In this paper, we present a fourth order method for computing simple roots of nonlinear equations by using suitable Taylor and weight function approximation. The method is based on Weerakoon-Fernando method [S. Weerakoon, G.I. Fernando, A variant of Newton's method with third-order convergence, Appl. Math. Lett. 17 (2000) 87-93]. The method is optimal, as it needs three evaluations per iterate,...

In this paper, we give a new and direct proof for the recently proved conjecture raised in Soltani and Roozegar (2012). The conjecture can be proved in a few lines via the integral representation of the Gauss-hypergeometric function unlike the long proof in Roozegar and Soltani (2013).

Let p be a prime k|p−1, t = (p−1)/k and γ(k, p) be the minimal value of s such that every number is a sum of s k-th powers (mod p). We prove Heilbronn’s conjecture that γ(k, p) k1/2 for t > 2. More generally we show that for any positive integer q, γ(k, p) ≤ C(q)k1/q for φ(t) ≥ q. A comparable lower bound is also given. We also establish exact values for γ(k, p) when φ(t) = 2. For instance, whe...

We examine the problem of writing every sufficiently large even number as the sum of two primes and at most K powers of 2. We outline an approach that only just falls short of improving the current bounds on K. Finally, we improve the estimates in other Waring–Goldbach problems.

Let K be a number field and let S be a finite set of places of K which contains all the Archimedean places. For any φ(z) ∈ K(z) of degree d ≥ 2 which is not a d-th power in K(z), Siegel’s theorem implies that the image set φ(K) contains only finitely many S-units. We conjecture that the number of such S-units is bounded by a function of |S| and d (independently of K, S and φ). We prove this con...

‎an  oriented perfect path double cover (oppdc) of a‎ ‎graph $g$ is a collection of directed paths in the symmetric‎ ‎orientation $g_s$ of‎ ‎$g$ such that‎ ‎each arc‎ ‎of $g_s$ lies in exactly one of the paths and each‎ ‎vertex of $g$ appears just once as a beginning and just once as an‎ ‎end of a path‎. ‎maxov{'a} and ne{v{s}}et{v{r}}il (discrete‎ ‎math‎. ‎276 (2004) 287-294) conjectured that ...

Let F be a number field, and (ρ, V ) a continuous, n-dimensional representation of the absolute Galois group Gal(F/F ) on a finite-dimensional C-vector space V . Denote by L(s, ρ) the associated L-function, which is known to be meromorphic with a functional equation. Artin’s conjecture predicts that L(s, ρ) is holomorphic everywhere except possibly at s = 1, where its order of pole is the multi...

The purpose of this note is to prove the conjecture. The ingredients of the argument are: (a) Taubes’ theorem [11] on the non-vanishing of the SeibergWitten invariants for symplectic 4-manifolds; (b) the theorem of Gabai [7] on the existence of taut foliations on 3-manifolds with non-zero betti number; (c) the construction of Eliashberg and Thurston [4], which produces a contact structure from ...

(2) ‖Hα(f1, f2)‖p ≤ Cα,p1,p2‖f1‖p1‖f2‖p2 with constants Cα,p1,p2 depending only on α, p1, p2 and p := p1p2 p1+p2 hold. The first result of this type is proved in [4], and the purpose of the current paper is to extend the range of exponents p1 and p2 for which (2) is known. In particular, the case p1 = 2, p2 = ∞ is solved to the affirmative. This was originally considered to be the most natural ...

Remark. Apropos of reduction mod p: If V is a Qp-vector space and G ⊂ GL(V ) is a compact subgroup, then there exists a G-fixed lattice in V for the following reason. Pick any lattice L ⊂ V . Then the G-stabilizer of L is open and of finite index. So Λ = ∑ g∈G gL ⊂ V is also a lattice, and it is definitely G-stable. The same works with coefficients in any finite extension of Qp, or even in Qp (...