Mahler's conjecture in convex geometry: a summary and further numerical analysis

In this thesis we study Mahler's conjecture in convex geometry, give a short summary about its history, gather and explain different approaches that have been used to attack the conjecture, deduce formulas to calculate the Mahler volume and perform numerical analysis on it. The conjecture states that the Mahler volume of any symmetric convex body, i.e. the product of the volume of the symmetric convex body and the volume of its dual body, is minimized by the (hyper-)cube. The conjecture was stated and solved in 1938 for the 2-dimensional case by Kurt Mahler. While the maximizer for this problem is known (it is the ball), the conjecture about the minimizer is still open for all dimensions greater than 2. A lot of effort has benn made to solve this conjecture, and many different ways to attack the conjecture, from simple geometric attempts to ones using sophisticated results from functional analysis, have all been tried unsuccesfully. We will present and discuss the most important approaches. Given the support function of the body, we will then introduce several formulas for the volume of the dual and the original body and hence for the Mahler volume. These formulas are tested for their effectiveness and used to perform numerical work on the conjecture. We examine the conjectured minimizers of the Mahler volume by approximating them in different ways. First the spherical harmonic expansion of their support functions is calculated and then the bodies are analyzed with respect to the length of that expansion. Afterwards the cube is further examined by approximating its principal radii of curvature functions, which involve Dirac delta functions.M.S.Committee Chair: Harrell, Evans; Committee Member: Ghomi, Mohammad; Committee Member: Loss, Michae

Experimental assessment of the strengths of boron-boron triple bonds

Diborynes, molecules containing homoatomic boron-boron triple bonds, have been investigated by Raman spectroscopy in order to determine the stretching frequencies of their central B≡B unit as an experimental measure of homoatomic bond strength. The observed frequencies between 1600 cm-1 and 1750 cm-1 were assigned on the basis of DFT modelling and the characteristic pattern produced by the isotopic distribution of boron. This frequency completes the series of known stretches of homoatomic triple bonds, fitting into the trend established by the long-known stretching frequencies of C≡C and N≡N triple bonds in alkynes and dinitrogen, respectively. A quantitative analysis was carried out using the concept of relaxed force constants. The results support the classification of the diboryne as a true triple bond, and speak to the similarities of molecules constructed of the first-row elements of the p-block. The relaxed force constants of a recently reported diborabutatriene are also reported, which again fit into the trend established by the vibrational spectroscopy of organic cumulenes. As part of these studies, a new diboryne with decreased steric bulk was synthesized, and a computational study of the rotation of the stabilizing ligands indicated alkyne-like electronic isolation of the central B2-unit

Experimental Assessment of the Strengths of B–B Triple Bonds

Diborynes, molecules containing homo­atomic boron–boron triple bonds, have been investigated by Raman spectroscopy in order to determine the stretching frequencies of their central BB units as an experimental measure of homoatomic bond strengths. The observed frequencies between 1600 and 1750 cm^–1 were assigned on the basis of DFT modeling and the characteristic pattern produced by the isotopic distribution of boron. This frequency completes the series of known stretches of homoatomic triple bonds, fitting into the trend established by the long-known stretching frequencies of CC and NN triple bonds in alkynes and dinitrogen, respectively. A quantitative analysis was carried out using the concept of relaxed force constants. The results support the classification of the diboryne as a true triple bond and speak to the similarities of molecules constructed from first-row elements of the p block. Also reported are the relaxed force constants of a recently reported diborabutatriene, which again fit into the trend established by the vibrational spectroscopy of organic cumulenes. As part of these studies, a new diboryne with decreased steric bulk was synthesized, and a computational study of the rotation of the stabilizing ligands indicated alkyne-like electronic isolation of the central B₂ unit