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In this paper we study the b -chromatic number of the cartesian product of two graphs. The b -chromatic number of a graph G is defined as the maximum number k of colors that can be used to color the vertices of G , such that we obtain a proper coloring and each color i has at least one representative χ i adjacent to a vertex of every color j , 1 ≦ jik . In this paper we get ρ( G□H ) ≦ ρ( G )( n H + 1) + Δ( H ) + 1, when the girth of G is assumed to be greater than or equal to 7.

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The basis number of a graph G is defined to be the least positive integer d such that G has a d-fold basis for the cycle space of G. We investigate the basis number of the cartesian product of stars and wheels with ladders, circular ladders and Möbius ladders.

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Letp=(p 1,p2,...) be a vector with an infinite number of coordinates, 1≦p k≦,k=1,2,... On the set of random functions depending on infinite number of variables, a mixed norm ∥. p is introduced, and thus the spacesL p with mixed norm are defined. Part 1 contains observations of general properties of those spaces (in particular, convergence properties depending on the behaviour of the exponentsp k ask→ ∞). Part 2 contains the proof of infinite-dimensional version of S. L. Sobolev's theorem (in mixed norm) for potentials of Wiener semigroup on infinite dimensional torusT .

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We refine a method introduced in [1] and [2] for studying the number of distinct values taken by certain polynomials of two real variables on Cartesian products. We apply it to prove a "gap theorem", improving a recent lower bound on the number of distinct distances between two collinear point sets in the Euclidean space.

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An arbitrary linear relation (multivalued operator) acting from one Hilbert space to another Hilbert space is shown to be the sum of a closable operator and a singular relation whose closure is the Cartesian product of closed subspaces. This decomposition can be seen as an analog of the Lebesgue decomposition of a measure into a regular part and a singular part. The two parts of a relation are characterized metrically and in terms of Stone’s characteristic projection onto the closure of the linear relation.

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Cartesian product of graphs , Graphs Combin. , 30 ( 2014 ), 511 – 520 . [3] Balakrishnan , R. , Francis Raj , S. and Kavaskar , T. , b

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M. Kouider and Mahéo, M. , The b-chromatic number of the Cartesian product of two graphs, Studia Sci. Math. Hungar. , 44 (2007), 49–55. MR 2309686 ( 2008c :05066) Mahéo M

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finite and countable Cartesian products , Compositio Math. , 23 : 2 ( 1971 ), 199 – 214 . [16] Ostaszewski , A. J. , On

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Aaronson, J., Lin, M. and Weiss, B. , Mixing properties of Markov operators and ergodic transformations, and ergodicity of Cartesian products . A collection of invited papers on ergodic

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