The bidirectional reflectance distribution function has a four dimensional parameter space and such high dimensionality makes it impractical to use it directly in hardware rendering. When a BRDF has no analytical representation, common solutions to overcome this problem include expressing it as a sum of basis functions or factorizing it into several functions of smaller dimensions. This thesis describes factorization extensions that significantly improve factor computation speed and eliminate drawbacks of previous techniques that overemphasize low sample values. The improved algorithm is used to calculate factorizations and material maps from colored images. The technique presented in this thesis allows interactive definition of arbitrary materials, and although this method is based on physical parameters, it can be also used for achieving a variety of non-photorealistic effects.

This thesis is an investigation of some of the properties of polynomial rings, unique factorization domains, Euclidean domains, and principal ideal domains. The nature of some of the relationships between each of the above systems is also developed in this paper.

Title: NP search problems Author: Tomáš Jirotka Department: Department of Algebra Supervisor: Prof. RNDr. Jan Krajíček, DrSc. Abstract: The thesis summarizes known results in the field of NP search pro- blems. We discuss the complexity of integer factoring in detail, and we propose new results which place the problem in known classes and aim to separate it from PLS in some sense. Furthermore, we define several new search problems. Keywords: Computational complexity, TFNP, integer factorization. 1

In this work we are going to investigate a factorization method that was invented by John Pollard. It makes possible to factorize medium large integers into a product of prime numbers. We will run a C++ program and test how do different parameters affect the results. There will be a connection drawn between the Pollard's rho method, the Birthday paradox and the Floyd's cycle finding algorithm. In results we will find a polynomial function that has the best effectiveness and performance for Pollard's rho method.

by Chiu Chak Lam. / Thesis submitted in: June, 1997. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1998. / Includes bibliographical references (leaves 67-69). / Abstract also in Chinese. / Chapter 1 --- Basic Knowledge of Elliptic Curve --- p.2 / Chapter 1.1 --- Elliptic Curve Group Law --- p.2 / Chapter 1.2 --- Discriminant and j-invariant --- p.7 / Chapter 1.3 --- Elliptic Curve over C --- p.10 / Chapter 1.4 --- Complex Multiplication --- p.15 / Chapter 2 --- Order of Elliptic Curve Group Over Finite Fields and the Endo- morphism Ring --- p.18 / Chapter 2.1 --- Hasse's Theorem --- p.18 / Chapter 2.2 --- The Torsion Group --- p.23 / Chapter 2.3 --- The Weil Conjectures --- p.33 / Chapter 3 --- Computing the Order of an Elliptic Curve over a Finite Field --- p.35 / Chapter 3.1 --- Schoof's Algorithm --- p.35 / Chapter 3.2 --- Computation Formula --- p.38 / Chapter 3.3 --- Recent Works --- p.42 / Chapter 4 --- Primality Test Using Elliptic Curve --- p.43 / Chapter 4.1 --- Goldwasser-Kilian Test --- p.43 / Chapter 4.2 --- Atkin's Test --- p.44 / Chapter 4.3 --- Binary Quadratic Form --- p.49 / Chapter 4.4 --- Practical Consideration --- p.51 / Chapter 5 --- Elliptic Curve Factorization Method --- p.54 / Chapter 5.1 --- Lenstra's method --- p.54 / Chapter 5.2 --- Worked Example --- p.56 / Chapter 5.3 --- Practical Considerations --- p.56 / Chapter 6 --- Elliptic Curve Public Key Cryptosystem --- p.59 / Chapter 6.1 --- Outline of the Cryptosystem --- p.59 / Chapter 6.2 --- Index Calculus Method --- p.61 / Chapter 6.3 --- Weil Pairing Attack --- p.63

The goal of the thesis is to extend the kernel methods to matrix factorization(MF) for collaborative ltering(CF). In current literature, MF methods usually assume that the correlated data is distributed on a linear hyperplane, which is not always the case. The best known member of kernel methods is support vector machine (SVM) on linearly non-separable data. In this thesis, we apply kernel methods on MF, embedding the data into a possibly higher dimensional space and conduct factorization in that space. To improve kernelized matrix factorization, we apply multi-kernel learning methods to select optimal kernel functions from the candidates and introduce L2-norm regularization on the weight learning process. In our empirical study, we conduct experiments on three real-world datasets. The results suggest that the proposed method can improve the accuracy of the prediction surpassing state-of-art CF methods.

