Local operator multipliers and positivity

We establish an unbounded version of Stinespring's Theorem and a lifting result for Stinespring representations of completely positive modular maps defined on the space of all compact operators. We apply these results to study positivity for Schur multipliers. We characterise positive local Schur multipliers, and provide a description of positive local Schur multipliers of Toeplitz type. We introduce local operator multipliers as a non-commutative analogue of local Schur multipliers, and characterise them extending both the characterisation of operator multipliers from [16] and that of local Schur multipliers from [27]. We provide a description of the positive local operator multipliers in terms of approximation by elements of canonical positive cones.

Quantum Chromatic Numbers via Operator Systems

We define several new types of quantum chromatic numbers of a graph and characterize them in terms of operator system tensor products. We establish inequalities between these chromatic numbers and other parameters of graphs studied in the literature and exhibit a link between them and non-signalling correlation boxes.

Operator and Workflow Optimization for High-Performance Analytics

We make a case for studying the impact of intra-node parallelism on the performance of data analytics. We identify four performance optimizations that are enabled by an increasing number of processing cores on a chip. We discuss the performance impact of these opimizations on two analytics operators and we identify how these optimizations affect each another.

Network virtualisation from an operator perspective

Network virtualisation is seen as a promising approach to overcome the so-called “Internet impasse” and bring innovation back into the Internet, by allowing easier migration towards novel networking approaches as well as the coexistence of complementary network architectures on a shared infrastructure in a commercial context. Recently, the interest from the operators and mainstream industry in network virtualisation has grown quite significantly, as the potential benefits of virtualisation became clearer, both from an economical and an operational point of view. In the beginning, the concept has been mainly a research topic and has been materialized in small-scale testbeds and research network environments. This PhD Thesis aims to provide the network operator with a set of mechanisms and algorithms capable of managing and controlling virtual networks. To this end, we propose a framework that aims to allocate, monitor and control virtual resources in a centralized and efficient manner. In order to analyse the performance of the framework, we performed the implementation and evaluation on a small-scale testbed. To enable the operator to make an efficient allocation, in real-time, and on-demand, of virtual networks onto the substrate network, it is proposed a heuristic algorithm to perform the virtual network mapping. For the network operator to obtain the highest profit of the physical network, it is also proposed a mathematical formulation that aims to maximize the number of allocated virtual networks onto the physical network. Since the power consumption of the physical network is very significant in the operating costs, it is important to make the allocation of virtual networks in fewer physical resources and onto physical resources already active. To address this challenge, we propose a mathematical formulation that aims to minimize the energy consumption of the physical network without affecting the efficiency of the allocation of virtual networks. To minimize fragmentation of the physical network while increasing the revenue of the operator, it is extended the initial formulation to contemplate the re-optimization of previously mapped virtual networks, so that the operator has a better use of its physical infrastructure. It is also necessary to address the migration of virtual networks, either for reasons of load balancing or for reasons of imminent failure of physical resources, without affecting the proper functioning of the virtual network. To this end, we propose a method based on cloning techniques to perform the migration of virtual networks across the physical infrastructure, transparently, and without affecting the virtual network. In order to assess the resilience of virtual networks to physical network failures, while obtaining the optimal solution for the migration of virtual networks in case of imminent failure of physical resources, the mathematical formulation is extended to minimize the number of nodes migrated and the relocation of virtual links. In comparison with our optimization proposals, we found out that existing heuristics for mapping virtual networks have a poor performance. We also found that it is possible to minimize the energy consumption without penalizing the efficient allocation. By applying the re-optimization on the virtual networks, it has been shown that it is possible to obtain more free resources as well as having the physical resources better balanced. Finally, it was shown that virtual networks are quite resilient to failures on the physical network.

An algebraic operator approach to the analysis of Gerber-Shiu functions

We introduce an algebraic operator framework to study discounted penalty functions in renewal risk models. For inter-arrival and claim size distributions with rational Laplace transform, the usual integral equation is transformed into a boundary value problem, which is solved by symbolic techniques. The factorization of the differential operator can be lifted to the level of boundary value problems, amounting to iteratively solving first-order problems. This leads to an explicit expression for the Gerber-Shiu function in terms of the penalty function.

