Polynomial Equations: Theory and Practice

Solving polynomial equations is a subtask of polynomial optimization. This article introduces systems of such equations and the main approaches for solving them. We discuss critical point equations, algebraic varieties, and solution counts. The theory is illustrated by many examples using different software packages.Comment: This article will appear as a chapter of a forthcoming book presenting research acitivies conducted in the European Network POEMA. It discusses polynomial equations, with optimization as point of entry. 24 pages, 7 figure

Computer Science for Continuous Data:Survey, Vision, Theory, and Practice of a Computer Analysis System

Building on George Boole's work, Logic provides a rigorous foundation for the powerful tools in Computer Science that underlie nowadays ubiquitous processing of discrete data, such as strings or graphs. Concerning continuous data, already Alan Turing had applied "his" machines to formalize and study the processing of real numbers: an aspect of his oeuvre that we transform from theory to practice.The present essay surveys the state of the art and envisions the future of Computer Science for continuous data: natively, beyond brute-force discretization, based on and guided by and extending classical discrete Computer Science, as bridge between Pure and Applied Mathematics

Particle Swarm Optimization of Low-Thrust, Geocentric-to-Halo-Orbit Transfers

Missions to Lagrange points are becoming increasingly popular amongst spacecraft mission planners. Lagrange points are locations in space where the gravity force from two bodies, and the centrifugal force acting on a third body, cancel. To date, all spacecraft that have visited a Lagrange point have done so using high-thrust, chemical propulsion. Due to the increasing availability of low-thrust (high efficiency) propulsive devices, and their increasing capability in terms of fuel efficiency and instantaneous thrust, it has now become possible for a spacecraft to reach a Lagrange point orbit without the aid of chemical propellant. While at any given time there are many paths for a low-thrust trajectory to take, only one is optimal. The traditional approach to spacecraft trajectory optimization utilizes some form of gradient-based algorithm. While these algorithms offer numerous advantages, they also have a few significant shortcomings. The three most significant shortcomings are: (1) the fact that an initial guess solution is required to initialize the algorithm, (2) the radius of convergence can be quite small and can allow the algorithm to become trapped in local minima, and (3) gradient information is not always assessable nor always trustworthy for a given problem. To avoid these problems, this dissertation is focused on optimizing a low-thrust transfer trajectory from a geocentric orbit to an Earth-Moon, L1, Lagrange point orbit using the method of Particle Swarm Optimization (PSO). The PSO method is an evolutionary heuristic that was originally written to model birds swarming to locate hidden food sources. This PSO method will enable the exploration of the invariant stable manifold of the target Lagrange point orbit in an effort to optimize the spacecraft\u27s low-thrust trajectory. Examples of these optimized trajectories are presented and contrasted with those found using traditional, gradient-based approaches. In summary, the results of this dissertation find that the PSO method does, indeed, successfully optimize the low-thrust trajectory transfer problem without the need for initial guessing. Furthermore, a two-degree-of-freedom PSO problem formulation significantly outperformed a one-degree-of-freedom formulation by at least an order of magnitude, in terms of CPU time. Finally, the PSO method is also used to solve a traditional, two-burn, impulsive transfer to a Lagrange point orbit using a hybrid optimization algorithm that incorporates a gradient-based shooting algorithm as a pre-optimizer. Surprisingly, the results of this study show that fast transfers outperform slow transfers in terms of both delta-V and time of flight

Q(sqrt(-3))-Integral Points on a Mordell Curve

We use an extension of quadratic Chabauty to number fields,recently developed by the author with Balakrishnan, Besser and M ̈uller,combined with a sieving technique, to determine the integral points overQ(√−3) on the Mordell curve y2 = x3 − 4

Computation and Physics in Algebraic Geometry

Physics provides new, tantalizing problems that we solve by developing and implementing innovative and effective geometric tools in nonlinear algebra. The techniques we employ also rely on numerical and symbolic computations performed with computer algebra. First, we study solutions to the Kadomtsev-Petviashvili equation that arise from singular curves. The Kadomtsev-Petviashvili equation is a partial differential equation describing nonlinear wave motion whose solutions can be built from an algebraic curve. Such a surprising connection established by Krichever and Shiota also led to an entirely new point of view on a classical problem in algebraic geometry known as the Schottky problem. To explore the connection with curves with at worst nodal singularities, we define the Hirota variety, which parameterizes KP solutions arising from such curves. Studying the geometry of the Hirota variety provides a new approach to the Schottky problem. We investigate it for irreducible rational nodal curves, giving a partial solution to the weak Schottky problem in this case. Second, we formulate questions from scattering amplitudes in a broader context using very affine varieties and D-module theory. The interplay between geometry and combinatorics in particle physics indeed suggests an underlying, coherent mathematical structure behind the study of particle interactions. In this thesis, we gain a better understanding of mathematical objects, such as moduli spaces of point configurations and generalized Euler integrals, for which particle physics provides concrete, non-trivial examples, and we prove some conjectures stated in the physics literature. Finally, we study linear spaces of symmetric matrices, addressing questions motivated by algebraic statistics, optimization, and enumerative geometry. This includes giving explicit formulas for the maximum likelihood degree and studying tangency problems for quadric surfaces in projective space from the point of view of real algebraic geometry

