Let m,n, r be nonzero fixed positive integers, R a 2-torsion free prime ring, Q its right Martindale quotient ring, and L a non-central Lie ideal of R. Let D : R −→ R be a skew derivation of R and E(x) = D(xm+n+r)−D(xm)xn+r − xmD(xn)xr − xm+nD(xr). We prove that if E(x) = 0 for all x ∈ L, then D is a usual derivation of R or R satisfies s4(x1, . . . , x4), the standard identity of degree 4.

4 B r o m o x a n t h o t o x i n r e a c t s w i t h c y a n o a c e t a m i d e t o f o r m 4 b r o m o 5 , 6 d i h y d r o x a n t h o t o x i n , w h i c h o n h y d r o l y s i s w i t h c o l d h y d r o c h l o r i c a c i d l e a d s t o t h e f o r m a t i o n o f c y a n o a c e t i c a c i d d e r i v a t i v e , b u t h y d r o l y s i s b y b o i l e d h y d r o c h l o r i c a c i...

P r e p a r a t i v e a n d E l e c t r o c h e m i c a l R e d o x R e a c t i o n s , M e t a l M e t a l B o n d s T h e o x i d a t i o n s a n d r e d u c t i o n s o f 2 7 s i n g l y a r s e n i c b r i d g e d d i n u c l e a r m e t a l c a r b o n y l c o m p l e x e s w e r e i n v e s t i g a t e d . T h e r e a r e i n d i c a t i o n s t h a t t h e c o m p l e x e s a s a w h o l...

Let XR be a geometrically irreducible smooth projective curve, defined over R, such that XR does not have any real points. Let X = XR×R C be the complex curve. We show that there is a universal real algebraic line bundle over XR × Pic d(XR) if and only if χ(L) is odd for L ∈ Picd(XR). There is a universal quaternionic algebraic line bundle over X × Pic(X) if and only if the degree d is odd. Tak...

Let R be a prime ring with center Z, I nonzero ideal of and D: × → symmetric bi–(α, α)-derivation d the trace D. In present paper, we have considered following conditions: i) [d(x), x]α,α = 0, ii)[d(x), ⊆ Cα,α, iii)(d(x), x)α,α iv)D1(d2(x), x) v)d1(d2(x)) f(x), for all x, y ∈ I, where D1 D2 are two bi-(α, α)-derivations, d1, d2 traces D1, respectively, B: is bi-additive mapping, f B.

We study the principal eigenvalues (i.e., eigenvalues corresponding to positive eigenfunctions) for the boundary value problem: −∆u(x) = λg(x)u(x), x ∈ D; (∂u/∂n)(x) + αu(x) = 0, x ∈ ∂D, where ∆ is the standard Laplace operator, D is a bounded domain with smooth boundary, g : D → R is a smooth function which changes sign on D and α∈R. We discuss the relation between α and the principal eigenval...

Let π(x; d, a) denote the number of primes p ≤ x with p ≡ a(mod d). Chebyshev’s bias is the phenomenon that ‘more often’ π(x; d, n) > π(x; d, r), than the other way around, where n is a quadratic non-residue mod d and r is a quadratic residue mod d. If π(x; d, n) ≥ π(x; d, r) for every x up to some large number, then one expects that N(x; d, n) ≥ N(x; d, r) for every x. Here N(x; d, a) denotes ...

Let π(x; d, a) denote the number of primes p ≤ x with p ≡ a(mod d). Chebyshev’s bias is the phenomenon for which “more often” π(x; d, n) > π(x; d, r), than the other way around, where n is a quadratic nonresidue mod d and r is a quadratic residue mod d. If π(x; d, n) ≥ π(x; d, r) for every x up to some large number, then one expects that N(x; d, n) ≥ N(x; d, r) for every x. Here N(x; d, a) deno...

Let p ( · ) $p(\cdot )$ be a measurable function defined on R d ${\mathbb {R}}^d$ and − : = inf x ∈ $p_-:=\inf _{x\in {\mathbb {R}}^d}p(x)$ . In this paper, we generalize the Hardy–Littlewood maximal operator. definition, instead of cubes or balls, take supremum over all rectangles side lengths which are in cone-like set by given ψ. Moreover, integral means, consider L q $L_{q(\cdot )}$ -means....