An elementary proof of Ramanujan’s circular summation formula and its generalizations

In this paper, we give a completely elementary proof of Ramanujan’s circular summation formula of theta functions and its generalizations given by S.H. Chan and Z.-G. Liu, who used the theory of elliptic functions. In contrast to all other proofs, our proofs are elementary. An application of this summation formula is given.     

A simple proof of Olatinwo’s theorems

In this paper we provide a simple proof of the existence coupled fixed point theorem in complete cone metric spaces due to Sabetghadam et al. (Fixed Point Theory Appl 2009:8, 2009) and due to Olatinwo (Annali Dell’Universita’Di Ferrara 57:173–180, 2011). In particular we prove that these results are spacial cases of Rezapour and Hamlbarani’s theorems (J Math Anal Appl 345(2):719–724, 2008).     

A simple proof of Veech’s Theorem

We present a simple proof of Veech’s Theorem.     

A simple proof of Schmidt–Summerer’s inequality

In this paper we give a simple proof of an inequality for intermediate Diophantine exponents obtained recently by W.M. Schmidt and L. Summerer.     

A new proof of two identities involving Ramanujan’s cubic continued fraction

In this paper, we give a new proof of two identities involving Ramanujan’s cubic continued fraction. These identities are the key ingredients to an analog of Ramanujan’s “Most Beautiful Identity” discovered recently.     

A very simple proof of Pascal’s hexagon theorem and some applications

In this article we present a simple and elegant algebraic proof of Pascal’s hexagon theorem which requires only knowledge of basics on conic sections without theory of projective transformations. Also, we provide an efficient algorithm for finding an equation of the conic containing five given points and a criterion for verification whether a set of points is a subset of the conic.     

A proof of snevily’s conjecture

We prove Snevily’s conjecture, which states that for any positive integer k and any two k-element subsets {a ₁, …, a _k } and {b ₁, …, b _k } of a finite abelian group of odd order there exists a permutation π ∈ S _k such that all sums a _i + b _π(i) (i ∈ [1, k]) are pairwise distinct.     

A simple proof of the Rogers–Selberg identities

A short and simple proof of the Rogers–Selberg identities is obtained from the work of Andrew Sills on identities of the Rogers–Ramanujan type.     

A refinement of Ramanujan’s factorial approximation

We present a new proof of the monotonicity of the correction term θ _n in Ramanujan’s refinement of Stirling’s formula. Moreover we prove that θ _n is concave.     

Some extensions for Ramanujan’s circular summation formulas and applications

The Ramanujan Journal - In this paper, we give some extensions for Ramanujan’s circular summation formulas with the mixed products of two Jacobi’s theta functions. As applications, we...     

A direct proof of Gromov’s theorem

A new proof of a theorem by Gromov is given: for any positive C and any integer n greater than 1, there exists a function Δ _C,n (δ) such that if the Gromov–Hausdorff distance between two complete Riemannian n-manifolds V and W is at most δ, their sectional curvaturcs |K _σ | do not exceed C, and their injectivity radii are at least 1/C, than the Lipschitz distance between V and W is less than Δ _C,n (δ), and Δ _C,n (δ) → 0 as δ → 0. Bibliography: 6 titles.     

A new proof of the Askey–Wilson integral via a five-variable Ramanujan’s reciprocity theorem

In this paper, we give an easy and short proof of the well-known Askey–Wilson integral by means of the five-variable Ramanujan’s reciprocity theorem.     

A new proof of Stetsenko’s theorem

A short proof of V.A. Stetsenko’s theorem on weakly read-many Boolean functions is given, which is based on the technique of representing read-once functions in the form of trees.     

A motivated proof of Gordon’s identities

We generalize the ??motivated proof?? of the Rogers?CRamanujan identities given by Andrews and Baxter to provide an analogous ??motivated proof?? of Gordon??s generalization of the Rogers?CRamanujan identities. Our main purpose is to provide insight into certain vertex-algebraic structure being developed.     

A short proof of Levinson’s theorem

We give a short proof of Levinson’s result that over 1/3 of the zeros of the Riemann zeta function are on the critical line.     

A topological proof of Sklar’s theorem

We present a proof of Sklar’s Theorem that uses topological arguments, namely compactness (under the weak topology) of the class of copulas and some density properties of the class of distribution functions.     

A simple proof of the generalized Craig–Sakamoto theorem

The Craig–Sakamoto theorem establishes the independence of two quadratic forms in normal variates. In this article, we provide a simple proof of a generalized Craig–Sakamoto theorem.