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Dans cette thèse, nous explorons de nouvelles méthodes d’analyse d’images pour la détection précoce des changements métaboliques cérébraux causés par la maladie d’Alzheimer. Nous introduisons deux apports méthodologiques que nous appliquons à un ensemble de données réelles. Le premier est basé sur l’apprentissage automatique afin de créer une carte des informations pertinentes pour la classification d'un ensemble d’images. Pour cela nous échantillonnons des blocs de Voxels selon un algorithme de Monte-Carlo. La mise en œuvre d’une classification basée sur ces patchs 3d a pour conséquence la réduction significative du volume de patchs à traiter et l’extraction de caractéristiques dont l’importance est statistiquement quantifiable. Cette méthode s’applique à différentes caractéristiques et est adaptée à des types d’images variés. La résolution des cartes produites par cette méthode peut être affinée à volonté et leur contenu informatif est cohérent avec des résultats antérieurs obtenus dans la littérature. Le second apport méthodologique porte sur la conception d’un nouvel algorithme de décomposition de tenseur d’ordre important, adapté à notre application. Cet algorithme permet de réduire considérablement la consommation de mémoire et donc en évite la surcharge. Il autorise la décomposition rapide de tenseurs, y compris ceux de dimensions très déséquilibrées. Nous appliquons cet algorithme en tant que méthode d’extraction de caractéristiques dans une situation où le clinicien doit diagnostiquer des stades précoces de la maladie d'Alzheimer en utilisant la TEP-FDG seule. Les taux de classification obtenus sont souvent au-dessus des niveaux de l’état de l’art. / In this thesis we want to explore novel image analysis methods for the early detection of metabolic changes in the human brain caused by Alzheimer's disease (AD). We will present two methodological contributions and present their application to a real life data set. We present a machine learning based method to create a map of local distribution of classification relevant information in an image set. The presented method can be applied using different image characteristics which makes it possible to adapt the method to many kinds of images. The maps generated by this method are very localized and fully consistent with prior findings based on Voxel wise statistics. Further we preset an algorithm to draw a sample of patches according to a distribution presented by means of a map. Implementing a patch based classification procedure using the presented algorithm for data reduction we were able to significantly reduce the amount of patches that has to be analyzed in order to obtain good classification results. We present a novel non-negative tensor factorization (NTF) algorithm for the decomposition of large higher order tensors. This algorithm considerably reduces memory consumption and avoids memory overhead. This allows the fast decomposition even of tensors with very unbalanced dimensions. We apply this algorithm as feature extraction method in a computer-aided diagnosis (CAD) scheme, designed to recognize early-stage ad and mild cognitive impairment (MCI) using fluorodeoxyglucose (FDG) positron emission tomography (PET) scans only. We achieve state of the art classification rates.

Factorization theory is concerned with the decomposition of mathematical objects. Such an object could be a polynomial, a number in the set of integers, or more generally an element in a ring. A classic example of a ring is the set of integers. If we take any two integers, for example 2 and 3, we know that $2 \cdot 3=3\cdot 2$, which shows that multiplication is commutative. Thus, the integers are a commutative ring. Also, if we take any two integers, call them $a$ and $b$, and their product $a\cdot b=0$, we know that $a$ or $b$ must be $0$. Any ring that possesses this property is called an integral domain. If there exist two nonzero elements, however, whose product is zero we call such elements zero divisors. This thesis focuses on factorization in commutative rings with zero divisors. In this work we extend the theory of factorization in commutative rings to polynomial rings with zero divisors. For a commutative ring $R$ with identity and its polynomial extension $R[X]$ the following questions are considered: if one of these rings has a certain factorization property, does the other? If not, what conditions must be in place for the answer to be yes? If there are no suitable conditions, are there counterexamples that demonstrate a polynomial ring can possess one factorization property and not another? Examples are given with respect to the properties of atomicity and ACCP. The central result is a comprehensive characterization of when $R[X]$ is a unique factorization ring.

The bidirectional reflectance distribution function has a four dimensional parameter space and such high dimensionality makes it impractical to use it directly in hardware rendering. When a BRDF has no analytical representation, common solutions to overcome this problem include expressing it as a sum of basis functions or factorizing it into several functions of smaller dimensions. This thesis describes factorization extensions that significantly improve factor computation speed and eliminate drawbacks of previous techniques that overemphasize low sample values. The improved algorithm is used to calculate factorizations and material maps from colored images. The technique presented in this thesis allows interactive definition of arbitrary materials, and although this method is based on physical parameters, it can be also used for achieving a variety of non-photorealistic effects.