Algebraic geometry is the study of systems of polynomial equations in one or more variables. Thinking of polynomials as functions reveals a close connection between affine varieties, which are geometric structures, and ideals, which are algebraic objects. An affine variety is a collection of tuples that represents the solutions to a system of equations. An ideal is a special subset of a ring and is what provides the tools to prove geometric theorems algebraically. In this thesis, we establish that a variety depends on the ideal generated by its defining equations. The ability to change the basis of an ideal without changing the variety is a powerful tool in determining a variety. In general, the quotient and remainder on division of polynomials in more than one variable are not unique. One property of a Groebner basis is that it yields a unique remainder on division. To prove geometric theorems algebraically, we first express the hypotheses and conclusions as polynomials. Then, with the aid of a computer, apply the Groebner Basis Algorithm to determine if the conclusion polynomial(s) vanish on the same variety as the hypotheses.

Abstract not available

Bibliography: p. 155-161.

Bibliography: p. 155-161. / x, 161 p. ; 30 cm. / Title page, contents and abstract only. The complete thesis in print form is available from the University Library. / Thesis (Ph.D.)--University of Adelaide, Dept. of Pure Mathematics, 2001

Les fonctions analytiques généralisées sont définies par des séries convergentes de monômes à coeficients réels et exposants réels positifs. Nous étudions l'extension de la géométrie analytique réelle associée à ces algèbres de fonctions. Nous introduisons pour cela la notion de variété analytique réelle généralisée. Il s'agit de variétés topologiques à bord munies de la structure du faisceau des fonctions analytiques réelles généralisées. Notre résultat principal est un théorème de monomialisation locale de ces fonctions.

<p>Den kvasilinjära formen av differential-algebraiska ekvationer är både en mycket allmängiltig generalisering av den linjära tidsinvarianta formen, och en form som visar sig lämpa sig väl för indexreduktionsmetoder som vi hoppas ska komma att bli både praktiskt tillämpbara och väl förstådda i framtiden.</p><p>Kuperingsalgoritmen (engelska: the shuffle algorithm) användes ursprungligen för att bestämma konsistenta initialvillkor för linjära tidsinvarianta differential-algebraiska ekvationer, men har även andra tillämpningar, till exempel det grundläggande problemet numerisk integration. I syfte att förstå hur kuperingsalgoritmen kan tillämpas på kvasilinjära differential-algebraiska ekvationer som inte låter sig analyseras utifrån mönstret av nollor, har problemet att förstå singulära perturbationer i differential-algebraiska ekvationer uppstått. Den här avhandlingen presenterar en indexreduktionsmetod där behovet framgår tydligt, och visar att algoritmen inte bara generaliserar kuperingsalgoritmen, utan även är ett specialfall av den mer allmänna strukturalgoritmen (engelska: the structure algorithm) för att invertera system av Li och Feng.</p><p>Ett kapitel av den här avhandlingen söker av en klass av ekvations-former efter former som är mindre generella än den kvasilinjära, men som en algoritm lik vår kan anpassas till. Det visar sig att indexreduktionen ofta förstör strukturella egenskaper hos ekvationerna, och att det därför är naturligt att arbeta med den mest allmänna kvasilinjära formen.</p><p>Avhandlingen innehåller också några tidiga resultat gällande hur perturbationerna kan hanteras. Huvudresultaten är inspirerade av den modellering i skilda tidskalor som görs i teorin om singulära perturbationer (engelska: singular perturbation theory). Medan teorin om singulära perturbationer betraktar inverkan av en försvinnande skalär i ekvationerna, betraktar analysen häri en okänd matris vars norm begränsas av en liten skalär. Resultaten är begränsade till linjära tidsinvarianta ekvationer av index inte högre än 1, men det är värt att notera att index 0-fallet självt innebär en intressant generalisering av teorin för singulära perturbationer för ordinära differentialekvationer.</p> / <p>The quasilinear form of differential-algebraic equations is at the same time both a very versatile generalization of the linear time-invariant form, and a form which turns out to suit methods for index reduction which we hope will be practically applicable and well understood in the future.</p><p>The shuffle algorithm was originally a method for computing consistent initial conditions for linear time-invariant differential algebraic equations, but has other applications as well, such as the fundamental task of numerical integration. In the prospect of understanding how the shuffle algorithm can be applied to quasilinear differential-algebraic equations that cannot be analyzed by zero-patterns, the question of understanding singular perturbation in differential-algebraic equations has arose. This thesis details an algorithm for index reduction where this need is evident, and shows that the algorithm not only generalizes the shuffle algorithm, but also specializes the more general structure algorithm for system inversion by Li and Feng.</p><p>One chapter of this thesis surveys a class of forms of equations, searching less general forms than the quasilinear, to which an algorithm like ours can be tailored. It is found that the index reduction process often destroys structural properties of the equations, and hence that it is natural to work with the quasilinear form in its full generality.</p><p>The thesis also contains some early results on how the perturbations can be handled. The main results are inspired by the separate timescale modeling found in singular perturbation theory. While the singular perturbation theory considers the influence of a vanishing scalar in the equations, the analysis herein considers an unknown matrix bounded in norm by a small scalar. Results are limited to linear time-invariant equations of index at most 1, but it is worth noting that the index 0 case in itself holds an interesting generalization of the singular perturbation theory for ordinary differential equations.</p> / Report code: LiU-TEK-LIC-2007:27.

We propose in this paper an algebraic approach to hard-ware/software partitioning in Verilog Hardware Description Language (HDL). We explore a collection of algebraic laws for Verilog programs, from which we design a set of syntax-based algebraic rules to conduct hardware/software partitioning. The co-specification language and the target hardware and software description languages are specific subsets of Verilog. Through this, we confirm successful verification for the correctness of the partitioning process by an algebra of Verilog. Facilitated by Verilog’s rich features, we have also successfully studied hw/sw partitioning for environment-driven systems. / Singapore-MIT Alliance (SMA)

Thesis (M.S.)--Brigham Young University. Dept of Mathematics, 2006. / Includes bibliographical references (p. 30).

Thesis (Ph. D.)--University of Idaho, 2006. / Abstract. "April 2006." Includes bibliographical references (leaf 69). Also available online in PDF format.

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