Degeneracy in Linear Programming: A Simple Geometric Interpretation

1957 ◽  
Vol 39 (4) ◽  
pp. 402 ◽  
Author(s):  
Richard R. Nelson
Keyword(s):  
Linear Programming ◽  
Geometric Interpretation ◽  
Simple Geometric Interpretation

The ellipsoid of alinement and its precessional motion in magnetic resonance

1972 ◽  
Vol 330 (1581) ◽  
pp. 271-280 ◽  
Keyword(s):  
Magnetic Resonance ◽  
Field Strength ◽  
Geometric Interpretation ◽  
Static Field ◽  
Relaxation Processes ◽  
At Resonance ◽  
Principal Axes ◽  
Precessional Motion ◽  
Simple Geometric Interpretation ◽  
Limiting Value

A surface of the second degree is constructed from the five components of the second rank tensor which describes the alinement of a spin-assembly undergoing magnetic resonance. The functions which characterize alinement are given a simple geometric interpretation in terms of radii vectores of this ellipsoid. The time-dependence of the different resonance functions at frequencies 0, ω and 2 ω is easily understood in terms of the rotation of the ellipsoid. In the absence of an r.f. field, and with pumping and relaxation processes only, the ellipsoid is uniaxial with its axes in the direction of the static field ( Z axis). With a weak r.f. field the shape of the ellipsoid is unchanged, but it is tilted and precesses round the Z axis at the frequency of the driving field. With stronger r.f. fields the shape of the ellipsoid changes, but at resonance one of the principal axes is always in the direction of the r.f. field and the length of this axis is independent of the field strength. At resonance also, the tilt increases to a limiting value of 1/4π with increasing r.f. field strength and the lengths of the axes in the plane perpendicular to the r.f. field tend to equality.

scholarly journals Dual Approach in the Application of Geometric Interpretation of Linear Programming on the Organisation of Goods Distribution

2021 ◽  
Vol 33 (6) ◽  
pp. 847-858
Author(s):  
Bogdan Marković ◽  
Milan Marković
Keyword(s):  
Linear Programming ◽  
Geometric Interpretation ◽  
Manufacturing Companies ◽  
Mathematical Aspect ◽  
Calculation Methods ◽  
Dual Approach ◽  
The Core ◽  
Primal And Dual ◽  
Core Activity ◽  
Research Studies

The topic of the paper is the application of dual approach in formulation and resolution of goods distribution tasks problems. The gap in previous goods distribution research is the absence of the methodologies and goods transportation calculation methods for manufacturing companies with not too large amount of goods distribution whereby goods distribution is not the core activity. The goal of this paper is to find a solution for transportation in such companies. In such cases it is not rational to procure a specific software for the improvement of goods transportation but rather apply the calculation presented in this paper. The aim of this paper from mathematical aspect is to show the convenience of switching from the basic geometric interpretation of linear programming applied on transportation tasks to dual approach for the companies with too many costs limitations per transport task but not enough available transportation means. Recent research studies that use dual approach in linear programming are generally not applied to transportation tasks although such approach is very convenient. The goal of the paper is also to resolve transportation tasks by both primal and dual approach in order to prove the correctness of the method.

scholarly journals Stability of Multi-Dimensional Birth-and-Death Processes with State-Dependent 0-Homogeneous Jumps

2014 ◽  
Vol 46 (01) ◽  
pp. 59-75 ◽  
Author(s):  
Matthieu Jonckheere ◽  
Seva Shneer
Keyword(s):  
Stochastic Systems ◽  
Direct Proof ◽  
Geometric Interpretation ◽  
Stability Conditions ◽  
Birth And Death Processes ◽  
Piecewise Constant ◽  
State Dependent ◽  
The Stability ◽  
Deterministic Dynamical System ◽  
Simple Geometric Interpretation

We study the conditions for positive recurrence and transience of multi-dimensional birth-and-death processes describing the evolution of a large class of stochastic systems, a typical example being the randomly varying number of flow-level transfers in a telecommunication wire-line or wireless network. First, using an associated deterministic dynamical system, we provide a generic method to construct a Lyapunov function when the drift is a smooth function on ℝN. This approach gives an elementary and direct proof of ergodicity. We also provide instability conditions. Our main contribution consists of showing how discontinuous drifts change the nature of the stability conditions and of providing generic sufficient stability conditions having a simple geometric interpretation. These conditions turn out to be necessary (outside a negligible set of the parameter space) for piecewise constant drifts in dimension two.

