Using a simple transfer matrix approach we have derived very long series expansions for the perimeter generating function of three-choice polygons. We find that all the terms in the generating function can be reproduced from a linear Fuchsian differential equation of order 8. We perform an analysis of the properties of the differential equation.

This paper explores links between basis construction methods in Markov decision processes and power series expansions of value functions. This perspective provides a useful framework to analyze properties of existing bases, as well as provides insight into constructing more effective bases. Krylov and Bellman error bases are based on the Neumann series expansion. These bases incur very large in...

We derive a measure of firm speed of price adjustment that is directly inversely related to market power and compare this to the measure derived by Martin (1993). However, both measures are incorrect when firms have non-zero price conjectural variations and treat competing price levels as exogenous. This is because Taylor series expansions of the demand function implicitly assume that firms inf...

The electromagnetic field that an overhead infinitely line current produces at the surface of the earth can be expressed as inverse Fourier integrals over a horizontal wave number in terms of the Neumann and Struve functions. These functions have known mathematical properties, including the series expansions. The latter are utilized in this work to derive expressions for the electromagnetic fie...

Suppose that K ⊂ U(n) is a compact Lie group acting on the (2n+1)dimensional Heisenberg group Hn. We say that (K,Hn) is a Gelfand pair if the convolution algebra LK(Hn) of integrable K-invariant functions on Hn is commutative. In this case, the Gelfand space ∆(K,Hn) is equipped with the GodementPlancherel measure, and the spherical transform ∧ : LK(Hn)→ L(∆(K,Hn)) is an isometry. The main resul...

The successful quasi-particle model is compared with recent lattice data of the coefficients in the Taylor series expansion of the excess pressure at finite temperature and baryon density. A chain of approximations, starting from QCD to arrive at the model expressions for the entropy density, is presented.

We consider a minimal scalar in the presence of a three-brane in ten dimensions. The linearized equation of motion, which is just the wave equation in the three-brane metric, can be solved in terms of associated Mathieu functions. An exact expression for the reflection and absorption probabilities can be obtained in terms of the characteristic exponent of Mathieu’s equation. We describe an algo...

We consider the question whether all the coefficients in the series expansions of some specific rational functions are positive, and we demonstrate how computer algebra can help answering questions arising in this context. By giving partial computer proofs, we provide new evidence in support of some longstanding open conjectures. Also two new conjectures are made.

Considerable work has been devoted to the question of how best to parameterize the properties of dark energy, in particular its equation of state w. We argue that, in the absence of a compelling model for dark energy, the parameterizations of functions about which we have no prior knowledge, such as w(z), should be determined by the data rather than by our ingrained beliefs or familiar series e...

We present a novel surface smoothing framework using the Laplace-Beltrami eigenfunctions. The Green’s function of an isotropic diffusion equation on a manifold is analytically represented using the eigenfunctions of the Laplace-Beltraimi operator. The Green’s function is then used in explicitly constructing heat kernel smoothing as a series expansion of the eigenfunctions. Unlike many previous ...