Optimization and dynamics of protein-protein complexes using B-splines

A moving-grid approach for optimization and dynamics of protein-protein complexes is introduced, which utilizes cubic B-spline interpolation for rapid energy and force evaluation. The method allows for the efficient use of full electrostatic potentials joined smoothly to multipoles at long distance so that multiprotein simulation is possible. Using a recently published benchmark of 58 protein complexes, we examine the performance and quality of the grid approximation, refining cocrystallized complexes to within 0.68 A RMSD of interface atoms, close to the optimum 0.63 A produced by the underlying MMFF94 force field. We quantify the theoretical statistical advantage of using minimization in a stochastic search in the case of two rigid bodies, and contrast it with the underlying cost of conjugate gradient minimization using B-splines. The volumes of conjugate gradient minimization basins of attraction in cocrystallized systems are generally orders of magnitude larger than well volumes based on energy thresholds needed to discriminate native from nonnative states; nonetheless, computational cost is significant. Molecular dynamics using B-splines is doubly efficient due to the combined advantages of rapid force evaluation and large simulation step sizes. Large basins localized around the native state and other possible binding sites are identifiable during simulations of protein-protein motion. In addition to providing increased modeling detail, B-splines offer new algorithmic possibilities that should be valuable in refining docking candidates and studying global complex behavior.     

Natural Geometry of Nonzero Quaternions

It is shown that the group of nonzero quaternions carries a family of natural closed Friedmann-Lemaître-Robertson-Walker metrics.     

四元数矩阵的极分解及其GL偏序

本文给出了四元数矩阵的唯一极分解定理和两个四元数矩阵可同时极分解的两种刻画；进而引进了四元数矩阵的GL偏序的概念，它是重要的Lǒwner偏序的一般化，并得到这个新偏序的6种刻画．     

Gaussian Integral Formula for Quaternions Based on Class Operator＇s Formula of Rotation Group

Abstract There exists rare integration formulas over quaternions due to the noncommutativity of quarternions multplication. Based on the class operator＇s formula of rotation group we derive a Gaussian integral formula for quaternions, which is similar in form to the integration for coherent state＇s completeness relation.     

Letter: The Energy Density of the Quaternionic Field as Dark Energy in the Universe

In this article we describe a model of the universe consisting of a mixture of the ordinary matter and a so-called cosmic quaternionic field. The basic idea here consists in an attempt to interpret as the energy density of the quaternionic field whose source is any form of energy including the proper energy density of this field. We set the energy density of this field to and show that the ratio of ordinary dark matter energy density assigned to is constant during the cosmic evolution. We investigate the interaction of the quaternionic field with the ordinary dark matter and show that this field exerts a force on the moving dark matter which might possible create the dark matter in the early universe. Such determined fulfils the requirements asked from the dark energy. In this model of the universe, the cosmological constant, the fine-tuning and the age problems might be solved. Finally, we sketch the evolution of the universe with the cosmic quaternionic field and show that the energy density of the cosmic quaternionic field might be a possible candidate for the dark energy.     

The Geometry of Polygons in ℝ5 and Quaternions

We consider the moduli space _r of polygons with fixed side lengths in five-dimensional Euclidean space. We analyze the local structure of its singularities and exhibit a real-analytic equivalence between _r and a weighted quotient of n-fold products of the quaternionic projective line ¹ by the diagonal PSL(2, )-action. We explore the relation between _r and the fixed point set of an anti-symplectic involution on a GIT quotient Gr_ℂ(2, 4) ⁿ /SL(4, ℂ). We generalize the Gel'fand—MacPherson correspondence to more general complex Grassmannians and to the quaternionic context, and realize our space _r as a quotient of a subspace in the quaternionic Grassmannian Gr_ℍ(2, n) by the action of the group Sp(1) ⁿ . We also give analogues of the Gel'fand—Tsetlin coordinates on the space of quaternionic Hermitean marices and briefly describe generalized action—angle coordinates on _r .     

The Hecke-algebras related to the unimodular and modular group over the Hurwitz order of integral quaternions

In the present paper the elementary divisor theory over the Hurwitz order of integral quaternions is applied in order to determine the structure of the Hecke-algebras related to the attached unimodular and modular group of degreen. In the casen = 1 the Hecke-algebras fail to be commutative. Ifn > 1 the Hecke-algebras prove to be commutative and coincide with the tensor product of their primary components. Each primary component turns out to be a polynomial ring inn resp.n + 1 resp. 2n resp. 2n+1 algebraically independent elements. In the case of the modular group of degreen, the law of interchange with the Siegel ϕ-operator is described. The induced homomorphism of the Hecke-algebras is surjective except for the weightsr = 4n-4 andr = 4n-2.     

Function representation of a noncommutative uniform algebra

We construct a Gelfand type representation of a real noncommutative Banach algebra satisfying , for all

     

Solution methods to the nearest rotation matrix problem in ℝ3: A comparative survey

Nowadays, the singular value decomposition (SVD) is the standard method of choice for solving the nearest rotation matrix problem. Nevertheless, many other methods are available in the literature for the 3D case. This article reviews the most representative ones, proposes alternative ones, and presents a comparative analysis to elucidate their relative computational costs and error performances. This analysis leads to the conclusion that some algebraic closed-form methods are as robust as the SVD, but significantly faster and more accurate.