1 
Theory of integrable latticesCheng, Y. January 1987 (has links)
This thesis deals with the theory of integrable lattices in "solitons" throughout. Chapter 1 is a general introduction, which includes an historical survey and a short surrunary of the "solitons" theory and the present work. In Chapter 2, we discuss the equivalence between two kinds of lattice AKNS spectral problems  one includes two potentials, while the other includes four. The two nonlinear lattice systems associated with those two spectral problems, respectively is also proved to be equivalent to each other. In Chapter 3, we derive a class of nonlinear differentialdifference equations (NDDEs) and put them into the Hamiltonian systems. Their complete integrability are proved in terms of so called "rmatrix". In the end of this Chapter, we study the symmetry properties and the related topics for lattice systems. In particular, we give detail for the Toda lattice systems. Chapter 4 is concerned with the Backlund transformations (BTs) and nonlinear superposition formulae (NSFs) for a class of NDDEs. A new method is presented to derive the generalized BTs and to prove that these BTs are precisely and really the autoBTs. The three kinds of NSFs are derived by analysis of so called "elementary BTs". In Chapter 5, we investigate some relations between our lattices and the wellstudied continuous systems. The continuum limits of our lattice systems and the discretizations of the continuous systems are discussed. The other study is about how we can consider a BT of continuous systems as a NDDE and then how a BT of such a NDDE can be reduced to the three kinds of NSFs of the continuous systems. The last Chapter is a study of integrable lattices under periodic boundary conditions. It provides a mathematical foundation for the study of integrable models in statistical mechanics. We are particularly interested in the lattice sineGordon and sinhGordon models. We not only prove the integrability of these models but also derive all kinds of classical phase shifts and some other physically interesting relations.

2 
Symplectic topology of some Stein and rational surfacesEvans, J. D. January 2010 (has links)
A <i>symplectic manifold </i>is a 2<i>n</i>dimensional smooth manifold endowed with a closed, nondegenerate 2form. This picks out the set of <i>Lagrangian submanifolds, n</i>dimensional submanifolds on which the 2form vanishes, and the group of <i>symplectomorphisms, </i>diffeomorphisms which preserve the symplectic form. In this thesis I study the homotopy type of the (compactlysupported) symplectomorphism group and the connectivity of the space of Lagrangian spheres for an array of symplectic 4manifolds comprising some Stein surfaces and some Del Prezzo surfaces. In part I of the thesis, concerning Stein surfaces, I calculate the homotopy type of the compactlysupported symplectomorphism group for C* x C with its split symplectic form and <i>T</i>*RP<sup>2</sup> with its canonical symplectic form. More significantly, I show that the compactlysupported symplectomorphism group of the 4dimensional <i>A<sub>n</sub></i>Milnor fibre {<i>x</i><sup>2</sup> + <i>y</i><sup>2</sup> + <i>z<sup>n</sup></i><sup>+1</sup> = 1} is homotopy equivalent to a discrete group which injects naturally into the braid group on <i>n</i> + 1strands. In part II of the thesis, concerning Del Pezzo surfaces: I show that the isotopy class of a Lagrangian sphere in the monotone 2, 3 or 4point blowup of CP<sup>2</sup> is determined by its homology class; I calculate the homotopy type of the symplectomorphism group for the monotone 3, 4 and 5point blowups of CP<sup>2</sup>. The calculations of homotopy groups of symplectomorphism groups rely on nothing more than the standard technology of pseudoholomorphic curves and some involved topological arguments to prove the fibration property of various maps between infinitedimensional spaces. The new idea is the compactification of the Milnor fibres by a configuration of holomorphic spheres which puts the calculation in a context familiar from the world of LalondePinsonnault and Abreu. The classification of Lagrangian spheres is based on an argument of Richard Hind.

3 
Topics in arithmetic combinatoricsGreen, B. January 2003 (has links)
No description available.

4 
Combinatorial embeddings and isotopiesHudson, J. F. P. January 1964 (has links)
No description available.

5 
Allied subsets of topological groups and linear spacesJameson, G. J. O. January 1969 (has links)
No description available.

