<p> In this thesis we study the representation theorems for evenly positive definite functions on Euclidean spaces. A generalization of the concept of evenness on R^n to a concept of symmetry on any locally compact abelian group is given. In addition, a result analogous to the Weil-Povzner-Raikov Theorem is obtained for the representation of symmetrically positive definite functions on locally compact abelian groups.</p> / Thesis / Master of Science (MSc)

No description available.

The topic of this paper is the logical analysis and translation of definite descriptions (structures of the form ‘the F’), in particular Bertrand Russell’s Theory of Descriptions, as put forth in “On Denoting” (1905). I argue in favor of an opposing theory, a presuppositional analysis of definite descriptions that fits in the tradition of Frege and Strawson, building upon the recent work of Heim and Elbourne. I argue that a definite description has a referential function that is supported by presuppositions of existence and uniqueness located outside of the analyzed sentence. Using a series of example sentences, I show that a presuppositional analysis handles the logic of ordinary language in a manner superior to a Russellian analysis, produces more natural readings of embedded sentences containing definite descriptions, explains why definite descriptions function in a consistent way across different types of sentences, and provides a much better account of the logical commitments of using referential terms. After providing background on Russell’s theory and its criticisms, I review and analyze Elbourne’s examples of sentences embedding definite descriptions in non-doxastic propositional attitudes or the antecedents of conditionals. I then present and analyze my own examples involving embedding within disjunctions and within two kinds of non-statement: questions and commands. I compare the effects of embedding sentences using proper names—another kind of referential term. I then analyze some logical consequences of Russellian analysis, and answer a potential Russellian objection.

This paper is a partial replication study of research to investigate an aspect of English articles: the acquisition of 4 nongeneric uses of the definite article in English. Analysis of article use in these four categories (i.e. cultural, situational, structural, and textual) was investigated by Liu and Gleason (2002), and a hierarchy of difficulty and acquisition was proposed based upon the initial results. This partial replication used the same testing instrument as Liu and Gleason (2002) with additional items included to further investigate the category, cultural use of the article. The participants, 17 low-intermediate, 20 high-intermediate, and 34 advanced ESL learners completed a 100 item test with sentences containing deleted obligatory uses of the as well as distractor items. The results partially supported Liu and Gleason‟s original study, and raised many concerns with the test instrument.

This paper is a partial replication study of research to investigate an aspect of English articles: the acquisition of 4 nongeneric uses of the definite article in English. Analysis of article use in these four categories (i.e. cultural, situational, structural, and textual) was investigated by Liu and Gleason (2002), and a hierarchy of difficulty and acquisition was proposed based upon the initial results. This partial replication used the same testing instrument as Liu and Gleason (2002) with additional items included to further investigate the category, cultural use of the article. The participants, 17 low-intermediate, 20 high-intermediate, and 34 advanced ESL learners completed a 100 item test with sentences containing deleted obligatory uses of the as well as distractor items. The results partially supported Liu and Gleason‟s original study, and raised many concerns with the test instrument.

Thesis (PH. D.) - University of Michigan. / Reprinted from Duke mathematical journal, vol. 4, no. 3, September, 1938.

The aim of this work is to describe and to analyse one type of anaphoric nominal syntagma found in the articles of French sports newspapers. The work is divided in two parts theoretical and practical. The theoretical part describes the basic concepts of text linguistics. Then it focuses on the concept of anaphora and nominal syntagma with brief reference to journalistic-style writing. In the practical part, an analysis of anaphoric nominal syntagmas and the description of their semantico-pragmatic relations is carried out.

