Limit points for the positive eigenvalues of the normalized Laplacian matrix of a graph are considered. Specifically, it is shown that the set of limit points for the j-th smallest such eigenvalues is equal to [0, 1], while the set of limit points for the j-th largest such eigenvalues is equal to [1, 2]. Limit points for certain functions of the eigenvalues, motivated by considerations for rand...

The research described in this paper was motivated by an enigmatic entry in Ramanujan’s lost notebook [11, p. 45] in which he claimed, in an unorthodox fashion, that a certain q-continued fraction possesses three limit points. More precisely, he claimed that as n tends to ∞ in the three residue classes modulo 3, the nth partial quotients tend, respectively, to three distinct limits, which he ex...

Let M(P (z1, . . . , zn)) denote Mahler’s measure of the polynomial P (z1, . . . , zn). Measures of polynomials in n variables arise naturally as limiting values of measures of polynomials in fewer variables. We describe several methods for searching for polynomials in two variables with integer coefficients having small measure, demonstrate effective methods for computing these measures, and i...

Taking line tting to points in two dimensions as a typical example, we point out the inherent di erence between statistical inference and geometric tting. We describe their duality in the sense that the asymptotic properties of statistical inference in the limit of an in nite number of observations hold for geometric tting in the limit of in nitesimal perturbations. We contrast stochastic model...