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Generic Algebras and Kazhdan-Lusztig Theory for Monomial Groups: The Iwahori-Hecke algebras of Coxeter groups play a central role in the study of representations of semisimple Lie-type groups. An important tool is the combinatorial approach to representations of Iwahori-Hecke algebras introduced by Kazhdan and Lusztig in 1979. In this dissertation, I discuss a generalization of the Iwahori-Hecke algebra of the symmetric group that is instead based on the complex reflection group G(r,1,n). Using the analogues of Kazhdan and Lusztig's R-polynomials, I show that this algebra determines a partial order on G(r,1,n) that generalizes the Chevalley-Bruhat order on the symmetric group. I also consider possible analogues of Kazhdan-Lusztig polynomials.
Mathematical Modeling of Charged Liquid Droplets: Numerical Simulation and Stability Analysis: The goal of this thesis is to study of the evolution of 3D electrically charged liquid droplets of fluid evolving under the influence of surface tension and electrostatic forces. In the first part of the thesis, an appropriate mathematical model of the problem is introduced and the linear stability analysis is developed by perturbing a sphere with spherical harmonics. In the second part, the numerical solution of the problem is described with the use of the boundary elements method (BEM) on an adaptive mesh of triangular elements. The numerical method is validated by comparison with exact solutions. Finally, various numerical results are presented. These include neck formation in droplets, the evolution of surfaces with holes, singularity formation on droplets with various symmetries and numerical evidence that oblate spheroids are unstable.
Applications in Fixed Point Theory: Banach's contraction principle is probably one of the most important theorems in fixed point theory. It has been used to develop much of the rest of fixed point theory. Another key result in the field is a theorem due to Browder, Göhde, and Kirk involving Hilbert spaces and nonexpansive mappings. Several applications of Banach's contraction principle are made. Some of these applications involve obtaining new metrics on a space, forcing a continuous map to have a fixed point, and using conditions on the boundary of a closed ball in a Banach space to obtain a fixed point. Finally, a development of the theorem due to Browder et al. is given with Hilbert spaces replaced by uniformly convex Banach spaces.
Dynamics, Thermodynamic formalism and Perturbations of Transcendental Entire Functions of Finite Singular Type: In this dissertation, we study the dynamics, fractal geometry and the topology of the Julia set of functions in the family H which is a set in the class S, the Speiser class of entire transcendental functions which have only finitely many singular values. One can think of a function from H as a generalized expanding function from the cosh family. We shall build a version of thermodynamic formalism for functions in H and we shall show among others, the existence and uniqueness of a conformal measure. Then we prove a Bowen's type formula, i.e. we show that the Hausdorff dimension of the set of returning points, is the unique zero of the pressure function. We shall also study conjugacies in the family H, perturbation of functions in the family and related dynamical properties. We define Perron-Frobenius operators for some functions naturally associated with functions in the family H and then, using fundamental properties of these operators, we shall prove the important result that the Hausdorff dimension of the subset of returning points depends analytically on the parameter taken from a small open subset of the n-dimensional parameter space.
Hamiltonian cycles in subset and subspace graphs.: In this dissertation we study the Hamiltonicity and the uniform-Hamiltonicity of subset graphs, subspace graphs, and their associated bipartite graphs. In 1995 paper "The Subset-Subspace Analogy," Kung states the subspace version of a conjecture. The study of this problem led to a more general class of graphs. Inspired by Clark and Ismail's work in the 1996 paper "Binomial and Q-Binomial Coefficient Inequalities Related to the Hamiltonicity of the Kneser Graphs and their Q-Analogues," we defined subset graphs, subspace graphs, and their associated bipartite graphs. The main emphasis of this dissertation is to describe those graphs and study their Hamiltonicity. The results on subset graphs are presented in Chapter 3, on subset bipartite graphs in Chapter 4, and on subspace graphs and subspace bipartite graphs in Chapter 5. We conclude the dissertation by suggesting some generalizations of our results concerning the panciclicity of the graphs.
Lyapunov Exponents, Entropy and Dimension: We consider diffeomorphisms of a compact Riemann Surface. A development of Oseledec's Multiplicative Ergodic Theorem is given, along with a development of measure theoretic entropy and dimension. The main result, due to L.S. Young, is that for certain diffeomorphisms of a surface, there is a beautiful relationship between these three concepts; namely that the entropy equals dimension times expansion.