Dynamics of model-reference and hill climbing systems

In this thesis the theory of both linear differential equations with periodic coefficients and linear differential equations with random coefficients is applied to investigate the stability and accuracy of parameter adaptation of sinusoidal perturbation and model reference adaptive control systems. Throughout dimensional analysis applied so that all the results are presented in a non-dimensional form. The first part of the thesis t s devoted to investigating the stability of such differential equations. In chapter 1 a system of linear homogeneous differential equations with periodic coefficients is considered and a numerical procedure, based on Floquet theory and well suited for use on a digital computer, is presented for obtaining necessary and sufficient conditions for asymptotic stability of the null solution. Also considered in this chapter is the so called infinite determinant method of obtaining the stability boundaries for a restricted class of linear differential equations with periodic coefficients. Chapter 2 is devoted to reviewing the current state of the stability theory of linear differential equations with random coefficients. In chapter 3 a theoretical analysis of the stability and accuracy of parameter adaptation of a single input, sinusoidal perturbation, extremum control system with output lag is considered. Using the principle of harmonic balance it is shown that various stable harmonic and sub-harmonic steady state solutions are possible in certain regions of the parameter space. By examining the domains of attraction, corresponding to the stable solutions, regions in three dimensional space are obtained within .which initial conditions will lead to a given steady state stable oscillation. It is also shown that the subharmonic steady state solutions do not correspond to the optimum solution, so that, for certain initial conditions and parameter values, it is possible for the system to reach a steady state solution which is not the optimum solution. All the theoretical results are verified by direct analogue computer simulation of the system. The remainder of the thesis is devoted to investigating the stability and accuracy of parameter adapt.at ion of model reference adaptive control systems. In order to develop a mathematical analysis, and to illustrate the difficulties involved, a stability analysis of a first order M.I.T. type system with controllable gain, when the input varies with time in both a periodic and random manner, is first carried out. Also considered are the effects of (a) random disturbances at the system output (b) and (b) periodic and random variations, with time, of the controlled process environmental parameters, on the stability of the system and the accuracy of its parameter adaptation. When the input varies sinusoidally with time stability boundaries are obtained using both a numerical implementation of Floquet theory and the infinite determinant method; the relative merits of the two methods is discussed. The theoretical results are compared with stability boundaries obtained by analogue computer simulation of the system. It is shown that the stability boundaries are complex in nature and that some knowledge of such boundaries is desirable before embarking on an analogue computer investigation of the system. When the input varies randomly with time the stability problem reduces to one of investigating the stability of a system of linear differential equations with random coefficients. Both the theory of Markov processes, involving use of the Fokker-Planck equation, and the second method of Liapunov are used to investigate the problem; limitations and difficulty of applications of the theory is discussed. The theoretical results obtained are compared with those obtained by digital simulation of the system. If the controlled process environmental parameter is allowed to become time varying then it is shown that this effects both the stability of the system and the accuracy of its parameter adaptation. Theoretical results are obtained for the cases of the parameter varying both sinusoidally and randomly with time; some of the results are compared with those obtained by digital simulation of the system. It is also shown that noise disturbance at the system output has no effect on the system stability but does effect the accuracy of the parameter adaptation. The doubts concerning the stability and the difficulty of analysis of the M.I.T. , type system have led.:researchers to think about redesigning the model reference system from the point of view of stability. In particular we have the Liapunov synthesis method where the resulting system is guaranteed stable for all possible inputs. However, in designing such systems the controlled process environmental parameters are assumed constant and, by considering the Liapunov redesign scheme of the first order M.I.T. system previously discussed, it will be shown that the effect of making such parameters time varying is to introduce a stability problem. In chapter 6 the methods developed for analysing the first order system are extended to examine the stability of a higher order M.I.T. type system. The system considered has a third order process and a second order model and a stability analysis is presented for both sinusoidal and random input. Steady state values of the adapting parameters are first obtained and the linearized variational equations , for small disturbances about such steady states, examined to answer the stability problem. Theoretical results are compared with those obtained by direct analogue computer simulation of the system. The effect, on the mathematical analysis , of replacing the system multipliers by diode switching units is also considered in this chapter. The chapter concludes by presenting a method of obtaining a Liapunov redesign scheme for the system under discussion

