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It is known that the smallest complete bipartite graph which is not 3-choosable has 14 vertices. We show that the extremal configuration is unique.
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The usual tight-binding (TB) approximation, employed in electronic structure calculations, is extended to include the more-distant neighbor sites than the first. In doing so, a simple power law is adopted to describe the bond and overlap contributions, which enables the energy dispersion relation to...
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The following questions are studied: Under what conditions does the existence of a (nonzero) fixed point for every member of a semigroup of matrices imply a common fixed point for the entire semigroup? What is the smallest number k such that the existence of a common fixed point for every k members ...
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An approximate solution of the Klein-Gordon equation for the general Hulthen-type potentials in D -dimensions within the framework of an approximation to the centrifugal term is obtained. The bound state energy eigenvalues and the normalized eigenfunctions are obtained in terms of hypergeometric pol...
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The distance from an arbitrary rank-one projection to the set of nilpotent operators, in the space of k x k matrices with the usual operator norm, is shown to be sec(pi/(k+2))/2. This gives improved bounds for the distance between the set of all non-zero projections and the set of nilpotents in the ...
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We analyze the polynomial solutions of the linear differential equation p2(x)y″+p1(x)y′+p0(x)y=0 where pj(x) is a jth-degree polynomial. We discuss all the possible polynomial solutions and their dependence on the parameters of the polynomials pj(x). Special cases are related to known differenti...
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We prove that it is relatively consistent with ZFC that in any perfect Polish space, for every nonmeager set A there exists a nowhere dense Cantor set C such that A boolean AND C is nonmeager in C. We also examine variants of this result and establish a measure theoretic analog.
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We construct an irreducible multiplicative semigroup of non-negative square-zero operators acting on L-P[0, 1), for 1 less than or equal to p < infinity.
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Let P be a poset, consisting of all sets X subset of or equal to [n] = {1, 2, ..., n} which contain at least one of a given collection F of 2-subsets of [n], ordered by inclusion. By modifying a construction of Greene and Kleitman, we show that if F is hamiltonian, that is, contains {1, 2}, {2, 3}, ...
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The asymptotic iteration method is used to find exact and approximate solutions of Schrödinger’s equation for a number of one-dimensional trigonometric potentials (sine-squared, double-cosine, tangent-squared, and complex cotangent). Analytic and approximate solutions are obtained by first using ...
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Recursive procedures are found for calculating the kth-order entropy of non-periodic sequences. These entropies are used as measures of disorder of the sequences and we treat: the Fibonacci sequence and generalizations; the Thue-Morse sequence together with generalizations and the period-doubling se...
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Shannon entropy for the position and momentum eigenstates of an asymmetric trigonometric Rosen–Morse potential for the ground and first excited states is evaluated. The position and momentum information entropies Sx and Sp are calculated numerically. Also, we find that S1 x is obtained analyticall...