A proof of Saitoh's conjecture for conjugate Hardy $H^{2}$ kernels

Notes on the Proof of Second Hardy-Littlewood Conjecture

In this paper a slightly stronger version of the Second Hardy-Littlewood Conjecture (see [1]), namely that inequality π(x)+π(y) > π(x+y) is examined, where π(x) denotes the number of primes not exceeding x. It is shown that the inequality holds for all su ciently large x and y. It has also been shown that for a given value of y ≥ 55 the inequality π(x)+π(y) > π(x+y) holds for all su ciently lar...

Partial proof of Graham Higman's conjecture related to coset diagrams

Graham Higman has defined coset diagrams for PSL(2,ℤ). These diagrams are composed of fragments, and the fragments are further composed of two or more circuits. Q. Mushtaq has proved in 1983 that existence of a certain fragment γ of a coset diagram in a coset diagram is a polynomial f in ℤ[z]. Higman has conjectured that, the polynomials related to the fragments are monic and for a fixed degree...

A short proof of the maximum conjecture in CR dimension one

In this paper and by means of the extant results in the Tanaka theory, we present a very short proof in the specific case of CR dimension one for Beloshapka's maximum conjecture. Accordingly, we prove that each totally nondegenerate model of CR dimension one and length >= 3 has rigidity. As a result, we observe that the group of CR automorphisms associated with each of such models contains onl...

On the Closed-Form Solution of a Nonlinear Difference Equation and Another Proof to Sroysang’s Conjecture

The purpose of this paper is twofold. First we derive theoretically, using appropriate transformation on x(n), the closed-form solution of the nonlinear difference equation x(n+1) = 1/(±1 + x(n)), n ∈ N_0. The form of solution of this equation, however, was first obtained in [10] but through induction principle. Then, with the solution of the above equation at hand, we prove a case ...

A proof of the Hardy - Ramanujan formulaDan