Consider long-range Bernoulli percolation on $\mathbb{Z}^d$ in which we connect each pair of distinct points $x$ and $y$ by an edge with probability $1-\exp(-\beta\|x-y\|^{-d-\alpha})$, where $\alpha>0$ is fixed $\beta\geq 0$ a parameter. We prove that if $0<\alpha<d$ then the critical two-point function satisfies \[ \frac{1}{|\Lambda_r|}\sum_{x\in \Lambda_r} \mathbf{P}_{\beta_c}(0\leftrightarr...

In this paper, we introduce the notions of (l, r)-derivations, (r, l)-derivations, and derivations Hilbert algebras investigate some related properties. addition, define two subsets for a derivation d algebra X, Ker d(X) Fix d(X), also take look at their characteristics.

1.1 Topology Let us first recall some basic Euclidean topology. For x0 = (x0, y0, z0) ∈ R3 and r > 0 we define the open ball of radius r > 0 centered at x0 to be B(x0, r) = {x ∈ R | ‖x− x0‖ < r}. Recall that ‖x‖ = √ x2 + y2 + z2 is the Euclidean norm, or length, of the vector x. Definition 1 (Open set). We say a set D ⊂ R3 is open if for every x0 ∈ D, there exists a radius r > 0 such that B(x0,...

1. The Hardy-Littlewood maximal inequality Let us work in Euclidean space R d with Lebesgue measure; we write |E| instead of µ(E) for the Lebesgue measure of a set E. For any x ∈ R d and r > 0 let B(x, r) := {y ∈ R d : |x − y| < r} denote the open ball of radius r centred at x. Thus for instance |B(x, r)| ∼ d r d. For any c > 0, we use cB(x, r) = B(x, cr) to denote the dilate of B(x, r) around ...

i n s u p p o r t i n g e n v i r o n m e n t a l s u s t a i n a b i l i t y, m a n a g i n g p r o d u c t r e t u r n s h a s b e c o m e a v e r y i m p o r t a n t a n d c h a l l e n g i n g i s s u e. r e s p o n d i n g t o t h i s t r e n d, r e s e a r c h e r s i n m a n y p a r t s o f t h e w o r l d h a v e c o n d u c t e d n u m e r o u s s t u d i e s i n r e v e r s e l o g i ...

Let G = (V,E) be a graph and let r ≥ 1 be an integer. For a set D ⊆ V , define Nr[x] = {y ∈ V : d(x, y) ≤ r} and Dr(x) = Nr[x] ∩ D, where d(x, y) denotes the number of edges in any shortest path between x and y. D is known as an r-identifying code (r-locating-dominating set, respectively), if for all vertices x ∈ V (x ∈ V \D, respectively), Dr(x) are all nonempty and different. In this paper, w...

We obtain uniqueness and existence of a solution u to the following second-order stochastic partial differential equation: 1 $$\begin{aligned} du= \left( {\bar{a}}^{ij}(\omega ,t)u_{x^ix^j}+ f \right) dt + g^k dw^k_t, \quad t \in (0,T); u(0,\cdot )=0, \end{aligned}$$ where $$T (0,\infty )$$ , $$w^k$$ $$(k=1,2,\ldots are independent Wiener processes, $$({\bar{a}}^{ij}(\omega ,t))$$ is (predictab...

Theorem ([11, Theorem 2, page 282]). Let R be a prime ring, L a noncommutative Lie ideal of R and d 6= 0 a derivation of R. If [d(x), x] ∈ Z(R), for all x ∈ L, then either R is commutative, or char(R) = 2 and R satisfies s4, the standard identity in 4 variables. Here we will examine what happens in case [d(x), x]n ∈ Z(R), for any x ∈ L, a noncommutative Lie ideal of R and n ≥ 1 a fixed integer....

We consider a connected, locally compact topological space X. We suppose that a pseudo-distance d is defined on X that is, d : X × X 7−→ R+ such that d (x, y) > 0 if and only if x 6= y; d (x, y) = d (y, x) ; d (x, z) ≤ γ [d (x, y) + d (y, z)] for all x, y, z ∈ X, where γ ≥ 1 is some given constant and we suppose that the pseudo-balls B (x, r) = {y ∈ X : d (x, y) < r} , r > 0, form a basis of op...