Thesis (doctoral)--Friedrich-Wilhelms-Universität zu Berlin, 1882. / Vita.

Circles, parabolas, ellipses and hyperbolas are conic sections and have many unique properties. The properties of the tangents to conic sections prove quite interesting. Dandelin spheres are tangent to ellipses inside a cone and support the geometric definition of an ellipse. Tangent lines to parabolas, ellipses and hyperbolas in the form of families of folds are shown to create conic sections in unique ways. The equations of these tangent lines to conic sections and their equations can be found without using calculus. The equations of the tangent lines are also used to prove the bisection theorem for all conic sections and prove uniqueness for the bisection theorem in connection to conic sections. / text

published_or_final_version / abstract / toc / Mathematics / Master / Master of Philosophy

Proefschrift--Utrecht. / "Stellingen": [2] p. inserted. "Summary": 1 p. at end.

Proefschrift--Utrecht. / "Stellingen": [2] p. inserted. "Summary": 1 p. at end.

Thesis (Ph. D.)--Catholic University of America, 1927. / Biographical sketch.

<p> In this thesis parametric analysis for conic quadratic optimization problems is studied. In parametric analysis, which is often referred to as parametric optimization or parametric programming, a perturbation parameter is introduced into the optimization problem, which means that the coefficients in the objective function of the problem and in the right-hand-side of the constraints are perturbed. First, we describe linear, convex quadratic and second order cone optimization problems and their parametric versions. Second, the theory for finding solutions of the parametric problems is developed. We also present algorithms for solving such problems. Third, we demonstrate how to use parametric optimization techniques to solve multiobjective optimization problems and compute Pareto efficient surfaces. </p> <p> We implement our novel algorithm for hi-parametric quadratic optimization. It utilizes existing solvers to solve auxiliary problems. We present numerical results produced by our parametric optimization package on a number of practical financial and non-financial computational problems. In the latter we consider problems of drug design and beam intensity optimization for radiation therapy. </p> <p> In the financial applications part, two risk management optimization models are developed or extended. These two models are a portfolio replication framework and a credit risk optimization framework. We describe applications of multiobjective optimization to existing financial models and novel models that we have developed. We solve a number of examples of financial multiobjective optimization problems using our parametric optimization algorithms. </p> / Thesis / Doctor of Philosophy (PhD)

The discovery of conic sections was made by Menaechmus (375-325 B.C.) an associate of Plato and a pupil of Eudoxus. This discovery, in the course of only a century, raised geometry to the loftiest height which it was destined to reach during antiquity.

An affine transformation such as T(P)=Q is a locus of an affine conic. Any affine conic can be produced from this incidence construction. The affine type of conic (ellipse, parabola, hyperbola) is determined by the invariants of T, the determinant and trace of its linear part. The purpose of this thesis is to obtain a corresponding classification in the hyperbolic plane of conics defined by this construction.

The purpose of this thesis is to give a detailed group theoretic proof of the rank formula in a more general setting. By using the proof of Mordell-Weil theorem, a formula for the rank of the elliptical curves in certain cases over algebraic number fields can be obtained and computable.