scholarly journals A novel bibliometric index with a simple geometric interpretation

PLoS ONE ◽  
2018 ◽  
Vol 13 (7) ◽  
pp. e0200098 ◽  
Author(s):  
Trevor Fenner ◽  
Martyn Harris ◽  
Mark Levene ◽  
Judit Bar-Ilan
Keyword(s):  
Geometric Interpretation ◽  
Simple Geometric Interpretation ◽  
Bibliometric Index

scholarly journals Simple geometric interpretation of signal evolution in phase-sensitive fibre optic parametric amplifier

2017 ◽  
Vol 25 (1) ◽  
pp. 223 ◽  
Author(s):  
A. A. Redyuk ◽  
A. E. Bednyakova ◽  
S. B. Medvedev ◽  
M. P. Fedoruk ◽  
S. K. Turitsyn
Keyword(s):  
Geometric Interpretation ◽  
Parametric Amplifier ◽  
Signal Evolution ◽  
Fibre Optic ◽  
Phase Sensitive ◽  
Simple Geometric Interpretation

An Engineer's Approach to a General Algorithm for Finding the Sum of Powers of Natural Numbers

2003 ◽  
Vol 31 (1) ◽  
pp. 1-14 ◽  
Author(s):  
R. Venkatachalam ◽  
Umesh Chandra Sharma
Keyword(s):  
Positive Integer ◽  
Geometric Interpretation ◽  
General Algorithm ◽  
Natural Numbers ◽  
Engineering Applications ◽  
Engineering Approach ◽  
Simple Geometric Interpretation

Convenient formulae for finding the sums of kth power of the first n natural numbers may be useful in some engineering applications. The formulae for k = 1, 2, and 3, are commonly found in the literature. In this paper, an attempt is made to develop a general algorithm for finding the sum for any positive integer value of k. The development of the algorithm is entirely an engineering approach, based purely on a simple geometric interpretation and does not involve any deep mathematics. This algorithm may be used to derive the formulae for different values of k. Some of the possible engineering applications of these formulae are also discussed.

A Geometric Interpretation of the Simplex Method of Linear Programming

1973 ◽  
Vol 66 (3) ◽  
pp. 257-264
Author(s):  
Maurice Nadler
Keyword(s):  
Linear Programming ◽  
Simplex Method ◽  
Mathematical Knowledge ◽  
Matrix Algebra ◽  
Elementary Mathematics ◽  
Geometric Interpretation ◽  
Analytic Geometry ◽  
Mathematical Basis ◽  
Straight Line ◽  
Linear Programming Problems

Concepts and techniques of matrix algebra, sometimes relatively sophisticated ones, are the mathematical basis of the simplex method of solving linear programming problems. The procedures, however, can be learned without advanced mathematical knowledge and can to an extent be explained on the basis of elementary mathematics. Simple analytic geometry of the straight line will be used for an explanation of the strategy of the simplex method.

A Simple Tool for Engineers for Generating Formulae for the Sum of Powers of Natural Numbers

2005 ◽  
Vol 33 (3) ◽  
pp. 278-282
Author(s):  
R. Venkatachalam ◽  
Umesh Chandra Sharma
Keyword(s):  
Automatic Generation ◽  
Geometric Interpretation ◽  
Natural Numbers ◽  
Simple Tool ◽  
Close Study ◽  
Simplified Algorithm ◽  
Simple Geometric Interpretation

The formulae for the sum of powers of natural numbers are in the form of polynomials. These formulae are derived using an algorithm which was developed from a simple geometric interpretation. It has been found that these polynomials are interdependent. In this paper, a close study is made and many interesting features are brought out. A method is proposed to generate a new polynomial based on the features of these polynomials. Another simplified algorithm is also presented which is found to be more suitable for the automatic generation of the polynomials. The latter may be realized as a convenient and handy tool for generating the formulae, especially for engineers.

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