scholarly journals Differential Equation and Recursive Formulas of Sheffer Polynomial Sequences

2011 ◽  
Vol 2011 ◽  
pp. 1-16 ◽  
Author(s):  
Heekyung Youn ◽  
Yongzhi Yang
Keyword(s):  
Differential Equation ◽  
Generating Functions ◽  
Matrix Algebra ◽  
Polynomial Sequences ◽  
Recursive Formulas ◽  
The Relationship ◽  
Sheffer Polynomial

We derive a differential equation and recursive formulas of Sheffer polynomial sequences utilizing matrix algebra. These formulas provide the defining characteristics of, and the means to compute, the Sheffer polynomial sequences. The tools we use are well-known Pascal functional and Wronskian matrices. The properties and the relationship between the two matrices simplify the complexity of the generating functions of Sheffer polynomial sequences. This work extends He and Ricci's work (2002) to a broader class of polynomial sequences, from Appell to Sheffer, using a different method. The work is self-contained.

Developed Plate Expansion Using Geodesics

1984 ◽  
Vol 21 (04) ◽  
pp. 384-388
Author(s):  
John C. Clements
Keyword(s):  
Differential Equation ◽  
Numerical Quadrature ◽  
Developable Surface ◽  
Accuracy Control ◽  
Isometric Mapping ◽  
Variable Stepsize ◽  
Mapping Algorithm ◽  
Equation Solving ◽  
Expansion Procedure ◽  
The Relationship

This work is concerned with the application of a new isometric mapping algorithm to hull plate expansion procedures for ships with all or portions of the hull consisting of developable surfaces. The expansion procedure is based on the relationship between the ruling lines r⇀(s) generating the developable surface S⇀(s,t) and one additional geodesic g⇀(s) constructed within the surface as the solution of the differential equation det(g⇀'g⇀"n⇀) = 0 where n⇀ is the unit normal to S⇀ at g⇀. Precise accuracy control is achieved through the use of adaptive numerical quadrature and a variable stepsize differential equation solving routine.

scholarly journals Relation between Small Functions with Differential Polynomials Generated by Solutions of Linear Differential Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-11
Author(s):  
Zhigang Huang
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Linear Differential Equation ◽  
Entire Functions ◽  
Linear Differential Equations ◽  
Differential Polynomials ◽  
Small Functions ◽  
Linear Differential ◽  
The Relationship ◽  
Growth Of Solutions

This paper is devoted to studying the growth of solutions of second-order nonhomogeneous linear differential equation with meromorphic coefficients. We also discuss the relationship between small functions and differential polynomialsL(f)=d2f″+d1f′+d0fgenerated by solutions of the above equation, whered0(z),d1(z),andd2(z)are entire functions that are not all equal to zero.

Generating Functions for Hermite Functions

1959 ◽  
Vol 11 ◽  
pp. 141-147 ◽  
Author(s):  
Louis Weisner
Keyword(s):  
Differential Equation ◽  
Entire Function ◽  
Generating Functions ◽  
Hermite Functions ◽  
Kummer’S Function ◽  
Image Position

Hermite's function Hn(x) is denned for all complex values of x and n bywhere F (α; γ; x) is Kummer's function with the customary indices omitted. It satisfies the differential equation1.1of whichis a second solution. Every solution of (1.1) is an entire function.

Notes on the Riccati Equation

2005 ◽  
Vol 70 (7) ◽  
pp. 941-950 ◽  
Author(s):  
Eugene S. Kryachko
Keyword(s):  
Differential Equation ◽  
Normal Form ◽  
Riccati Equation ◽  
Second Order ◽  
Order Linear Differential Equation ◽  
Riccati Transformation ◽  
Rational Solutions ◽  
Order Polynomial ◽  
Linear Differential ◽  
The Relationship

The relationship between the Riccati and Schrödinger equations is discussed. It is shown that the transformation converting the Riccati equation into its normal form is expressed in terms of the roots of its algebraic part treated as a second-order polynomial. Together with the well-known Riccati transformation, a new transformation which also links the Riccati equation to the second-order linear differential equation is introduced. The latter is actually the Riccati transformation applied to an "inverse" Riccati equation. Two specific forms of the Riccati equation admitting the explicit particular rational solutions are obtained.