Thermodynamical Formalism: Thermodynamical formalism is a relatively recent area of pure mathematics owing a lot to some classical notions of thermodynamics. On this thesis we state and prove some of the main results in the area of thermodynamical formalism. The first chapter is an introduction to ergodic theory. Some of the main theorems are proved and there is also a quite thorough study of the topology that arises in Borel probability measure spaces. In the second chapter we introduce the notions of topological pressure and measure theoretic entropy and we state and prove two very important theorems, Shannon-McMillan-Breiman theorem and the Variational Principle. Distance expanding maps and their connection with the calculation of topological pressure cover the third chapter. The fourth chapter introduces Gibbs states and the very important Perron-Frobenius Operator. The fifth chapter establishes the connection between pressure and geometry. Topological pressure is used in the calculation of Hausdorff dimensions. Finally the sixth chapter introduces the notion of conformal measures.
A Detailed Proof of the Prime Number Theorem for Arithmetic Progressions: We follow a research paper that J. Elstrodt published in 1998 to prove the Prime Number Theorem for arithmetic progressions. We will review basic results from Dirichlet characters and L-functions. Furthermore, we establish a weak version of the Wiener-Ikehara Tauberian Theorem, which is an essential tool for the proof of our main result.
Exhaustivity, continuity, and strong additivity in topological Riesz spaces.: In this paper, exhaustivity, continuity, and strong additivity are studied in the setting of topological Riesz spaces. Of particular interest is the link between strong additivity and exhaustive elements of Dedekind s-complete Banach lattices. There is a strong connection between the Diestel-Faires Theorem and the Meyer-Nieberg Lemma in this setting. Also, embedding properties of Banach lattices are linked to the notion of strong additivity. The Meyer-Nieberg Lemma is extended to the setting of topological Riesz spaces and uniform absolute continuity and uniformly exhaustive elements are studied in this setting. Counterexamples are provided to show that the Vitali-Hahn-Saks Theorem and the Brooks-Jewett Theorem cannot be extended to submeasures or to the setting of Banach lattices.
Spaces of Compact Operators: In this dissertation we study the structure of spaces of operators, especially the space of all compact operators between two Banach spaces X and Y. Work by Kalton, Emmanuele, Bator and Lewis on the space of compact and weakly compact operators motivates much of this paper. Let L(X,Y) be the Banach space of all bounded linear operators between Banach spaces X and Y, K(X,Y) be the space of all compact operators, and W(X,Y) be the space of all weakly compact operators. We study problems related to the complementability of different operator ideals (the Banach space of all compact, weakly compact, completely continuous, resp. unconditionally converging) operators in the space of all bounded linear operators. The structure of Dunford-Pettis sets, strong Dunford-Pettis sets, and certain spaces of operators is studied in the context of the injective and projective tensor products of Banach spaces. Bibasic sequences are used to study relative norm compactness of strong Dunford-Pettis sets. Next, we use Dunford-Pettis sets to give sufficient conditions for K(X,Y) to contain c0.
Complemented Subspaces of Bounded Linear Operators: For many years mathematicians have been interested in the problem of whether an operator ideal is complemented in the space of all bounded linear operators. In this dissertation the complementation of various classes of operators in the space of all bounded linear operators is considered. This paper begins with a preliminary discussion of linear bounded operators as well as operator ideals. Let L(X, Y ) be a Banach space of all bounded linear operator between Banach spaces X and Y , K(X, Y ) be the space of all compact operators, and W(X, Y ) be the space of all weakly compact operators. We denote space all operator ideals by O.
The Study of Translation Equivalence on Integer Lattices: This paper is a contribution to the study of countable Borel equivalence relations on standard Borel spaces. We concentrate here on the study of the nature of translation equivalence. We study these known hyperfinite spaces in order to gain insight into the approach necessary to classify certain variables as either being hyperfinite or not. In Chapter 1, we will give the basic definitions and examples of spaces used in this work. The general construction of marker sets is developed in this work. These marker sets are used to develop several invariant tilings of the equivalence classes of specific variables . Some properties that are equivalent to hyperfiniteness in the certain space are also developed. Lastly, we will give the new result that there is a continuous injective embedding from certain defined variables.
