This paper is devoted to finding integral solutions of algebraic equations. Only algebraic equations with integral coefficients are considered. The elementary properties of integers are assumed.

Power system dynamics are commonly modeled by parameter dependent nonlinear differential-algebraic equations (DAE) x p y x f ) and 0 = p y x g ) . Due to (,, (,, the algebraic constraints, we cannot directly perform integration based on the DAE. Traditionally, we use implicit function theorem to solve for fast variables y to get a reduced model in terms of slow dynamics locally around x or we compute y numerically at each x . However, it is well known that solving nonlinear algebraic equations analytically is quite difficult and numerical solution methods also face many uncertainties since nonlinear algebraic equations may have many solutions, especially around bifurcation points. In this thesis, we apply the singular perturbation method to model power system dynamics in a singularly perturbed ODE (ordinary-differential equation) form, which makes it easier to observe time responses and trace bifurcations without reduction process. The requirements of introducing the fast dynamics are investigated and the complexities in the procedures are explored. Finally, we propose PTE (Perturb and Taylor’s expansion) technique to carry out our goal to convert a DAE to an explicit state space form of ODE. A simplified unreduced Jacobian matrix is also introduced. A dynamic voltage stability case shows that the proposed method works well without complicating the applications.

The aim of this work is to get acquainted with the issue of solving word problems by means of equations, to state and to analyse various ways of solving them. The thesis also includes the comparison of strategies for solving word problems that are selected by students who have been already taught the given type of the equation, and students who have not yet mastered these procedures. The introductory part of the work deals with word problems in general. There the terms such as the word problems, procedures, methods and strategies used for solving word problems are specified. In the next part of the text, the basic types of equations are characterized. For each type of the equation, one typical word problem was selected and it was experimentally found out which solving strategies were used by students most often. Keywords: word problem, algebraic equation, non-algebraic equation, system of equations

Differential-algebraic equations (DAEs) result in many areas of science and engineer- ing. In this thesis, numerical methods for solving DAEs are compared for two prob- lems, energy-economic models and traffic flow models. An energy-economic model is presented based on the Hubbert model of oil production and is extended to include economic factors for the first time. Using numerical methods to simulate the DAE model, the resulting graphs break the symmetry of the traditional Hubbert curve. For the traffic flow models, a numerical method is developed to solve the steady-state flow pattern including the linearly unstable regime, i.e. solutions which cannot be found with an initial value solver. / UOIT

Differential-algebraic equations (DAEs) result in many areas of science and engineer- ing. In this thesis, numerical methods for solving DAEs are compared for two prob- lems, energy-economic models and traffic flow models. An energy-economic model is presented based on the Hubbert model of oil production and is extended to include economic factors for the first time. Using numerical methods to simulate the DAE model, the resulting graphs break the symmetry of the traditional Hubbert curve. For the traffic flow models, a numerical method is developed to solve the steady-state flow pattern including the linearly unstable regime, i.e. solutions which cannot be found with an initial value solver.

This paper was presented at CJK-OSM 7, 18–21 June 2012, Huangshan, China.

No description available.

This report studies polynomial equations and how one solves them using only the coefficients of the polynomial. It examines why it is impossible to solve equations of degree greater than four using only radicals and how instead one can solve them using elliptic functions. Although the quintic equation is the main area of our investigation, we also present parts of the history of algebraic equations, Galois theory, and elliptic functions.

