Avoiding the Use of Lagrange Multipliers: I—Evaluating the Constrained Extrema of Functions with Projection Matrices

2021 ◽  
Vol 2 (4) ◽  
Author(s):  
David S. Corti ◽  
Ricardo Fariello
2014 ◽  
Vol 11 (1) ◽  
pp. 15
Author(s):  
Set Foong Ng ◽  
Pei Eng Ch’ng ◽  
Yee Ming Chew ◽  
Kok Shien Ng

Soil properties are very crucial for civil engineers to differentiate one type of soil from another and to predict its mechanical behavior. However, it is not practical to measure soil properties at all the locations at a site. In this paper, an estimator is derived to estimate the unknown values for soil properties from locations where soil samples were not collected. The estimator is obtained by combining the concept of the ‘Inverse Distance Method’ into the technique of ‘Kriging’. The method of Lagrange Multipliers is applied in this paper. It is shown that the estimator derived in this paper is an unbiased estimator. The partiality of the estimator with respect to the true value is zero. Hence, the estimated value will be equal to the true value of the soil property. It is also shown that the variance between the estimator and the soil property is minimised. Hence, the distribution of this unbiased estimator with minimum variance spreads the least from the true value. With this characteristic of minimum variance unbiased estimator, a high accuracy estimation of soil property could be obtained.


Forests ◽  
2021 ◽  
Vol 12 (8) ◽  
pp. 989
Author(s):  
Basáñez-Muñoz Agustín de Jesús ◽  
Jordán-Garza Adán Guillermo ◽  
Serrano Arturo

Mangrove forests have declined worldwide and understanding the key drivers of regeneration at different perturbation levels can help manage and preserve these critical ecosystems. For example, the Ramsar site # 1602, located at the Tampamachoco lagoon, Veracruz, México, consists of a dense forest of medium-sized trees composed of three mangrove species. Due to several human activities, including the construction of a power plant around the 1990s, an area of approximately 2.3 km2 has suffered differential levels of perturbation: complete mortality, partial tree loss (divided into two sections: main and isolated patch), and apparently undisturbed sites. The number and size of trees, from seedlings to adults, were measured using transects and quadrats. With a matrix of the abundance of trees by size categories and species, an ordination (nMDS) showed three distinct groups corresponding to the degree of perturbation. Projection matrices based on the size structure of Avicennia germinans showed transition probabilities that varied according to perturbation levels. Lambda showed growing populations except on the zone that showed partial tree loss; a relatively high abundance of seedlings is not enough to ensure stable mangrove dynamics or start regeneration; and the survival of young trees and adult trees showed high sensitivity.


Author(s):  
Wooyong Han ◽  
Dong-Won Jung ◽  
Jungil Lee ◽  
Chaehyun Yu
Keyword(s):  

PAMM ◽  
2014 ◽  
Vol 14 (1) ◽  
pp. 473-474 ◽  
Author(s):  
Seyedmohammad Zinatbakhsh ◽  
Wolfgang Ehlers

1964 ◽  
Vol 68 (638) ◽  
pp. 111-116 ◽  
Author(s):  
D. J. Bell

SummaryThe problem of maximising the range of a given unpowered, air-launched vehicle is formed as one of Mayer type in the calculus of variations. Eulers’ necessary conditions for the existence of an extremal are stated together with the natural end conditions. The problem reduces to finding the incidence programme which will give the greatest range.The vehicle is assumed to be an air-to-ground, winged unpowered vehicle flying in an isothermal atmosphere above a flat earth. It is also assumed to be a point mass acted upon by the forces of lift, drag and weight. The acceleration due to gravity is assumed constant.The fundamental constraints of the problem and the Euler-Lagrange equations are programmed for an automatic digital computer. By considering the Lagrange multipliers involved in the problem a method of search is devised based on finding flight paths with maximum range for specified final velocities. It is shown that this method leads to trajectories which are sufficiently close to the “best” trajectory for most practical purposes.It is concluded that such a method is practical and is particularly useful in obtaining the optimum incidence programme during the initial portion of the flight path.


1982 ◽  
Vol 14 (7) ◽  
pp. 869-888 ◽  
Author(s):  
P F Lesse

This paper deals with a class of models which describe spatial interactions and are based on Jaynes's principle. The variables entering these models can be partitioned in four groups: (a) probability density distributions (for example, relative traffic flows), (b) expected values (average cost of travel), (c) their duals (Lagrange multipliers, traffic impedance coefficient), and (d) operators transforming probabilities into expected values. The paper presents several dual formulations replacing the problem of maximizing entropy in terms of the group of variables (a) by equivalent extreme problems involving groups (b)-(d). These problems form the basis of a phenomenological theory. The theory makes it possible to derive useful relationships among groups (b) and (c). There are two topics discussed: (1) practical application of the theory (with examples), (2) the relationship between socioeconomic modelling and statistical mechanics.


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