Convergence of partial-wave expansions for energies, scattering amplitudes and positron annihilation rates

We use many-body theory to find the asymptotic behaviour of second-order correlation corrections to the energies and positron annihilation rates in many- electron systems with respect to the angular momenta l of the single-particle orbitals included. The energy corrections decrease as 1/(l+1/2)4, in agreement with the result of Schwartz, whereas the positron annihilation rate has a slower 1/(l+1/2)2 convergence rate. We illustrate these results by numerical calculations of the energies of Ne and Kr and by examining results from extensive con?guration-interaction calculations of PsH binding and annihilation.

A splitting result for the algebraic K-theory of projective toric schemes

Suppose X is a projective toric scheme defined over a ring R and equipped with an ample line bundle L . We prove that its K-theory has a direct summand of the form K(R)(k+1) where k = 0 is minimal such that L?(-k-1) is not acyclic. Using a combinatorial description of quasi-coherent sheaves we interpret and prove this result for a ring R which is either commutative, or else left noetherian.

Mixed radix Reed-Muller expansions

The choice of radix is crucial for multi-valued logic synthesis. Practical examples, however, reveal that it is not always possible to find the optimal radix when taking into consideration actual physical parameters of multi-valued operations. In other words, each radix has its advantages and disadvantages. Our proposal is to synthesise logic in different radices, so it may benefit from their combination. The theory presented in this paper is based on Reed-Muller expansions over Galois field arithmetic. The work aims to firstly estimate the potential of the new approach and to secondly analyse its impact on circuit parameters down to the level of physical gates. The presented theory has been applied to real-life examples focusing on cryptographic circuits where Galois Fields find frequent application. The benchmark results show the approach creates a new dimension for the trade-off between circuit parameters and provides information on how the implemented functions are related to different radices.

Distinguished algebraic semantics for t-norm based fuzzy logics:Methods and algebraic equivalencies

This paper is a contribution to Mathematical fuzzy logic, in particular to the algebraic study of t-norm based fuzzy logics. In the general framework of propositional core and ?-core fuzzy logics we consider three properties of completeness with respect to any semantics of linearly ordered algebras. Useful algebraic characterizations of these completeness properties are obtained and their relations are studied. Moreover, we concentrate on five kinds of distinguished semantics for these logics-namely the class of algebras defined over the real unit interval, the rational unit interval, the hyperreals (all ultrapowers of the real unit interval), the strict hyperreals (only ultrapowers giving a proper extension of the real unit interval) and finite chains, respectively-and we survey the known completeness methods and results for prominent logics. We also obtain new interesting relations between the real, rational and (strict) hyperreal semantics, and good characterizations for the completeness with respect to the semantics of finite chains. Finally, all completeness properties and distinguished semantics are also considered for the first-order versions of the logics where a number of new results are proved. © 2009 Elsevier B.V. All rights reserved.

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.

Many valued algebraic structures as measures of comparison

This thesis studies the properties and usability of operators called t-norms, t-conorms, uninorms, as well as many valued implications and equivalences. Into these operators, weights and a generalized mean are embedded for aggregation, and they are used for comparison tasks and for this reason they are referred to as comparison measures. The thesis illustrates how these operators can be weighted with a differential evolution and aggregated with a generalized mean, and the kinds of measures of comparison that can be achieved from this procedure. New operators suitable for comparison measures are suggested. These operators are combination measures based on the use of t-norms and t-conorms, the generalized 3_-uninorm and pseudo equivalence measures based on S-type implications. The empirical part of this thesis demonstrates how these new comparison measures work in the field of classification, for example, in the classification of medical data. The second application area is from the field of sports medicine and it represents an expert system for defining an athlete's aerobic and anaerobic thresholds. The core of this thesis offers definitions for comparison measures and illustrates that there is no actual difference in the results achieved in comparison tasks, by the use of comparison measures based on distance, versus comparison measures based on many valued logical structures. The approach has been highly practical in this thesis and all usage of the measures has been validated mainly by practical testing. In general, many different types of operators suitable for comparison tasks have been presented in fuzzy logic literature and there has been little or no experimental work with these operators.

An Abstract Algebraic Theory of L-Fuzzy Relations for Relational Databases

Classical relational databases lack proper ways to manage certain real-world situations including imprecise or uncertain data. Fuzzy databases overcome this limitation by allowing each entry in the table to be a fuzzy set where each element of the corresponding domain is assigned a membership degree from the real interval [0…1]. But this fuzzy mechanism becomes inappropriate in modelling scenarios where data might be incomparable. Therefore, we become interested in further generalization of fuzzy database into L-fuzzy database. In such a database, the characteristic function for a fuzzy set maps to an arbitrary complete Brouwerian lattice L. From the query language perspectives, the language of fuzzy database, FSQL extends the regular Structured Query Language (SQL) by adding fuzzy specific constructions. In addition to that, L-fuzzy query language LFSQL introduces appropriate linguistic operations to define and manipulate inexact data in an L-fuzzy database. This research mainly focuses on defining the semantics of LFSQL. However, it requires an abstract algebraic theory which can be used to prove all the properties of, and operations on, L-fuzzy relations. In our study, we show that the theory of arrow categories forms a suitable framework for that. Therefore, we define the semantics of LFSQL in the abstract notion of an arrow category. In addition, we implement the operations of L-fuzzy relations in Haskell and develop a parser that translates algebraic expressions into our implementation.

A Cryptosystem Using the Concepts of Algebraic Geometric Code

Cryptosystem using linear codes was developed in 1978 by Mc-Eliece. Later in 1985 Niederreiter and others developed a modified version of cryptosystem using concepts of linear codes. But these systems were not used frequently because of its larger key size. In this study we were designing a cryptosystem using the concepts of algebraic geometric codes with smaller key size. Error detection and correction can be done efficiently by simple decoding methods using the cryptosystem developed. Approach: Algebraic geometric codes are codes, generated using curves. The cryptosystem use basic concepts of elliptic curves cryptography and generator matrix. Decrypted information takes the form of a repetition code. Due to this complexity of decoding procedure is reduced. Error detection and correction can be carried out efficiently by solving a simple system of linear equations, there by imposing the concepts of security along with error detection and correction. Results: Implementation of the algorithm is done on MATLAB and comparative analysis is also done on various parameters of the system. Attacks are common to all cryptosystems. But by securely choosing curve, field and representation of elements in field, we can overcome the attacks and a stable system can be generated. Conclusion: The algorithm defined here protects the information from an intruder and also from the error in communication channel by efficient error correction methods.

Spectral Analysis of Bounded Self-adjoint operators - A Linear Algebraic Approach

This thesis Entitled Spectral theory of bounded self-adjoint operators -A linear algebraic approach.The main results of the thesis can be classified as three different approaches to the spectral approximation problems. The truncation method and its perturbed versions are part of the classical linear algebraic approach to the subject. The usage of block Toeplitz-Laurent operators and the matrix valued symbols is considered as a particular example where the linear algebraic techniques are effective in simplifying problems in inverse spectral theory. The abstract approach to the spectral approximation problems via pre-conditioners and Korovkin-type theorems is an attempt to make the computations involved, well conditioned. However, in all these approaches, linear algebra comes as the central object. The objective of this study is to discuss the linear algebraic techniques in the spectral theory of bounded self-adjoint operators on a separable Hilbert space. The usage of truncation method in approximating the bounds of essential spectrum and the discrete spectral values outside these bounds is well known. The spectral gap prediction and related results was proved in the second chapter. The discrete versions of Borg-type theorems, proved in the third chapter, partly overlap with some known results in operator theory. The pure linear algebraic approach is the main novelty of the results proved here.