Complementary variational principles in neutron diffusion theory

1967 ◽  
Vol 298 (1452) ◽  
pp. 97-101 ◽  
Keyword(s):  
Lower Bounds ◽  
Variational Principles ◽  
Diffusion Theory ◽  
Upper And Lower Bounds ◽  
Neutron Diffusion ◽  
Diffusion In Solids ◽  
Absorption Probability

Complementary variational principles associated with neutron diffusion in solids are presented. The resulting formulae are used to derive new expressions which provide upper and lower bounds for the absorption probability.

Complementary variational principles for free molecular channel flow

1979 ◽  
Vol 82 (3-4) ◽  
pp. 211-223 ◽  
Author(s):  
R. J. Cole
Keyword(s):  
Integral Equations ◽  
Channel Flow ◽  
Lower Bounds ◽  
Variational Principles ◽  
Transmission Probability ◽  
Numerical Calculations ◽  
Upper And Lower Bounds ◽  
Gas Flows ◽  
Rectangular Duct ◽  
Molecular Channel

SynopsisA collisionless gas flows through a finite rectangular duct which reflects molecules diffusely. The transmission probability Q of the duct involves the solution of a pair of coupled integral equations. Complementary variational principles have been employed which supply upper and lower bounds to Q. Numerical calculations have been made for a variety of duct shapes and compared, where appropriate, to those of other authors.

Growth rates of bending KdV solitons

1982 ◽  
Vol 28 (3) ◽  
pp. 469-484 ◽  
Author(s):  
E. W. Laedke ◽  
K. H. Spatschek
Keyword(s):  
Growth Rate ◽  
Lower Bounds ◽  
Acoustic Waves ◽  
Variational Principles ◽  
Soliton Solutions ◽  
Type Equation ◽  
Maximum Growth ◽  
Maximum Growth Rate ◽  
Upper And Lower Bounds ◽  
Ion Acoustic Waves

Nonlinear ion-acoustic waves in magnetized plasmas are investigated. In strong magnetic fields they can be described by a Korteweg-de Vries (KdV) type equation. It is shown here that these plane soliton solutions become unstable with respect to bending distortions. Variational principles are derived for the maximum growth rate γ as a function of the transverse wavenumber k of the perturbations. Since the variational principles are formulated in complementary form, the numerical evaluation yields upper and lower bounds for γ. Choosing appropriate test functions and increasing the accuracy of the computations we find very close upper and lower bounds for the γ(k) curve. The results show that the growth rate peaks at a certain value of k and a cut-off kc exists. In the region where the γ(k) curve was not predicted numerically with high accuracy, i.e. near the cut-off, we find very precise analytical estimates. These findings are compared with previous results. For k≥kc, stability with respect to transverse perturbations is proved.

Complementary variational principles in perturbation theory

1968 ◽  
Vol 303 (1475) ◽  
pp. 503-509 ◽  
Keyword(s):  
Perturbation Theory ◽  
Lower Bound ◽  
Harmonic Oscillator ◽  
Lower Bounds ◽  
Upper Bound ◽  
Variational Principles ◽  
Upper And Lower Bounds ◽  
Quantum Mechanical ◽  
Simple Application ◽  
Mechanical Perturbation

Complementary upper and lower bounds are derived for second-order quantum-mechanical perturbation energies. The upper bound is equivalent to that of Hylleraas. The lower bound appears to be new, but reduces to that of Prager & Hirschfelder if a certain constraint is applied. A simple application to a perturbed harmonic oscillator is presented.

Extremum principles for electrostatic and magnetostatic problems

1969 ◽  
Vol 66 (2) ◽  
pp. 433-436 ◽  
Author(s):  
A. M. Arthurs ◽  
P. D. Robinson
Keyword(s):  
Lower Bounds ◽  
Variational Principles ◽  
Upper And Lower Bounds ◽  
Adjoint Operators

AbstractComplementary variational principles are derived for a class of electrostatic and magnetostatic problems using the pairs of adjoint operators (grad, – div) and (curl, curl). This theory unifies the principles of Dirichlet and Thomson in electrostatics and of Schrader in magnetostatics. The results are illustrated by deriving upper and lower bounds for the capacity of a surface, and it is shown how such bounds can be systematically improved by Ritz procedures.

Upper and lower bounds for the torsional stiffness of a prismatic bar

1972 ◽  
Vol 328 (1573) ◽  
pp. 295-299 ◽  
Keyword(s):  
Lower Bound ◽  
Cross Section ◽  
Lower Bounds ◽  
Upper Bound ◽  
Variational Principles ◽  
Circular Cross Section ◽  
Torsional Stiffness ◽  
Upper And Lower Bounds ◽  
Steady Creep

Upper and lower bounds for the torsional stiffness of a prismatic bar in steady creep are derived in a unified manner from the theory of complementary variational principles. The lower bound is known in the literature, but the upper bound appears to be new. The results are illustrated with calculations for a bar with circular cross-section.

