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 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.

scholarly journals ON THE DENSITY OF INTEGERS OF THE FORM (p−1)2−n IN ARITHMETIC PROGRESSIONS

2008 ◽  
Vol 78 (3) ◽  
pp. 431-436 ◽  
Author(s):  
XUE-GONG SUN ◽  
JIN-HUI FANG
Keyword(s):  
Positive Integer ◽  
Prime Number ◽  
Arithmetic Progression ◽  
Arithmetic Progressions ◽  
Asymptotic Density ◽  
Natural Numbers ◽  
Positive Proportion ◽  
Covering System

AbstractErdős and Odlyzko proved that odd integers k such that k2n+1 is prime for some positive integer n have a positive lower density. In this paper, we characterize all arithmetic progressions in which natural numbers that can be expressed in the form (p−1)2−n (where p is a prime number) have a positive proportion. We also prove that an arithmetic progression consisting of odd numbers can be obtained from a covering system if and only if those integers in such a progression which can be expressed in the form (p−1)2−n have an asymptotic density of zero.

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 Evaluation of the convolution sum involving the sum of divisors function for 22, 44 and 52

2017 ◽  
Vol 15 (1) ◽  
pp. 446-458 ◽  
Author(s):  
Ebénézer Ntienjem
Keyword(s):  
Positive Integer ◽  
Modular Forms ◽  
Quadratic Forms ◽  
Modular Space ◽  
Natural Numbers ◽  
Convolution Sums ◽  
Sum Of Divisors

Abstract The convolution sum, $ \begin{array}{} \sum\limits_{{(l\, ,m)\in \mathbb{N}_{0}^{2}}\atop{\alpha \,l+\beta\, m=n}} \sigma(l)\sigma(m), \end{array} $ where αβ = 22, 44, 52, is evaluated for all natural numbers n. Modular forms are used to achieve these evaluations. Since the modular space of level 22 is contained in that of level 44, we almost completely use the basis elements of the modular space of level 44 to carry out the evaluation of the convolution sums for αβ = 22. We then use these convolution sums to determine formulae for the number of representations of a positive integer by the octonary quadratic forms $a\,(x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2})+b\,(x_{5}^{2}+x_{6}^{2}+x_{7}^{2}+x_{8}^{2}),$ where (a, b) = (1, 11), (1, 13).

Independent Gödel sentences and independent sets

1975 ◽  
Vol 40 (2) ◽  
pp. 159-166
Author(s):  
A. M. Dawes ◽  
J. B. Florence
Keyword(s):  
Positive Integer ◽  
Independent Sets ◽  
Peano Arithmetic ◽  
Natural Numbers ◽  
Simple Set ◽  
Logical Notion ◽  
Infinite Set ◽  
Element Set

In this paper we investigate some of the recursion-theoretic problems which are suggested by the logical notion of independence.A set S of natural numbers will be said to be k-independent (respectively, ∞-independent) if, roughly speaking, in every correct system there is a k-element set (respectively, an infinite set) of independent true sentences of the form x ∈ S. S will be said to be effectively independent (respectively, absolutely independent) if given any correct system we can generate an infinite set of independent (respectively, absolutely independent) true sentences of the form x ∈ S.We prove that(a) S is absolutely independent ⇔S is effectively independent ⇔S is productive;(b) for every positive integer k there is a Π1 set which is k-independent but not (k + 1)-independent;(c) there is a Π1 set which is k-independent for all k but not ∞-independent;(d) there is a co-simple set which is ∞-independent.We also give two new proofs of the theorem of Myhill [1] on the existence of an infinite set of Σ1 sentences which are absolutely independent relative to Peano arithmetic. The first proof uses the existence of an absolutely independent Π1 set of natural numbers, and the second uses a modification of the method of Gödel and Rosser.

Theories incomparable with respect to relative interpretability

1962 ◽  
Vol 27 (2) ◽  
pp. 195-211 ◽  
Author(s):  
Richard Montague
Keyword(s):  
Positive Integer ◽  
The Other ◽  
Natural Numbers ◽  
Infinite Set

The present paper concerns the relation of relative interpretability introduced in [8], and arises from a question posed by Tarski: are there two finitely axiomatizable subtheories of the arithmetic of natural numbers neither of which is relatively interpretable in the other? The question was answered affirmatively (without proof) in [3], and the answer was generalized in [4]: for any positive integer n, there exist n finitely axiomatizable subtheories of arithmetic such that no one of them is relatively interpretable in the union of the remainder. A further generalization was announced in [5] and is proved here: there is an infinite set of finitely axiomatizable subtheories of arithmetic such that no one of them is relatively interpretable in the union of the remainder. Several lemmas concerning the existence of self-referential and mutually referential formulas are given in Section 1, and will perhaps be of interest on their own account.

