Let E/Q be an elliptic curve and let p be an odd supersingular prime for E. In this article, we study the simplest case of Iwasawa theory for elliptic curves, namely when E(Q) is finite, X(E/Q) has no p-torsion and the Tamagawa factors for E are all prime to p. Under these hypotheses, we prove that E(Qn) is finite and make precise statements about the size and structure of the p-power part of X...

In this article, by utilizing the theory of quantum (or q-) calculus, we define a new subclass analytic and multivalent p-valent) functions class Ap, where Ap is invariant symmetric) under rotations. The well-known Janowski are used with help principle subordination between in order to p-valent functions. This function generalizes various other subclasses functions, not only classical Geometric...

We explain how to combine deep results from Iwasawa theory with explicit computation to obtain information about p-parts of Tate-Shafarevich groups of elliptic curves over Q. This method provides a practical way to compute #X(E/Q)(p) in many cases when traditional p-descent methods are completely impractical and also in situations where results of Kolyvagin and Kato do not apply.

The metrical theory of Diophantine approximation for quaternions is developed using recent results in the general theory. In particular, Quaternionic analogues of the classical theorems of Khintchine, Jarnı́k and Jarnı́k-Besicovitch are established. Introduction Diophantine approximation begins with a more quantitative understanding of the density of the rationals Q in the reals R. The starting p...

In this paper, we develop the theory of a p, q-analogue of the binomial coefficients. Some properties and identities parallel to those of the usual and q-binomial coefficients will be established including the triangular, vertical, and the horizontal recurrence relations, horizontal generating function, and the orthogonality and inverse relations. The construction and derivation of these result...

Given a positive sequence a = (an) ∈ lp,q, for 0 < p < 2 and 0 < q ≤ ∞, and a finite set A = {x1, . . . , xm} ⊂ l2 with |xi| ≤ a for all i = 1, . . . ,m we prove ‖(en(acoA))‖p,q ≤ cp,q √ log(m + 1) ‖a‖p,q , where en(acoA) is the n th dyadic entropy number of the absolutely convex hull acoA of A and cp,q > 0 is a suitable constant only depending on p and q. Moreover we show that this is asymptot...

Two elements $g$ and $h$ of a permutation group $G$ acting on set $V$ are said to be intersecting if $g(v) = h(v)$ for some $v \in V$. More generally, subset ${\cal F}$ is an every pair intersecting. The intersection density $\rho(G)$ transitive the maximum value quotient $|{\cal F}|/|G_v|$ where $G_v$ stabilizer $v\in V$ runs over all sets in $G$. Intersection densities groups degree $pq$, $p>...