A Lagrange series approach to the spectrum of the Kite

A Lagrange series around adjustable expansion points to compute the eigenvalues of graphs, whose characteristic polynomial is analytically known, is presented. The computations for the kite graph P_nK_m, whose largest eigenvalue was studied by Stevanovic and Hansen [D. Stevanovic and P. Hansen. The minimum spectral radius of graphs with a given clique number. Electronic Journal of Linear Algebra, 17:110–117, 2008.], are illustrated. It is found that the first term in the Lagrange series already leads to a better approximation than previously published bounds.

On generalizations of the series of Taylor, Lagrange, Laurent and Teixeira

The classical theorems of Taylor, Lagrange, Laurent and Teixeira, are extended from the representation of a complex functionF(z), to its derivativeF(ν)(z)of complex orderν, understood as either a ‘Liouville’ (1832) or a ‘Rieman (1847)’ differintegration (Campos 1984, 1985); these results are distinct from, and alternative to, other extensions of Taylor's series using differintegrations (Osler 1972, Lavoie & Osler & Tremblay 1976). We consider a complex functionF(z), which is analytic (has an isolated singularity) atζ, and expand its derivative of complex orderF(ν)(z), in an ascending (ascending-descending) series of powers of an auxiliary functionf(z), yielding the generalized Teixeira (Lagrange) series, which includes, forf(z)=z−ζ, the generalized Taylor (Laurent) series. The generalized series involve non-integral powers and/or coefficients evaluated by fractional derivatives or integrals, except in the caseν=0, when the classical theorems of Taylor (1715), Lagrange (1770), Laurent (1843) and Teixeira (1900) are regained. As an application, these generalized series can be used to generate special functions with complex parameters (Campos 1986), e.g., the Hermite and Bessel types.

Formule d'Inversion de Lagrange et Son Application a La Theorie Des Perturbations

The article studies the application of the Lagrange series inversion formula to perturbation theories. One of the main problems is to generalize the Lagrange formula to several variables. The article also shows the need for introducing some operators to simplify the notation. To terminate, an application to the Von Zeipel-Brouwer satellite theory is given.

The transient behaviour of the queueing system Gi/M/1

SUMMARYWe consider a single server queue for which the interarrival times are identically and independently distributed with distribution function A(x) and whose service times are distributed independently of each other and of the interarrival times with distribution function B(x) = 1 − e−x, x ≧ 0. We suppose that the system starts from emptiness and use the results of P. D. Finch [2] to derive an explicit expression for qnj, the probability that the (n + 1)th arrival finds more than j customers in the system. The special cases M/M/1 and D/M/1 are considerend and it is shown in the general case that qnj is a partial sum of the usual Lagrange series for the limiting probability .