The Experts below are selected from a list of 282 Experts worldwide ranked by ideXlab platform
Jacek Marchwicki - One of the best experts on this subject based on the ideXlab platform.
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levy steinitz theorem and achievement sets of Conditionally Convergent series on the real plane
Journal of Mathematical Analysis and Applications, 2018Co-Authors: Szymon Glab, Jacek MarchwickiAbstract:Abstract Levy–Steinitz theorem characterizes the sum range of Conditionally Convergent series, that is a set of all its Convergent rearrangements; in finitely dimensional spaces – it is an affine subspace. An achievement of a series is a set of all its subsums. We study the properties of achievement sets of series whose sum range is the whole plane. It turns out that it varies on the number of Levy vectors of a series.
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Levy–Steinitz theorem and achievement sets of Conditionally Convergent series on the real plane
Journal of Mathematical Analysis and Applications, 2018Co-Authors: Szymon Gła̧b, Jacek MarchwickiAbstract:Abstract Levy–Steinitz theorem characterizes the sum range of Conditionally Convergent series, that is a set of all its Convergent rearrangements; in finitely dimensional spaces – it is an affine subspace. An achievement of a series is a set of all its subsums. We study the properties of achievement sets of series whose sum range is the whole plane. It turns out that it varies on the number of Levy vectors of a series.
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Subsums of Conditionally Convergent series in finite dimensional spaces
arXiv: Functional Analysis, 2018Co-Authors: Jacek Marchwicki, Václav VlasákAbstract:An achievement set of a series is a set of all its subsums. We study the properties of achievement sets of Conditionally Convergent series in finite dimensional spaces. The purpose of the paper is to answer some of the open problems formulated in \cite{GM}. We obtain general results for series with harmonic-like coordinates, that is $A((-1)^{n+1}n^{-\alpha_1},\dots,(-1)^{n+1}n^{-\alpha_d})=\mathbb{R}^d$ for pairwise distinct numbers $\alpha_1,\dots,\alpha_d\in(0,1]$. For $d=2$, $\alpha_1=1, \alpha_2=\frac{1}{2}$ it was stated as an open problem in \cite{GM}, that is $A(\frac{(-1)^n}{n},\frac{(-1)^n}{\sqrt{n}})=\mathbb{R}^2$.
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Subsums of Conditionally Convergent series in finite dimensional spaces
Filomat, 2018Co-Authors: Jacek Marchwicki, Václav VlasákAbstract:An achievement set of a series is a set of all its subsums. We study the properties of achievement sets of Conditionally Convergent series in finite dimensional spaces. The purpose of the paper is to answer some of the open problems formulated in [10]. We obtain general result for series with harmonic-like coordinates, that is A((-1)n+1n-?1,..., (-1)n+1n-?d) = Rd for pairwise distinct numbers ?1,..., ?d ? (0,1]. For d = 2, ?1 = 1, ?2 = 1/2 this problem was stated in [10].
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Achievement sets and sum ranges with ideal supports
arXiv: Functional Analysis, 2017Co-Authors: Jacek MarchwickiAbstract:We introduce the notion of ideally supported achievement sets for a series of real numbers. We analize their complexity and topological properties. We compare the notion of ideal achievement sets with the notion of ideally supported sum range of real series, considered by Filip\'ow and Szuca. We complete Filip\'ow and Szuca characterization of ideal sum ranges, [R. Filip\'ow, P. Szuca, Rearrangement of Conditionally Convergent series on a small set, J. Math. Anal. Appl. 362 (2010), no. 1, 64-71.], and we obtain some generalization of Riemann's Theorem.
Szymon Glab - One of the best experts on this subject based on the ideXlab platform.