6 
Metalevel and reflexive extension in mechanical theorem provingMatthews, S. January 1994 (has links)
In spite of many years of research into mechanical assistance for mathematics it is still much more difficult to construct a proof on a machine than on paper. Of course this is partly because, unlike a proof on paper, a machine checked proof must be formal in the strictest sense of that word, but it is also because usually the ways of going about building proofs on a machine are limited compared to what a mathematician is used to. This thesis looks at some possible extensions to the range of tools available on a machine that might lend a user more flexibility in proving theorems, complementing whatever is already available. In particular, it examines what is possible in a framework theorem prover. Such a system, if it is configured to prove theorems in a particular logic T, must have a formal description of the proof theory of T written in the framework theory F of the system. So it should be possible to use whatever facilities are available in F not only to prove theorems of T, but also theorems about T that can then be used in their turn to aid the user in building theorems of T. The thesis is divided into three parts. The first describes the theory FS0, which has been suggested by Feferman as a candidate for a framework theory suitable for doing metatheory. The second describes some experiments with FS0, proving metatheorems. The third describes an experiment in extending the theory PRA, declared in FS0, with a reflection facility.

7 
Tensor products of Banach spacesMilne, H. K. January 1975 (has links)
No description available.

8 
Within and without  the geometry of knotsBall, A. A. January 1972 (has links)
The aim of this thesis is to evolve simple answers to the questions 'What is a knot? ' and 'when are two knots the same?' The work is far more in the spirit of, for example, Reidemeister's 'Knotentheorie' rather than the strictly topological approach of more recent writers, like Crowell and Fox. There are few prerequisites for understanding the thesis. Generally the work is elementary and complete in itself. Occasionally I have referred to works in the Bibliography; enclosing the author's name in brackets, but a knowledge of these works is not necessary. They are listed primarily to acknowledge my continual reference to them during my research. I have used a number of special terms, and have underlined each where it is introduced. This draws attention to the new term and later makes the definition easier to find. If the term is to be used in another section then it is listed in the Index of Definitions.

9 
Matrices of nonnegative elementsSchneider, H. January 1952 (has links)
No description available.

10 
Topics in seminearring theorySamman, M. S. January 1998 (has links)
The idea of a seminearring was introduced in [8], as an algebraic system that can be constructed from a set <I>S</I> with two binary operations : addition + and multiplication ., such that (<I>S</I>, +) and (<I>S</I>, .) are semigroups and one distributive law is satisfied. A seminearring <I>S</I> is called distributively generated (d.g.) if <I>S</I> contains a multiplicative subsemigroup (<I>T</I>, .) of distributive elements which generates (<I>S</I>, +). Unlike the nearrings case for which a rich theory has already been developed, very little seems to be known about seminearrings. The aim of this dissertation consists mainly of two goals. The first is to generalize some results which are known in the theory of nearrings. The second goal of this thesis appears mainly in the last 6 chapters in which we obtain some results about seminearrings of endomorphisms. In chapter 1, the definitions and basic concepts about seminearrings are given; e.g. an arbitrary seminearring can be embedded in a seminearring of the form <I>M</I>(<I>S</I>). Fröhlich [1], [2] and Meldrum [5] have given some results concerning free d.g. nearrings in a variety <I>V</I>. In chapter 2, we generalize some of these results to free d.g. seminearrings and we can prove the existence of free (<I>S,T</I>)semigroups on a set <I>X</I> in a variety <I>V</I>. In section 2.4, we prove a theorem which asserts that not every d.g. seminearring has a faithful representation. This would generalize the result which was given by Meldrum [5] for the nearring case. Chapter 3 gives an overview of strong semilattices of nearrings and of rings. In this context we show that a strong semilattice of nearrings is a seminearring while a strong semilattice of rings is a nearring. Chapter 4 is designed to be a preparatory chapter for the remaining part of the thesis. It explains the main plan which will be followed in all the last 6 chapters. It also includes some basic ideas and results which are of great use in the remaining work.

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