Discrete commutative difference operator theory

The object of this thesis is to formulate a basic commutative difference operator theory for functions defined on a basic sequence, and a bibasic commutative difference operator theory for functions defined on a bibasic sequence of points, which can be applied to the solution of basic and bibasic difference equations. in this thesis a brief survey of the work done in this field in the classical case, as well as a review of the development of q~difference equations, q—analytic function theory, bibasic analytic function theory, bianalytic function theory, discrete pseudoanalytic function theory and finally a summary of results of this thesis

IRIS BIOMETRIC RECOGNITION SYSTEM EMPLOYING CANNY OPERATOR

Biometrics has become important in security applications. In comparison with many other biometric features, iris recognition has very high recognition accuracy because it depends on iris which is located in a place that still stable throughout human life and the probability to find two identical iris's is close to zero. The identification system consists of several stages including segmentation stage which is the most serious and critical one. The current segmentation methods still have limitation in localizing the iris due to circular shape consideration of the pupil. In this research, Daugman method is done to investigate the segmentation techniques. Eyelid detection is another step that has been included in this study as a part of segmentation stage to localize the iris accurately and remove unwanted area that might be included. The obtained iris region is encoded using haar wavelets to construct the iris code, which contains the most discriminating feature in the iris pattern. Hamming distance is used for comparison of iris templates in the recognition stage. The dataset which is used for the study is UBIRIS database. A comparative study of different edge detector operator is performed. It is observed that canny operator is best suited to extract most of the edges to generate the iris code for comparison. Recognition rate of 89% and rejection rate of 95% is achieved

Orthogonal polynomials and recurrence equations, operator equations and factorization

This article surveys the classical orthogonal polynomial systems of the Hahn class, which are solutions of second-order differential, difference or q-difference equations. Orthogonal families satisfy three-term recurrence equations. Example applications of an algorithm to determine whether a three-term recurrence equation has solutions in the Hahn class - implemented in the computer algebra system Maple - are given. Modifications of these families, in particular associated orthogonal systems, satisfy fourth-order operator equations. A factorization of these equations leads to a solution basis.

Contribution of the exact Breit operator plus the electron state-dependent screening to the multiplet and fine structure of muon-electron atoms

We investigate for very general cases the multiplet and fine structure splitting of muonelectron atoms arising from the coupling of the electron and muon angular momenta, including the effect of the Breit operator plus the electron state-dependent screening. Although many conditions have to be fulfilled simultaneously to observe these effeets, it should be possible to measure them in the 6h- 5g muonic transition in the Sn region.

On Oseen Resolvent Estimates: A Negative Result

We consider the resolvent problem for the scalar Oseen equation in the whole space R^3. We show that for small values of the resolvent parameter it is impossible to obtain an L^2-estimate analogous to the one which is valid for the Stokes resolvent, even if the resolvent parameter has positive real part.

Elimination in Operator Algebras

A large class of special functions are solutions of systems of linear difference and differential equations with polynomial coefficients. For a given function, these equations considered as operator polynomials generate a left ideal in a noncommutative algebra called Ore algebra. This ideal with finitely many conditions characterizes the function uniquely so that Gröbner basis techniques can be applied. Many problems related to special functions which can be described by such ideals can be solved by performing elimination of appropriate noncommutative variables in these ideals. In this work, we mainly achieve the following: 1. We give an overview of the theoretical algebraic background as well as the algorithmic aspects of different methods using noncommutative Gröbner elimination techniques in Ore algebras in order to solve problems related to special functions. 2. We describe in detail algorithms which are based on Gröbner elimination techniques and perform the creative telescoping method for sums and integrals of special functions. 3. We investigate and compare these algorithms by illustrative examples which are performed by the computer algebra system Maple. This investigation has the objective to test how far noncommutative Gröbner elimination techniques may be efficiently applied to perform creative telescoping.

Operator expansion: definition and molecular integral applications

Es defineix l'expansió general d'operadors com una combinació lineal de projectors i s'exposa la seva aplicació generalitzada al càlcul d'integrals moleculars. Com a exemple numèric, es fa l'aplicació al càlcul d'integrals de repulsió electrònica entre quatre funcions de tipus s centrades en punts diferents, i es mostren tant resultats del càlcul com la definició d'escalat respecte a un valor de referència, que facilitarà el procés d'optimització de l'expansió per uns paràmetres arbitraris. Es donen resultats ajustats al valor exacte