The problem under consideration is that of determining a simple and effective invariant of knots. To this end, the Conway polynomial is defined as a generalization of Alexander's original knot polynomial. It is noted, however, that the Conway polynomial is not a complete invariant. If two knots are equivalent, as defined in this investigation, then they receive identical polynomials. Yet, if two knots have identical polynomials, no information about their equivalence may be obtained. To define the Conway polynomial, the Axioms for Computation are given and many examples of their use are included. A major result of this investigation is the proof of topological invariance of these polynomials and the proof that the axioms are sufficient for the calculation of the knot polynomial for any given knot or link.

Current cryptographic algorithms pose high standards of security yet they are susceptible to side-channel analysis (SCA). When it comes to implementation, the hardness of cryptography dangles on the weak link of side-channel information leakage. The widely adopted AES encryption algorithm, and others, can be easily broken when they are implemented without any resistance to SCA. This work applies state of the art techniques, namely Secret Sharing and Secure Multiparty Computation (SMC), on AES-128 encryption as a countermeasure to those attacks. This embedded C implementation explores multiple time-memory trade-offs for the design of its fundamental components, SMC and field arithmetic, to meet a variety of execution and storage demands. The performance and leakage assessment of this implementation for an ARM based micro-controller demonstrate the capabilities of masking schemes and prove their feasibility on embedded software.

Over the past few decades various models of different formats have been developed to correctly evaluate and predict the strength of materials. However, these models are limited in certain environmental conditions in implementing the effect of material's thickness into their models. As such an there was a need to consider the basics of mechanical engineering and to try and define the trend, thickness has upon the behaviour of materials with respect to environmental conditions. The work consisted of a representation of tensile testing testing of common engineering alloys across a wide range of temperature, strain rate and thickness. Acquisition of high strain rate data and extended strain data (split-hopkinson, bulge forming and plane strain compression). A review of existing graphical techniques and limited applications using strength reduction factors, as well as applying the accepted empirical formulae, Johnson-Cook, Armstrong-Zerrili, Ramberg-Osgood and Hollomon. Later, recognising a need for a new approach as with a universal (quartic) polynomial fit to all plastic flow curves in which coefficients are T, ε̇ and t̄ dependant. Adoptation of a common numerical procedure for strain intercept ε0 and cut-off instability co-ordinates (σi, εi)- each as the solution to the roots of a quartic. Therefore, a proposal of the flow curve tables allowing interpolation and extrapolation, a numerical representation of any previous "Atlas of Curves". Subsequently, leading to reconstruction of the full stress-strain curve with the addition of elastic strain calculated from the modulus applicable to the specific test condition by further testing of these data from literature; both improving the existing and producing new empirical and simulation based models to analyse the materials, which will be subjected to dynamic loading as well as temperature and strain rates variations. The main objective of the work, was involved in creating a polynomial fit to describe the three physical conditions in terms of coefficients and to verify the findings in a FEA package, ABAQUS. A new process in reading the stress-strain data. By means of such development an instability study of strain limits based on Considére criteria was developed which illustrated the ways to prolong the instability limit. A secondary study of this work relates to creating a bridge between the micro-structure and macro-structure of the tested materials. A series of correlations and trends were developed to further signify the shift in micro-structural restructuring, whilst the material is under load. Another important aspect of the work consists, of carrying out an analytical study on Ramberg-Osgood equation. Ramberg-Osgood equation has been at the forefront of many engineering advancement. However it can yet be improved and reformatted by means of defining a set value for its variable constants. As such a fix ƞt value based on a best-fit approach was developed which was analytically tested.

Dans ma thèse j'etude les variétés abéliennes ordinaires définies avec multiplication réelle maximale. Je définis des polynômes modulaires dans ce situation et je donne un algorithme pour calculer sur les nombres complexes et pour les surfaces sur des corps finis. Je donne aussi un théorème de structure pour les graphs des isogénies dans ce contexte. Je donne une généralisation de Schoof-Elkies-Atkin aux courbes de genre 2 avec multiplication réelle maximale fixe en utilisant les polynômes modulaires. / My thesis looks at ordinary abelian varieties defined with maximal real multiplication. I define modular polynomials in this setting and give an algorithm to compute them over the complex numbers, and for surfaces over finite fields. I also give a structure theorem for isogeny graphs in this setting. I give a generalisation of Schoof-Elkies-Atkin to genus 2 curves with fixed maximal real multiplication using the modular polynomials.

