scholarly journals A proof of the “Theorem of the Means.”

1943 ◽  
Vol 33 ◽  
pp. 17-18
Author(s):  
C. E. Walsh
Keyword(s):  
Elementary Proof ◽  
Image Position ◽  
Proof By Induction

Numerous proofs have been given of this familiar theorem, which states that if a1, a2, …, an are positive, and not all equal, thenThe following is an elementary proof by induction, which I have not seen used before. It is, of course, not claimed to be novel, and not likely to be so.

scholarly journals An Elementary Proof of the Theorems of Cauchy and Mayer

1935 ◽  
Vol 4 (3) ◽  
pp. 112-117
Author(s):  
A. J. Macintyre ◽  
R. Wilson
Keyword(s):  
Differential Equation ◽  
Existence Theorem ◽  
Elementary Proof ◽  
Theoretical Discussion ◽  
Practical Advantage ◽  
Total Differential ◽  
Method Of Solution ◽  
The Subject ◽  
Image Position ◽  
Natural Way

Attention has recently been drawn to the obscurity of the usual presentations of Mayer's method of solution of the total differential equationThis method has the practical advantage that only a single integration is required, but its theoretical discussion is usually based on the validity of some other method of solution. Mayer's method gives a result even when the equation (1) is not integrable, but this cannot of course be a solution. An examination of the conditions under which the result is actually an integral of equation (1) leads to a proof of the existence theorem for (1) which is related to Mayer's method of solution in a natural way, and which moreover appears to be novel and of value in the presentation of the subject.

An elementary proof and an extension of Thas' theorem on k-arcs

1989 ◽  
Vol 105 (3) ◽  
pp. 459-462 ◽  
Author(s):  
Hitoshi Kaneta ◽  
Tatsuya Maruta
Keyword(s):  
Finite Field ◽  
Projective Space ◽  
Elementary Proof ◽  
Rational Curve ◽  
Image Position

Let q be the finite field of q elements. Denote by Sr q the projective space of dimension r over q. In Sr,q, where r ≥ 2, a k-arc is defined (see [4]) as a set of k points such that no j + 2 lie in a Sj,q, for j = 1,2,…, r−1. (For a k-arc with k > r, this last condition holds for all j when it holds for j = r−1.) A rational curve Cn of order n in Sr,q, is the set

scholarly journals An Euler-type volume identity

1999 ◽  
Vol 59 (3) ◽  
pp. 495-508
Author(s):  
Kevin Callahan ◽  
Kathy Hann
Keyword(s):  
Elementary Proof ◽  
Three Dimensional ◽  
Convex Polytope ◽  
Image Position

In this paper we present an elementary proof of a congruence by subtraction relation. In order to prove congruence by subtraction, we produce a dissection relating equal sub-polytopes. An immediate consequence of this relation is an Euler-type volume identity in ℝ3 which appeared in the Unsolved Problems section of the December 1996 MAA Monthly.This Euler-type volume identity relates the volumes of subsets of a polytope called wedges that correspond to its faces, edges, and vertices. A wedge consists of the inward normal chords of the polytope emanating from a face, vertex, or edge. This identity is stated in the theorem below.Euler Volume Theorem. For any three dimensional convex polytope PThis identity follows immediately from

scholarly journals A note on the zeros of the Bessel function

1939 ◽  
Vol 6 (1) ◽  
pp. 17-18 ◽  
Author(s):  
R. Wilson
Keyword(s):  
Bessel Function ◽  
Infinite Number ◽  
Elementary Proof ◽  
Real Parameter ◽  
Image Position

This note contains an elementary proof of the well-known result that every solution of Besse's equationwith a real parameter n has an infinite number of real positive zeros.

On a Theorem of Rav Concerning Egyptian Fractions

1975 ◽  
Vol 18 (1) ◽  
pp. 155-156 ◽  
Author(s):  
William A. Webb
Keyword(s):  
Positive Integer ◽  
Elementary Proof ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Egyptian Fractions ◽  
Image Position ◽  
Complete Bibliography ◽  
Necessary And Sufficient

Problems involving Egyptian fractions (rationals whose numerator is 1 and whose denominator is a positive integer) have been extensively studied. (See [1] for a more complete bibliography). Some of the most interesting questions, many still unsolved, concern the solvability ofwhere k is fixed.In [2] Rav proved necessary and sufficient conditions for the solvabilty of the above equation, as a consequence of some other theorems which are rather complicated in their proofs. In this note we give a short, elementary proof of this theorem, and at the same time generalize it slightly.