Quantization Of Spin Direction For Solitary Waves in a Uniform Magnetic Field: It is known that there are nonlinear wave equations with localized solitary wave solutions. Some of these solitary waves are stable (with respect to a small perturbation of initial data)and have nonzero spin (nonzero intrinsic angular momentum in the centre of momentum frame). In this paper we consider vector-valued solitary wave solutions to a nonlinear Klein-Gordon equation and investigate the behavior of these spinning solitary waves under the inﬂuence of an externally imposed uniform magnetic ﬁeld. We ﬁnd that the only stationary spinning solitary wave solutions have spin parallel or antiparallel to the magnetic ﬁeld direction.
Analysis Of Sequential Barycenter Random Probability Measures via Discrete Constructions: Hill and Monticino (1998) introduced a constructive method for generating random probability measures with a prescribed mean or distribution on the mean. The method involves sequentially generating an array of barycenters that uniquely defines a probability measure. This work analyzes statistical properties of the measures generated by sequential barycenter array constructions. Specifically, this work addresses how changing the base measures of the construction affects the statististics of measures generated by the SBA construction. A relationship between statistics associated with a finite level version of the SBA construction and the full construction is developed. Monte Carlo statistical experiments are used to simulate the effect changing base measures has on the statistics associated with the finite level construction.
Understanding Ancient Math Through Kepler: A Few Geometric Ideas from The Harmony of the World: Euclid's geometry is well-known for its theorems concerning triangles and circles. Less popular are the contents of the tenth book, in which geometry is a means to study quantity in general. Commensurability and rational quantities are first principles, and from them are derived at least eight species of irrationals. A recently republished work by Johannes Kepler contains examples using polygons to illustrate these species. In addition, figures having these quantities in their construction form solid shapes (polyhedra) having origins though Platonic philosophy and Archimedean works. Kepler gives two additional polyhedra, and a simple means for constructing the “divine” proportion is given.
Dimensions in Random Constructions.: We consider random fractals generated by random recursive constructions, prove zero-one laws concerning their dimensions and find their packing and Minkowski dimensions. Also we investigate the packing measure in corresponding dimension. For a class of random distribution functions we prove that their packing and Hausdorff dimensions coincide.
Borel Determinacy and Metamathematics: Borel determinacy states that if G(T;X) is a game and X is Borel, then G(T;X) is determined. Proved by Martin in 1975, Borel determinacy is a theorem of ZFC set theory, and is, in fact, the best determinacy result in ZFC. However, the proof uses sets of high set theoretic type (N1 many power sets of ω). Friedman proved in 1971 that these sets are necessary by showing that the Axiom of Replacement is necessary for any proof of Borel Determinacy. To prove this, Friedman produces a model of ZC and a Borel set of Turing degrees that neither contains nor omits a cone; so by another theorem of Martin, Borel Determinacy is not a theorem of ZC. This paper contains three main sections: Martin's proof of Borel Determinacy; a simpler example of Friedman's result, namely, (in ZFC) a coanalytic set of Turing degrees that neither contains nor omits a cone; and finally, the Friedman result.
Quantization Dimension for Probability Definitions: The term quantization refers to the process of estimating a given probability by a discrete probability supported on a finite set. The quantization dimension Dr of a probability is related to the asymptotic rate at which the expected distance (raised to the rth power) to the support of the quantized version of the probability goes to zero as the size of the support is allowed to go to infinity. This assumes that the quantized versions are in some sense ``optimal'' in that the expected distances have been minimized. In this dissertation we give a short history of quantization as well as some basic facts. We develop a generalized framework for the quantization dimension which extends the current theory to include a wider range of probability measures. This framework uses the theory of thermodynamic formalism and the multifractal spectrum. It is shown that at least in certain cases the quantization dimension function D(r)=Dr is a transform of the temperature function b(q), which is already known to be the Legendre transform of the multifractal spectrum f(a). Hence, these ideas are all closely related and it would be expected that progress in one area could lead to new results in another. It would also be expected that the results in this dissertation would extend to all probabilities for which a quantization dimension function exists. The cases considered here include probabilities generated by conformal iterated function systems (and include self-similar probabilities) and also probabilities generated by graph directed systems, which further generalize the idea of an iterated function system.
