10 research outputs found

    The Generalised Raychaudhuri Equations : Examples

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    Specific examples of the generalized Raychaudhuri Equations for the evolution of deformations along families of DD dimensional surfaces embedded in a background NN dimensional spacetime are discussed. These include string worldsheets embedded in four dimensional spacetimes and two dimensional timelike hypersurfaces in a three dimensional curved background. The issue of focussing of families of surfaces is introduced and analysed in some detail.Comment: 8 pages (Revtex, Twocolumn format). Corrected(see section on string worldsheets), reorganised and shortened slightl

    Compositional Reasoning for Explicit Resource Management in Channel-Based Concurrency

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    We define a pi-calculus variant with a costed semantics where channels are treated as resources that must explicitly be allocated before they are used and can be deallocated when no longer required. We use a substructural type system tracking permission transfer to construct coinductive proof techniques for comparing behaviour and resource usage efficiency of concurrent processes. We establish full abstraction results between our coinductive definitions and a contextual behavioural preorder describing a notion of process efficiency w.r.t. its management of resources. We also justify these definitions and respective proof techniques through numerous examples and a case study comparing two concurrent implementations of an extensible buffer.Comment: 51 pages, 7 figure

    The development of algorithms in mathematical programming

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    This thesis was submitted for the degree of Doctor of Philosophy and was awarded by Brunel University.In this thesis some problems in mathematical programming have been studied. Chapter 1 contains a brief review of the problems studied and the motivation for choosing these problems for further investigation. The development of two algorithms for finding all the vertices of a convex polyhedron and their applications are reported in Chapter 2. The linear complementary problem is studied in Chapter 3 and an algorithm to solve this problem is outlined. Chapter 4 contains a description of the plant location problem (uncapacited). This problem has been studied in some depth and an algorithm to solve this problem is presented. By using the Chinese representation of integers a new algorithm has been developed for transforming a nonsingular integer matrix into its Smith Normal Form; this work is discussed in Chapter 5. A hybrid algorithm involving the gradient method and the simplex method has also been developed to solve the linear programming problem. Chapter 6 contains a description of this method. The computer programs written in FORTRAN IV for these algorithms are set out in Appendices Rl to R5. A report on study of the group theory and its application in mathematical programming is presented as supplementary material. The algorithms in Chapter 2 are new. Part one of Chapter 3 is a collection of published material on the solution of the linear complementary problem; however the algorithm in Part two of this Chapter is original. The formulation of the plant location problem (uncapacited) together with some simplifications are claimed to be original. The use of Chinese representation of integers to transform an integer matrix into its Smith Normal Form is a new technique. The algorithm in Chapter 6 illustrates a new approach to solve the linear programming problem by a mixture of gradient and simplex method

    Unramified Extensions of the Cyclotomic Z_2-Extension of Q(sqrt(d),i)

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    Let F0 = Q((-d)½), K0 = Q(d½), and L0 = Q(d½, i) with d a square-free positive integer such that 2 does not divide d. Let Lj = L0(zeta22+j) so that the fields Lj are the cyclotomic Z2-extension of L0. We determine when fourth roots of certain elements of K0 generate unramified extensions of Lj. In particular, for elements of K0 that are relatively prime to 2 and are generators of principal ideals that are fourth powers, we give explicit congruence conditions under which the fourth root of the element gives an unramified extension. For any such element gamma, we show that if there is some j such that Lj(gamma1/4)/Lj is unramified, then L2(gamma1/4)/L2 is unramified. We also show that when (2) is split in F0, L2(gamma1/4)/L2 is unramified for any such gamma. This result is analogous to a result by Hubbard and Washington in which they work with the cyclotomic Z3-extension of Q((-d)½, zeta3) when 3 does not divide d and consider extensions generated by cube roots of elements in Q((3d)½). However, many more technical problems arise in the present work because the degree of the extension Lj/Kj is not relatively prime to the degrees of the extensions being generated. In order to prove our main results, we also give a congruence condition, which, for any number field K containing i and for any element gamma in K with gamma relatively prime to 2 and gamma a generator of an ideal that is a fourth power, dictates whether or not adjoining a fourth root of gamma to K gives an unramified extension