Em 1968, J. Milnor introduziu a monodromia local de Picard-Lefschetz de uma hipersuperfície complexa com singularidade isolada. Em seguida, E. Brieskorn perguntou se esta monodromia é sempre finita. Em 1972, Lê Dúng Trâng provou que a resposta é positiva no caso de germes de curvas planas analíticas irredutíveis. Na época, já eram conhecidos exemplos de curvas planas com dois ramos e monodromia finita. Em 1973, N. A?Campo produziu o primeiro exemplo de germe de curva plana com dois ramos e monodromia infinita. Portanto, a questão mais simples, e ainda em aberto, que se coloca neste contexto, é a determinação da finitude da monodromia para germes de curvas planas com dois ramos. O presente trabalho, consiste em determinar, em várias situações, o polinômio mínimo da monodromia de germes de curvas analíticas planas com dois ramos, cujos gêneros são menores ou iguais a dois, o que permite decidir a sua finitude / In 1968, J. Milnor introduced the Picard-Lefschetz monodromy of a complex hypersurface with an isolated singularity. Subsequently, E. Brieskorn asked if this monodromy is always finite. In 1972, Lê Dúng Trâng proved that the answer is positive in the case of irreducible analytic germs of plane curves. At this time, examples of plane curves with two branches and finite monodromy were known. In 1973, N. A?Campo produced the first example of a germ of plane curve with two branches and infinite monodromy. Therefore, the simplest and still open problem in this context is to determine whether the monodromy of a plane curve with two branches is finite or infinite. The present work consists in determining, in several situations, the minimal polynomial of the monodromy for germs of plane analytic curves with two branches, whose genera are less or equal than two, wich allows us to decide its finiteness

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.

La méthode des éléments finis stochastiques (MEFS) a été développée pour modéliser l'aléa sous la forme de variables aléatoires de type quelconque dans le cadre de la mécanique linéaire élastique. Elle consiste à écrire les composantes de la réponse aléatoire du système sous la forme d'une série polynomiale de variables aléatoires (baptisée chaos polynomial), dont les coefficients sont obtenus par une méthode de type Galerkin. Le champ d'application de cette méthode étant limité, de nouvelles méthodes, dites non intrusives, permettant le calcul du développement de la réponse dans la base du chaos polynomial ont été recherchées.<br />Les méthodes MEFS et non intrusive ont été testées et comparées sur des exemples de mécanique élastique linéaire. Enfin les approches non intrusives ont été utilisées dans un cas de mécanique de la rupture non linéaire.

<p>In some applications polynomials should be evaluated, e.g., polynomial approximation of elementary function and Farrow filter for arbitrary re-sampling. For polynomial evaluation Horner’s scheme uses the minimum amount of hardware resources, but it is sequential. Many algorithms were developed to introduce parallelism in polynomial evaluation. This parallelism is achieved at the cost of hardware, but ensures evaluation in less time.</p><p>This work examines the trade-off between hardware cost and the critical path for different level of parallelism for polynomial evaluation. The trade-offs in generating powers in polynomial evaluation using different building blocks(squarers and multipliers) are also discussed. Wordlength requirements of the polynomial evaluation and the effect of power generating schemes on the timing of operations is also discussed. The area requirements are calculated by using Design Analyzer from Synopsys (tool for logic synthesis) and the GLPK (GNU Linear Programming Kit) is used to calculate the bit requirements.</p>

This paper develops a potential reduction algorithm for solving a linear-programming problem directly from a "warm start" initial point that is neither feasible nor optimal. The algorithm is of an "interior point" variety that seeks to reduce a single potential function which simultaneously coerces feasibility improvement (Phase I) and objective value improvement (Phase II). The key feature of the algorithm is the ability to specify beforehand the desired balance between infeasibility and nonoptimality in the following sense. Given a prespecified balancing parameter /3 > 0, the algorithm maintains the following Phase I - Phase II "/3-balancing constraint" throughout: (cTx- Z*) < /3TX, where cTx is the objective function, z* is the (unknown) optimal objective value of the linear program, and Tx measures the infeasibility of the current iterate x. This balancing constraint can be used to either emphasize rapid attainment of feasibility (set large) at the possible expense of good objective function values or to emphasize rapid attainment of good objective values (set /3 small) at the possible expense of a lower infeasibility gap. The algorithm exhibits the following advantageous features: (i) the iterate solutions monotonically decrease the infeasibility measure, (ii) the iterate solutions satisy the /3-balancing constraint, (iii) the iterate solutions achieve constant improvement in both Phase I and Phase II in O(n) iterations, (iv) there is always a possibility of finite termination of the Phase I problem, and (v) the algorithm is amenable to acceleration via linesearch of the potential function.

Nonlinear, local and highly parallel algorithms can perform several simple but important visual computations. Specific classes of algorithms can be considered in an abstract way. I study here the class of polynomial algorithms to exemplify some of the important issues for visual processing like linear vs. nonlinear and local vs. global. Polynomial algorithms are a natural extension of Perceptrons to time dependent grey level images.. Although they share most of the limitations of Perceptrons, they are powerful parallel computational devices. Several of their properties are characterized and especially (a) their equivalence with Perceptrons for geometrical figures and (b) the synthesis of non-linear algorithms (mappings) via associative learning. Finally, the paper considers how algorithms of this type could be implemented in nervous hardware, in terms of synaptic interactions strategically located in a dendritic tree.