Hypersurfaces in Hyperbolic Poincar\'e Manifolds and Conformally Invariant PDEs

We derive a relationship between the eigenvalues of the Weyl-Schouten tensor of a conformal representative of the conformal infinity of a hyperbolic Poincar\'e manifold and the principal curvatures on the level sets of its uniquely associated defining function with calculations based on [9] [10]. This relationship generalizes the result for hypersurfaces in {\H}^{n+1} and their connection to the conformal geometry of {\SS}^n as exhibited in [7] and gives a correspondence between Weingarten hypersurfaces in hyperbolic Poincar\'e manifolds and conformally invariant equations on the conformal infinity. In particular, we generalize an equivalence exhibited in [7] between Christoffel-type problems for hypersurfaces in {\H}^{n+1} and scalar curvature problems on the conformal infinity {\SS}^n to hyperbolic Poincar\'e manifolds.Comment: 16 page

Sweeping of three-dimensional objects

Evaluating the volume swept out by a three-dimensional (3D) object as it moves along an arbitrary path is of interest to many areas of CAD and CAM, such as mechanism design and robot path planning. This paper shows how envelope theory from differential geometry can be used to find the volumes swept out by the individual surfaces of a solid body, and how computer algebra methods may be of use to perform the computations involved. Finally, a new algorithm is presented which shows how the results of sweeping the individual surfaces of a solid body can be combined to form a new 3D model of the swept volume. This algorithm has strong resemblance to hidden line algorithms, but works in one dimension higher

Holomorphic extension of CR functions, envelopes of holomorphy and removable singularities

This is an extensive (published) survey on CR geometry, whose major themes are: formal analytic reflection principle; generic properties of Systems of (CR) vector fields; pairs of foliations and conjugate reflection identities; Sussmann's orbit theorem; local and global aspects of holomorphic extension of CR functions; Tumanov's solution of Bishop's equation in Hoelder classes with optimal loss of smoothness; wedge-extendability on C^2,a generic submanifolds of C^n consisting of a single CR orbit; propagation of CR extendability and edge-of-the-wedge theorem; Painlev\'{e} problem; metrically thin singularities of CR functions; geometrically removable singularities for solutions of the induced d-barre. Selected theorems are fully proved, while surveyed results are put in the right place in the architecture.Comment: 283 pages ; 33 illustrations ; 16 open problems http://www.hindawi.com/journals/imrs

Developing methods to construct ring pucker free energy hypersurfaces applied to the analysis of glycosidase enzyme catalytic mechanisms

Includes bibliographical references.Carbohydrates consist of one or more sub-units usually various 5- and 6-membered cycles (furanoses and pyranoses) which can twist, bend or flip into a variety of conformers that differ in strain - this is ring puckering. These puckers notably the strained puckering conformers are observed during enzymatically assisted bond formation or cleavage of the glycosidic bonds of carbohydrate substrates. In this thesis, the free energy of ring puckering is calculated by implementing the Hill-Reilly reduced coordinate pucker description into the sampling enhancing Free Energies from Adaptive Reaction Coordinate Forces (FEARCF) method. FEARCF non-Boltzmann simulations of prototypical sugars β-Dribose and β-D-glucose converged to yield free energy pucker surfaces and volumes when using several semi-empirical QM methods - AM1, PM3, PM3CARB-1 and SCC-DFTB. From this, the accessible puckering conformations and minimum free energy paths of puckering were reasoned An analysis of the furanose and pyranose free energy pucker surfaces and volumes compared with both Density Functional Theory RB3LYP/6-311++G** optimised structures and a Hartree-Fock free energy surface revealed that SCC-DFTB provides the best semi-empirical description of 5- and 6- membered carbohydrate ring deformation. This illustrates that necessary high energy ring conformations observed in enzymatic binding sites requires the enzyme to induce and preserve high energy conformations required for successful hydrolyses and synthesis of the glycosidic bond. To further test this hypothesis, a 5- and 6-membered cycle were studied within enzymatic environments. The polysaccharide cellulose contains β 1-4 linked glucose subunit and is degraded by cellulase, a glycosidase. Specifically, the retaining cellobiohydrolase I (CBHI) of Trichoderma Reesei which cleaves cellobiose units from crystalline cellulose.The free energy volumes of puckering for the glucose sub-unit (in the catalytic position of an 8 unit cellulosic fragment - cellooctaose) were calculated and explored in vacuum, water and in the active site of CBHI. It was observed that the binding pocket of enzymes limits the ring pucker and that the active site amino acids preferentially stabilise certain puckering conformations. For CBHI, the first part of the glycosidase reaction is the glycosylation step. This was driven to completion during SCC-DFTB QM/MD FEARCF calculations where GLU212, ASP214 and GLU217 and part of the substrate were treated quantum mechanically. The general hybrid orbital method was used to connect the QM and MM regions. The free energy barriers of glycosylation were computed and the puckering statistics during the conversion of cellooctaose to products were correlated with this. Guanosine, a 5-membered ribose derivative is phosphorylated by Purine Nucleoside Phosphorylase (PNP) in order to salvage the guanine base. The effect of the PNP protein environment on ring pucker was studied by using FEARCF SCC-DFTB QM/MD non Boltzmann free energy calculations to quantify the pucker change induced in guanosine when changing environment from vacuum, to water and to the protein. In vacuo, the E4 and E1 pucker conformers were observed as minima. Upon solvation, the puckering phase space became less restricted with the 3T4 and 2T3 pucker conformers as minima. In the PNP active site pucker became restricted with only the 4E conformer observed