Let G denote a connected, quasi-split reductive group over a field F that is complete with respect to a discrete valuation and that has a perfect residue field. Under mild hypotheses, we produce a subset of the Lie algebra g(F ) that picks out a G(F )-conjugacy class in every stable, regular, topologically nilpotent conjugacy class in g(F ). This generalizes an earlier result obtained by DeBack...

Let $V$ be a valuation domain with quotient field $K$. We show how to describe all extensions of $K(X)$ when the $V$-adic completion $\widehat{K}$ is algebraically closed, generalizing similar result obtained by Ostrowski in case one-dimensional domains. This accomplished realizing such means pseudo-monotone sequences, generalization pseudo-convergent sequences introduced Chabert. also that rin...

Recently, many mathematicians have studied in the area of the Euler numbers and polynomials see 1–15 . The Euler numbers and polynomials possess many interesting properties and arising in many areas of mathematics and physics. In 14 , we introduced that Euler equation En x 0 has symmetrical roots for x 1/2 see 14 . It is the aim of this paper to observe an interesting phenomenon of “scattering”...

The divisor function σ(n) denotes the sum of divisors positive integer n. For a prime p and m∈N, p-adic valuation m is highest power which divides m. Formulas for νp(σ(n)) are established. p=2, these involve only odd primes dividing These expressions used to establish bound ν2(σ(n))≤⌈log2⁡(n)⌉, with equality if n product distinct Mersenne primes, an p, νp(σ(n))≤⌈logp⁡(n)⌉, related solutions Lju...

We introduced the continuous and discrete $p$-adic shearlet systems. We restrict ourselves to a brief description of the $p$-adic theory and shearlets in real case. Using the group $G_p$ consist of all $p$-adic numbers that all of its elements have a square root, we defined the continuous $p$-adic shearlet system associated with $L^2left(Q_p^{2}right)$. The discrete $p$-adic shearlet frames for...

1.1. The p-adic measure. Let p be a prime number. We consider a field K with a valuation ord : K× → Z, extended to K by ord(0) = ∞. We denote by OK the valuation ring OK = {x ∈ K|ord(x) ≥ 0} and we fix an uniformizing parameter $, that is, an element of valuation 1 in OK . The ring OK is a local ring with maximal ideal MK of OK generated by $. We shall assume the residue field k := OK/MK is fin...

Let $G$ be a reductive $p$-adic group‎. ‎We consider the general question‎ ‎of whether the reducibility of an induced representation can be detected in a‎ ‎``co-rank one‎" ‎situation‎. ‎For smooth complex representations induced from supercuspidal‎ ‎representations‎, ‎we show that a sufficient condition is the existence of a subquotient‎ ‎that does not appear as a subrepresentation‎. ‎An import...