Mathematical aspects of the LCAO MO first order density function (1): atomic partition, metric structure and practical applications

2007 ◽  
Vol 43 (3) ◽  
pp. 1076-1101 ◽  
Author(s):  
Ramon Carbó-Dorca
Keyword(s):  
Density Function ◽  
Practical Applications ◽  
First Order ◽  
Lcao Mo

scholarly journals On the integrability problem for systems of partial differential equations in one unknown function, I

2019 ◽  
Vol 11 (4) ◽  
pp. 35-71 ◽  
Author(s):  
Antonio Kumpera
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Unknown Function ◽  
Contact Structures ◽  
Partial Differential ◽  
Practical Applications ◽  
First Order ◽  
Integrability Problem ◽  
Integration Problem ◽  
In The Beginning

We discuss the integration problem for systems of partial differential equations in one unknown function and special attention is given to the first order systems. The Grassmannian contact structures are the basic setting for our discussion and the major part of our considerations inquires on the nature of the Cauchy characteristics in view of obtaining the necessary criteria that assure the existence of solutions. In all the practical applications of partial differential equations, what is mostly needed and what in fact is hardest to obtains are the solutions of the system or, occasionally, some specific solutions. This work is based on four most enlightening Mémoires written by Élie Cartan in the beginning of the last century.

Weight, size and shape relationships and their uses in the study of graded yield data

1983 ◽  
Vol 100 (1) ◽  
pp. 211-220 ◽  
Author(s):  
G. E. L. Morris ◽  
I. E. Currah
Keyword(s):  
Probability Density Function ◽  
Probability Density ◽  
Density Function ◽  
Published Data ◽  
Simple Experiment ◽  
Yield Data ◽  
Practical Applications ◽  
Size And Shape ◽  
Horticultural Crops ◽  
Using Data

SUMMARYFor many horticultural crops the distribution of weight over size grades is of more importance than the total weight. This paper shows how simply determined features of interrelationships of the weight, size and shape of an individual in the crop can be combined to provide estimates of various aspects of the distribution of crop weight over size grades. The two relationships required are (i) the probability density function of the grading variable for the crop; (ii) a function relating the weight of an individual to the corresponding value of the grading variable.The paper shows how each of these can be determined either from published data or by simple experiment. Examples using data on onions and carrots are given to illustrate this and also to show some of the more important practical applications of the methods. For example, they allow the results of grading with one set of size grades to be extrapolated to a different set of grades without recourse to further measurement or experimentation and this is illustrated using published data on carrots. Other possible uses are also discussed and outlined.

On some distributional properties of a first-order nonnegative bilinear time series model

2001 ◽  
Vol 38 (3) ◽  
pp. 659-671 ◽  
Author(s):  
Zhiqiang Zhang ◽  
Howell Tong
Keyword(s):  
Time Series ◽  
Probability Density Function ◽  
Numerical Method ◽  
Probability Density ◽  
Density Function ◽  
Stationary Distribution ◽  
Time Series Model ◽  
First Order ◽  
Distributional Properties ◽  
Tail Behaviour

We study a simple first-order nonnegative bilinear time-series model and give conditions under which the model is stationary. The probability density function of the stationary distribution (when it exists) is found. We also discuss the tail behaviour of the stationary distribution and calculate the probability density function by a numerical method. Simulation is used to check the calculation.

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