On two integral inequalities

1977 ◽  
Vol 77 (3-4) ◽  
pp. 325-328 ◽  
Author(s):  
E. T. Copson
Keyword(s):  
Elementary Proof ◽  
Integral Inequalities ◽  
Image Position

SynopsisIn 1932, Hardy and Littlewood [1] proved the inequalityThe constant 4 is best possible; equality occurs when f(x) = A Y(Bx), wherey(x) = e−½x sin (x sin y−y) (y = ⅓π), (x ≧ o)and A and B (>0) are constants. In [2], three proofs are given. The inequality has also been discussed in [3, 4]. A very elementary proof in which the function Y(x) emerges naturally is given in this paper.

scholarly journals NONEXISTENCE OF A CIRCULANT EXPANDER FAMILY

2010 ◽  
Vol 83 (1) ◽  
pp. 87-95
Author(s):  
KA HIN LEUNG ◽  
VINH NGUYEN ◽  
WASIN SO
Keyword(s):  
Regular Graph ◽  
Elementary Proof ◽  
Explicit Construction ◽  
Simple Graph ◽  
Regular Graphs ◽  
Circulant Graph ◽  
Circulant Graphs ◽  
Largest Eigenvalue ◽  
Image Position ◽  
Second Largest Eigenvalue

AbstractThe expansion constant of a simple graph G of order n is defined as where $E(S, \overline {S})$ denotes the set of edges in G between the vertex subset S and its complement $\overline {S}$. An expander family is a sequence {Gi} of d-regular graphs of increasing order such that h(Gi)>ϵ for some fixed ϵ>0. Existence of such families is known in the literature, but explicit construction is nontrivial. A folklore theorem states that there is no expander family of circulant graphs only. In this note, we provide an elementary proof of this fact by first estimating the second largest eigenvalue of a circulant graph, and then employing Cheeger’s inequalities where G is a d-regular graph and λ2(G) denotes the second largest eigenvalue of G. Moreover, the associated equality cases are discussed.

scholarly journals Linear Partial Differential Equations with Constant Coefficients: an Elementary Proof of an Existence Theorem

1959 ◽  
Vol 42 ◽  
pp. 3-5
Author(s):  
D. H. Parsons
Keyword(s):  
Differential Equation ◽  
Elementary Proof ◽  
Linear Partial Differential Equation ◽  
Partial Differential ◽  
Independent Variables ◽  
Linear Partial Differential Equations ◽  
Two Factors ◽  
Constant Coefficients ◽  
Image Position ◽  
The Right

We consider a linear partial differential equation with constant coefficients in one dependent and m independent variables, the right-hand side being zero,P being a symbolic polynomial in D1, …, Dm. If P can be decomposed into two factors, so that the equation can be writtenit is evident that the sum of any solution ofand any solution ofis also a solution of (1).

The most elementary proof that ?

2016 ◽  
Vol 100 (549) ◽  
pp. 429-434
Author(s):  
Nick Lord
Keyword(s):  
Elementary Proof ◽  
New Approach ◽  
Image Position

Here we present a simplification of one of the standard proofs that . We then look at extensions of the new approach, and add comments on the nature of the simplification (which relates to Step 1 below) and finally on the literature.As in most proofs of the result, we shall actually prove that ; becausethis is equivalent to the result sought.

An elementary proof of the Birkhoff-Hopf theorem

1995 ◽  
Vol 117 (1) ◽  
pp. 31-55 ◽  
Author(s):  
Simon P. Eveson ◽  
Roger D. Nussbaum
Keyword(s):  
Elementary Proof ◽  
Linear Operators ◽  
Positive Linear Operators ◽  
Important Work ◽  
Large Classes ◽  
Contraction Mappings ◽  
Measurable Functions ◽  
Almost Everywhere ◽  
Image Position ◽  
Hopf Theorem

In important work some thirty years ago, G. Birkhoff[2, 3] and E. Hopf [16, 17] showed that large classes of positive linear operators behave like contraction mappings with respect to certain ‘almost’ metrics. Hopf worked in a space of measurable functions and took as his ‘almost’ metric the oscillation ω(y/x) of functions y and x with x(t) > 0 almost everywhere, defined by

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