Inaug.-diss.--Giessen. / Lebenslauf.

Thesis (Ph. D.)--University of Wisconsin--Madison, 1963. / Typescript. eContent provider-neutral record in process. Description based on print version record. Includes bibliographical references (leaves 52-53).

The object of this thesis is to cover the results of [1] from a piecewise linear point of view. The principal result of [1] is the theorem on the homogeneity of spheres, i.e. the complement of a combinatorial n-cell in a combinatorial n-sphere is a combinatorial n-cell. A piecewise linear proof of this theorem by a "long induction" using regular neighbourhoods and collapsing was given in [4]. A direct piecewise linear proof appeared recently in [2]; it is based on the existence of a "collar" for the boundary of a combinatorial manifold with boundary. Our proof is similar to the proof in [2]. We proceed by induction on dimensions, proving simultaneously the existence of a collar for the boundary of a combinatorial manifold with boundary and the homogeneity theorem. From [2] we adopted an argument which eliminates a certain combinatorial technique applied in [1] and involving induction on the length of stellar subdivisions. The results of [1] were previously interpreted in piecewise linear topology by use of a theorem in [3] stating that piecewise linearly homeomorphic simplicial complexes have subdivisions which are combinatorially equivalent in the sense of [1]. The thesis is divided into three parts. The first gives definitions and basic properties relating to simplicial complexes. The second concerns combinatorial manifolds, and in the third we present our proof of the piecewise linear homogeneity of spheres. / Science, Faculty of / Mathematics, Department of / Graduate

No description available.

Thesis (Ph. D.)--New York University, 1935. / Planographed. Bibliography: p. 38-41.

No description available.

Thesis (Ph. D.)--New York University, 1935. / Planographed. Bibliography: p. 38-41.

Thesis (Ph. D.)--University of Wisconsin--Madison, 1968. / Typescript. Vita. eContent provider-neutral record in process. Description based on print version record. Includes bibliography.

Some inequalities of H. J. Ryser with combinatorial applications are generalized. Let f be a non-negative concave symmetric function on v-tuples of non-negative reals. If f has the property that when θa + (1- θ)b ∈ G[subscript f] = f[power -1] ({t:t > 0}), 0 < θ < 1, then f(θa + (1- θ)b) = θf(a) + (1-θ)f(b), then we say that f is strictly concave. (Similarly, if f is convex and has the property just mentioned, then we say that f is strictly convex). Let H be a non-negative hermitian matrix with eigenvalues λ₁, ..., λ[subscript v], where λ₁ ≧ ... ≧λ[subscript e] > λ[subscript e+1] = … = λ[subscript v] = 0. Let h be an integer, 1 < h, such that e ≦h ≦ v and define k and λ by k = trace (H)/h, λ[subscript h] ≦k + (h-1) λ ≦λ₁. Define the matrix B of order h by B = (k- λ)I + λJ, where I is the identity matrix all of whose entries are 1's. Let B₀ = B ∔ 0, where the matrix B₀ of order v is the direct sum of the matrix B of order h and the (v-h)-order zero matrix. Let f(H) denote f(λ₁, … , λ[subscript v]). Then we prove theorems of the following nature. THEOREM: The matrices H and B₀ satisfy f(H) ≦ f(B₀). If f is strictly concave and if (λ₁, ..., λ[subscript v]) ∈ G[subscript f] then equality holds if and only if H and B₀ have the same eigenvalues. If f is strictly concave and if for some integer z, G[subscript f] is the set of non-negative vectors with at least z positive coordinates and if k + (h-1) λ ≠ 0 and z ≦ h or k + (h-1)λ = 0 and z < h, then f(H) = f(B₀) if and only if H and B₀ have the same eigenvalues. If f is convex a similar theorem with the inequality reversed can be proved. We discuss various choices of the function f and indicate some applications of the results to some combinatorial problems. / Science, Faculty of / Mathematics, Department of / Graduate

A dissertation submitted to the Faculty of Science, University of the Witwatersrand, Johannesburg, in fulfilment of requirements for the degree of Master of Science. 9 December 2014. / The aim of this research project is to survey and elaborate on various generalizations and re nements of Euler's celebrated distinct-odd partition theorem which asserts the equality of the numbers of partitions of a positive integer into distinct summands and into odd summands. Although the work is not originally my own, I give clarity where there is obscurity by bridging the gaps on the already existing work. I touch on combinatorial proofs, which are either bijective or involutive. In some cases I give both combinatorial and analytic proofs. The main source of this dissertation is [22, 5, 6, 8]. I start by rst summarizing some methods and techniques used in partition theory.