scholarly journals Series Representation of Power Function

2018 ◽  
Author(s):  
Petro Kolosov
Keyword(s):  
Numerical Methods ◽  
Power Function ◽  
Exponential Function ◽  
Finite Differences ◽  
Series Representation ◽  
Discrete Case ◽  
Natural Numbers ◽  
Binomial Theorem ◽  
Binomial Expansion

In this paper described numerical expansion of natural-valued power function xn, in point x = x0 where n, x0 - natural numbers. Apply- ing numerical methods, that is calculus of finite differences, namely, discrete case of Binomial expansion is reached. Received results were compared with solutions according to Newton’s Binomial theorem and MacMillan Double Bi- nomial sum. Additionally, in section 4 exponential function’s ex representation is shown.

scholarly journals Series Representation of Power Function

2018 ◽  
Author(s):  
Petro Kolosov
Keyword(s):  
Numerical Methods ◽  
Power Function ◽  
Exponential Function ◽  
Finite Differences ◽  
Series Representation ◽  
Discrete Case ◽  
Natural Numbers ◽  
Binomial Theorem ◽  
Binomial Expansion

In this paper described numerical expansion of natural-valued power function xn, in point x = x0 where n, x0 - natural numbers. Apply- ing numerical methods, that is calculus of finite differences, namely, discrete case of Binomial expansion is reached. Received results were compared with solutions according to Newton’s Binomial theorem and MacMillan Double Bi- nomial sum. Additionally, in section 4 exponential function’s ex representation is shown.

scholarly journals Series Representation of Power Function

2017 ◽  
Author(s):  
Kolosov Petro
Keyword(s):  
Numerical Methods ◽  
Power Function ◽  
Exponential Function ◽  
Finite Differences ◽  
Series Representation ◽  
Discrete Case ◽  
Natural Numbers ◽  
Binomial Theorem ◽  
Binomial Expansion

In this paper described numerical expansion of natural-valued power function xn, in point x = x0 where n, x0 - natural numbers. Apply- ing numerical methods, that is calculus of finite differences, namely, discrete case of Binomial expansion is reached. Received results were compared with solutions according to Newton’s Binomial theorem and MacMillan Double Bi- nomial sum. Additionally, in section 4 exponential function’s ex representation is shown.

scholarly journals Series Representation of Power Function

2018 ◽  
Author(s):  
Petro Kolosov
Keyword(s):  
Numerical Methods ◽  
Power Function ◽  
Series Representation ◽  
Positive Integers

In this paper described numerical expansion of natural-valued power function xn, in point x = x0, where n; x0 - positive integers. Applying numerical methods, the calculus of nite dierences, particular pattern, that is sequence A287326 in OEIS, which shows the expansion of perfect cube n as row sum over k; 0 ≤ k ≤ n − 1 is generalized, obtained results are applied to show expansion of monomial n2m+1; m = 0; 1; 2; ..., N. Additionally, relation between Faulhaber's sum ∑nm and nite dierences of power are shown in section 4.

The Wind-Sand Flux Structure of Grassland on the Northern Foot of Yinshan Mountain

2014 ◽  
Vol 668-669 ◽  
pp. 1530-1537
Author(s):  
Hong Tao Jiang ◽  
Chun Rong Guo ◽  
Chun Xing Hai ◽  
Shan Shan Sun ◽  
Yun Hu Xie ◽  
...  
Keyword(s):  
Vertical Distribution ◽  
Power Function ◽  
Exponential Function ◽  
Vertical Direction ◽  
Power Functions ◽  
Height Range ◽  
Sand Flux ◽  
Soil Flux ◽  
The Vertical Distribution ◽  
Better Than

