Let Nm(f(x)) denote the number of solutions of the congruence equation f(x)≡0 (modm), where m≥2 is any composite integer and f(x) is a cubic polynomial. In this thesis, we use different theorems and corollaries to find a number of solutions of the congruence equations without solving then we also construct the general expression of corresponding congruence equations to demonstrate the solutions of the equations. In this thesis, we use Mathematica software as a tool.

In this paper we prove a topological lemma on real valued random variables which implies the basic ingredients for the proof of the Fundamental Theorem of Asset Pricing in the two period case. In particular, previous results of Stricker and of Schachermayer are special cases of our result. Our proof is considerably shorter and more transparent than previous proofs of related special cases. / Series: Working Papers SFB "Adaptive Information Systems and Modelling in Economics and Management Science"

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Nous montrons un résultat du type Closing Lemma pour les mesures non uniformément hyperboliques associées à des applications méromorphes. Nous prouvons aussi qu'il est possible d'approximer la dynamique de telles mesures par des codages du type Bernoulli. / We prove a Closing Lemma for nonuniformly hyperbolic measures of meromor-phic maps. We prove also a theorem of approxiamtion of the dynamics of such measures by Bernoulli coding maps.

The Regularity Lemma of Szemeredi is a fundamental tool in extremal graph theory with a wide range of applications in theoretical computer science. Partly as a recognition of his work on the Regularity Lemma, Endre Szemeredi has won the Abel Prize in 2012 for his outstanding achievement. In this thesis we present both practical and theoretical applications of the Regularity Lemma. The practical applications are concerning the important problem of data clustering, the theoretical applications are concerning the monochromatic vertex partition problem. In spite of its numerous applications to establish theoretical results, the Regularity Lemma has a drawback that it requires the graphs under consideration to be astronomically large, thus limiting its practical utility. As stated by Gowers, it has been ``well beyond the realms of any practical applications', the existing applications have been theoretical, mathematical. In the first part of the thesis, we propose to change this and we propose some modifications to the constructive versions of the Regularity Lemma. While this affects the generality of the result, it also makes it more useful for much smaller graphs. We call this result the practical regularity partitioning algorithm and the resulting clustering technique Regularity Clustering. This is the first integrated attempt in order to make the Regularity Lemma applicable in practice. We present results on applying regularity clustering on a number of benchmark data-sets and compare the results with k-means clustering and spectral clustering. Finally we demonstrate its application in Educational Data Mining to improve the student performance prediction. In the second part of the thesis, we study the monochromatic vertex partition problem. To begin we briefly review some related topics and several proof techniques that are central to our results, including the greedy and absorbing procedures. We also review some of the current best results before presenting ours, where the Regularity Lemma has played a critical role. Before concluding we discuss some future research directions that appear particularly promising based on our work.

O objetivo deste trabalho é provar um Closing Lema Parcial para variedades bidimensionais compactas, orientáveis ou não--orientáveis. Para enunciá--lo, considere um campo vetorial \\linebreak $X\\in\\mathfrak^r(M)$, $r\\ge 2$, de classe $C^r$ em uma variedade bidimensional compacta $M$, e seja $\\Sigma$ um segmento transversal a $X$ passando por um ponto recorrente não--trivial $p$ de $X$. Seja $P:\\Sigma\\to\\Sigma$ a correspondente transformação de primeiro retorno. O primeiro resultado deste trabalho consiste em mostrar que se $P$ tem a propriedade de que para todo $n\\ge N$ e $x\\in{m dom}\\,(P^n)$, $\\vert DP^n(x)\\vert<\\lambda$, onde $N\\in\\N$ e $0<\\lambda<1$, então existe um campo vetorial $Y$ arbitrariamente próximo de $X$ na topologia $C^r$ tendo uma trajetória periódica passando por $p$. O segundo resultado consiste em apresentar condições, sobre os expoentes de Lyapunov de $P$, para que $\\vert DP^n\\vert<\\lambda$ para todo $n\\ge N$. Nesta tese, também incluímos um resultado sobre a estabilidade assintótica no infinito de campos planares diferenciáveis, mas não necessariamente de classe $C^1$. / The aim of this work is to provide a Partial $C^r$ Closing Lemma for compact surfaces, orientable or non--orientable. To state it, let $X\\in\\mathfrak^r(M)$, $r\\ge 2$, be a $C^r$ vector field on a compact surface $M$ and let $\\Sigma$ be a transverse segment to $X$ passing through a non--trivial recurrent point $p$ of $X$. Let $P:\\Sigma\\to\\Sigma$ be the corresponding first return map. The first result of this work consists in showing that if $P^n$ has the property that for all $n\\ge N$ and $x\\in{m dom}\\,(P^n)$, $\\vert DP^n(x)\\vert<\\lambda$, where $N\\in\\N$ e $0<\\lambda<1$, then there exists a vector field $Y$ arbitrarily close to $X$ in the $C^r$ topology such that $p$ is a periodic point of $Y$. The second result consists in presenting sufficient conditions, upon the Lyapunov exponents of $P$, so that $\\vert DP^n\\vert<\\lambda$ for all $n\\ge N$. In this thesis, we also include a result concerning the asymptotic stability at infinity of planar differentiable vector fields, not necessarily of class $C^1$.