Symmetries of Riemann surfaces and magnetic monopoles

This thesis studies, broadly, the role of symmetry in elucidating structure. In particular, I investigate the role that automorphisms of algebraic curves play in three specific contexts; determining the orbits of theta characteristics, influencing the geometry of the highly-symmetric Bring’s curve, and in constructing magnetic monopole solutions. On theta characteristics, I show how to turn questions on the existence of invariant characteristics into questions of group cohomology, compute comprehensive tables of orbit decompositions for curves of genus 9 or less, and prove results on the existence of infinite families of curves with invariant characteristics. On Bring’s curve, I identify key points with geometric significance on the curve, completely determine the structure of the quotients by subgroups of automorphisms, finding new elliptic curves in the process, and identify the unique invariant theta characteristic on the curve. With respect to monopoles, I elucidate the role that the Hitchin conditions play in determining monopole spectral curves, the relation between these conditions and the automorphism group of the curve, and I develop the theory of computing Nahm data of symmetric monopoles. As such I classify all 3-monopoles whose Nahm data may be solved for in terms of elliptic functions

Das Spektrum zeitverzögerter Differentialgleichungen: numerische Methoden, Stabilität und Störungstheorie

Three types of problems related to delay-differential equations (DDEs) are treated in this thesis. We first consider the problem of numerically computing the eigenvalues of a DDE. Here, we present an application of a projection method for nonlinear eigenvalue problems (NLEPs). We compare this projection method with other methods, suggested in the literature, and used in software packages. The projection method is computationally superior to all of the other tested method for the presented large-scale examples. We give interpretations of methods based on discretizations in terms of rational approximations. Some notes regarding a special case where the spectrum can be explicitly expressed with a formula containing a matrix version of the are Lambert W function are presented. We clarify its range of applicability, and, by counter-example, show that it does not hold in general. The second part of this thesis is related to exact stability conditions of the DDE. All those combinations of the delays such that there is a purely imaginary eigenvalue (called critical delays) are parameterized. In general, an evaluation of the parameterization map consists of solving a quadratic eigenvalue problem of squared dimension. We show how the computational cost for one evaluation of the map can be reduced by exploiting a relation to a Lyapunov equation. The third and last part of this thesis is about generalizations of perturbation results for NLEPs. A sensitivity formula for the movement of the eigenvalues extends to NLEPs. We introduce a fixed point form for the NLEP, and show that some methods in the literature can be interpreted as set-valued fixed point iterations for which asymptotic convergence can be established. We also show how the Bauer-Fike theorem can be generalized to the NLEP under special conditions.In dieser Arbeit werden drei verschiedene Problemklassen im Bezug zu delay-differential equations (DDEs) behandelt. Als erstes gehen wir auf die Berechnung der Eigenwerte von DDEs ein. In dieser Arbeit wenden wir eine Projektionsmethode für nichtlineare Eigenwertprobleme (NLEPe) an. Wir vergleichen diese mit anderen bereits bekannten Verfahren, wobei die hier vorgestellte Methode bedeutend bessere numerische Eigenschaften für die verwendeten Beispiele hat. Zusätzlich treffen wir Aussagen über Diskretisierungsmethoden zur rationalen Approximation. Desweiteren betrachten wir einen Spezialfall, bei welchem das Spektrum explizit mit Hilfe einer Matrix-Version der Lambert W-Funktion dargestellt werden kann. Für diese Formel bestimmen wir einen möglichen Anwendungsbereich. Im zweiten Teil der Arbeit werden exakte Stabilitätsbedingungen von DDEs betrachtet. Die Menge der Delays, für welche die DDE einen imaginären Eigenwert hat (sogenannte kritische Delays), wird parameterisiert. Im Allgemeinen ist zur Auswertung der Parametrisierungsabbildung das Lösen eines quadratischen Eigenwertproblems nötig, dessen Größe dem Quadrat der Dimension der DDE entspricht. Wir zeigen wie der Rechenaufwand durch Ausnutzung einer Lyapunov-Gleichung reduziert werden kann. Der letzte Teil dieser Arbeit befasst sich mit der Verallgemeinerung der Störungstheorie auf NLEPe. Unter anderem lässt sich eine Sensitivitätsformel auf NLEPe erweitern. Desweiteren wird eine Fixpunktform für NLEPe vorgestellt, und gezeigt dass einige Methoden aus der Literatur als mengenwertige Fixpunktiterationen dargestellt werden können, für welche wir asymptotische Konvergenz feststellen. Wir zeigen zusätzlich, dass das Bauer-Fike Theorem unter bestimmten Bedingungen auf NLEPe verallgemeinert werden kann

Generalized averaged Gaussian quadrature and applications

A simple numerical method for constructing the optimal generalized averaged Gaussian quadrature formulas will be presented. These formulas exist in many cases in which real positive GaussKronrod formulas do not exist, and can be used as an adequate alternative in order to estimate the error of a Gaussian rule. We also investigate the conditions under which the optimal averaged Gaussian quadrature formulas and their truncated variants are internal

MS FT-2-2 7 Orthogonal polynomials and quadrature: Theory, computation, and applications

Quadrature rules find many applications in science and engineering. Their analysis is a classical area of applied mathematics and continues to attract considerable attention. This seminar brings together speakers with expertise in a large variety of quadrature rules. It is the aim of the seminar to provide an overview of recent developments in the analysis of quadrature rules. The computation of error estimates and novel applications also are described