65 research outputs found
Les pavages, les quasi-cristaux et le 18th problème de Hilbert
Le 18e problème de Hilbert est constitué de trois questions vaguement liées : Le nombre de groupes a région fondamentale (bornée) dans E mest-il fini ? Existet-il un pavage sur les paves duquel aucun groupe n’agisse de façon transitive ? Quels sont les juxtapositions les plus denses
de corps congruents dans E3 ? Ces questions ont orienté la cristallographie mathématique vers de nouvell es directions et ont été excessivement efficaces: de nos jours, les quasicristaux posent des problèmes mathématiques qui se situent précisément dans les champs indiqués par Hilbert. En effet, plusieurs des nouveaux problèmes sont des reformulations de ceux de Hilbert. On a fait de considérables progrès dans les demières années, mais une question clé - comment les parties du problème sont liées entre elles - n’e st pas encore complètement comprise.Hilbert’s 18th problem consisted of three loosely related questions: Is the number of groups in En with (bounded) fundamental region finite? Does there exist a tiling on whose tiles no group acts transitively? What are the densest packings of congruent bodies in E3? These questions pointed mathematical crystallography in new directions and have been unreasonably effective: in our time quasicrystals pose mathematical problems in precisely the areas indicated by Hilbert. Indeed, many of the new problems are reformulations of Hilbert’s. Considerable progress has been made in the last few years, but a key issue-how the parts of the problem are related to one another-is still
not completely understood.Peer Reviewe
On the Hausdorff Dimension of the Boundary of a Self-Similar Tile
In our everyday experiences, we have developed a concept of dimension, neatly expressed as integers, i.e. a point, line, square and cube as 0-, 1-, 2-, and 3-dimensional, respectively. Less intuitive are dimensions of sets such as the Koch Curve and Cantor Set. The formal definition of toplogical dimension in a metric space conforms to our intuitive concept of dimension, but it is inadequate to describe the dimension of fractals. The purpose of this thesis is to develop notions of fractal dimension and in particular, to explore Hausdorff dimension with regard to self-similar tiles in detail. Methods of calculation of Hausdorff dimension will be discussed
Lattice setup for quantum field theory in AdS2
Holographic conformal field theories (CFTs) are usually studied in a limit where the gravity description is weakly coupled. By contrast, lattice quantum field theory can be used as a tool for doing computations in a wider class of holographic CFTs where nongravitational interactions in AdS become strong, and gravity is decoupled. We take preliminary steps for studying such theories on the lattice by constructing the discretized theory of a scalar field in
AdS
2
and investigating its approach to the continuum limit in the free and perturbative regimes. Our main focus is on finite sublattices of maximally symmetric tilings of hyperbolic space. Up to boundary effects, these tilings preserve the triangle group as a large discrete subgroup of
AdS
2
, but have a minimum lattice spacing that is comparable to the radius of curvature of the underlying spacetime. We quantify the effects of the lattice spacing as well as the boundary effects, and find that they can be accurately modeled by modifications within the framework of the continuum limit description. We also show how to do refinements of the lattice that shrink the lattice spacing at the cost of breaking the triangle group symmetry of the maximally symmetric tilings.Published versio
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Predicting multibody assembly of proteins
textThis thesis addresses the multi-body assembly (MBA) problem in the context of protein assemblies. [...] In this thesis, we chose the protein assembly domain because accurate and reliable computational modeling, simulation and prediction of such assemblies would clearly accelerate discoveries in understanding of the complexities of metabolic pathways, identifying the molecular basis for normal health and diseases, and in the designing of new drugs and other therapeutics. [...] [We developed] F²Dock (Fast Fourier Docking) which includes a multi-term function which includes both a statistical thermodynamic approximation of molecular free energy as well as several of knowledge-based terms. Parameters of the scoring model were learned based on a large set of positive/negative examples, and when tested on 176 protein complexes of various types, showed excellent accuracy in ranking correct configurations higher (F² Dock ranks the correcti solution as the top ranked one in 22/176 cases, which is better than other unsupervised prediction software on the same benchmark). Most of the protein-protein interaction scoring terms can be expressed as integrals over the occupied volume, boundary, or a set of discrete points (atom locations), of distance dependent decaying kernels. We developed a dynamic adaptive grid (DAG) data structure which computes smooth surface and volumetric representations of a protein complex in O(m log m) time, where m is the number of atoms assuming that the smallest feature size h is [theta](r[subscript max]) where r[subscript max] is the radius of the largest atom; updates in O(log m) time; and uses O(m)memory. We also developed the dynamic packing grids (DPG) data structure which supports quasi-constant time updates (O(log w)) and spherical neighborhood queries (O(log log w)), where w is the word-size in the RAM. DPG and DAG together results in O(k) time approximation of scoring terms where k << m is the size of the contact region between proteins. [...] [W]e consider the symmetric spherical shell assembly case, where multiple copies of identical proteins tile the surface of a sphere. Though this is a restricted subclass of MBA, it is an important one since it would accelerate development of drugs and antibodies to prevent viruses from forming capsids, which have such spherical symmetry in nature. We proved that it is possible to characterize the space of possible symmetric spherical layouts using a small number of representative local arrangements (called tiles), and their global configurations (tiling). We further show that the tilings, and the mapping of proteins to tilings on arbitrary sized shells is parameterized by 3 discrete parameters and 6 continuous degrees of freedom; and the 3 discrete DOF can be restricted to a constant number of cases if the size of the shell is known (in terms of the number of protein n). We also consider the case where a coarse model of the whole complex of proteins are available. We show that even when such coarse models do not show atomic positions, they can be sufficient to identify a general location for each protein and its neighbors, and thereby restricts the configurational space. We developed an iterative refinement search protocol that leverages such multi-resolution structural data to predict accurate high resolution model of protein complexes, and successfully applied the protocol to model gp120, a protein on the spike of HIV and currently the most feasible target for anti-HIV drug design.Computer Science
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