional Lévy area with Hurst parameter , considering the mean square error at a single point as error criterion. We derive the optimal rate of convergence that can be achieved by arbitrary approximation methods that are based on an equidistant discretization of the driving fractional Brownian motion. This rate is , where denotes the number of evaluations of the fractional Brownian motion, and is obtained by a trapezoidal rule. 相似文献
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Antal Joós 《Discrete Mathematics》2018,341(9):2544-2552
It is known that . In 1968, Meir and Moser (1968) asked for finding the smallest such that all the rectangles of sizes , , can be packed into a square or a rectangle of area . First we show that in Paulhus (1997), the key lemma, as a statement, in the proof of the smallest published upper bound of the minimum area is false, then we prove a different new upper bound. We show that if the rectangles are packed into a square and if the rectangles are packed into a rectangle. 相似文献
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《Journal of Complexity》2016,32(6):867-884
We are interested in approximation of a multivariate function by linear combinations of products of univariate functions , . In the case it is the classical problem of bilinear approximation. In the case of approximation in the space the bilinear approximation problem is closely related to the problem of singular value decomposition (also called Schmidt expansion) of the corresponding integral operator with the kernel . There are known results on the rate of decay of errors of best bilinear approximation in under different smoothness assumptions on . The problem of multilinear approximation (nonlinear tensor product approximation) in the case is more difficult and much less studied than the bilinear approximation problem. We will present results on best multilinear approximation in under mixed smoothness assumption on . 相似文献
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Ts.Ch-D. Batueva O.V. Borodin M.A. Bykov A.O. Ivanova O.N. Kazak D.V. Nikiforov 《Discrete Mathematics》2017,340(11):2659-2664
The weight of an edge in a normal plane map (NPM) is the degree-sum of its end-vertices. An edge is of type if and . In 1940, Lebesgue proved that every NPM has an edge of one of the types , , or , where 7 and 6 are best possible. In 1955, Kotzig proved that every 3-connected planar graph has an edge with , which bound is sharp. Borodin (1989), answering Erd?s’ question, proved that every NPM has either a -edge, or -edge, or -edge.A vertex is simplicial if it is completely surrounded by 3-faces. In 2010, Ferencová and Madaras conjectured (in different terms) that every 3-polytope without simplicial 3-vertices has an edge with . Recently, we confirmed this conjecture by proving that every NPM has either a simplicial 3-vertex adjacent to a vertex of degree at most 10, or an edge of types , , or .By a -vertex we mean a -vertex incident with precisely triangular faces. The purpose of our paper is to prove that every NPM has an edge of one of the following types: , , , , , , or , where all bounds are best possible. In particular, this implies that the bounds in , , and can be attained only at NPMs having a simplicial 3-, 4-, or 5-vertex, respectively. 相似文献
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We consider the problem of determining , the smallest possible length for which an code of minimum distance over the field of order 4 exists. We prove the nonexistence of codes for and the nonexistence of a code for using the geometric method through projective geometries, where . This yields to determine the exact values of for these values of . We also give the updated table for for all except some known cases. 相似文献
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