Algebro-Differentialgleichungen (engl. differential-algebraic equations - DAEs) sind implizite singuläre gewöhnliche Differentialgleichungen, die restringierte dynamische Prozesse beschreiben. Sie unterscheiden sich von expliziten gewöhnlichen Differentialgleichungen dahingehend, dass Anfangswerte nicht beliebig vorgegeben werden können. Weiterhin sind in einer DAE neben Integrations- auch Differentiationsaufgaben involviert. Der Differentiationsindex gibt an, wieviele Differentiationen zur Lösung notwendig sind. Seit den 1980er Jahren wird vorwiegend an der Charakterisierung und Klassifizierung regulärer DAEs und der Konstruktion nebst Fundierung von Integrationsmethoden gearbeitet. I. Higueras, R. März und C. Tischendorf haben gezeigt, dass man lineare DAEs mit properem Hauptterm, A(t)(D(t)x(t))'' + B(t)x(t) = q(t), die regulär mit Traktabilitätsindex 2 sind, zuverlässig numerisch integrieren kann - im Unterschied zu linearen DAEs in Standardform. In Publikationen von R. Riaza und R. März wird die Klassifizierungen kritischer Punkten von linearen DAEs an die Verletzung bestimmter Rangbedingungen von Matrixfunktionen im Rahmen des Traktabilitätsindexes geknüpft. Im wesentlichen heißt ein kritischer Punkt harmlos, wenn der durch die inhärente Differentialgleichung beschriebene Fluß nicht tangiert ist. Gegenstand der vorliegenden Arbeit sind lineare quasi-proper formulierte DAEs. Es werden Index 2 DAEs mit harmlosen kritischen Punkten charakterisiert. Unter Verwendung von quasi-zulässigen Projektorfunktionen können neben DAEs, die fast überall gleiche charakteristische Werte haben, nun erstmalig auch solche mit Indexwechseln behandelt werden. Der Hauptteil der Arbeit besteht im Nachweis von Durchführbarkeit, Konvergenz und nur schwacher Instabilität von numerischen Integrationsmethoden (BDF, IRK(DAE)) für lineare Index 2 DAEs mit harmlosen kritischen Punkten, sowie in der Entwicklung von Fehlerschätzern und Schrittweitensteuerung. / Differential-algebraic equations (DAEs) are implicit singular ordinary differential equations, which describe dynamical processes that are restricted by some constraints. In contrast to explicit regular ordinary differential equations, for a DAE not any value can be imposed as an initial condition. Furthermore, DAEs involve not only integration problems but also differentiation problems. The differentiation index of a DAE indicates the number of differentiations required in order to solve a DAE. Since the 1980th, research focuses primarily on the characterization and classification of regular problem classes and the construction and foundation of integration methods for simulation software. I. Higueras, R. Maerz, and C. Tischendorf have shown that one can reliably integrate a general linear DAE with a properly stated leading term, A(t)(D(t)x(t))'' + B(t)x(t) = q(t), which is regular with tractability index 2 - in contrast to linear standard form DAEs. The first classification of critical points of linear DAEs has been published by R. Riaza and R. Maerz. Based on the tractability index, critical points are classified according to failures of certain rank conditions of matrix functions. Essentially, a critical point is said to be harmless, if the flow described by the inherent differential equation is not affected. The subject of this work are quasi-proper linear DAEs. Index-2 DAEs with harmless critical points are characterized. Under the application of quasi-admissible projector functions. Besides DAEs which have almost everywhere the same characteristic values, DAEs with index changes can now be discussed for the first time. The main part of the work is to provide a proof of feasibility, convergence, and only weak instability of numerical integration methods (BDF, IRK (DAE)) for linear index-2 DAEs with harmless critical points, as well as the development and testing of error estimators and stepsize control.

In this research, the focus is on improving a designer's capability to determine near-optimal sizes of components for a given system architecture. Component sizing is a hard problem to solve because of the presence of competing objectives, requirements from multiple disciplines, and the need for finding a solution quickly for the architecture being considered. In current approaches, designers rely on heuristics and iterate over the multiple objectives and requirements until a satisfactory solution is found. To improve on this state of practice, this research introduces advances in the following two areas: a.) Formulating a component sizing problem in a manner that is convenient to designers and b.) Solving the component sizing problem in an efficient manner so that all of the imposed requirements are satisfied simultaneously and the solution obtained is mathematically optimal. In particular, an acausal, algebraic, equation-based, declarative modeling approach is taken to solve component sizing problems efficiently. This is because global optimization algorithms exist for algebraic models and the computation time is considerably less as compared to the optimization of dynamic simulations. In this thesis, the mathematical programming language known as GAMS (General Algebraic Modeling System) and its associated global optimization solvers are used to solve component sizing problems efficiently. Mathematical programming languages such as GAMS are not convenient for formulating component sizing problems and therefore the Systems Modeling Language developed by the Object Management Group (OMG SysML ) is used to formally capture and organize models related to component sizing into libraries that can be reused to compose new models quickly by connecting them together. Model-transformations are then used to generate low-level mathematical programming models in GAMS that can be solved using commercial off-the-shelf solvers such as BARON (Branch and Reduce Optimization Navigator) to determine the component sizes that satisfy the requirements and objectives imposed on the system. This framework is illustrated by applying it to an example application for sizing a hydraulic log splitter.