    On Subgroups and Epimorphic Images of Finite p-Groups

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    Über Struktur- und SensitivitĂ€tsaussagen in Ganzzahligen Programmen und deren Anwendung in Kombinatorischer Optimierung

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    In this thesis we investigate properties of integer linear programs (ILPs) and their algorithmic use. Our main focus are ILP-formulations that come from concrete algorthmic problems like the bin packing problem or the scheduling problem on identical machines. Especially for this kind of ILPs we study structural properties as well as properties for their sensitivity. As a result, we are able to answer open algorithmical questions in the area of approximation and parameterized complexity. In the context of sensitivity we analyze how much an ILP solution has to be adjusted when the parameters of the ILP change. There is a classical results by Cook et al. which gave bounds for that question when optimal solutions are considered. However, in this thesis we investigate the sensitivity of ILPs when approximate solutions are allowed, i.e. solutions that differ by a factor of at most (1+ \epsilon) from the optimum value. We could apply the obtained results to the online bin packing problem, when an approximation guarantee with ratio 1+Ï”1+ \epsilon has to be fulfilled and repacking of already assigned items (limited by the so called migration factor) is allowed. In the context of structural results, we prove the existence (assuming the ILP is feasible) of solutions of a certain class of ILPs with a certain simplified structure. Specifically, in this thesis, we prove structure properties for ILPs that arise from formulations of bin packing or scheduling problems and natural generalization of those formulations. Based on the those structure properties, we develop an efficient approximation scheme for the scheduling problem on identical machines with a running time of 2^{\tilde{O}(1/\epsilon)} + poly}(n) and furthermore, we develop a structure theorem, which is applied to the bin packing problem when the number of different item sizes d is bounded.In dieser Dissertation werden Eigenschaften von ganzzahligen linearen Programmen (engl. integer linear programs, kurz: ILPs) untersucht. Von Interesse sind dabei hauptsĂ€chlich ILP-Formulierungen, welche sich aus dem Kontext von algorithmischen Problemstellungen ergeben, wie beispielsweise dem Bin Packing-Problem und dem Scheduling-Problem auf identischen Maschinen. Insbesondere fĂŒr diese ILPs zeigen wir Strukturaussagen, sowie Aussagen ĂŒber die SensitivitĂ€t und können so offene algorithmische Fragestellungen im Bereich von Approximation und parametrisierter KomplexitĂ€t lösen. Im Kontext von SensitivitĂ€tsaussagen wird untersucht, inwiefern Lösung des ILPs angepasst werden können, wenn sich die Parameter des ILPs leicht Ă€ndern. Ein klassisches Resultat von Cook u.a. gibt dabei fĂŒr optimale Lösungen des ILPs AbschĂ€tzungen an. In dieser Arbeit betrachten wir AbschĂ€tzungen fĂŒr die SenstivitĂ€t wenn approximative Lösungen erlaubt sind, d.h. Lösungen deren Zielfunktionswert höchstens um einen Faktor 1+ \epsilon ĂŒber dem optimalen Zielfunktionswert liegt. Diese Ergebnisse konnten wir auf das Online-Bin Packing-Problem anwenden, wenn eine approximative Lösung mit GĂŒte 1+ \epsilon erreicht werden soll und in beschrĂ€nktem Maße Items umgepackt werden dĂŒrfen. Im Kontext von Strukturaussagen wird in dieser Dissertation die Existenz von ILP-Lösungen bewiesen, welche eine bestimmte vereinfachte Struktur aufweisen. Insbesondere, konnten wir Strukturaussagen fĂŒr ILPs entwickeln, welche sich aus Formulierungen des Bin Packing-Problems ergeben bzw. natĂŒrliche Verallgemeinerungen dieser Formulierung. Dadurch ist es uns zum einen gelungen ein effizientes Approximationsschemata fĂŒr das Scheduling-Problem auf identischen Maschinen mit einer Laufzeit von 2^{\tilde{O}(1/\epsilon)} + poly(n) zu entwicklen und außerdem konnten wir eine Strukturaussage entwickeln, welche unter anderem Anwendung im Bin Packung-Problem fand, wenn die Anzahl der unterschiedlichen ItemgrĂ¶ĂŸen d beschrĂ€nkt ist