Flow in random porous media: mathematical formulation, variational principles, and rigorous bounds

1989 ◽  
Vol 206 ◽  
pp. 25-46 ◽  
Author(s):  
Jacob Rubinstein ◽  
S. Torquato
Keyword(s):  
Lower Bounds ◽  
Variational Principles ◽  
Mathematical Formulation ◽  
Upper And Lower Bounds ◽  
Random Boundary ◽  
Slow Viscous Flow ◽  
Fluid Permeability ◽  
Different Types ◽  
Rigorous Bounds ◽  
Random Porous Medium

The problem of the slow viscous flow of a fluid through a random porous medium is considered. The macroscopic Darcy's law, which defines the fluid permeability k, is first derived in an ensemble-average formulation using the method of homogenization. The fluid permeability is given explicitly in terms of a random boundary-value problem. General variational principles, different to ones suggested earlier, are then formulated in order to obtain rigorous upper and lower bounds on k. These variational principles are applied by evaluating them for four different types of admissible fields. Each bound is generally given in terms of various kinds of correlation functions which statistically characterize the microstructure of the medium. The upper and lower bounds are computed for flow interior and exterior to distributions of spheres.

Upper and lower bounds for the overall properties of a nonlinear composite dielectric. II. Periodic microgeometry

1994 ◽  
Vol 447 (1930) ◽  
pp. 385-396 ◽  
Keyword(s):  
Variational Principle ◽  
Electric Field ◽  
Lower Bounds ◽  
Variational Principles ◽  
Effective Properties ◽  
Upper And Lower Bounds ◽  
New Approach ◽  
Simple Cubic ◽  
Nonlinear Composite ◽  
Nonlinear Comparison

The study of the effective properties of a nonlinear composite dielectric begun in the companion to this paper is continued in the context of a simple cubic array of inclusions embedded in a matrix. Use of a nonlinear comparison material in the nonlinear Hashin-Shtrikman variational principles permits the generation of both upper and lower bounds for any composite. It is straightforward to apply the new approach to the simple composite considered here as the trial electric field which is substituted into the Hashin-Shtrikman variational principle is finite everywhere. The results obtained indicate the likely improvement over the simple classical bounds that can be obtained using the Hashin-Shtrikman principles.

scholarly journals On the Generalized Distance Energy of Graphs

Mathematics ◽  
2019 ◽  
Vol 8 (1) ◽  
pp. 17 ◽  
Author(s):  
Abdollah Alhevaz ◽  
Maryam Baghipur ◽  
Hilal A. Ganie ◽  
Yilun Shang
Keyword(s):  
Lower Bounds ◽  
Complete Graph ◽  
Connected Graph ◽  
Distance Matrix ◽  
Wiener Index ◽  
Diagonal Matrix ◽  
Upper And Lower Bounds ◽  
Star Graph ◽  
Extremal Graphs ◽  
Generalized Distance

The generalized distance matrix D α ( G ) of a connected graph G is defined as D α ( G ) = α T r ( G ) + ( 1 − α ) D ( G ) , where 0 ≤ α ≤ 1 , D ( G ) is the distance matrix and T r ( G ) is the diagonal matrix of the node transmissions. In this paper, we extend the concept of energy to the generalized distance matrix and define the generalized distance energy E D α ( G ) . Some new upper and lower bounds for the generalized distance energy E D α ( G ) of G are established based on parameters including the Wiener index W ( G ) and the transmission degrees. Extremal graphs attaining these bounds are identified. It is found that the complete graph has the minimum generalized distance energy among all connected graphs, while the minimum is attained by the star graph among trees of order n.

scholarly journals Hadamard k-fractional inequalities of Fejér type for GA-s-convex mappings and applications

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Hui Lei ◽  
Gou Hu ◽  
Zhi-Jie Cao ◽  
Ting-Song Du
Keyword(s):  
Lower Bounds ◽  
Hypergeometric Functions ◽  
Upper And Lower Bounds ◽  
Weighted Inequalities ◽  
Convex Mappings ◽  
Special Means

Abstract The main aim of this paper is to establish some Fejér-type inequalities involving hypergeometric functions in terms of GA-s-convexity. For this purpose, we construct a Hadamard k-fractional identity related to geometrically symmetric mappings. Moreover, we give the upper and lower bounds for the weighted inequalities via products of two different mappings. Some applications of the presented results to special means are also provided.

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