Topological uniqueness results for the special linear and other classical Lie Algebras.: Suppose L is a complete separable metric topological group (ring, field, etc.). L is topologically unique if the Polish topology on L is uniquely determined by its underlying algebraic structure. More specifically, L is topologically unique if an algebraic isomorphism of L with any other complete separable metric topological group (ring, field, etc.) induces a topological isomorphism. A local field is a locally compact topological field with non-discrete topology. The only local fields (up to isomorphism) are the real, complex, and p-adic numbers, finite extensions of the p-adic numbers, and fields of formal power series over finite fields. We establish the topological uniqueness of the special linear Lie algebras over local fields other than the complex numbers (for which this result is not true) in the context of complete separable metric Lie rings. Along the way the topological uniqueness of all local fields other than the field of complex numbers is established, which is derived as a corollary to more general principles which can be applied to a larger class of topological fields. Lastly, also in the context of complete separable metric Lie rings, the topological uniqueness of the special linear Lie algebra over the real division algebra of quaternions, the special orthogonal Lie algebras, and the special unitary Lie algebras is proved.
Hyperspace Topologies: In this paper we study properties of metric spaces. We consider the collection of all nonempty closed subsets, Cl(X), of a metric space (X,d) and topologies on C.(X) induced by d. In particular, we investigate the Hausdorff topology and the Wijsman topology. Necessary and sufficient conditions are given for when a particular pseudo-metric is a metric in the Wijsman topology. The metric properties of the two topologies are compared and contrasted to show which also hold in the respective topologies. We then look at the metric space R-n, and build two residual sets. One residual set is the collection of uncountable, closed subsets of R-n and the other residual set is the collection of closed subsets of R-n having n-dimensional Lebesgue measure zero. We conclude with the intersection of these two sets being a residual set representing the collection of uncountable, closed subsets of R-n having n-dimensional Lebesgue measure zero.
The Pettis Integral and Operator Theory: Let (Ω, Σ, µ) be a finite measure space and X, a Banach space with continuous dual X*. A scalarly measurable function f: Ω→X is Dunford integrable if for each x* X*, x*f L1(µ). Define the operator Tf. X* → L1(µ) by T(x*) = x*f. Then f is Pettis integrable if and only if this operator is weak*-to-weak continuous. This paper begins with an overview of this function. Work by Robert Huff and Gunnar Stefansson on the operator Tf motivates much of this paper. Conditions that make Tf weak*-to-weak continuous are generalized to weak*-toweak continuous operators on dual spaces. For instance, if Tf is weakly compact and if there exists a separable subspace D X such that for each x* X*, x*f = x*fχDµ-a.e, then f is Pettis integrable. This nation is generalized to bounded operators T: X* → Y. To say that T is determined by D means that if x*| D = 0, then T (x*) = 0. Determining subspaces are used to help prove certain facts about operators on dual spaces. Attention is given to finding determining subspaces far a given T: X* → Y. The kernel of T and the adjoint T* of T are used to construct determining subspaces for T. For example, if T*(Y*) ∩ X is weak* dense in T*(Y*), then T is determined by T*(Y*) ∩ X. Also if ker(T) is weak* closed in X*, then the annihilator of ker(T) (in X) is the unique minimal determining subspace for T.
A Collapsing Result Using the Axiom of Determinancy and the Theory of Possible Cofinalities: Assuming the axiom of determinacy, we give a new proof of the strong partition relation on ω1. Further, we present a streamlined proof that J<λ+(a) (the ideal of sets which force cof Π α < λ) is generated from J<λ+(a) by adding a singleton. Combining these results with a polarized partition relation on ω1
Examples and Applications of Infinite Iterated Function Systems: The aim of this work is the study of infinite conformal iterated function systems. More specifically, we investigate some properties of a limit set J associated to such system, its Hausdorff and packing measure and Hausdorff dimension. We provide necessary and sufficient conditions for such systems to be bi-Lipschitz equivalent. We use the concept of scaling functions to obtain some result about 1-dimensional systems. We discuss particular examples of infinite iterated function systems derived from complex continued fraction expansions with restricted entries. Each system is obtained from an infinite number of contractions. We show that under certain conditions the limit sets of such systems possess zero Hausdorff measure and positive finite packing measure. We include an algorithm for an approximation of the Hausdorff dimension of limit sets. One numerical result is presented. In this thesis we also explore the concept of positively recurrent function. We use iterated function systems to construct a natural, wide class of such functions that have strong ergodic properties.
