The classification of tensor surface harmonic functions for clusters and coordination compounds

1989 ◽  
Vol 75 (1) ◽  
pp. 11-32 ◽  
Author(s):  
Roy L. Johnston ◽  
D. Michael P. Mingos
Keyword(s):  
Coordination Compounds ◽  
Harmonic Functions ◽  
Surface Harmonic

scholarly journals Boundary Regularity for p-Harmonic Functions and Solutions of Obstacle Problems on Unbounded Sets in Metric Spaces

2019 ◽  
Vol 7 (1) ◽  
pp. 179-196
Author(s):  
Anders Björn ◽  
Daniel Hansevi
Keyword(s):  
Harmonic Functions ◽  
Obstacle Problem ◽  
Metric Spaces ◽  
Local Property ◽  
Boundary Regularity ◽  
Obstacle Problems ◽  
Regular Boundary ◽  
Complete Metric Spaces ◽  
Regularity Results

Abstract The theory of boundary regularity for p-harmonic functions is extended to unbounded open sets in complete metric spaces with a doubling measure supporting a p-Poincaré inequality, 1 < p < ∞. The barrier classification of regular boundary points is established, and it is shown that regularity is a local property of the boundary. We also obtain boundary regularity results for solutions of the obstacle problem on open sets, and characterize regularity further in several other ways.

scholarly journals Fast Simulation of Lipid Vesicle Deformation Using Spherical Harmonic Approximation

2016 ◽  
Vol 21 (1) ◽  
pp. 40-64
Author(s):  
Michael Mikucki ◽  
Yongcheng Zhou
Keyword(s):  
Harmonic Functions ◽  
Degrees Of Freedom ◽  
Finite Difference Methods ◽  
Harmonic Approximation ◽  
Lipid Vesicles ◽  
Membrane Curvature ◽  
Lipid Vesicle ◽  
Nonlinear Conjugate Gradient Method ◽  
Surface Harmonic ◽  
Curvature Energy

AbstractLipid vesicles appear ubiquitously in biological systems. Understanding how the mechanical and intermolecular interactions deform vesicle membranes is a fundamental question in biophysics. In this article we develop a fast algorithm to compute the surface configurations of lipid vesicles by introducing surface harmonic functions to approximate themembrane surface. This parameterization allows an analytical computation of the membrane curvature energy and its gradient for the efficient minimization of the curvature energy using a nonlinear conjugate gradient method. Our approach drastically reduces the degrees of freedom for approximating the membrane surfaces compared to the previously developed finite element and finite difference methods. Vesicle deformations with a reduced volume larger than 0.65 can be well approximated by using as small as 49 surface harmonic functions. The method thus has a great potential to reduce the computational expense of tracking multiple vesicles which deform for their interaction with external fields.

scholarly journals Manifolds Without Green’s Formula

1972 ◽  
Vol 45 ◽  
pp. 129-138
Author(s):  
Moses Glasner
Keyword(s):  
Differential Equations ◽  
Weak Solutions ◽  
Harmonic Functions ◽  
Green’S Formula ◽  
Global Properties ◽  
Green's Formula

Recently attention has been focused on manifolds that carry covariant tensors that are merely bounded measurable. In terms of these tensors global differential equations are defined and their weak solutions are called harmonic functions. Nakai initiated the classification of these manifolds with respect to the global properties of the harmonic functions that they carry.

scholarly journals Transformations of Galton-Watson processes and linear fractional reproduction

2007 ◽  
Vol 39 (4) ◽  
pp. 1036-1053 ◽  
Author(s):  
F. C. Klebaner ◽  
U. Rösler ◽  
S. Sagitov
Keyword(s):  
Markov Chain ◽  
Time Reversal ◽  
Harmonic Functions ◽  
Invariant Measures ◽  
Graphical Representation ◽  
Simple Random Walk ◽  
Time Process ◽  
Offspring Distribution ◽  
Watson Process

By establishing general relationships between branching transformations (Harris-Sevastyanov, Lamperti-Ney, time reversals, and Asmussen-Sigman) and Markov chain transforms (Doob's h-transform, time reversal, and the cone dual), we discover a deeper connection between these transformations with harmonic functions and invariant measures for the process itself and its space-time process. We give a classification of the duals into Doob's h-transform, pathwise time reversal, and cone reversal. Explicit results are obtained for the linear fractional offspring distribution. Remarkably, for this case, all reversals turn out to be a Galton-Watson process with a dual reproduction law and eternal particle or some kind of immigration. In particular, we generalize a result of Klebaner and Sagitov (2002) in which only a geometric offspring distribution was considered. A new graphical representation in terms of an associated simple random walk on N2 allows for illuminating picture proofs of our main results concerning transformations of the linear fractional Galton-Watson process.

Classification of Solutions for Harmonic Functions With Neumann Boundary Value

2018 ◽  
Vol 61 (2) ◽  
pp. 438-448 ◽  
Author(s):  
Tao Zhang ◽  
Chunqin Zhou
Keyword(s):  
Harmonic Functions ◽  
Boundary Value ◽  
Neumann Boundary ◽  
All Solutions ◽  
Classification Of Solutions

AbstractIn this paper, we classify all solutions ofwith the finite conditionsHere c is a positive number and β > −1.

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