A new proof of a generalization of Gerzon’s bound

Periodica Mathematica Hungarica - In this paper we give a short, new proof of a natural generalization of Gerzon’s bound. This bound improves the Delsarte, Goethals and Seidel’s upper...     

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.     

New proof of a Calabi’s theorem

A Calabi’s theorem says that on a compact Riemann surface, an extremal metric is a constant scalar curvature metric. In this paper, we use a new method to prove this theorem. Then we give an interesting corollary.     

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 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 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 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 simple proof of Veech’s Theorem

We present a simple proof of Veech’s Theorem.     

On a generalization of Gross’s formula

We prove a relation between Whittaker functionals of cusp forms on ${{\widetilde{SL}}_2}$ and the toric periods of forms on the quaternion algebra. As an application we prove a generalization of a formula of Gross.     

Geometric proof of R?dseth’s formula for Frobenius numbers

Using a geometric interpretation of continued fractions, we give a new proof of R?dseth’s formula for Frobenius numbers.     

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 short proof of Halin’s grid theorem

We give a short proof of Halin’s theorem that every thick end of a graph contains the infinite grid.     

A proof of Ewell’s octuple product identity

We give a new proof of Ewell’s octuple product identity in [7] by using a general theorem developed by the first author in [3].     

On a generalization of Kronecker’s limit formula

In this paper, we first generalize the Kronecker limit formula for a class of Epstein zeta functions using new approximation formulas. This enables us to derive some applications to the class number of quadratic imaginary number fields K and the period ratios of elliptic curves with complex multiplication.     

A proof of Price’s law for the collapse of a self-gravitating scalar field

A well-known open problem in general relativity, dating back to 1972, has been to prove Price’s law for an appropriate model of gravitational collapse. This law postulates inverse-power decay rates for the gravitational radiation flux through the event horizon and null infinity with respect to appropriately normalized advanced and retarded time coordinates. It is intimately related both to astrophysical observations of black holes and to the fate of observers who dare cross the event horizon. In this paper, we prove a well-defined (upper bound) formulation of Price’s law for the collapse of a self-gravitating scalar field with spherically symmetric initial data. We also allow the presence of an additional gravitationally coupled Maxwell field. Our results are obtained by a new mathematical technique for understanding the long-time behavior of large data solutions to the resulting coupled non-linear hyperbolic system of p.d.e.’s in 2 independent variables. The technique is based on the interaction of the conformal geometry, the celebrated red-shift effect, and local energy conservation; we feel it may be relevant for the problem of non-linear stability of the Kerr solution. When combined with previous work of the first author concerning the internal structure of charged black holes, which had assumed the validity of Price’s law, our results can be applied to the strong cosmic censorship conjecture for the Einstein-Maxwell-real scalar field system with complete spacelike asymptotically flat spherically symmetric initial data. Under Christodoulou’s C⁰-formulation, the conjecture is proven to be false.