Qualitative Symbolic Perturbation: a new geometry-based perturbation framework

In a classical Symbolic Perturbation scheme,degeneracies are handled by substituting some polynomials in$\varepsilon$ to the input of a predicate. Instead of a singleperturbation, we propose to use a sequence of (simpler)perturbations. Moreover, we look at their effects geometricallyinstead of algebraically; this allows us to tackle cases that werenot tractable with the classical algebraic approach.Avec les méthodes de perturbations symboliques classiques,les dégénérescences sont résolues en substituant certains polynômes en $\varepsilon$ aux entrées du prédicat.Au lieu d'une seule perturbation compliquée, nous proposons d'utiliser unesuite de perturbation plus simple. Et nous regardons les effets deces perturbations géométriquement plutôt qu'algébriquementce qui permet de traiter des cas inatteignables par les méthodesalgébriques classiques

Numerical Evidence that the Perturbation Expansion for a Non-Hermitian PT\mathcal{PT}-Symmetric Hamiltonian is Stieltjes

Recently, several studies of non-Hermitian Hamiltonians having $\mathcal{PT}$ symmetry have been conducted. Most striking about these complex Hamiltonians is how closely their properties resemble those of conventional Hermitian Hamiltonians. This paper presents further evidence of the similarity of these Hamiltonians to Hermitian Hamiltonians by examining the summation of the divergent weak-coupling perturbation series for the ground-state energy of the $\mathcal{PT}$-symmetric Hamiltonian $H=p^2+{1/4}x^2+i\lambda x^3$ recently studied by Bender and Dunne. For this purpose the first 193 (nonzero) coefficients of the Rayleigh-Schr\"odinger perturbation series in powers of $\lambda^2$ for the ground-state energy were calculated. Pad\'e-summation and Pad\'e-prediction techniques recently described by Weniger are applied to this perturbation series. The qualitative features of the results obtained in this way are indistinguishable from those obtained in the case of the perturbation series for the quartic anharmonic oscillator, which is known to be a Stieltjes series.Comment: 20 pages, 0 figure

A Symbolic Algorithm for the Computation of Periodic Orbits in Non–Linear Differential Systems

The Poincaré–Lindstedt method in perturbation theory is used to compute periodic solutions in perturbed differential equations through a nearby periodic orbit of the unperturbed problem. The adaptation of this technique to systems of differential equations of first order could produce meaningful advances in the qualitative analysis of many dynamical systems. In this paper, we present a new symbolic algorithm as well as a new symbolic computation tool to calculate periodic solutions in systems of differential equations of first order. The algorithm is based on an optimized adaptation of the Poincaré–Lindstedt technique to differential systems. This algorithm is applied to compute a periodic solution in a Lotka–Volterra system

Symbolic-numeric interface: A review

A survey of the use of a combination of symbolic and numerical calculations is presented. Symbolic calculations primarily refer to the computer processing of procedures from classical algebra, analysis, and calculus. Numerical calculations refer to both numerical mathematics research and scientific computation. This survey is intended to point out a large number of problem areas where a cooperation of symbolic and numerical methods is likely to bear many fruits. These areas include such classical operations as differentiation and integration, such diverse activities as function approximations and qualitative analysis, and such contemporary topics as finite element calculations and computation complexity. It is contended that other less obvious topics such as the fast Fourier transform, linear algebra, nonlinear analysis and error analysis would also benefit from a synergistic approach

Evaluation of stochastic effects on biomolecular networks using the generalised Nyquist stability criterion

Abstract—Stochastic differential equations are now commonly used to model biomolecular networks in systems biology, and much recent research has been devoted to the development of methods to analyse their stability properties. Stability analysis of such systems may be performed using the Laplace transform, which requires the calculation of the exponential matrix involving time symbolically. However, the calculation of the symbolic exponential matrix is not feasible for problems of even moderate size, as the required computation time increases exponentially with the matrix order. To address this issue, we present a novel method for approximating the Laplace transform which does not require the exponential matrix to be calculated explicitly. The calculation time associated with the proposed method does not increase exponentially with the size of the system, and the approximation error is shown to be of the same order as existing methods. Using this approximation method, we show how a straightforward application of the generalized Nyquist stability criterion provides necessary and sufficient conditions for the stability of stochastic biomolecular networks. The usefulness and computational efficiency of the proposed method is illustrated through its application to the problem of analysing a model for limit-cycle oscillations in cAMP during aggregation of Dictyostelium cells

Tridiagonal PT-symmetric N by N Hamiltonians and a fine-tuning of their observability domains in the strongly non-Hermitian regime

A generic PT-symmetric Hamiltonian is assumed tridiagonalized and truncated to N dimensions, and its up-down symmetrized special cases with J=[N/2] real couplings are considered. In the strongly non-Hermitian regime the secular equation gets partially factorized at all N. This enables us to reveal a fine-tuned alignment of the dominant couplings implying an asymptotically sharply spiked shape of the boundary of the J-dimensional quasi-Hermiticity domain in which all the spectrum of energies remains real and observable.Comment: 28 pp., 4 tables, 1 figur

Parking a Spacecraft near an Asteroid Pair

This paper studies the dynamics of a spacecraft moving in the field of a binary asteroid. The asteroid pair is modeled as a rigid body and a sphere moving in a plane, while the spacecraft moves in space under the influence of the gravitational field of the asteroid pair, as well as that of the sun. This simple model captures the coupling between rotational and translational dynamics. By assuming that the binary dynamics is in a relative equilibrium, a restricted model for the spacecraft in orbit about them is constructed that also includes the direct effect of the sun on the spacecraft dynamics. The standard restricted three-body problem (RTBP) is used as a starting point for the analysis of the spacecraft motion. We investigate how the triangular points of the RTBP are modified through perturbations by taking into account two perturbations, namely, that one of the primaries is no longer a point mass but is an extended rigid body, and second, taking into account the effect of orbiting the sun. The stable zones near the modified triangular equilibrium points of the binary and a normal form of the Hamiltonian around them are used to compute stable periodic and quasi-periodic orbits for the spacecraft, which enable it to observe the asteroid pair while the binary orbits around the sun

Spatial Aggregation: Theory and Applications

Visual thinking plays an important role in scientific reasoning. Based on the research in automating diverse reasoning tasks about dynamical systems, nonlinear controllers, kinematic mechanisms, and fluid motion, we have identified a style of visual thinking, imagistic reasoning. Imagistic reasoning organizes computations around image-like, analogue representations so that perceptual and symbolic operations can be brought to bear to infer structure and behavior. Programs incorporating imagistic reasoning have been shown to perform at an expert level in domains that defy current analytic or numerical methods. We have developed a computational paradigm, spatial aggregation, to unify the description of a class of imagistic problem solvers. A program written in this paradigm has the following properties. It takes a continuous field and optional objective functions as input, and produces high-level descriptions of structure, behavior, or control actions. It computes a multi-layer of intermediate representations, called spatial aggregates, by forming equivalence classes and adjacency relations. It employs a small set of generic operators such as aggregation, classification, and localization to perform bidirectional mapping between the information-rich field and successively more abstract spatial aggregates. It uses a data structure, the neighborhood graph, as a common interface to modularize computations. To illustrate our theory, we describe the computational structure of three implemented problem solvers -- KAM, MAPS, and HIPAIR --- in terms of the spatial aggregation generic operators by mixing and matching a library of commonly used routines.Comment: See http://www.jair.org/ for any accompanying file