A simple geometric interpretation of increasing associated multiplicity with increasing transverse momentum

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.

Download Full-text

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

Advances in Applied Probability ◽

10.1017/s0001867800006935 ◽

2014 ◽

Vol 46 (01) ◽

pp. 59-75 ◽

Cited By ~ 2

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.

Download Full-text

A novel bibliometric index with a simple geometric interpretation

PLoS ONE ◽

10.1371/journal.pone.0200098 ◽

2018 ◽

Vol 13 (7) ◽

pp. e0200098 ◽

Cited By ~ 15

Author(s):

Trevor Fenner ◽

Martyn Harris ◽

Mark Levene ◽

Judit Bar-Ilan

Keyword(s):

Geometric Interpretation ◽

Simple Geometric Interpretation ◽

Bibliometric Index

Download Full-text

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

Optics Express ◽

10.1364/oe.25.000223 ◽

2017 ◽

Vol 25 (1) ◽

pp. 223 ◽

Cited By ~ 1

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

Download Full-text

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

International Journal of Mechanical Engineering Education ◽

10.7227/ijmee.31.1.1 ◽

2003 ◽

Vol 31 (1) ◽

pp. 1-14 ◽

Cited By ~ 1

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.

Download Full-text

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

International Journal of Mechanical Engineering Education ◽

10.7227/ijmee.33.3.9 ◽

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.

Download Full-text

Estimation of the Form of the Best Fitting Model

The Canadian Surveyor ◽

10.1139/tcs-1965-0010 ◽

1965 ◽

Vol 19 (1) ◽

pp. 64-69

Author(s):

J. Vlcek

Keyword(s):

Statistical Theory ◽

Linear Models ◽

Geometric Interpretation ◽

Method Of Least Squares ◽

Normal Equations ◽

Method Of Solution ◽

Best Fitting ◽

Main Ideas ◽

Simple Geometric Interpretation ◽

Fitting Model

Fitting of linear models occupies a large portion of the adjustments of surveying data. In this paper, an attempt is made to present the main ideas of the statistical theory connected with the estimation of linear models, based on the method of least squares. A method of solution of normal equations is given, which lends itself to a simple geometric interpretation and has particular statistical advantages. Also, some possibilities for testing the adequacy of the fitted model are discussed.

Download Full-text

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

Advances in Applied Probability ◽

10.1239/aap/1396360103 ◽

2014 ◽

Vol 46 (1) ◽

pp. 59-75 ◽

Cited By ~ 3

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.

Download Full-text