Thesis (doctoral)--Universität Karlsruhe. / Vita. Includes bibliographical references (p. 91-92).

Thesis (doctoral)--Rheinische Friedrich-Wilhelms-Universität Bonn, 1995. / Includes bibliographical references (p. 53-[55]).

Thesis (Ph. D.)--University of Chicago, 1918. / "Reprinted from Dickson's History of the Theory of Numbers, Vol. III, Ch. VI. The Carnegie Institution of Washington." Includes bibliographical references and index.

Thesis (Ph.D.)--University of Chicago, 1918. / "Reprinted from Dickson's History of the theory of numbers, vol. III, ch. VI. The Carnegie Institution of Washington."

Thesis (Ph. D.)--University of Chicago, 1918. / "Reprinted from Dickson's History of the Theory of Numbers, Vol. III, Ch. VI. The Carnegie Institution of Washington." Includes bibliographical references and index. Also available on the Internet.

We study sums of squares formulae from the perspective of normed bilinear maps and their Hopf constructions. We begin with the geometric properties of quadratic forms between euclidean spheres. Let F: Sm → Sn be a quadratic form. For every point q in the image, the inverse image F⁻¹ (q) is the intersection of Sm with a linear subspace wq, whose dimension can be determined easily. In fact, for every k ≤ m+1 with nonempty Yk = {q ∈ Sn: dim Wq = k}, the restriction F⁻¹ (Yk) → Yk is a great (k-1) - sphere bundle. The quadratic form F is the Hopf construction of a normed bilinear map if and only if it admits a pair of "poles" ±p such that dim Wp + dim W₋p = m+1. In this case, the inverse images of points on a "meridian", save possibly the poles, are mutually isoclinic. Furthermore, the collection of all poles forms a great sphere of relatively low dimension. We also prove that the classical Hopf fibrations are the only nonconstant quadratic forms which are harmonic morphisms in the sense that the composite with every real valued harmonic function is again harmonic. Hidden in a quadratic form F: Sm → Sn are nonsingular bilinear maps Rk x Rm-k⁺¹ → Rn, one for each point in the image, all representing the homotopy class of F, which lies in Im J. Moreover, every hidden nonsingular bilinear map can be homotoped to a normed bilinear map. The existence of one sums of squares formula, therefore, anticipates others which cannot be obtained simply by setting some of the indeterminates to zero. These geometric and topological properties of quadratic forms are then used, together with homotopy theory results in the literature, to deduce that certain sums of squares formulae cannot exist, notably of types [12,12,20] and [16,16,24]. We also prove that there is no nonconstant quadratic form S²⁵ → S²³. Sums of squares formulae with integer coefficients are equivalent to "intercalate matrices of colors with appropriate signs". This combinatorial nature enables us to establish a stronger nonexistence result: no sums of squares formula of type [16,16, 28] can exist if only integer coefficients are permitted. We also classify integral [10,10,16] formulae, and show that they all represent ±2Ʋ∈ π [s over 3]. With the aid of the KO theory of real projective spaces, we determine, for given δ ≤ 5 and s, the greatest possible r for which there exists an [r,s,s+δ] formula. An explicit solution of the classical Hurwitz-Radon matrix equations is also recorded. / Science, Faculty of / Mathematics, Department of / Graduate

Thesis (Ph. D.)--Carleton University, 1992. / Also available in electronic format on the Internet.

This paper shall be mostly concerned with the development and the properties of three quadratic polynomials. The primary interest will by with n-ary quadratic polynomials, called forms.

This paper examines the probability that a random polynomial of specific degree over a field has a specific number of distinct roots in that field. Probabilities are found for random quadratic polynomials with respect to various probability measures on the real numbers and p-adic numbers. In the process, some properties of the p-adic integer uniform random variable are explored. The measure Witt ring, a generalization of the canonical Witt ring, is introduced as a way to link quadratic forms and measures, and examples are found for various fields and measures. Special properties of the Haar measure in connection with the measure Witt ring are explored. Higher-degree polynomials are explored with the aid of numerical methods, and some conjectures are made regarding higher-degree p-adic polynomials. Other open questions about measure Witt rings are stated. / Graduation date: 1999

Thesis (doctoral)--Universitate Litteraria Friderica Guilelma, 1835. / Vita.