    Generating sequences from the sums of binomial coefficients in a residue class modulo q.

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    For non-negative integers r we examine four families of alternating and non-alternating sign closed form binomial sums, Fs;ab(r; t; q), in a generalised congruence modulo q. We explore sums of squares and divisibility properties such as those determined by Weisman (and Fleck). Extending r to all integers we express the sequences in terms of closed form roots of unity and subsequently cosines. By a renumbering of these sequences we build eight new \diagonalised" sequences, Ls;abc(r; t; q), and construct equivalent closed forms and sums of squares relations. We modify Fibonacci type polynomials to construct order m recurrence polynomials that satisfy these diagonalised sequences. These recurrence polynomial sequences are shown to satisfy second order differential equations and exhibit orthogonal relations. From these latter relations we establish three term recurrence relations both between and within sequences. By the application of the reciprocal recurrence polynomial and hypergeometric functions, generating functions for these renumbered sequences are determined. Then employing these latter functions, we establish theorems that enable us to express each of the new sequences in terms of a Minor Corner Layered (MCL) determinant. When r is a negative integer and q = 2m+b is unspecified, the MCL determinants produce sequences of polynomials in m. For particular sequences we truncate these polynomials to contain only the leading coefficient and find that the truncated polynomial is equal to that of a Dirichlet series of the form zeta, lambda, beta or eta. From this relationship, recurrence polynomials for these latter functions are established Finally we develop a congruence for the denominator of the uncancelled modified Bernoulli numbers of the first kind, Bn=n!, and consequently a similar congruence for the zeta function at positive even valued integers. Furthermore we determine that these congruences obey the Fleck congruence

    Algebraic Approaches to State Complexity of Regular Operations

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    The state complexity of operations on regular languages is an active area of research in theoretical computer science. Through connections with algebra, particularly the theory of semigroups and monoids, many problems in this area can be simplified or completely reduced to combinatorial problems. We describe various algebraic techniques for attacking state complexity problems. We present a general method for constructing witness languages for operations -- languages that attain the worst-case state complexity when used as the argument(s) of the operation. Our construction is based on full transformation monoids, which contain all functions from a finite set into itself. When a witness for an operation is known, determining the state complexity essentially becomes a counting problem. These counting problems, however, are not necessarily easy, and the witness languages produced by this method are not ideal in the sense that they have extremely large alphabets. We thus investigate some commonly used operations in detail, and look for algebraic techniques to simplify the combinatorial side of state complexity problems and to simplify the search for small-alphabet witnesses. For boolean operations (e.g., union, intersection, difference) we show that these combinatorial problems can be solved easily in special cases by studying the subgroup of permutations in the syntactic monoid of a witness candidate. If the subgroup of permutations is known to have some strong transitivity property, such as primitivity or 2-transitivity, we can draw conclusions about the worst-case state complexity when this language is used in a boolean operation. For the operations of concatenation and Kleene star (an iterated version of concatenation), we describe a “construction set” method to simplify state complexity lower-bound proofs, and determine some algebraic conditions under which this method can be applied. For the reversal operation, we show that the state complexity of the reverse of a language is closely related to the syntactic monoid of the language, and use this fact to investigate a generalized version of the reversal state complexity problem. After describing our techniques, we demonstrate them by applying them to some classical state complexity problems. We obtain complex generalizations of the classical results that would be difficult to prove without the machinery we develop
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