Implementation of the Combined--Nonlinear Condensation Transformation

We discuss several applications of the recently proposed combined nonlinear-condensation transformation (CNCT) for the evaluation of slowly convergent, nonalternating series. These include certain statistical distributions which are of importance in linguistics, statistical-mechanics theory, and biophysics (statistical analysis of DNA sequences). We also discuss applications of the transformation in experimental mathematics, and we briefly expand on further applications in theoretical physics. Finally, we discuss a related Mathematica program for the computation of Lerch's transcendent.Comment: 23 pages, 1 table, 1 figure (Comput. Phys. Commun., in press

Combinatorial and Stochastic Approach to Parallelization of the Kangaroo Method of Solving the Discrete Logarithm Problem

The kangaroo method for the Pollard\u27s rho algorithm provides a powerful way to solve discrete log problems. There exist parameters for it that allow it to be optimized in such a way as to prevent what are known as useless collisions in exchange for the limitation that the number of parallel resources used must be both finite and known ahead of time. This thesis puts forward an analysis of the situation and examines the potential acceleration that can be gained through the use of parallel resources beyond those initially utilized by an algorithm so configured. In brief, the goal in doing this is to reconcile the rapid rate of increase in parallel processing capabilities present in consumer level hardware with the still largely sequential nature of a large portion of the algorithms used in the software that is run on that hardware. The core concept, then, would be to allow spare parallel resources to be utilized in an advanced sort of guess-and-check to potentially produce occasional speedups whenever, for lack of a better way to put it, those guesses are correct. The methods presented in this thesis are done so with an eye towards expanding and reapplying them to this broadly expressed problem, however herein the discrete log problem has been chosen to be utilized as a suitable example of how such an application can proceed. This is primarily due to the observation that Pollard\u27s parameters for the avoidance of so-called useless collisions generated from the kangaroo method of solving said problem are restrictive in the number of kangaroos used at any given time. The more relevant of these restrictions to this point is the fact that they require the total number of kangaroos to be odd. Most consumer-level hardware which provides more than a single computational core provides an even number of such cores, so as a result it is likely the utilization of such hardware for this purpose will leave one or more cores idle. While these idle compute cores could also potentially be utilized for other tasks given that we are expressly operating in the context of consumer-level hardware, such considerations are largely outside the scope of this thesis. Besides, with the rate of change consumer computational hardware and software environments have historically changed it seems to be more useful to address the topic on a more purely algorithmic level; at the very least, it is more efficient as less effort needs to be expended future-proofing this thesis against future changes to its context than might have otherwise been necessary

Scaling law in the Standard Map critical function. Interpolating hamiltonian and frequency map analysis

We study the behaviour of the Standard map critical function in a neighbourhood of a fixed resonance, that is the scaling law at the fixed resonance. We prove that for the fundamental resonance the scaling law is linear. We show numerical evidence that for the other resonances $p/q$, $q \geq 2$, $p \neq 0$ and $p$ and $q$ relatively prime, the scaling law follows a power--law with exponent $1/q$.Comment: AMS-LaTeX2e, 29 pages with 8 figures, submitted to Nonlinearit

Exact scaling functions for one-dimensional stationary KPZ growth

We determine the stationary two-point correlation function of the one-dimensional KPZ equation through the scaling limit of a solvable microscopic model, the polynuclear growth model. The equivalence to a directed polymer problem with specific boundary conditions allows one to express the corresponding scaling function in terms of the solution to a Riemann-Hilbert problem related to the Painleve II equation. We solve these equations numerically with very high precision and compare our, up to numerical rounding exact, result with the prediction of Colaiori and Moore [1] obtained from the mode coupling approximation.Comment: 24 pages, 6 figures, replaced with revised versio

On Buffon Machines and Numbers

The well-know needle experiment of Buffon can be regarded as an analog (i.e., continuous) device that stochastically "computes" the number 2/pi ~ 0.63661, which is the experiment's probability of success. Generalizing the experiment and simplifying the computational framework, we consider probability distributions, which can be produced perfectly, from a discrete source of unbiased coin flips. We describe and analyse a few simple Buffon machines that generate geometric, Poisson, and logarithmic-series distributions. We provide human-accessible Buffon machines, which require a dozen coin flips or less, on average, and produce experiments whose probabilities of success are expressible in terms of numbers such as, exp(-1), log 2, sqrt(3), cos(1/4), aeta(5). Generally, we develop a collection of constructions based on simple probabilistic mechanisms that enable one to design Buffon experiments involving compositions of exponentials and logarithms, polylogarithms, direct and inverse trigonometric functions, algebraic and hypergeometric functions, as well as functions defined by integrals, such as the Gaussian error function.Comment: Largely revised version with references and figures added. 12 pages. In ACM-SIAM Symposium on Discrete Algorithms (SODA'2011

The Dynamical Functional Particle Method for Multi-Term Linear Matrix Equations

Recent years have seen a renewal of interest in multi-term linear matrix equations, as these have come to play a role in a number of important applications. Here, we consider the solution of such equations by means of the dynamical functional particle method, an iterative technique that relies on the numerical integration of a damped second order dynamical system. We develop a new algorithm for the solution of a large class of these equations, a class that includes, among others, all linear matrix equations with Hermitian positive definite or negative definite coefficients. In numerical experiments, our MATLAB implementation outperforms existing methods for the solution of multi-term Sylvester equations. For the Sylvester equation AX + XB = C, in particular, it can be faster and more accurate than the built-in implementation of the Bartels–Stewart algorithm, when A and B are well conditioned and have very different size