Every countable transitive model M of ZF (without choice) has an ordinal preserving extension satisfying ZF, of power ^MnO/i" An aPPucation t0 infinitary logic is given. Any transitive model M of ZFC with countably many ordinals must be countable. The situation is quite different when the axiom of choice is dropped. The first examples of transitive models of ZF of power Wj with countably many o...

abstractlet w be a non-empty subset of a free group. the automorphism of a group g is said to be a marginal automorphism, if for all x in g,x^−1alpha(x) in w^*(g), where w^*(g) is the marginal subgroup of g.in this paper, we give necessary and sufficient condition for a purelynon-abelian p-group g, such that the set of all marginal automorphismsof g forms an elementary abelian p-group.

Let C be a q-ary code of length n and X ⊆ C, then d is called a descendant of X if di ∈ {xi : x ∈ X} for all 1 ≤ i ≤ n. C is said to be a w-identifiable parent property code (w-IPP code for short) if whenever d is a descendant of w (or fewer) codewords, one can always identify at least one of the parent codewords in C. In this paper, we give constructions for w-IPP codes of length w + 1. Furthe...

and Applied Analysis 3 We are interested in the following optimal control problem: min u∈Uad⊂X J ( u, y u ) 1 2 {∫T 0 ∥ ∥y − zd ∥ ∥2 0,Ωdt ∫T 0 ‖u‖0,ΩUdt } , 2.1 subject to yt − div ( A∇yt D∇y ∫ t 0 C t, τ ∇y x, τ dτ ) f Bu, in Ω × 0, T , y 0, on ∂Ω × 0, T , y|t 0 y0, in Ω, 2.2 where u is control, y is state, zd is the observation, Uad is a closed convex subset, f x, t ∈ L2 0, T ;L2 Ω , and zd ...

 Suppose $f$ is a map from a non-empty finite set $X$ to a finite group $G$. Define the map $zeta^f_G: Glongrightarrow mathbb{N}cup {0}$ by $gmapsto |f^{-1}(g)|$. In this article, we show that for a suitable choice of $f$, the map $zeta^f_G$ is a character. We use our results to show that the solution function for the word equation $w(t_1,t_2,dots,t_n)=g$ ($gin G$) is a character, where $w(t_1,...

We provide a new characterization of the Bivariate normal-Wishart distribution. Let ~ x = fx 1 x 2 g have a non-singular Bivariate normal pdf f(~ x) = N(~ W) with unknown mean vector ~ and unknown precision matrix W. L e t f(~ x) = f(x 1)f(x 2 jx 1) where f(x 1) = N(m 1 1=v 1) and f(x 2 jx 1) = N(m 2j1 +b 12 x 1 1=v 2j1). Similarly, deene fv 2 v 1j2 b 21 m 2 m 1j2 g using the factorization f(~ ...

Let $\Bbb F_q$ be a finite field with $q$ elements and $n$ a positive integer. Mart\'inez, Vergara and Oliveira \cite{MVO} explicitly factorized $x^{n} - 1$ over $\Bbb F_q$ under the condition of $rad(n)|(q-1)$. In this paper, suppose that $rad(n)\nmid (q-1)$ and $rad(n)|(q^w-1)$, where $w$ is a prime, we explicitly factorize $x^{n}-1$ into irreducible factors in $\Bbb F_q[x]$ and count the num...

This paper concerns the global in time existence of solutions for a semilinear heat equation (P)∂tu=Δu+f(u),x∈RN,t>0,u(x,0)=u0(x)≥0,x∈RN,where N≥1, u0 is nonnegative initial function and f∈C1([0,∞))∩C2((0,∞)) denotes superlinear nonlinearity problem. We consider nonexistence problem (P). The main purpose this to determine critical decay rate functions solutions. In particular, we show that it c...

We study the diameter of graph G ( w stretchy="false">) G(w) <...