Commutative Gröbner bases theory is well known and widely used. In this thesis, we will discuss thoroughly its generalization to noncommutative polynomial ring <em>k</em><<em>X</em>> which is also an associative free algebra. We introduce some results on monomial orders due to John Lawrence and the author. We show that a noncommutative monomial order is a well order while a one-sided noncommutative monomial order may not be. Then we discuss the generalization of polynomial reductions, S-polynomials and the characterizations of noncommutative Gröbner bases. Some results due to Mora are also discussed, such as the generalized Buchberger's algorithm and the solvability of ideal membership problem for homogeneous ideals. At last, we introduce Newman's diamond lemma and Bergman's diamond lemma and show their relations with Gröbner bases theory.

Commutative Gröbner bases theory is well known and widely used. In this thesis, we will discuss thoroughly its generalization to noncommutative polynomial ring <em>k</em><<em>X</em>> which is also an associative free algebra. We introduce some results on monomial orders due to John Lawrence and the author. We show that a noncommutative monomial order is a well order while a one-sided noncommutative monomial order may not be. Then we discuss the generalization of polynomial reductions, S-polynomials and the characterizations of noncommutative Gröbner bases. Some results due to Mora are also discussed, such as the generalized Buchberger's algorithm and the solvability of ideal membership problem for homogeneous ideals. At last, we introduce Newman's diamond lemma and Bergman's diamond lemma and show their relations with Gröbner bases theory.

Teilw. zugl.: Chemnitz, Techn. Univ., Masterarbeit, 2006.

O objetivo deste trabalho é provar um Closing Lema Parcial para variedades bidimensionais compactas, orientáveis ou não--orientáveis. Para enunciá--lo, considere um campo vetorial \\linebreak $X\\in\\mathfrak^r(M)$, $r\\ge 2$, de classe $C^r$ em uma variedade bidimensional compacta $M$, e seja $\\Sigma$ um segmento transversal a $X$ passando por um ponto recorrente não--trivial $p$ de $X$. Seja $P:\\Sigma\\to\\Sigma$ a correspondente transformação de primeiro retorno. O primeiro resultado deste trabalho consiste em mostrar que se $P$ tem a propriedade de que para todo $n\\ge N$ e $x\\in{m dom}\\,(P^n)$, $\\vert DP^n(x)\\vert<\\lambda$, onde $N\\in\\N$ e $0<\\lambda<1$, então existe um campo vetorial $Y$ arbitrariamente próximo de $X$ na topologia $C^r$ tendo uma trajetória periódica passando por $p$. O segundo resultado consiste em apresentar condições, sobre os expoentes de Lyapunov de $P$, para que $\\vert DP^n\\vert<\\lambda$ para todo $n\\ge N$. Nesta tese, também incluímos um resultado sobre a estabilidade assintótica no infinito de campos planares diferenciáveis, mas não necessariamente de classe $C^1$. / The aim of this work is to provide a Partial $C^r$ Closing Lemma for compact surfaces, orientable or non--orientable. To state it, let $X\\in\\mathfrak^r(M)$, $r\\ge 2$, be a $C^r$ vector field on a compact surface $M$ and let $\\Sigma$ be a transverse segment to $X$ passing through a non--trivial recurrent point $p$ of $X$. Let $P:\\Sigma\\to\\Sigma$ be the corresponding first return map. The first result of this work consists in showing that if $P^n$ has the property that for all $n\\ge N$ and $x\\in{m dom}\\,(P^n)$, $\\vert DP^n(x)\\vert<\\lambda$, where $N\\in\\N$ e $0<\\lambda<1$, then there exists a vector field $Y$ arbitrarily close to $X$ in the $C^r$ topology such that $p$ is a periodic point of $Y$. The second result consists in presenting sufficient conditions, upon the Lyapunov exponents of $P$, so that $\\vert DP^n\\vert<\\lambda$ for all $n\\ge N$. In this thesis, we also include a result concerning the asymptotic stability at infinity of planar differentiable vector fields, not necessarily of class $C^1$.