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levy steinitz theorem and achievement sets of Conditionally Convergent series on the real plane
Journal of Mathematical Analysis and Applications, 2018Co-Authors: Szymon Glab, Jacek MarchwickiAbstract:Abstract Levy–Steinitz theorem characterizes the sum range of Conditionally Convergent series, that is a set of all its Convergent rearrangements; in finitely dimensional spaces – it is an affine subspace. An achievement of a series is a set of all its subsums. We study the properties of achievement sets of series whose sum range is the whole plane. It turns out that it varies on the number of Levy vectors of a series.
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levy steinitz theorem and achievement sets of Conditionally Convergent series on the real plane
arXiv: Functional Analysis, 2017Co-Authors: Szymon Glab, Jacek MarchwickiAbstract:Levy-Steinitz theorem characterize sum range of Conditionally Convergent series, that is a set of all its Convergent rearrangements; in finitely dimensional spaces -- it is an affine subspace. An achievement of a series is a set of all its subsums. We study the properties of achievement sets of series whose sum range is the whole plane. It turns out that it varies on the number of Levy vectors of a series.
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achievement sets of Conditionally Convergent series
arXiv: Functional Analysis, 2016Co-Authors: Artur Bartoszewicz, Szymon Glab, Jacek MarchwickiAbstract:Considering the sets of subsums of series (or achievement sets) we show that for Conditionally Convergent series the multidimensional case is much more complicated than that of the real line. Although we are far from the full topological classification of such sets, we present many surprising examples and catch the ideas standing behind them in general theorems.
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algebrability of Conditionally Convergent series with cauchy product
Journal of Mathematical Analysis and Applications, 2012Co-Authors: Artur Bartoszewicz, Szymon GlabAbstract:Abstract We show that the set of Conditionally Convergent real series considered with Cauchy product is ( ω , 1 ) -algebrable.
Aljoša Volčič - One of the best experts on this subject based on the ideXlab platform.
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A Riemann-type theorem for a Riemann-type integral
Ricerche di Matematica, 2019Co-Authors: Aljoša VolčičAbstract:For functions which are Henstock–Kurzweil integrable but not Lebesgue-integrable we prove a theorem which resembles the Riemann theorem on the rearrangement of Conditionally Convergent series.
Vladimir Mityushev - One of the best experts on this subject based on the ideXlab platform.
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Effective conductivity of a random suspension of highly conducting spherical particles
arXiv: Mathematical Physics, 2019Co-Authors: Vladimir Mityushev, Wojciech NawalaniecAbstract:Randomly distributed non-overlapping perfectly conducting spheres are embedded in a conducting matrix with the concentration of inclusions $f$. Jeffrey (1973) suggested an analytical formula valid up to $O(f^3)$ for macroscopically isotropic random composites. A Conditionally Convergent sum arose in the spatial averaging. In the present paper, we apply a method of functional equations to random composites and correct Jeffrey's formula. The main revision concerns the proper investigation of the Conditionally Convergent sum and correction the $f^2$-term. A new model of symbolic computations is developed in order to compute the effective conductivity tensor. The corresponding algorithm is realized up to $O(f^{\frac{10}3})$. The obtained formulae explicitly demonstrate the dependence of the effective conductivity tensor on the deterministic and probabilistic distributions of inclusions in the $f^2$-term, and in the $f^3$-term. This leads to the conclusion that some previous formulae presented as universal, i.e., valid for all random composites, may be actually applied only to dilute or to special composites when interaction between inclusions do not matter.
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Effective conductivity of a random suspension of highly conducting spherical particles
Applied Mathematical Modelling, 2019Co-Authors: Vladimir Mityushev, Wojciech NawalaniecAbstract:Abstract Randomly distributed non-overlapping perfectly conducting spheres are embedded in a conducting matrix with the concentration of inclusions f. Jeffrey (1973) suggested an analytical formula valid up to O(f3) for macroscopically isotropic random composites. A Conditionally Convergent sum arose in the spatial averaging. In the present paper, we apply a method of functional equations to random composites and correct Jeffrey’s formula. The main revision concerns the proper investigation of the Conditionally Convergent sum and correction the f2-term. An algorithm for symbolic computations of the effective conductivity tensor is developed and performed up to O ( f 10 3 ) . The obtained formulae explicitly demonstrate the dependence of the effective conductivity tensor on the deterministic and probabilistic distributions of inclusions in the f2-term, and in the f3-term. This leads to the conclusion that some previous formulae presented as universal, i.e., valid for all random composites, may be actually applied only to dilute or to special composites when interaction between inclusions do not matter.
