Let p be a prime number and F an algebraic closure of the finite field Fp with p elements. Let n and N denote positive integers, N prime to p. We are interested in representations of the Galois group GQ = Gal(Q̄/Q) into GL(n,F), unramified at all finite primes not dividing pN . (We shall say “ρ is unramified outside pN .”) In this paper, “representation” will always mean “continuous, semisimple ...

We studied the central region of bipolar nebula M\,2-9 using radio-continuum observations obtained from Jansky Very Large Array (JVLA) and Atacama Millimeter (ALMA) interferometers. This work presents new images at $\sim$ 43 GHz ($\sim$ 7.0 mm) 345 0.9 with angular resolutions 0$\rlap{.}^{\prime\prime}$047 0$\rlap{.}^{\prime\prime}$09, respectively. The continuum emission shows an elongated jet...

For nonnegative real numbers α, β, γ , A, B, and C such that B C > 0 and α β γ > 0, the difference equation xn 1 α βxn γxn−1 / A Bxn Cxn−1 , n 0, 1, 2, . . . has a unique positive equilibrium. A proof is given here for the following statements: 1 For every choice of positive parameters α, β, γ , A, B, and C, all solutions to the difference equation xn 1 α βxn γxn−1 / A Bxn Cxn−1 , n 0, 1, 2, . ...

Let $P$ be a complex polynomial of the form $P(z)=zdisplaystyleprod_{k=1}^{n-1}(z-z_{k})$,where $|z_k|ge 1,1le kle n-1$ then $ P^prime(z)ne 0$. If $|z|

This paper introduce and study weakly 1-absorbing prime ideals in commutative rings. Let $$A $$ be a ring with nonzero identity $$1\ne 0.$$ A proper ideal $$P of is said to if for every nonunits $$x,y,z\in $$0\ne xyz\in P, then $$xy\in P$$ or $$z\in P. In addition give many properties characterizations ideals, we also determine rings which prime. Furthermore, investigate C(X), the continuous...

‎‎in this paper‎, ‎we define fuzzy subnexuses over a nexus $n$‎. ‎define and study the notions of the prime fuzzy subnexuses and the fractions‎‎induced by them‎. ‎finally‎, ‎we show that if s is a meet‎‎closed subset of the set fsub(n), ‎of fuzzy subnexuses of a nexus n‎, ‎and‎‎h= ⋀s ϵ s, ‎then the fractions s^-1 n and h^-1 n are isomorphic as meet-semilattices‎.

Two characterizations of hypercubes are given: 1) A graph is a hypercube if and only if it is antipodal and bipartite (0, 2)-graph. 2) A graph is an nhypercube if and only if there are n pairs of prime convexes, the graph is a prime convex intersection graph, and each intersection of n prime convexes (no one of which is from the same pair) is a vertex.

0.1.3 Unique Factorization Theorem If a is an integer, not 0 or ±1, then (1) a can be written as a product p1 · · · pn of primes. (2) If a = p1 · · · pn = q1 · · · qm, where the pi and qj are prime, then n = m and, after renumbering, pi = ±qi for all i. [We allow negative primes, so that, for example, −17 is prime. This is consistent with the general definition of prime element in an integral d...

We call a module M , quasi-catenary if for each pair of quasi-prime submodules K and L of M with K L all saturated chains of quasi-prime submodules of M from K to L have a common finite length. We show that any homomorphic image of a quasi-catenary module is quasi-catenary. We prove that if M   is a module with following properties: (i) Every quasi-prime submodule of M has finite quasi-height;...

‎this article presents a systematic study for structure of finite wavelet frames‎ ‎over prime fields‎. ‎let $p$ be a positive prime integer and $mathbb{w}_p$‎ ‎be the finite wavelet group over the prime field $mathbb{z}_p$‎. ‎we study theoretical frame aspects of finite wavelet systems generated by‎ ‎subgroups of the finite wavelet group $mathbb{w}_p$.