The Recursion Formula

1970 ◽  
Vol 63 (2) ◽  
pp. 121-125
Author(s):  
J. Garfunkel
Keyword(s):  
Recursion Formula ◽  
Equivalent Fraction

In Trying to find the sum of the fractions and , we usually replace each fraction by an equivalent fraction, e.g., . What is the purpose of this simple illustration? It shows clearly how mathematicians work. Given a prob lem, they try to solve it by solving an easier problem.

A Recursion Formula for the Coefficients of Entire Functions Satisfying an ODE with Polynomial Coefficients

2004 ◽  
Vol 11 (3) ◽  
pp. 409-414
Author(s):  
C. Belingeri
Keyword(s):  
Differential Equations ◽  
Sum Rules ◽  
Entire Functions ◽  
Recursion Formula ◽  
Linear Differential Equations ◽  
Bell Polynomials ◽  
Polynomial Coefficients ◽  
Linear Differential

Abstract A recursion formula for the coefficients of entire functions which are solutions of linear differential equations with polynomial coefficients is derived. Some explicit examples are developed. The Newton sum rules for the powers of zeros of a class of entire functions are constructed in terms of Bell polynomials.

A Greenwood-Williamson Model of Small-Scale Friction

2005 ◽  
Vol 74 (1) ◽  
pp. 31-40 ◽  
Author(s):  
Reese E. Jones
Keyword(s):  
Numerical Method ◽  
Recursion Formula ◽  
Three Dimensional ◽  
Interfacial Friction ◽  
Small Scale ◽  
Analytical Estimate ◽  
Normal Loading ◽  
Proposed Model ◽  
Loading Case ◽  
Transition Length

A Greenwood and Williamson based model for interfacial friction is presented that incorporates the presliding transition phenomenon that can significantly affect small devices. This work builds on previous similar models by developing: an analytical estimate of the transition length in terms of material and surface parameters, a general recursion formula for the case of slip in one direction with multiple reversals and constant normal loading, and a numerical method for the general three-dimensional loading case. In addition, the proposed model is developed within a plasticity-like framework and is shown to have qualitative similarities with published experimental observations. A number of model problems illustrate the response of the proposed model to various loading conditions.

Subgroups of Finite Index in Free Groups

1949 ◽  
Vol 1 (2) ◽  
pp. 187-190 ◽  
Author(s):  
Marshall Hall
Keyword(s):  
Free Group ◽  
Finite Index ◽  
Free Groups ◽  
Recursion Formula ◽  
Independent Interest ◽  
Finiteness Conditions ◽  
Total Length ◽  
Free Generators

This paper has as its chief aim the establishment of two formulae associated with subgroups of finite index in free groups. The first of these (Theorem 3.1) gives an expression for the total length of the free generators of a subgroup U of the free group Fr with r generators. The second (Theorem 5.2) gives a recursion formula for calculating the number of distinct subgroups of index n in Fr.Of some independent interest are two theorems used which do not involve any finiteness conditions. These are concerned with ways of determining a subgroup U of F.

A digital phase shift method for phase compensation of electronic transformer

2017 ◽  
Vol 40 (13) ◽  
pp. 3690-3695 ◽  
Author(s):  
Wei Wei ◽  
Han-miao Cheng ◽  
Fan Li ◽  
Deng-ping Tang ◽  
Shui-bin Xia
Keyword(s):  
Phase Shift ◽  
Phase Error ◽  
Recursion Formula ◽  
Rogowski Coil ◽  
Shift Method ◽  
Electronic Transformer ◽  
Digital Phase ◽  
Fitting Algorithm ◽  
Fixed Phase ◽  
Phase Shift Method

When sampling analog signal, the electronic transformer generally produces a fixed phase error that will compromise the measurement accuracy and require a phase shift method for correction. In this paper, we propose a digital phase shift method based on least squares fitting algorithm and derive the recursion formula of digital phase shift. The simulation has also been done to analysis its performance. The result shows that the method has high phase shift resolution and precision. By applying the method to an electronic transformer based on Rogowski coil, we have experimentally verified the feasibility and validity of the method.

Electromagnetic Induction in a Stratified Conducting Half-Space by an Arbitrary Periodic Source

1973 ◽  
Vol 51 (10) ◽  
pp. 1064-1074 ◽  
Author(s):  
D. M. Summers ◽  
J. T. Weaver
Keyword(s):  
Half Space ◽  
Recursion Formula ◽  
Electric And Magnetic Fields ◽  
Hertz Potential ◽  
Infinite Line ◽  
The Fourier Transform ◽  
The One ◽  
External Time ◽  
Entire Theory ◽  
Time Periodic

A general theory of induction in a horizontally stratified plane conductor by an external, time-periodic, magnetic source is presented. The analysis is a generalization to the case of an N-layered conductor of a previously published theory for induction in a uniform conducting half-space, in which the electromagnetic field was expressed in terms of electric and magnetic Hertz vectors oriented normally to the surface of the conductor. With the aid of this representation the entire theory is developed in terms of the one scalar component of the magnetic Hertz vector. Solutions for the electric and magnetic fields above and within the conductor are obtained in the form of double integrals whose integrands are related through a recursion formula to the Fourier transform of the magnetic Hertz potential of the source evaluated at the surface of the conductor. Special formulas applicable to 1- and 2-layer conductors are derived and the form of solution for some elementary sources is also discussed. As an illustration of the theory, numerical calculations are given for an infinite line current above a 10-layer conductor whose conductivity increases (i) linearly and (ii) exponentially with depth.

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