Positivity of polynomials, as a key notion in real algebra, is one of the oldest topics. In a given context, some polynomials can be represented in a form that reveals their positivity immediately, like sums of squares. A large body of literature deals with the question which positive polynomials can be represented in such a way.<p>The milestone in this development was Schm"udgen's solution of the moment problem for compact semi-algebraic sets. In 1991, Schm"udgen proved that if the associated basic closed semi-algebraic set $K_{S}$ is compact, then any polynomial which is strictly positive on $K_{S}$ is contained in the preordering $T_{S}$.<p>Putinar considered a further question: when are `linear representations' possible? He provided the first step in answering this question himself in 1993. Putinar proved if the quadratic module $M_{S}$ is archimedean, any polynomial which is strictly positive on $K_{S}$ is contained in $M_{S}$, i.e., has a linear representation.<p>In the present thesis, we concentrate on the linear representations in the one variable polynomial ring. We first investigate the relationship of the two conditions in Schm"udgen's Theorem and Putinar's Criterion: $K_{S}$ compact and $M_{S}$ archimedean. They are actually equivalent. We find another proof for this result and hereby we can improve Schm"udgen's Theorem in the one variable case.<p>Secondly, we investigate the relationship of $M_{S}$ and $T_{S}$. We use elementary arguments to prove in the one variable case when $K_{S}$ is compact, they are equal.<p>Thirdly, we present Scheiderer's Main Theorem with a detailed proof. Scheiderer established a local-global principle for the polynomials non-negative on $K_{S}$ to be contained in $M_{S}$ in 2003. This principle which we call Scheiderer's Main Theorem here extends Putinar's Criterion.<p>Finally, we consider Scheiderer's Main Theorem in the one variable case, and give a simplified version of this theorem. We also apply this Simple Version of the Main Theorem to give some elementary proofs for existing results.

Positivity of polynomials, as a key notion in real algebra, is one of the oldest topics. In a given context, some polynomials can be represented in a form that reveals their positivity immediately, like sums of squares. A large body of literature deals with the question which positive polynomials can be represented in such a way.<p>The milestone in this development was Schm"udgen's solution of the moment problem for compact semi-algebraic sets. In 1991, Schm"udgen proved that if the associated basic closed semi-algebraic set $K_{S}$ is compact, then any polynomial which is strictly positive on $K_{S}$ is contained in the preordering $T_{S}$.<p>Putinar considered a further question: when are `linear representations' possible? He provided the first step in answering this question himself in 1993. Putinar proved if the quadratic module $M_{S}$ is archimedean, any polynomial which is strictly positive on $K_{S}$ is contained in $M_{S}$, i.e., has a linear representation.<p>In the present thesis, we concentrate on the linear representations in the one variable polynomial ring. We first investigate the relationship of the two conditions in Schm"udgen's Theorem and Putinar's Criterion: $K_{S}$ compact and $M_{S}$ archimedean. They are actually equivalent. We find another proof for this result and hereby we can improve Schm"udgen's Theorem in the one variable case.<p>Secondly, we investigate the relationship of $M_{S}$ and $T_{S}$. We use elementary arguments to prove in the one variable case when $K_{S}$ is compact, they are equal.<p>Thirdly, we present Scheiderer's Main Theorem with a detailed proof. Scheiderer established a local-global principle for the polynomials non-negative on $K_{S}$ to be contained in $M_{S}$ in 2003. This principle which we call Scheiderer's Main Theorem here extends Putinar's Criterion.<p>Finally, we consider Scheiderer's Main Theorem in the one variable case, and give a simplified version of this theorem. We also apply this Simple Version of the Main Theorem to give some elementary proofs for existing results.

Prof. Marcel Wild completed his PhD with Zurick University and this is a copy of the original works / The original works can be found at http://www.hbz.uzh.ch/ / ABSTRACT: A linear representation of a modular lattice L is a homomorphism from L into the lattice Sub(V) of all subspaces of a vector space V. The representation theory of lattices was initiated by the Darmstadt school (Wille, Herrmann, Poguntke, et al), to large extent triggered by the linear representations of posets (Gabriel, Gelfand-Ponomarev, Nazarova, Roiter, Brenner, et al). Even though posets are more general than lattices, none of the two theories encompasses the other. In my thesis a natural type of finite lattice is identified, i.e. triangle lattices, and their linear representation theory is advanced. All of this was triggered by a more intricate setting of quadratic spaces (as opposed to mere vector spaces) and the question of how Witt’s Theorem on the congruence of finite-dimensional quadratic spaces lifts to spaces of uncountable dimensions. That issue is dealt with in the second half of the thesis.

This thesis studies grammar-based approaches in the application of Estimation of Distribution Algorithms (EDA) to the tree representation widely used in Genetic Programming (GP). Although EDA is becoming one of the most active fields in Evolutionary computation (EC), the solution representation in most EDA is a Genetic Algorithms (GA) style linear representation. The more complex tree representations, resembling GP, have received only limited exploration. This is unfortunate, because tree representations provide a natural and expressive way of representing solutions for many problems. This thesis aims to help fill this gap, exploring grammar-based approaches to extending EDA to GP-style tree representations. This thesis firstly provides a comprehensive survey of current research on EDA with emphasis on EDA with GP-style tree representation. The thesis attempts to clarify the relationship between EDA with conventional linear representations and those with a GP-style tree representation, and to reveal the unique difficulties which face this research. Secondly, the thesis identifies desirable properties of probabilistic models for EDA with GP-style tree representation, and derives the PRODIGY framework as a consequence. Thirdly, following the PRODIGY framework, three methods are proposed. The first method is Program Evolution with Explicit Learning (PEEL). Its incremental general-to-specific grammar learning method balances the effectiveness and efficiency of the grammar learning. The second method is Grammar Model-based Program Evolution (GMPE). GMPE realises the PRODIGY framework by introducing elegant inference methods from the formal grammar field. GMPE provides good performance on some problems, but also provides a means to better understand some aspects of conventional GP, especially the building block hypothesis. The third method is Swift GMPE (sGMPE), which is an extension of GMPE, aiming at reducing the computational cost. Fourthly, a more accurate Minimum Message Length metric for grammar learning in PRODIGY is derived in this thesis. This metric leads to improved performance in the GMPE system, but may also be useful in grammar learning in general. It is also relevant to the learning of other probabilistic graphical models.