When making an estimate of the total of some quantity, sampling at carefully selected points will frequently be preferable to employing a method which involves randomization. The estimation of the total stand of timber on a given area or the amount of energy being released in a given time and space are examples of problems where specified points for sampling should result in a reduction the error of estimate. Problems such as these naturally lead us to to numerical integration methods. In the case of single integrals, the Newton-Cotes formulae can be applied directly to experimentally determined ordinates at equally-spaced abscissa points and are of great practical importance. Gauss’ formulae yield maximum efficiency with respect to controlling the polynomial error and can be used appropriately when an analytical expression for the curve in question is available but defies exact integration, or if for some other reason the statistical error is of minor importance. Tchebichef’s a formulae give maximum efficiency with respect to controlling the statistical or observational error. The basic elements in the development of numerical integration formulae like Newton-Cotes, Gauss' and Tchebichef's, can be extended to developing formulae for the approximate evaluation of multiple integrals. In the case of double integrals, an eight point and a thirteen point formula for fifth degree accuracy and a twelve point and a twenty-one point formula for seventh degree accuracy have been developed for integrating over a rectangle and similar formulae have been developed for integrating over areas bounded by a parabola and a straight line or two parabolas. Formulae for the numerical evaluation of triple integrals taken over a rectangular parallelepiped are developed, including a twenty-one point formula with fifth degree accuracy. It is shown that comparable formulae can be developed for integrating functions of more than three variables and a 2n /- 1 point formula with third degree accuracy for integrating a function of n variables over a rectangular n-space is obtained. In many problems involving statistical estimation, the dominant source of inaccuracies will be the error of observation. The magnitude of this error can be estimated by subjecting the observations to an orthogonal transformation which will isolate the trends and leave the residual variance free from these effects. This treatment is most easily carried out in terms of orthogonal polynomials and it is shown that this type analysis can be extended to functions of several variables. / Ph. D.

Using the representation theory of positive definite sequences some propositions in additive number theory are obtained and H. Bohr's approximation theorem is deduced. A unified approach to theorems by S. Bochner, S, N, Bernstein and H. Hamburger is discussed and some operator versions of numerical moment problems are studied. The Appendix contains comments to J. von Neumann's spectral theorem for self-adjoint operators in Hilbert space. / Thesis / Doctor of Philosophy (PhD)

In this PhD thesis we study some weighted regular Sturm-Liouville problems in which the weight function takes on both positive and negative signs in an appropriate interval [a,b]. With such  problems there is the possible existence of non-real eigenvalues, unlike in the definite case (i.e. left or right definite) in which only real eigenvalues exist. This PhD thesis consists of five papers (papers A-E) and an introduction to this area, which puts these papers into a more general frame. In paper A we give some precise estimates on the Richardson number for the two turning point case, thereby complementing the work of Jabon and Atkinson from 1984 in an essential way. We also give a corrected version of their result since there seems to be a typographical error in their paper. In paper B we show that the interlacing property, which holds in the one turning point case, does not hold in the two turning point case. The paper consists of a detailed presentation of numerical results of the case in which the weight function is allowed to change its sign twice in the interval (-1, 2). We also present some theoretical results which support the numerical results. Moreover, a number of new open questions are raised. We also observe that the real and imaginary parts of a non-real eigenfunction either have the same number of zeros in the interval (-1,2) or the numbers of zeros differ by two. In paper C, we obtain bounds on real and imaginary parts of non-real eigenvalues of a non-definite Sturm-Liouville problem, with Dirichlet boundary conditions, thus complementing the results obtained in a paper byBehrndt et.al. from 2013 in an essential way. In paper D we obtain a lower bound on the eigenvalue of the smallest modulus associated with a Dirichlet problem in the general case of a regular Sturm-Liouville problem. In paper E we expand upon the basic oscillation theory for general boundary problems of the form   -y''+q(x)y=λw(x)y,  on  I = [a,b], where  q(x) and w(x) are real-valued continuous functions on [a,b] and y is required to satisfy a pair of homogeneous separated boundary conditions at the end-points. Already in 1918 Richardson proved that, in the case of the Dirichlet problem,  if w(x) changes its sign exactly once and the boundary problem is  non-definite, then the zeros of the real and imaginary parts of any non-real eigenfunction interlace. We show that, unfortunately, this result is false in the case of two turning points, thus removing any hope for a general separation theorem for the zeros of the non-real eigenfunctions. Furthermore, we show that when a non-real eigenfunction vanishes inside I, then the absolute value of the difference between the total number of zeros of its real and imaginary parts is exactly 2.