The analysis of solutions to algebraic equations is further simplified. A couple of functions and their analytic continuation or root findings are required.

For an abelian group , a formula to compute the characteristic polynomial of a -graph has been obtained by Lee and Kim [Characteristic polynomials of graphs having a semi-free action, Linear algebra Appl. 307 (2005) 35–46]. As a continuation of this work, we give a computational formula for generalized characteristic polynomial of a -graph when is a finite group. Moreover, after showing that th...

= 1 1− q1−s , where we have convergence for all s with <(s) > 1. This automatically gives us analytic continuation of ζ to all of C. We note the following simple observations: 1. The Riemann hypothesis: All zeroes of ζ lie on the line <(s) = 1/2, simply because there are no zeroes! 2. ζ has poles at s = 1 + 2πin log q . 3. Locally around s = 1, ζ has the Laurent series expansion: 1 (s− 1) log q...

We develop an algorithm to find all solutions of a generic system in a family of polynomial systems with parametric coefficients using numerical homotopy continuation and the action of the monodromy group. We argue that the expected number of homotopy paths that this algorithm needs to follow is roughly linear in the number of solutions. We demonstrate that our software implementation is compet...

CMPSm is a MATLAB program using a predictor-corrector path-following method for tracing homotopy curves to approximate all isolated solutions of a system of polynomial equations. The program was originally developed for the cheater's homotopy, but it can be also used for linear and polyhedral homotopies. The software CMPSm, this manual and some numerical examples are available at

Speedup measures how much faster we can solve the same problem using many cores. If we can afford to keep the execution time fixed, then quality up measures how much better the solution will be computed using many cores. In this paper we describe our multithreaded implementation to track one solution path defined by a polynomial homotopy. Limiting quality to accuracy and confusing accuracy with...

We investigate transmission problems between a (thermo-)viscoelastic system with Kelvin-Voigt damping, and a purely elastic system. It is shown that neither the elastic damping by Kelvin-Voigt mechanisms nor the dissipative effect of the temperature in one material can assure the exponential stability of the total system when it is coupled through transmission to a purely elastic system. The ap...

Green functions associated to complex reflection groups G(e, 1, n) were discussed in the author’s previous paper. In this paper, we consider the case of complex reflection groups W = G(e, p, n). Schur functions and Hall-Littlewood functions associated to W are introduced, and Green functions are described as the transition matrix between those two symmetric functions. Furthermore, it is shown t...

In this paper, we extend the method proposed by Cochelin and Vergez in [A high order purely frequency based harmonic balance formulation for continuation of periodic solutions, Journal of Sound and Vibration, 324:243–262, 2009] to the case of non polynomial non-linearities. This extension allows for the computation of branches of periodic solutions of a broader class of nonlinear dynamical syst...

tion “General Arithmetic” (Allgemeine Arithmetik). In his view, arithmetic is the building block of the whole edifice of mathematics. The aim of this paper is to show that Kronecker’s arithmetical philosophy and mathematical practice have exerted a permanent influence on a long tradition of mathematicians from Hilbert to Weil, Grothendieck and Langlands. The conclusion hints at a constructivist...