Thesis (doctoral)--Rheinische Friedrich-Wilhelms-Universität Bonn, 1999. / Includes bibliographical references (p. [85]-87).

Thesis (PH. D.)--University of Chicago, 1933. / Vita. Lithoprinted. "Private edition, distributed by the University of Chicago libraries, Chicago, Illinois." Bibliography: p. 28.

No description available.

Thesis (B.A.)--Haverford College, Dept. of Mathematics, 2004. / Includes bibliographical references.

Thesis (Ph. D.)--University of Wisconsin--Madison, 1970. / Typescript. Vita. eContent provider-neutral record in process. Description based on print version record. Includes bibliographical references.

Thesis (Ph. D.)--Waseda University, 2005. / Accompanied by summary (4 leaves ; 30 cm.) in Japanese. On t.p. "n̳" of "Cn-moves" is subscript. Includes bibliographical references (leaves 43-46). "List of papers by Harumi Yamada": leaf 47.

Thesis (doctoral)--Rheinische Friedrich-Wilhelms-Universität Bonn, 2000. / Includes bibliographical references (p. 127-131).

This paper explains the construction of Khovanov homology of which begins by un derstanding how Louis Kauffman generalizes the Jones polynomial using a state sum model of the bracket polynomial for an unoriented knot or link and in turn recovers the Jones polynomial, a knot invariant for an oriented knot or link. Kauffman associates the unknot by the polynomial (−A2 − A−2) whereas Khovanov associates the unknot by (q + q−1) through a change of variables. As an oriented knot or link K with n crossings produces 2n smoothings, Khovanov builds a commutative cube {0,1}n and associates a graded vector space to each smoothing in the cube. By defining a differential operator on the directed edges of the cube so that adjacent states differ by a type of smoothing for a fixed cross ing, we can form chain groups which are direct sums of these vector spaces. Naturally we get a bi-graded (co)chain complex which is called the Khovanov complex. The resulting (co)homology groups of these (co)chains turns out to be invariant under the Reidemeister moves and taking the Euler characteristic of the Khovanov complex returns the very same Jones polynomial that we started with.

The goal of this paper is to introduce a new algebraic structure for coloring regions in the planar complement of an oriented virtual knot or link diagram that I will refer to as virtual tribrackets. I will begin with an overview of classical knot theory where I introduce knot diagrams and ways of calculating knot invariants. This paper progresses into virtual knots and links, their geometric interpretations as well as their virtual moves, and some invariant examples for the virtual case. This informations allows me to introduce tribrackets, which is a labeling method used to define counting invariants for classical knots and link diagrams. Finally, this paper properly defines and proves the use of virtual tribackets in defining invariants for virtual knots as well as providing examples from [6] which more precisely show that these invariants can distinguish between certain virtual knots.

A spline function of degree k with knots S₀, S₁,...,Sr is a C[superscript]k-1 function which is a polynomial of degree at most k in each of the intervals (-∞, S₀), (S₀, S₁),…, (Sr,+∞). The Gauss-Markoff Theorem can be used to estimate by least squares the coefficients of a spline function of given degree and knots. Estimating a spline function of known knots without full knowledge of the degree entails an extension of the Gauss-Markoff technique. The estimation of the degree when the knots are also unknown has a possible solution in a method employing finite differences. The technique of minimizing sums of squared residuals forms the basis for a method of estimating the knots of a spline function of given degree. Estimates for the knots may also be obtained by a method of successive approximation, provided additional information about the spline function is known. / Science, Faculty of / Mathematics, Department of / Graduate