The phase transition in random catalytic sets

The notion of (auto) catalytic networks has become a cornerstone in understanding the possibility of a sudden dramatic increase of diversity in biological evolution as well as in the evolution of social and economical systems. Here we study catalytic random networks with respect to the final outcome diversity of products. We show that an analytical treatment of this longstanding problem is possible by mapping the problem onto a set of non-linear recurrence equations. The solution of these equations show a crucial dependence of the final number of products on the initial number of products and the density of catalytic production rules. For a fixed density of rules we can demonstrate the existence of a phase transition from a practically unpopulated regime to a fully populated and diverse one. The order parameter is the number of final products. We are able to further understand the origin of this phase transition as a crossover from one set of solutions from a quadratic equation to the other.Comment: 7 pages, ugly eps files due to arxiv restriction

Solving Vlasov Equations Using NRxx Method

In this paper, we propose a moment method to numerically solve the Vlasov equations using the framework of the NRxx method developed in [6, 8, 7] for the Boltzmann equation. Due to the same convection term of the Boltzmann equation and the Vlasov equation, it is very convenient to use the moment expansion in the NRxx method to approximate the distribution function in the Vlasov equations. The moment closure recently presented in [5] is applied to achieve the globally hyperbolicity so that the local well-posedness of the moment system is attained. This makes our simulations using high order moment expansion accessible in the case of the distribution far away from the equilibrium which appears very often in the solution of the Vlasov equations. With the moment expansion of the distribution function, the acceleration in the velocity space results in an ordinary differential system of the macroscopic velocity, thus is easy to be handled. The numerical method we developed can keep both the mass and the momentum conserved. We carry out the simulations of both the Vlasov-Poisson equations and the Vlasov-Poisson-BGK equations to study the linear Landau damping. The numerical convergence is exhibited in terms of the moment number and the spatial grid size, respectively. The variation of discretized energy as well as the dependence of the recurrence time on moment order is investigated. The linear Landau damping is well captured for different wave numbers and collision frequencies. We find that the Landau damping rate linearly and monotonically converges in the spatial grid size. The results are in perfect agreement with the theoretic data in the collisionless case

Algorithms for efficient vectorization of repeated sparse power system network computations

Cataloged from PDF version of article.Standard sparsity-based algorithms used in power system appllcations need to be restructured for efficient vectorization due to the extremely short vectors processed. Further, intrinsic architectural features of vector computers such as chaining and sectioning should also be exploited for utmost performance. This paper presents novel data storage schemes and vectorization alsorim that resolve the recurrence problem, exploit chaining and minimize the number of indirect element selections in the repeated solution of sparse linear system of equations widely encountered in various power system problems. The proposed schemes are also applied and experimented for the vectorization of power mismatch calculations arising in the solution phase of FDLF which involves typical repeated sparse power network computations. The relative performances of the proposed and existing vectorization schemes are evaluated, both theoretically and experimentally on IBM 3090ArF.Standard sparsity-based algorithms used in power system appllcations need to be restructured for efficient vectorization due to the extremely short vectors processed. Further, intrinsic architectural features of vector computers such as chaining and sectioning should also be exploited for utmost performance. This paper presents novel data storage schemes and vectorization alsorim that resolve the recurrence problem, exploit chaining and minimize the number of indirect element selections in the repeated solution of sparse linear system of equations widely encountered in various power system problems. The proposed schemes are also applied and experimented for the vectorization of power mismatch calculations arising in the solution phase of FDLF which involves typical repeated sparse power network computations. The relative performances of the proposed and existing vectorization schemes are evaluated, both theoretically and experimentally on IBM 3090ArF

Some investigations into the numerical solution of initial value problems in ordinary differential equations

PhD ThesisIn this thesis several topics in the numerical solution of the initial value problem in first-order ordinary diff'erentlal equations are investigated, Consideration is given initially to stiff differential equations and their solution by stiffly-stable linear multistep methods which incorporate second derivative terms. Attempts are made to increase the size of the stability regions for these methods both by particular choices for the third characteristic polynomial and by the use of optimization techniques while investigations are carried out regarding the capabilities of a high order method. Subsequent work is concerned with the development of Runge-Kutta methods which include second-derivative terms and are implicit with respect to y rather than k. Methods of order three and four are proposed which are L-stable. The major part of the thesis is devoted to the establishment of recurrence relations for operators associated with linear multistep methods which are based on a non-polynomial representation of the theoretical solution. A complete set of recurrence relations is developed for both implicit and explicit multistep methods which are based on a representation involving a polynomial part and any number of arbitrary functions. The amount of work involved in obtaining mulc iste, :ne::l'lJds by this technique is considered and criteria are proposed to Jecide when this particular method of derivation should be em~loyed. The thesis is conclud~d by using Prony's method to develop one-step methods and multistep methods which are exponentially adaptive and as such can be useful in obtaining solutions to problems which are exponential in nature

A toolbox to solve coupled systems of differential and difference equations

We present algorithms to solve coupled systems of linear differential equations, arising in the calculation of massive Feynman diagrams with local operator insertions at 3-loop order, which do {\it not} request special choices of bases. Here we assume that the desired solution has a power series representation and we seek for the coefficients in closed form. In particular, if the coefficients depend on a small parameter \ep (the dimensional parameter), we assume that the coefficients themselves can be expanded in formal Laurent series w.r.t.\ \ep and we try to compute the first terms in closed form. More precisely, we have a decision algorithm which solves the following problem: if the terms can be represented by an indefinite nested hypergeometric sum expression (covering as special cases the harmonic sums, cyclotomic sums, generalized harmonic sums or nested binomial sums), then we can calculate them. If the algorithm fails, we obtain a proof that the terms cannot be represented by the class of indefinite nested hypergeometric sum expressions. Internally, this problem is reduced by holonomic closure properties to solving a coupled system of linear difference equations. The underlying method in this setting relies on decoupling algorithms, difference ring algorithms and recurrence solving. We demonstrate by a concrete example how this algorithm can be applied with the new Mathematica package \texttt{SolveCoupledSystem} which is based on the packages \texttt{Sigma}, \texttt{HarmonicSums} and \texttt{OreSys}. In all applications the representation in $x$-space is obtained as an iterated integral representation over general alphabets, generalizing Poincar\'{e} iterated integrals

Formulas for Continued Fractions. An Automated Guess and Prove Approach

We describe a simple method that produces automatically closed forms for the coefficients of continued fractions expansions of a large number of special functions. The function is specified by a non-linear differential equation and initial conditions. This is used to generate the first few coefficients and from there a conjectured formula. This formula is then proved automatically thanks to a linear recurrence satisfied by some remainder terms. Extensive experiments show that this simple approach and its straightforward generalization to difference and $q$-difference equations capture a large part of the formulas in the literature on continued fractions.Comment: Maple worksheet attache