The present study deals with a spatially homogeneous and anisotropic Bianchi-II cosmological models representing massive strings by applying the variation law for generalized Hubble's parameter that yields a constant value of deceleration parameter. We find that the constant value of deceleration parameter is reasonable for the present day universe. The variation law for Hubble's parameter generates two types of solutions for the average scale factor, one is of power-law type and other is of the exponential form. Using these two forms, Einstein's field equations are solved separately that correspond to expanding singular and non-singular models of the universe respectively. The energy-momentum tensor for such string as formulated by Letelier (Phys. Rev. D 28:2414) is used to construct massive string cosmological models for which we assume that the expansion (θ ) in the model is proportional to the component σ 1 1 of the shear tensor σ j i . This condition leads to A = (BC) m , where A, B and C are the metric coefficients and m is proportionality constant. Our models are in accelerating phase which is consistent to the recent observations. The cosmological constant is found to be a decreasing function of time and it approaches a small positive value at present epoch which is in good agreement by the results from recent supernovae observations. Some physical and geometric behaviour of the models are also discussed.
Two Latin squares A, B of order n are called pseudo orthogonal if for any 1 ≤ i, j ≤ n there exists a k, 1 ≤ k ≤ n, such that A(i, k) = B(j, k). We prove that the existence of a family of m mutually pseudo orthogonal Latin squares of order n is equivalent to the existence of a family of m mutually orthogonal Latin squares of order n. We also obtain exact values of clique partition numbers of several classes of complete multipartite graphs and of the tensor product of complete graphs.
Два латинских квадрата $A,B$ порядка $n$ называются псевдоортогональными, если для любых $1\le i,j\le n$ существует такое $k,1\le k\le n$, что $A(i,k)=B(j,k)$. В статье доказано, что существование семейства из $m$ взаимно псевдоортогональных латинских квадратов порядка $n$ эквивалентно существованию семейства из $m$ взаимно ортогональных латинских квадратов порядка $n$. Найдены также точные значения минимальных мощностей кликовых разбиений для нескольких классов полных многодольных графов и для тензорного произведения полных графов.
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