Uniform symplicity of groups with proximal action

We prove that groups acting boundedly and order-primitively on linear orders or acting extremly proximality on a Cantor set (the class including various Higman-Thomson groups and Neretin groups of almost automorphisms of regular trees, also called groups of spheromorphisms) are uniformly simple. Explicit bounds are provided.Comment: 23 pages, appendix by Nir Lazarovich, corrected versio

A bound on the size of a graph with given order and bondage number

AbstractThe domination number of a graph is the minimum number of vertices in a set S such that every vertex of the graph is either in S or adjacent to a member of S. The bondage number of a graph G is the cardinality of a smallest set of edges whose removal results in a graph with domination number greater than that of G. We prove that a graph with p vertices and bondage number b has at least p(b + 1)/4 edges, and for each b there is at least one p for which this bound is sharp. © 1999 Elsevier Science B.V. All rights reserve

A review of Monte Carlo simulations of polymers with PERM

In this review, we describe applications of the pruned-enriched Rosenbluth method (PERM), a sequential Monte Carlo algorithm with resampling, to various problems in polymer physics. PERM produces samples according to any given prescribed weight distribution, by growing configurations step by step with controlled bias, and correcting "bad" configurations by "population control". The latter is implemented, in contrast to other population based algorithms like e.g. genetic algorithms, by depth-first recursion which avoids storing all members of the population at the same time in computer memory. The problems we discuss all concern single polymers (with one exception), but under various conditions: Homopolymers in good solvents and at the $\Theta$ point, semi-stiff polymers, polymers in confining geometries, stretched polymers undergoing a forced globule-linear transition, star polymers, bottle brushes, lattice animals as a model for randomly branched polymers, DNA melting, and finally -- as the only system at low temperatures, lattice heteropolymers as simple models for protein folding. PERM is for some of these problems the method of choice, but it can also fail. We discuss how to recognize when a result is reliable, and we discuss also some types of bias that can be crucial in guiding the growth into the right directions.Comment: 29 pages, 26 figures, to be published in J. Stat. Phys. (2011

Representability of algebraic topology for biomolecules in machine learning based scoring and virtual screening

This work introduces a number of algebraic topology approaches, such as multicomponent persistent homology, multi-level persistent homology and electrostatic persistence for the representation, characterization, and description of small molecules and biomolecular complexes. Multicomponent persistent homology retains critical chemical and biological information during the topological simplification of biomolecular geometric complexity. Multi-level persistent homology enables a tailored topological description of inter- and/or intra-molecular interactions of interest. Electrostatic persistence incorporates partial charge information into topological invariants. These topological methods are paired with Wasserstein distance to characterize similarities between molecules and are further integrated with a variety of machine learning algorithms, including k-nearest neighbors, ensemble of trees, and deep convolutional neural networks, to manifest their descriptive and predictive powers for chemical and biological problems. Extensive numerical experiments involving more than 4,000 protein-ligand complexes from the PDBBind database and near 100,000 ligands and decoys in the DUD database are performed to test respectively the scoring power and the virtual screening power of the proposed topological approaches. It is demonstrated that the present approaches outperform the modern machine learning based methods in protein-ligand binding affinity predictions and ligand-decoy discrimination

Hierarchical bounding structures for efficient virial computations: Towards a realistic molecular description of cholesterics

We detail the application of bounding volume hierarchies to accelerate second-virial evaluations for arbitrary complex particles interacting through hard and soft finite-range potentials. This procedure, based on the construction of neighbour lists through the combined use of recursive atom-decomposition techniques and binary overlap search schemes, is shown to scale sub-logarithmically with particle resolution in the case of molecular systems with high aspect ratios. Its implementation within an efficient numerical and theoretical framework based on classical density functional theory enables us to investigate the cholesteric self-assembly of a wide range of experimentally-relevant particle models. We illustrate the method through the determination of the cholesteric behaviour of hard, structurally-resolved twisted cuboids, and report quantitative evidence of the long-predicted phase handedness inversion with increasing particle thread angles near the phenomenological threshold value of $45^\circ$. Our results further highlight the complex relationship between microscopic structure and helical twisting power in such model systems, which may be attributed to subtle geometric variations of their chiral excluded-volume manifold

Constant Size Molecular Descriptors For Use With Machine Learning

A set of molecular descriptors whose length is independent of molecular size is developed for machine learning models that target thermodynamic and electronic properties of molecules. These features are evaluated by monitoring performance of kernel ridge regression models on well-studied data sets of small organic molecules. The features include connectivity counts, which require only the bonding pattern of the molecule, and encoded distances, which summarize distances between both bonded and non-bonded atoms and so require the full molecular geometry. In addition to having constant size, these features summarize information regarding the local environment of atoms and bonds, such that models can take advantage of similarities resulting from the presence of similar chemical fragments across molecules. Combining these two types of features leads to models whose performance is comparable to or better than the current state of the art. The features introduced here have the advantage of leading to models that may be trained on smaller molecules and then used successfully on larger molecules.Comment: 18 pages, 5 figure

Surface Induced Ordering on Model Liquid Crystalline Dendrimers

We present results from Monte Carlo simulations of liquid crystalline dendrimers (LCDrs) adsorbed on flat, impenetrable substrates. A tractable coarse grained force field for the inter-dendritic and the dendrimer-substrate interactions is introduced. We investigate the conformational and ordering properties of single, end-functionalized LCDrs under homeotropic, random (or degenerate) planar and unidirectional planar aligning substrates. Depending on the anchoring conditions of the mesogenic units of the LCDr and on temperature a variety of stable LCDr states, differing in their topology, are observed and analysed. The influence of the denritic generation and core functionality on the surface-induced ordering of the LCDrs are examined.Comment: 23 pages, 11 figure