Mathematical aspects of the LCAO MO first order density function (2): Relationships between density functions

2007 ◽  
Vol 43 (3) ◽  
pp. 1102-1118 ◽  
Author(s):  
Ramon Carbó-Dorca
Keyword(s):  
Density Function ◽  
Density Functions ◽  
First Order ◽  
Lcao Mo

scholarly journals Normalized multi-bump solutions for saturable Schrödinger equations

2019 ◽  
Vol 9 (1) ◽  
pp. 1259-1277
Author(s):  
Xiaoming Wang ◽  
Zhi-Qiang Wang
Keyword(s):  
Variational Method ◽  
Density Function ◽  
Global Maximum ◽  
Schrodinger Equations ◽  
Density Functions ◽  
Concentration Compactness ◽  
Schrödinger Equations ◽  
Compactness Method ◽  
Normalized Solutions

Abstract In this paper, we are concerned with the existence of multi-bump solutions for a class of semiclassical saturable Schrödinger equations with an density function: $$\begin{array}{} \displaystyle -{\it\Delta} v +{\it\Gamma} \frac{I(\varepsilon x) + v^2}{1+I(\varepsilon x) +v^2} v =\lambda v,\, x\in{{\mathbb{R}}^{2}}. \end{array}$$ We prove that, with the density function being radially symmetric, for given integer k ≥ 2 there exist a family of non-radial, k-bump type normalized solutions (i.e., with the L2 constraint) which concentrate at the global maximum points of density functions when ε → 0+. The proof is based on a variational method in particular on a convexity technique and the concentration-compactness method.

A Comparison of Some Multivariate Weibull Distributions

2010 ◽  
Author(s):  
Bernt J. Leira
Keyword(s):  
Probability Density ◽  
Density Function ◽  
Statistical Models ◽  
Mechanical Design ◽  
Bending Moment ◽  
Probability Density Functions ◽  
Density Functions ◽  
Weibull Distributions ◽  
Linear Combinations ◽  
Simplified Procedure

Three different possible choices of statistical models for multivariate Weibull distributions are considered and compared. The concept of “a correlation field” is introduced and is subsequently applied for the purpose of comparing the different models. Linear combinations of Weibull distributed random variables are considered, and expressions for the corresponding probability density functions are established. Furthermore, a simplified procedure for approximating the resulting density function is described. Comparison is made between the statistical moments of increasing order for the specific case of two Weibull components. This example of application arises e.g. in connection with mechanical design of a column which is subjected to a bi-axial bending moment.

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