The classical and the ω-complete arithmetic

1958 ◽  
Vol 23 (2) ◽  
pp. 188-206 ◽  
Author(s):  
A. Grzegorczyk ◽  
A. Mostowski ◽  
C. Ryll-Nardzewski
Keyword(s):  
Positive Integer ◽  
Second Order ◽  
Natural Numbers ◽  
Formal Systems ◽  
Two Systems ◽  
Distinct Individual

We consider two formal systems for the theory of (natural) numbers, both of which are applied second-order functional calculi with equality and the description operator. The two systems have the same primitive symbols, rules of formation, and axioms, differing only in the rules of inference.The primitive logical symbols of the systems are the improper symbols (,), the prepositional connectives ∨, &, ⊃, ≡, ~, the quantifiers ( ), (E), the equality symbol =, the description operator ι,-infinitely many distinct individual (or number) variables, and for each positive integer k infinitely many distinct k-place function variables. Our systems have in addition the following four primitive nonlogical (or arithmetical) constants:0, 1, +, ×.The classes of “number formulas” (nfs) and “propositional formulas” (pfs) are defined inductively as the least classes of formal expressions (i.e. of concatenations of primitive symbols) satisfying the following conditions:(1) 0, 1, and the number variables are nfs.

Countable Compagtifications

1966 ◽  
Vol 18 ◽  
pp. 616-620 ◽  
Author(s):  
Kenneth D. Magill
Keyword(s):  
Positive Integer ◽  
Compact Space ◽  
Real Line ◽  
Topological Spaces ◽  
Dense Subspace ◽  
Natural Numbers ◽  
Finite Sets ◽  
The Real ◽  
One To One

It is assumed that all topological spaces discussed in this paper are Hausdorff. By a compactification αX of a space X we mean a compact space containing X as a dense subspace. If, for some positive integer n, αX — X consists of n points, we refer to αX as an n-point compactification of X, in which case we use the notation αn X. If αX — X is countable, we refer to αX as a countable compactification of X. In this paper, the statement that a set is countable means that its elements are in one-to-one correspondence with the natural numbers. In particular, finite sets are not regarded as being countable. Those spaces with n-point compactifications were characterized in (3). From the results obtained there it followed that the only n-point compactifications of the real line are the well-known 1- and 2-point compactifications and the only n-point compactification of the Euclidean N-space, EN (N > 1), is the 1-point compactification.

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 COUNTING THE NUMBER OF SOLUTIONS TO THE ERDŐS–STRAUS EQUATION ON UNIT FRACTIONS

2013 ◽  
Vol 94 (1) ◽  
pp. 50-105 ◽  
Author(s):  
CHRISTIAN ELSHOLTZ ◽  
TERENCE TAO
Keyword(s):  
Positive Integer ◽  
Lower Bounds ◽  
Diophantine Equation ◽  
Upper And Lower Bounds ◽  
Natural Numbers ◽  
Number Of Solutions ◽  
Waring’S Problem ◽  
Unit Fractions ◽  
Image Position ◽  
Positive Integers

AbstractFor any positive integer $n$, let $f(n)$ denote the number of solutions to the Diophantine equation $$\begin{eqnarray*}\frac{4}{n} = \frac{1}{x} + \frac{1}{y} + \frac{1}{z}\end{eqnarray*}$$ with $x, y, z$ positive integers. The Erdős–Straus conjecture asserts that $f(n)\gt 0$ for every $n\geq 2$. In this paper we obtain a number of upper and lower bounds for $f(n)$ or $f(p)$ for typical values of natural numbers $n$ and primes $p$. For instance, we establish that $$\begin{eqnarray*}N\hspace{0.167em} {\mathop{\log }\nolimits }^{2} N\ll \displaystyle \sum _{p\leq N}f(p)\ll N\hspace{0.167em} {\mathop{\log }\nolimits }^{2} N\log \log N.\end{eqnarray*}$$ These upper and lower bounds show that a typical prime has a small number of solutions to the Erdős–Straus Diophantine equation; small, when compared with other additive problems, like Waring’s problem.

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