In this article, we move back almost 200 years to Christoph Gudermann, the great expert on elliptic functions, who successfully put the twelve Jacobi functions in a didactic setting. We prove the second hyperbolic series expansions for elliptic functions again, and express generalizations of many of Gudermann’s formulas in Carlson’s modern notation. The transformations between squares of ellipt...

Abundant research suggests that the study of lay theories helps to explain intergroup relations. Building on this work as well as ‘‘interactionist’’ theories for understanding social behaviour from developmental and social psychology, we propose an integrative social-developmental perspective examining how lay perceivers’ characteristics (e.g., age, race, psychological motivations) interact wit...

We give an algorithm to compute term by term multivariate Puiseux series expansions of series arising as local parametrizations of zeroes of systems of algebraic equations at singular points. The algorithm is an extension of Newton’s method for plane algebraic curves replacing the Newton polygon by the tropical variety of the ideal generated by the system.

The plane wave expansion method for calculating magneto-photonic crystals (MPCs) is modified via complex Fourier factorization whose basic principles are briefly explained. The new approach is used to simulate the photonic band structure of a twodimensional MPC made as Bi:YIG rods arranged with square periodicity. The considerable improvement of the convergence properties is demonstrated for se...

We develop the theory of Abelian functions defined using a tetragonal curve of genus six, discussing in detail the cyclic curve y 4 = x 5 + λ 4 x 4 + λ 3 x 3 + λ 2 x 2 + λ 1 x + λ 0. We construct Abelian functions using the multivariate σ-function associated to the curve, generalising the theory of the Weierstrass ℘-function. We demonstrate that such functions can give a solution to the KP-equa...

On the basis of Maxwell's equations and a plane wave expansion method, photonic band structure is computed for 3D chalcogenide photonic crystal with diamond lattice. The geometrical influence on the photonic bandgap is studied for three dimensional chalcogenide photonic crystal of diamond lattice for both chalcogenide spheres in air and air spheres in chalcogenide background. The air spheres in...

T. Erdélyi, A.P. Magnus and P. Nevai conjectured that for α, β ≥ − 1 2 , the orthonormal Jacobi polynomials P (α,β) k (x) satisfy the inequality max x∈[−1,1] (1− x) 1 2 (1 + x) 1 2 ( P (α,β) k (x) )2 = O (

In this paper, we extend the modulating functions method to estimate the state and the unknown input of a linear time-varying system defined by a linear differential equation. We first estimate the unknown input by taking a truncated Jacobi orthogonal series expansion with unknown coefficients which can be estimated by the modulating functions method. Then, we estimate the state by using extend...

Kautz series allow orthogonal series expansion of finiteenergy signals defined on a semi-infinite axis. The Kautz series consists of orthogonalized exponential functions or sequences. This series has as free parameters an ordered set of poles, each pole associated with an exponential function or sequence. For reasons of approximation and compact representation (coding), an appropriate set of or...

significance and objectives. A practical problem when integrating systems of linear ordinary differential equations, is that if we wish to sample the solution for a different value of an inherent parameter, then we must re-integrate. This is particularly inefficient when we want to accurately sample the solution over a continuous widespread set of parameter values. Though continuity methods res...