Related papers: A Dual-Radix Modular Division Algorithm for Comput…
We investigate numerically complex dynamical systems where a fixed point is surrounded by a disk or ball of quasiperiodic orbits, where there is a change of variables (or conjugacy) that converts the system into a linear map. We compute…
We investigate the evolution of families of periodic orbits in a bisymmetrical potential made up of a two-dimensional harmonic oscillator with only one quartic perturbing term, in a number of resonant cases. Our main objective is to compute…
We develop a method to evaluate integrals of non-holomorphic modular functions over the fundamental domain of the torus with modular parameter $\tau$ analytically. It proceeds in two steps: first the integral is transformed to a Lorentzian…
We present a quantum algorithm based on repeated measurement to solve initial-value problems for nonlinear ordinary differential equations (ODEs), which may be generated from partial differential equations in plasma physics. We map a…
This paper presents two modular grad-div algorithms for calculating solutions to the Navier-Stokes equations (NSE). These algorithms add to an NSE code a minimally intrusive module that implements grad-div stabilization. The algorithms do…
The main problem is to understand and to find periodic symmetric orbits in the $n$-body problem, in the sense of finding methods to prove or compute their existence, and more importantly to describe their qualitative and quantitative…
Classical (maximal) superintegrable systems in $n$ dimensions are Hamiltonian systems with $2n-1$ independent constants of the motion, globally defined, the maximum number possible. They are very special because they can be solved…
The derivation of the Hardy-Ramanujan-Rademacher formula for the number of partitions of $n$ is reviewed. Next, the steps for finding analogous formulas for certain restricted classes of partitions or overpartiitons is examined, bearing in…
Almost block diagonal linear systems of equations can be exemplified by two modules. This makes it possible to construct all sequential forms of band and/or block elimination methods, six old and fourteen new. It allows easy assessment of…
A (closed) dynamical system is a notion of how things can be, together with a notion of how they may change given how they are. The idea and mathematics of closed dynamical systems has proven incredibly useful in those sciences that can…
We develop an approach to finding upper bounds for the number of arithmetic operations necessary for doing harmonic analysis on permutation modules of finite groups. The approach takes advantage of the intrinsic orbital structure of…
This paper studies the distribution of characteristic multipliers, the structure of submanifolds, the phase diagram, bifurcations and chaotic motions in the potential field of rotating highly irregular-shaped celestial bodies (hereafter…
We establish the existence of non-stationary solutions to a symmetric system of second-order autonomous differential equations. Our technique is based on the equivariant degree theory and involves a novel characterization of orbit types of…
Consider a cohomologically hyperbolic birational self-map defined over the algebraic numbers, for example, a birational self-map in dimension two with the first dynamical degree greater than one, or in dimension three with the first and the…
We present a non-canonically symplectic integration scheme tailored to numerically computing the post-Newtonian motion of a spinning black-hole binary. Using a splitting approach we combine the flows of orbital and spin contributions. In…
In the present article, we investigate the properties of motion in Hamiltonian systems of two and three degrees of freedom, using the distribution of the values of two new dynamical parameters. The distribution functions of the new…
The purpose of this paper is to prove a new, more general version of the Morse index theorem for heteroclinic, homoclinic, and half-clinic solutions in general Lagrangian systems. In the final section, we compute the Morse index for…
We prove the existence of an effective universal upper bound for the order of any integral periodic orbit of any integral algebraic dynamical system in a fixed ambient space. Using this, we demonstrate the decidability of periodicity in…
We develop algorithms to turn quotients of rings of rings of integers into effective Euclidean rings by giving polynomial algorithms for all fundamental ring operations. In addition, we study normal forms for modules over such rings and…
Dual numbers and their higher order version are important tools for numerical computations, and in particular for finite difference calculus. Based upon the relevant algebraic rules and matrix realizations of dual numbers, we will present a…