We introduce a Random Energy Model on a hierarchical lattice where the interaction strength between variables is a decreasing function of their mutual hierarchical distance, making it a nonmean field model. Through small coupling series expansion and a direct numerical solution of the model, we provide evidence for a spin glass condensation transition similar to the one occurring in the usual m...

Selection of some of the numerous series expansion involving the famous constant π.

We show how to find series expansions for π of the form π = ∞ n=0 S(n) mn pn a n , where S(n) is some polynomial in n (depending on m, p, a). We prove that there exist such expansions for m = 8k, p = 4k, a = (−4) k , for any k, and give explicit examples for such expansions for small values of m, p and a.

The purpose of this paper is to initiate the study of a new kind of asymptotic series expansion for solutions of differential equations containing a parameter. We obtain uniform asymptotic solutions for certain equations of the form ê»y" = ait, e)y , ( )' = d/dt, where n is a positive integer, t and e are real variables ranging over \t\ S U, 0 < e ^ €o, and a is a function infinitely differenti...

We define an invariant of a multi-valued logic gate by considering the number of certain threshold functions associated with the gate. We call this invariant the continuous sensitivity of the gate. We discuss a method for analysing continuous sensitivity of a multi-valued logic gate by using experimental data about the gate. In particular, we will show that this invariant provides a lower bound...

The result was that, when guys at MIT or Princeton had trouble doing a certain integral, it was because they couldn’t do it with the standard methods they had learned in school. If it was contour integration, they would have found it; if it was a simple series expansion, they would have found it. Then I come along and try differentiating under the integral sign, and often it worked. So I got a ...

The massless one-loop three-point integral is obtained in terms of associated Clausen functions. The expression is manifestly symmetric in the three external variables. The main 'features of associated Clausen functions and their series expansions are presented. We also introduce the L-functions as analytical extensions of the associated Clausen functions.

In this paper, we present an estimation procedure which uses both option prices and highfrequency spot price feeds to estimate jointly the objective and risk-neutral parameters of stochastic volatility models. The procedure is based on a method of moments that uses analytical expressions for the moments of the integrated volatility and series expansions of option prices and implied volatilities...

Power series expansions for the even and odd angular Mathieu functions Sem(h, cos θ) and Som(h, cos θ), with small argument h, are derived for general integer values of m. The expansion coefficients that we evaluate are also useful for the calculation of the corresponding radial functions of any kind.

We survey algorithms for computing isogenies between elliptic curves defined over a field of characteristic either 0 or a large prime. We introduce a new algorithm that computes an isogeny of degree ` (` different from the characteristic) in time quasi-linear with respect to `. This is based in particular on fast algorithms for power series expansion of the Weierstrass ℘-function and related fu...