An Iterative Procedure for Computing the Minimum of a Quadratic Form on a Convex Set

1966 ◽  
Vol 4 (1) ◽  
pp. 61-80 ◽  
Author(s):  
Elmer G. Gilbert
Keyword(s):  
Quadratic Form ◽  
Iterative Procedure ◽  
Convex Set

A New Iterative Procedure for the Minimization of a Quadratic Form on a Convex Set

1970 ◽  
Vol 8 (3) ◽  
pp. 396-402 ◽  
Author(s):  
Thomas Pecsvaradi ◽  
Kumpati S. Narendra
Keyword(s):  
Quadratic Form ◽  
Iterative Procedure ◽  
Convex Set

On the construction of orthogonal curvilinear grids for regional oceanic modeling: algorithm and user guide

2020 ◽  
Vol 48 (4) ◽  
pp. 45-111
Author(s):  
A. F. Shepetkin
Keyword(s):  
Inverse Problem ◽  
Numerical Solution ◽  
Elliptic Problem ◽  
Software Package ◽  
Conformal Transformation ◽  
Iterative Procedure ◽  
Geometric Shape ◽  
Curvilinear Grids ◽  
Grid Points ◽  
Grid Nodes

A new algorithm for constructing orthogonal curvilinear grids on a sphere for a fairly general geometric shape of the modeling region is implemented as a “compile-once - use forever” software package. It is based on the numerical solution of the inverse problem by an iterative procedure -- finding such distribution of grid points along its perimeter, so that the conformal transformation of the perimeter into a rectangle turns this distribution into uniform one. The iterative procedure itself turns out to be multilevel - i.e. an iterative loop built around another, internal iterative procedure. Thereafter, knowing this distribution, the grid nodes inside the region are obtained solving an elliptic problem. It is shown that it was possible to obtain the exact orthogonality of the perimeter at the corners of the grid, to achieve very small, previously unattainable level of orthogonality errors, as well as make it isotropic -- local distances between grid nodes about both directions are equal to each other.

Spheric radial basis functions in Mahalanobis measuring for displaying local field features

2019 ◽  
Vol 952 (10) ◽  
pp. 2-9
Author(s):  
Yu.M. Neiman ◽  
L.S. Sugaipova ◽  
V.V. Popadyev
Keyword(s):  
Gravitational Field ◽  
Quadratic Form ◽  
Local Field ◽  
Euclidean Distance ◽  
Basis Functions ◽  
Spherical Functions ◽  
Local Models ◽  
Stationary Field ◽  
Modeling Process ◽  
The Earth

As we know the spherical functions are traditionally used in geodesy for modeling the gravitational field of the Earth. But the gravitational field is not stationary either in space or in time (but the latter is beyond the scope of this article) and can change quite strongly in various directions. By its nature, the spherical functions do not fully display the local features of the field. With this in mind it is advisable to use spatially localized basis functions. So it is convenient to divide the region under consideration into segments with a nearly stationary field. The complexity of the field in each segment can be characterized by means of an anisotropic matrix resulting from the covariance analysis of the field. If we approach the modeling in this way there can arise a problem of poor coherence of local models on segments’ borders. To solve the above mentioned problem it is proposed in this article to use new basis functions with Mahalanobis metric instead of the usual Euclidean distance. The Mahalanobis metric and the quadratic form generalizing this metric enables us to take into account the structure of the field when determining the distance between the points and to make the modeling process continuous.

Sharpening an Estimate of the Size of the Sumset of a Convex Set

2020 ◽  
Vol 107 (5-6) ◽  
pp. 984-987
Author(s):  
K. I. Ol’mezov
Keyword(s):  
Convex Set

EVALUATION OF CONVOLUTION SUMS AND FOR k = a · b = 21, 33, AND 35

2021 ◽  
pp. 1-20
Author(s):  
K. PUSHPA ◽  
K. R. VASUKI
Keyword(s):  
Positive Integer ◽  
Quadratic Form ◽  
Eisenstein Series ◽  
Modular Forms ◽  
Divisor Function ◽  
Convolution Sums

Abstract The article focuses on the evaluation of convolution sums $${W_k}(n): = \mathop \sum \nolimits_{_{m < {n \over k}}} \sigma (m)\sigma (n - km)$$ involving the sum of divisor function $$\sigma (n)$$ for k =21, 33, and 35. In this article, our aim is to obtain certain Eisenstein series of level 21 and use them to evaluate the convolution sums for level 21. We also make use of the existing Eisenstein series identities for level 33 and 35 in evaluating the convolution sums for level 33 and 35. Most of the convolution sums were evaluated using the theory of modular forms, whereas we have devised a technique which is free from the theory of modular forms. As an application, we determine a formula for the number of representations of a positive integer n by the octonary quadratic form $$(x_1^2 + {x_1}{x_2} + ax_2^2 + x_3^2 + {x_3}{x_4} + ax_4^2) + b(x_5^2 + {x_5}{x_6} + ax_6^2 + x_7^2 + {x_7}{x_8} + ax_8^2)$$ , for (a, b)=(1, 7), (1, 11), (2, 3), and (2, 5).

Effect of concentration dependence of viscosity on squeeze film lubrication

2020 ◽  
Vol 75 (6) ◽  
pp. 533-542
Author(s):  
Poosan Muthu ◽  
Vanacharla Pujitha
Keyword(s):  
Carrying Capacity ◽  
Iterative Procedure ◽  
Squeeze Film ◽  
Load Carrying Capacity ◽  
Convection Diffusion ◽  
Load Carrying ◽  
Newtonian Viscous Fluid ◽  
Nicolson Scheme ◽  
Human Joint ◽  
Film Lubrication

AbstractThe influence of concentration of solute particles on squeeze film lubrication between two poroelastic surfaces has been analyzed using a mathematical model. Newtonian viscous fluid is considered as a lubricant whose viscosity varies linearly with concentration of suspended solute particles. Convection-diffusion model is proposed to study the concentration of solute particles and is solved using finite difference method of Crank–Nicolson scheme. An iterative procedure is used to get the solution for concentration, pressure and velocity components in film region. It has been observed that load carrying capacity decreases as the concentration of solute particles in the fluid film decreases. Further, the concentration of suspended solute particles decreases as the permeability of the poroelastic plate increases and these results may be useful in understanding the mechanism of human joint.

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