Discussions: Relating to a Simple Proof by Induction of an Interesting Number Relation

1916 ◽  
Vol 23 (3) ◽  
pp. 89
Author(s):  
Charles R. Dines
Keyword(s):  
Simple Proof ◽  
Proof By Induction ◽  
Number Relation

scholarly journals Cauchy Mean Theorem

2014 ◽  
Vol 22 (2) ◽  
pp. 157-166
Author(s):  
Adam Grabowski
Keyword(s):  
Simple Proof ◽  
Geometric Mean ◽  
Jensen’S Inequality ◽  
Arithmetic Mean ◽  
Real Numbers ◽  
Jensen's Inequality ◽  
Mizar Mathematical Library ◽  
Proof By Induction

Summary The purpose of this paper was to prove formally, using the Mizar language, Arithmetic Mean/Geometric Mean theorem known maybe better under the name of AM-GM inequality or Cauchy mean theorem. It states that the arithmetic mean of a list of a non-negative real numbers is greater than or equal to the geometric mean of the same list. The formalization was tempting for at least two reasons: one of them, perhaps the strongest, was that the proof of this theorem seemed to be relatively easy to formalize (e.g. the weaker variant of this was proven in [13]). Also Jensen’s inequality is already present in the Mizar Mathematical Library. We were impressed by the beauty and elegance of the simple proof by induction and so we decided to follow this specific way. The proof follows similar lines as that written in Isabelle [18]; the comparison of both could be really interesting as it seems that in both systems the number of lines needed to prove this are really close. This theorem is item #38 from the “Formalizing 100 Theorems” list maintained by Freek Wiedijk at http://www.cs.ru.nl/F.Wiedijk/100/.

scholarly journals A simple proof of Andrews’s 5 F 4 evaluation

2013 ◽  
Vol 36 (1-2) ◽  
pp. 165-170 ◽  
Author(s):  
Ira M. Gessel
Keyword(s):  
Simple Proof

A simple proof of existence of weak and statistical solutions of Navier–Stokes equations

1992 ◽  
Vol 436 (1896) ◽  
pp. 1-11 ◽  
Keyword(s):  
Weak Solution ◽  
Initial Data ◽  
Natural Number ◽  
Simple Proof ◽  
Stokes Equations ◽  
Galerkin Approximation ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Existence Proofs ◽  
Statistical Solutions

The Galerkin approximation to the Navier–Stokes equations in dimension N , where N is an infinite non-standard natural number, is shown to have standard part that is a weak solution. This construction is uniform with respect to non-standard representation of the initial data, and provides easy existence proofs for statistical solutions.

scholarly journals A representation theorem for the linear quasi-differential equation(py′)′+qy=0

2000 ◽  
Vol 23 (8) ◽  
pp. 579-584
Author(s):  
J. G. O'Hara
Keyword(s):  
Differential Equation ◽  
Comparison Theorem ◽  
Simple Proof ◽  
Representation Theorem ◽  
Second Order ◽  
Order Linear

We establish a representation forqin the second-order linear quasi-differential equation(py′)′+qy=0. We give a number of applications, including a simple proof of Sturm's comparison theorem.

A simple proof of the kinematic formula

1988 ◽  
Vol 105 (4) ◽  
pp. 279-285 ◽  
Author(s):  
P. Mani-Levitska
Keyword(s):  
Simple Proof ◽  
Kinematic Formula

An argument of a function in H1/2

2012 ◽  
Vol 55 (2) ◽  
pp. 507-511
Author(s):  
Takahiko Nakazi ◽  
Takanori Yamamoto
Keyword(s):  
Hardy Space ◽  
Simple Proof ◽  
Unit Disc ◽  
Open Unit ◽  
Open Unit Disc

AbstractLet H1/2 be the Hardy space on the open unit disc. For two non-zero functions f and g in H1/2, we study the relation between f and g when f/g ≥ 0 a.e. on ∂D. Then we generalize a theorem of Neuwirth and Newman and Helson and Sarason with a simple proof.

A Simple Proof that ζ(2) =  $$\frac{\pi^{2}}{6}$$

2011 ◽  
Vol 33 (3) ◽  
pp. 81-82 ◽  
Author(s):  
Michael D. Hirschhorn
Keyword(s):  
Simple Proof
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