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This is a brief survey of certain constants associated with random lattice models, including self-avoiding walks, polyominoes, the Lenz-Ising model, monomers and dimers, ice models, hard squares and hexagons, and percolation models. 1. Introduction Random lattice models give rise to combinatorial problems that are often easily-stated but intractable. This(More)
A strictly increasing sequence of positive integers a-,, a 2 ? • • • is defined to be s-additive [1] if, for n > 2s, a n is the least integer greater than a n-\ having precisely s representations a^ + a.j = a n , i < j. The first 2s terms of an s-additive sequence are called the base of the sequence. An s-additive sequence, for a given base, may" be either(More)
When searching for a planar line, if given no further information , one should adopt a logarithmic spiral strategy (although unproven). This brief paper is concerned entirely with geometry in the plane and continues a thought in [1]. If a line intersects a circle in one or two points, we say that the line strikes the circle. If a line intersects a circle in(More)
Given a positive integer n, let P(n) denote the largest prime factor of n and S(n) denote the smallest integer m such that n divides m! The function S(n) is known as the Smarandache function and has been intensively studied [1]. Its behavior is quite erratic [2] and thus all we can reasonably hope for is a statistical approximation of its growth, e.g., an(More)
Given a positive integer n, let P(n) denote the largest prime factor of nand S(n) denote the smallest integer m such that n divides m! This paper extends earlier work [1] on the average value of the Smarandache function S(n) and is based on a recent asymptotic result [2]: (W J I{n ~ N:P(n) < S(n)}j = 01 • I ~ln(N)J for any positive integer j due to Ford. We(More)