Sand samplers were layed out in the grassland located in the northern foot of Yinshan Mountain for collecting soil flux samples from 0 to 1.5m height above the surface from Mar., 1, 2008 to Feb., 29, 2009.Exponential and Power functions were both used for describing vertical distribution of sand flux in the grassland, the results indicated that determination coefficient of Power function varied from 0.898 to 0.992 while 0.432 to 0.661 for exponential function. Power function is better than exponential function in describing the vertical distribution of both annual and seasonal soil flux, summer excluded. Annual cumulative percentage of each height was determined indirectly according to the power function mentioned above, the result indicated that up to 2m height,15-25% of soil flux concentrated with in 10cm above the surface,25-35% of soil flux concentrated within 20cm above the surface,30-40% of soil flux concentrated within 30 cm above the surface, 43-54% of soil flux concentrated within 50 cm above the surface,85-90% of soil flux concentrated within 150 cm above the surface, respectively. No significant differences of soil flux structures in spring, autumn, winter and in the whole year were found. The research on wind erosion of grassland in the vertical direction more dispersed, in the height range of sediment accumulated percentage was lower than that of the previous research.

scholarly journals Elementary methods in the theory of numbers

1939 ◽  
Vol 31 ◽  
pp. xvi-xxiii
Author(s):  
S. A. Scott
Keyword(s):  
Number Theory ◽  
Positive Integer ◽  
Exponential Function ◽  
Monotone Sequence ◽  
Binomial Theorem ◽  
Irrational Numbers ◽  
Early Stages ◽  
Elementary Methods ◽  
Usual Notation ◽  
Theory Of Numbers

§ 1. The importance of proving inequalities of an essentially algebraic nature by “elementary” methods has been emphasised by Hardy (Prolegomena to a Chapter on Inequalities), and by Hardy, Littlewood and Polya (Inequalities). The object of this Note is to show how some of the results in the early stages of Number Theory can be obtained by making a minimum appeal to irrational numbers and the notion of a limit. We use the elementary notion of a logarithm to a base “a” > 1, and make no appeal to the exponential function. The Binomial Theorem is only used for a positive integer index. Our minimum appeal rests in the assumption that a bounded monotone sequence tends to a limit. We adopt throughout the usual notation. Finally, it need scarcely be added that the methods employed are not claimed to be new.

The Method to Determine the Expression for Fatigue Curves

2007 ◽  
Vol 353-358 ◽  
pp. 331-334
Author(s):  
Hong Liang Yi ◽  
Ming Tu Ma ◽  
Zhi Gang Li ◽  
Hao Zhang
Keyword(s):  
Power Function ◽  
Exponential Function ◽  
Constant Term ◽  
Empirical Expressions ◽  
Fatigue Data ◽  
Function Expression ◽  
Two Parameter

There are three common empirical expressions used for the fatigue curves, which are power function, exponential function and three-parameter power function expression, respectively. The mathematical difference between the former two and the latter is whether there exists the constant term S0 in the equations. The S0 can be calculated to determine whether the two-paprameter expression or three-parameter expression should be used. If the two-parameter expression should be used, the power function and exponential function expressions can be compared to determine which one is the optimum one. Finally, the method has been validated by several groups of fatigue data.

scholarly journals A Reduction of Integer Factorization to Modular Tetration

2020 ◽  
Vol 31 (04) ◽  
pp. 461-481
Author(s):  
Markus Hittmeir
Keyword(s):  
Exponential Function ◽  
Polynomial Time ◽  
Integer Factorization ◽  
Natural Numbers ◽  
Prime Factorization

Let [Formula: see text]. By [Formula: see text] and [Formula: see text], we denote the [Formula: see text] th iterate of the exponential function [Formula: see text] evaluated at [Formula: see text], also known as tetration. We demonstrate how an algorithm for evaluating tetration modulo natural numbers [Formula: see text] could be used to compute the prime factorization of [Formula: see text] and provide heuristic arguments for the efficiency of this reduction. Additionally, we prove that the problem of computing the squarefree part of integers is deterministically polynomial-time reducible to modular tetration.

Delving Deeper: Derivatives of Generalized Power Functions

2009 ◽  
Vol 102 (7) ◽  
pp. 554-557
Author(s):  
John M. Johnson
Keyword(s):  
Power Function ◽  
Exponential Function ◽  
Power Functions ◽  
Exponential Functions ◽  
First Semester ◽  
Derivatives Of

After several years of teaching multiple sections of first-semester calculus, it was easy for me to think that I had nothing new to learn. But every year and every class bring a new group of students with their unique gifts and insights. In a recent class, after covering the derivative rules for power functions and exponential functions, I asked the class about the derivative of a function like y = xsinx, which is neither a power function (the power is not constant) nor an exponential function (the base is not constant).

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