The asymmetric unit of the title compound, C(14)H(12)F(8)N(2), contains one tetra-fluoro-pyrrolidine system and one half-mol-ecule of benzene; the latter, together with a second heterocyclic unit, are completed by symmetry, with a twofold crystallographic axis crossing through both the middle of the bond between the C atoms bearing the heterocyclic rings and the opposite C-C bonds of the whole ...

We introduce linear cyclic codes over the ring F 2 + uF 2 = f0; 1; u; u = u + 1g, where u 2 = 0. This ring shares many properties of Z 4 and F 4 and admits a linear "Gray map". Cyclic codes are described as modules over (F 2 + uF 2) n which may not be free. Self-dual codes of odd length exists as in the case of Z 4-codes. We exhibit some extremal codes of this very interesting family.

In this paper, we explicitly describe all the elements of sequence fractional parts {af(n)/n}, n=1,2,3,…, where f(x)∈Z[x] is a nonconstant polynomial with positive leading coefficient and a≥2 an integer. We also show that each value w={af(n)/n}, n≥nf nf least integer such f(n)≥n/2 for every n≥nf, attained by infinitely many terms sequence. These results combined some earlier estimates on gaps b...

Copyright q 2012 Kalyan Kumar Dey et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Let N be a 2-torsion free prime Γ-near-ring with center ZN. Let f, d and g, h be two generalized derivations on N. We prove the following res...

A Lie algebra g is Frobenius if it admits a linear functional F ∈ g * such that the Kirillov form BF (x, y) = F ([x, y]) is non-degenerate. If g is the mth maximal parabolic subalgebra P(n, m) of sl(n) this occurs precisely when (n, m) = 1. We define a cyclic functional F on P(n, m) and prove it is non-degenerate using properties of graphs associated to F. These graphs also provide in certain c...

let $r$ be a ring with unity. the undirected nilpotent graph of $r$, denoted by $gamma_n(r)$, is a graph with vertex set ~$z_n(r)^* = {0neq x in r | xy in n(r) for some y in r^*}$, and two distinct vertices $x$ and $y$ are adjacent if and only if $xy in n(r)$, or equivalently, $yx in n(r)$, where $n(r)$ denoted the nilpotent elements of $r$. recently, it has been proved that if $r$ is a left ar...

Let $R$ be a prime ring, $Q_r$ its right Martindale quotient $L$ non-central Lie ideal of $R$, $n\geq 1$ fixed integer, $F$ and $G$ two generalized skew derivations with the same associated automorphism, $p\in R$ element. If $p\bigl(F(x)F(y)-G(y)x\bigr)^n=0$, for any $x,y \in L$, then there exist $a,c\in Q_r$ such that $F(x)=ax$ $G(x)=cx$, $x\in R$, $pa=pc=0$, unless when satisfies standard pol...

Mat r x ( F q ) o f a l l r b y r m a t r i c e s o v e r F q , b y C t h e g r o u p G L t i ( F q ) o f t h e u n i t s of M, and by S the special linear subgroup SL , (F q ) o f C . F o r a n a r b i t r a r y field F containing F q , l e t U s t a n d f o r t h e ( c o m m u t a t i v e ) p o l y n o m i a l a l g e b r a F[xl, , x r ] , a n d c o n s i d e r U g r a d e d a s u s u a l : U...

We consider the following condition (*) on an associative ring R : (*). There exists a function f from R into R such that f is a group homomorphism of (R,+), f is injective on R2, and f(xy) = (xy)n(x,y) for some positive integer n(x,y) > 1. Commutativity and structure are established for Artinian rings R satisfying (*), and a counterexample is given for nonArtinian rings. The results generalize...

In the title compound, C(34)H(26)F(2)N(2)O(2)S, an intra-molecular O-H⋯N hydrogen bond forms an S(5) ring motif. The piperidine ring adopts a chair conformation. The thia-zolidine ring and one of the pyrrolidine rings adopt envolope conformations with methyl-ene C atoms at the flap, whereas the other pyrrolidine ring adopts a half-chair conformation. The fluoro-substituted benzene rings form di...