Infinite Planar Graphs: How many equivalence classes of geodesic rays does a graph contain? How many bounded automorphisms does a planar graph have? Neimayer and Watkins studied these two questions and answered them for a certain class of graphs. Using the concept of excess of a vertex, the class of graphs that Neimayer and Watkins studied are extended to include graphs with positive excess at each vertex. The results of this paper show that there are an uncountable number of geodesic fibers for graphs in this extended class and that for any graph in this extended class the only bounded automorphism is the identity automorphism.
Maximum-Sized Matroids with no Minors Isomorphic to U2,5, F7, F7¯, OR P7: Let M be the class of simple matroids which do not contain the 5-point line U2,5 , the Fano plane F7 , the non-Fano plane F7- , or the matroid P7 , as minors. Let h(n) be the maximum number of points in a rank-n matroid in M. We show that h(2)=4, h(3)=7, and h(n)=n(n+1)/2 for n>3, and we also find all the maximum-sized matroids for each rank.
A Presentation of Current Research on Partitions of Lines and Space: We present the results from three papers concerning partitions of vector spaces V over the set R of reals and of the set of lines in V.
The Computation of Ultrapowers by Supercompactness Measures: The results from this dissertation are a computation of ultrapowers by supercompactness measures and concepts related to such measures. The second chapter gives an overview of the basic ideas required to carry out the computations. Included are preliminary ideas connected to measures, and the supercompactness measures. Order type results are also considered in this chapter. In chapter III we give an alternate characterization of 2 using the notion of iterated ordinal measures. Basic facts related to this characterization are also considered here. The remaining chapters are devoted to finding bounds fwith arguments taking place both inside and outside the ultrapowers. Conditions related to the upper bound are given in chapter VI.
Countable Additivity, Exhaustivity, and the Structure of Certain Banach Lattices: The notion of uniform countable additivity or uniform absolute continuity is present implicitly in the Lebesgue Dominated Convergence Theorem and explicitly in the Vitali-Hahn-Saks and Nikodym Theorems, respectively. V. M. Dubrovsky studied the connection between uniform countable additivity and uniform absolute continuity in a series of papers, and Bartle, Dunford, and Schwartz established a close relationship between uniform countable additivity in ca(Σ) and operator theory for the classical continuous function spaces C(K). Numerous authors have worked extensively on extending and generalizing the theorems of the preceding authors. Specifically, we mention Bilyeu and Lewis as well as Brooks and Drewnowski, whose efforts molded the direction and focus of this paper. This paper is a study of the techniques used by Bell, Bilyeu, and Lewis in their paper on uniform exhaustivity and Banach lattices to present a Banach lattice version of two important and powerful results in measure theory by Brooks and Drewnowski. In showing that the notions of exhaustivity and continuity take on familiar forms in certain Banach lattices of measures they show that these important measure theory results follow as corollaries of the generalized Banach lattice versions. This work uses their template to generalize results established by Bator, Bilyeu, and Lewis.
On the Cohomology of the Complement of a Toral Arrangement: The dissertation uses a number of mathematical formula including de Rham cohomology with complex coefficients to state and prove extension of Brieskorn's Lemma theorem.
Natural Smooth Measures on the Leaves of the Unstable Manifold of Open Billiard Dynamical Systems: In this paper, we prove, for a certain class of open billiard dynamical systems, the existence of a family of smooth probability measures on the leaves of the dynamical system's unstable manifold. These measures describe the conditional asymptotic behavior of forward trajectories of the system. Furthermore, properties of these families are proven which are germane to the PYC programme for these systems. Strong sufficient conditions for the uniqueness of such families are given which depend upon geometric properties of the system's phase space. In particular, these results hold for a fairly nonrestrictive class of triangular configurations of scatterers.