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Cluster method in composites and its convergence
Applied Mathematics Letters, 2018Co-Authors: Vladimir MityushevAbstract:Abstract Extensions of Maxwell’s self-consistent approach from single- to n - inclusions problems lead to cluster methods applied to computation of the effective properties of composites. We describe applications of Maxwell’s formalism to finite clusters and explain the uncertainty arising when n tends to infinity by study of the corresponding Conditionally Convergent series.
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Closed-form evaluation of two-dimensional static lattice sums
Proceedings. Mathematical physical and engineering sciences, 2016Co-Authors: Semyon Yakubovich, P. Drygas, Vladimir MityushevAbstract:Closed-form formulae for the Conditionally Convergent two-dimensional (2D) static lattice sums S 2 (for conductivity) and T 2 (for elasticity) are deduced in terms of the complete elliptic integrals of the first and second kind. The obtained formulae yield asymptotic analytical formulae for the effective tensors of 2D composites with circular inclusions up to the third order in concentration. Exact relations between S 2 and T 2 for different lattices are established. In particular, the value S 2= π for the square and hexagonal arrays is discussed and T 2= π /2 for the hexagonal is deduced.
Wojciech Nawalaniec - One of the best experts on this subject based on the ideXlab platform.
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Effective conductivity of a random suspension of highly conducting spherical particles
arXiv: Mathematical Physics, 2019Co-Authors: Vladimir Mityushev, Wojciech NawalaniecAbstract:Randomly distributed non-overlapping perfectly conducting spheres are embedded in a conducting matrix with the concentration of inclusions $f$. Jeffrey (1973) suggested an analytical formula valid up to $O(f^3)$ for macroscopically isotropic random composites. A Conditionally Convergent sum arose in the spatial averaging. In the present paper, we apply a method of functional equations to random composites and correct Jeffrey's formula. The main revision concerns the proper investigation of the Conditionally Convergent sum and correction the $f^2$-term. A new model of symbolic computations is developed in order to compute the effective conductivity tensor. The corresponding algorithm is realized up to $O(f^{\frac{10}3})$. The obtained formulae explicitly demonstrate the dependence of the effective conductivity tensor on the deterministic and probabilistic distributions of inclusions in the $f^2$-term, and in the $f^3$-term. This leads to the conclusion that some previous formulae presented as universal, i.e., valid for all random composites, may be actually applied only to dilute or to special composites when interaction between inclusions do not matter.
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Effective conductivity of a random suspension of highly conducting spherical particles
Applied Mathematical Modelling, 2019Co-Authors: Vladimir Mityushev, Wojciech NawalaniecAbstract:Abstract Randomly distributed non-overlapping perfectly conducting spheres are embedded in a conducting matrix with the concentration of inclusions f. Jeffrey (1973) suggested an analytical formula valid up to O(f3) for macroscopically isotropic random composites. A Conditionally Convergent sum arose in the spatial averaging. In the present paper, we apply a method of functional equations to random composites and correct Jeffrey’s formula. The main revision concerns the proper investigation of the Conditionally Convergent sum and correction the f2-term. An algorithm for symbolic computations of the effective conductivity tensor is developed and performed up to O ( f 10 3 ) . The obtained formulae explicitly demonstrate the dependence of the effective conductivity tensor on the deterministic and probabilistic distributions of inclusions in the f2-term, and in the f3-term. This leads to the conclusion that some previous formulae presented as universal, i.e., valid for all random composites, may be actually applied only to dilute or to special composites when interaction between inclusions do not matter.