Computer simulation of the hydraulic milking method

1992 ◽  
Vol 59 (3) ◽  
pp. 253-264
Author(s):  
M. Clare Butler ◽  
Robert J. Grindal
Keyword(s):  
Differential Equation ◽  
Flow Rate ◽  
Equations Of Motion ◽  
Flow Rates ◽  
Air Inlet ◽  
Wall Movement ◽  
Linear Differential ◽  
Milking Machine ◽  
The Relationship ◽  
Return Valve

SummaryA mathematical model of the interactions within a milking machine teatcup has been developed, which describes the relationship between liner wall movement, pressures and flow rate when milking without an air inlet. It is based on equations of motion for a column of incompressible fluid and requires a second-order, non-linear differential equation to be solved. Incorporating a non-return valve allows hydraulic milking to be modelled, and the comparison between predicted and measured pressures, flow rates and liner wall movement when milking hydraulically is shown. The model can be used to optimize milking conditions to reduce vacuum peaks, improve liner opening and thus maximize flow rate.

Characteristics of GWrSn

1976 ◽  
Vol 79 (3) ◽  
pp. 433-441
Author(s):  
A. G. Williams
Keyword(s):  
Finite Group ◽  
Symmetric Group ◽  
Wreath Product ◽  
Generating Functions ◽  
Block Structure ◽  
Clifford Theory ◽  
A Finite Group ◽  
The Relationship

The ‘characteristics’ of the wreath product GWrSn, where G is a finite group, are certain polynomials (to be defined in section 2) which are generating functions for the simple characters of GWrSn. Schur (8) first used characteristics of the symmetric group. Specht (9) defined characteristics for GWrSn and found a relation between the characteristics of GWrSn and those of Sn which determined the simple characters of GWrSn. The object of this paper is to describe the p-block structure of GWrSn in the case where p is not a factor of the order of G. We use the relationship between the characteristics of GWrSn and those of Sn, which we deduce from a knowledge of the simple characters of GWrSn (these can be determined, independently of Specht's work, by using Clifford theory).

Imbibition in a cracked porous medium

1969 ◽  
Vol 47 (22) ◽  
pp. 2519-2524 ◽  
Author(s):  
A. P. Verma
Keyword(s):  
Differential Equation ◽  
Porous Medium ◽  
Viscosity Ratio ◽  
Perturbation Technique ◽  
Water Imbibition ◽  
Phase Saturation ◽  
Water Viscosity ◽  
Oil Water ◽  
The Relationship ◽  
Special Case

In this paper, one special case of oil–water imbibition phenomena in a cracked porous medium of a finite length is analytically discussed. The equation for the linear countercurrent imbibition is a nonlinear differential equation whose solution has been obtained by a perturbation technique. For definiteness, specific results have been used for the relationship between relative permeability and phase saturation) impregnation function, oil–water viscosity ratio, and capillary pressure dependence on phase saturation due to Jones, Bokserman et al., Evgen'ev, and Oroveanu, respectively. An expression for the wetting phase saturation has been derived.

An algebraic approach to Sheffer polynomial sequences

2013 ◽  
Vol 25 (4) ◽  
pp. 295-311 ◽  
Author(s):  
Francesco Aldo Costabile ◽  
Elisabetta Longo
Keyword(s):  
Algebraic Approach ◽  
Polynomial Sequences ◽  
Sheffer Polynomial

scholarly journals Calculation and Analysis of the Amber Interval in Traffic Flow

2019 ◽  
pp. 1-10
Author(s):  
Meng Xinyu ◽  
Zhao Jian ◽  
Zhang Wei ◽  
Meng Zhaoping
Keyword(s):  
Differential Equation ◽  
Equation Model ◽  
Related Factors ◽  
Traffic Condition ◽  
Amber Light ◽  
A Value ◽  
Differential Equation Models ◽  
Light Duration ◽  
Difficult Area ◽  
The Relationship

According to the relationship between the speed of vehicle and the amber light, we establish the differential equation model of the amber light duration. And based on the relevant conditions given in the title, three differential equation models of amber light duration under different conditions are obtained. Considering the traffic condition and driver's habit, we calculate a value that is most suitable to the actual demand. The sensitivity and stability of the model and its related factors are analyzed. We improve the model for the problem of difficult area.

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