Existence of Many Sign Changing Non Radial Solutions for Semilinear Elliptic Problems on Annular Domains: The aim of this work is the study of the existence and multiplicity of sign changing nonradial solutions to elliptic boundary value problems on annular domains.
A Generalization of Sturmian Sequences: Combinatorial Structure and Transcendence: We investigate a class of minimal sequences on a finite alphabet Ak = {1,2,...,k} having (k - 1)n + 1 distinct subwords of length n. These sequences, originally defined by P. Arnoux and G. Rauzy, are a natural generalization of binary Sturmian sequences. We describe two simple combinatorial algorithms for constructing characteristic Arnoux-Rauzy sequences (one of which is new even in the Sturmian case). Arnoux-Rauzy sequences arising from fixed points of primitive morphisms are characterized by an underlying periodic structure. We show that every Arnoux-Rauzy sequence contains arbitrarily large subwords of the form V^2+ε and, in the Sturmian case, arbitrarily large subwords of the form V^3+ε. Finally, we prove that an irrational number whose base b-digit expansion is an Arnoux-Rauzy sequence is transcendental.
Multifractal Analysis of Parabolic Rational Maps: The investigation of the multifractal spectrum of the equilibrium measure for a parabolic rational map with a Lipschitz continuous potential, φ, which satisfies sup φ < P(φ) x∈J(T) is conducted. More specifically, the multifractal spectrum or spectrum of singularities, f(α) is studied.
Plane Curves, Convex Curves, and Their Deformation Via the Heat Equation: We study the effects of a deformation via the heat equation on closed, plane curves. We begin with an overview of the theory of curves in R3. In particular, we develop the Frenet-Serret equations for any curve parametrized by arc length. This chapter is followed by an examination of curves in R2, and the resultant adjustment of the Frenet-Serret equations. We then prove the rotation index for closed, plane curves is an integer and for simple, closed, plane curves is ±1. We show that a curve is convex if and only if the curvature does not change sign, and we prove the Isoperimetric Inequality, which gives a bound on the area of a closed curve with fixed length. Finally, we study the deformation of plane curves developed by M. Gage and R. S. Hamilton. We observe that convex curves under deformation remain convex, and simple curves remain simple.
Primitive Substitutive Numbers are Closed under Rational Multiplication: Lehr (1991) proved that, if M(q, r) denotes the set of real numbers whose expansion in base-r is q-automatic i.e., is recognized by an automaton A = (Aq, Ar, ao, δ, φ) (or is the image under a letter to letter morphism of a fixed point of a substitution of constant length q) then M(q, r) is closed under addition and rational multiplication. Similarly if we let M(r) denote the set of real numbers α whose base-r digit expansion is ultimately primitive substitutive, i.e., contains a tail which is the image (under a letter to letter morphism) of a fixed point of a primitive substitution then in an attempt to generalize Lehr's result we show that the set M(r) is closed under multiplication by rational numbers. We also show that M(r) is not closed under addition.
Minimality of the Special Linear Groups: Let F denote the field of real numbers, complex numbers, or a finite algebraic extension of the p-adic field. We prove that the special linear group SLn(F) with the usual topology induced by F is a minimal topological group. This is accomplished by first proving the minimality of the upper triangular group in SLn(F). The proof for the upper triangular group uses an induction argument on a chain of upper triangular subgroups and relies on general results for locally compact topological groups, quotient groups, and subgroups. Minimality of SLn(F) is concluded by appealing to the associated Lie group decomposition as the product of a compact group and an upper triangular group. We also prove the universal minimality of homeomorphism groups of one dimensional manifolds, and we give a new simple proof of the universal minimality of S∞.
Polish Spaces and Analytic Sets: A Polish space is a separable topological space that can be metrized by means of a complete metric. A subset A of a Polish space X is analytic if there is a Polish space Z and a continuous function f : Z —> X such that f(Z)= A. After proving that each uncountable Polish space contains a non-Borel analytic subset we conclude that there exists a universally measurable non-Borel set.
A Topological Uniqueness Result for the Special Linear Groups: The goal of this paper is to establish the dependency of the topology of a simple Lie group, specifically any of the special linear groups, on its underlying group structure. The intimate relationship between a Lie group's topology and its algebraic structure dictates some necessary topological properties, such as second countability. However, the extent to which a Lie group's topology is an "algebraic phenomenon" is, to date, still not known.
Generalized Function Solutions to Nonlinear Wave Equations with Distribution Initial Data: In this study, we consider the generalized function solutions to nonlinear wave equation with distribution initial data. J. F. Colombeau shows that the initial value problem u_tt - Δu = F(u); m(x,0) = U_0; u_t (x,0) = i_1 where the initial data u_0 and u_1 are generalized functions, has a unique generalized function solution u. Here we take a specific F and specific distributions u_0, u_1 then inspect the generalized function representatives for the initial value problem solution to see if the generalized function solution is a distribution or is more singular. Using the numerical technics, we show for specific F and specific distribution initial data u_0, u_1, there is no distribution solution.
Tensor Products of Banach Spaces: Tensor products of Banach Spaces are studied. An introduction to tensor products is given. Some results concerning the reciprocal Dunford-Pettis Property due to Emmanuele are presented. Pelczyriski's property (V) and (V)-sets are studied. It will be shown that if X and Y are Banach spaces with property (V) and every integral operator from X into Y* is compact, then the (V)-subsets of (X⊗F)* are weak* sequentially compact. This in turn will be used to prove some stronger convergence results for (V)-subsets of C(Ω,X)*.
Descriptions and Computation of Ultrapowers in L(R): The results from this dissertation are an exact computation of ultrapowers by measures on cardinals $\aleph\sb{n},\ n\in w$, in $L(\IR$), and a proof that ordinals in $L(\IR$) below $\delta\sbsp{5}{1}$ represented by descriptions and the identity function with respect to sequences of measures are cardinals. An introduction to the subject with the basic definitions and well known facts is presented in chapter I. In chapter II, we define a class of measures on the $\aleph\sb{n},\ n\in\omega$, in $L(\IR$) and derive a formula for an exact computation of the ultrapowers of cardinals by these measures. In chapter III, we give the definitions of descriptions and the lowering operator. Then we prove that ordinals represented by descriptions and the identity function are cardinals. This result combined with the fact that every cardinal $<\delta\sbsp{5}{1}$ in $L(\IR$) is represented by a description (J1), gives a characterization of cardinals in $L(\IR$) below $\delta\sbsp{5}{1}. Concrete examples of formal computations are shown in chapter IV.
Existence of a Sign-Changing Solution to a Superlinear Dirichlet Problem: We study the existence, multiplicity, and nodal structure of solutions to a superlinear elliptic boundary value problem. Under specific hypotheses on the superlinearity, we show that there exist at least three nontrivial solutions. A pair of solutions are of one sign (positive and negative respectively), and the third solution changes sign exactly once. Our technique is variational, i.e., we study the critical points of the associated action functional to find solutions. First, we define a codimension 1 submanifold of a Sobolev space . This submanifold contains all weak solutions to our problem, and in our case, weak solutions are also classical solutions. We find nontrivial solutions which are local minimizers of our action functional restricted to various subsets of this submanifold. Additionally, if nondegenerate, the one-sign solutions are of Morse index 1 and the sign-changing solution has Morse index 2. We also establish that the action level of the sign-changing solution is bounded below by the sum of the two lesser levels of the one-sign solutions. Our results extend and complement the findings of Z. Q. Wang ([W]). We include a small sample of earlier works in the general area of superlinear elliptic boundary value problems.
A Numerical Method for Solving Singular Differential Equations Utilizing Steepest Descent in Weighted Sobolev Spaces: We develop a numerical method for solving singular differential equations and demonstrate the method on a variety of singular problems including first order ordinary differential equations, second order ordinary differential equations which have variational principles, and one partial differential equation.
On Groups of Positive Type: We describe groups of positive type and prove that a group G is of positive type if and only if G admits a non-trivial partition. We completely classify groups of type 2, and present examples of other groups of positive type as well as groups of type zero.
Physical Motivation and Methods of Solution of Classical Partial Differential Equations: We consider three classical equations that are important examples of parabolic, elliptic, and hyperbolic partial differential equations, namely, the heat equation, the Laplace's equation, and the wave equation. We derive them from physical principles, explore methods of finding solutions, and make observations about their applications.
Steepest Sescent on a Uniformly Convex Space: This paper contains four main ideas. First, it shows global existence for the steepest descent in the uniformly convex setting. Secondly, it shows existence of critical points for convex functions defined on uniformly convex spaces. Thirdly, it shows an isomorphism between the dual space of H^{1,p}[0,1] and the space H^{1,q}[0,1] where p > 2 and {1/p} + {1/q} = 1. Fourthly, it shows how the Beurling-Denny theorem can be extended to find a useful function from H^{1,p}[0,1] to L_{p}[1,0] where p > 2 and addresses the problem of using that function to establish a relationship between the ordinary and the Sobolev gradients. The paper contains some numerical experiments and two computer codes.
Sufficient Conditions for Uniqueness of Positive Solutions and Non Existence of Sign Changing Solutions for Elliptic Dirichlet Problems: In this paper we study the uniqueness of positive solutions as well as the non existence of sign changing solutions for Dirichlet problems of the form $$\eqalign{\Delta u + g(\lambda,\ u) &= 0\quad\rm in\ \Omega,\cr u &= 0\quad\rm on\ \partial\Omega,}$$where $\Delta$ is the Laplace operator, $\Omega$ is a region in $\IR\sp{N}$, and $\lambda>0$ is a real parameter. For the particular function $g(\lambda,\ u)=\vert u\vert\sp{p}u+\lambda$, where $p={4\over N-2}$, and $\Omega$ is the unit ball in $\IR\sp{N}$ for $N\ge3$, we show that there are no sign changing solutions for small $\lambda$ and also we show that there are no large sign changing solutions for $\lambda$ in a compact set. We also prove uniqueness of positive solutions for $\lambda$ large when $g(\lambda,\ u)=\lambda f(u)$, where f is an increasing, sublinear, concave function with f(0) $<$ 0, and the exterior boundary of $\Omega$ is convex. In establishing our results we use a number of methods from non-linear functional analysis such as rescaling arguments, methods of order, estimation near the boundary, and moving plane arguments.
Continuous, Nowhere-Differentiable Functions with no Finite or Infinite One-Sided Derivative Anywhere: In this paper, we study continuous functions with no finite or infinite one-sided derivative anywhere. In 1925, A. S. Beskovitch published an example of such a function. Since then we call them Beskovitch functions. This construction is presented in chapter 2, The example was simple enough to clear the doubts about the existence of Besicovitch functions. In 1932, S. Saks showed that the set of Besicovitch functions is only a meager set in C[0,1]. Thus the Baire category method for showing the existence of Besicovitch functions cannot be directly applied. A. P. Morse in 1938 constructed Besicovitch functions. In 1984, Maly revived the Baire category method by finding a non-empty compact subspace of (C[0,1], || • ||) with respect to which the set of Morse-Besicovitch functions is comeager.
Cycles and Cliques in Steinhaus Graphs: In this dissertation several results in Steinhaus graphs are investigated. First under some further conditions imposed on the induced cycles in steinhaus graphs, the order of induced cycles in Steinhaus graphs is at most [(n+3)/2]. Next the results of maximum clique size in Steinhaus graphs are used to enumerate the Steinhaus graphs having maximal cliques. Finally the concept of jumbled graphs and Posa's Lemma are used to show that almost all Steinhaus graphs are Hamiltonian.
Intuition versus Formalization: Some Implications of Incompleteness on Mathematical Thought: This paper describes the tension between intuition about number theory and attempts to formalize it. I will first examine the root of the dilemma, Godel's First Incompleteness Theorem, which demonstrates that in any reasonable formalization of number theory, there will be independent statements. After proving the theorem, I consider some of its consequences on intuition, focusing on Freiling's "Dart Experiment" which is based on our usual notion of the real numbers as a line. This experiment gives an apparent refutation of the Axiom of Choice and the Continuum Hypothesis; however, it also leads to an equally apparent paradox. I conclude that such paradoxes are inevitable as the formalization of mathematics takes us further from our initial intuitions.

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