out, R will represent an associative ring with center Z(R). As usual the commutator xy−yx will be denoted by [x,y]. We will use basic commutator identities [xy,z] = [x,z]y+x[y,z] and [x,yz]= [x,y]z+y[x,z]. Recall that a ring R is prime if aRb = (0) implies that a = 0 or b = 0, and is semiprime if aRa = (0) implies a = 0. An additive mapping x x∗ on a ring R is called involution in case (xy)∗ = ...

In the Hilbert space $H:=L_2(\mathbb{R})$, we consider impedance Sturm--Liouville operator $T:H\to H$ generated by differential expression $ -p\frac{d}{dx}{\frac1{p^2}}\frac{d}{dx}p$, where function $p:\mathbb{R}\to\mathbb{R}_+$ is of bounded variation on $\mathbb{R}$ and $\inf_{x\in\mathbb{R}} p(x)>0$. Existence transformation for $T$ its properties are studied.
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On the rectangle D = [a, b]× [c, d], the problem on the existence and uniqueness of a nonnegative solution of the characteristic initial value problem for the equation ∂u(t, x) ∂t ∂x = `(u)(t, x) + q(t, x) is considered, where ` : C(D;R) → L(D;R) is a linear bounded operator and q ∈ L(D;R+).

The main purpose of this paper is to prove the following result. Let X be a real or complex Banach space, let L(X) be the algebra of all bounded linear operators on X, let A(X) ⊆ L(X) be a standard operator algebra, and let T : A(X) → L(X) be an additive mapping satisfying the relation T (A2n+1) = 2n+1 ∑ i=1 (−1)i+1Ai−1T (A)A2n+1−i, for all A ∈ A(X) and some fixed integer n ≥ 1. In this case T ...

In this note, we study the volume of arithmetic linear series with base conditions. As an application, we consider the problemof Zariski decompositions on arithmetic varieties. Introduction LetXbeaprojective andflat integral schemeoverZ. Weassume thatX is normal and thegeneric fiber ofX → Spec(Z) is a d-dimensional smoothvariety overQ. Let Div(X) be the groupofCartier divisors onX and letDiv(X)...

We study the cover time of random geometric graphs. Let I(d) = [0, 1] denote the unit torus in d dimensions. Let D(x, r) denote the ball (disc) of radius r. Let Υd be the volume of the unit ball D(0, 1) in d dimensions. A random geometric graph G = G(d, r, n) in d dimensions is defined as follows: Sample n points V independently and uniformly at random from I(d). For each point x draw a ball D(...

Suppose that α ∈ (0, 2) and that X is an α-stable-like process on R d. Let µ be a signed measure on R d belonging to the class K d,α and A µ t be the continuous additive functional of X associated with µ. In this paper we show that the Feynman-Kac semigroup {T µ t : t ≥ 0} defined by T µ t f (x) = E x e −A µ t f (X t) has a density q µ and that there exist positive constants c 1 , c 2 , c 3 , c...

Let R be an associative ring not necessarily with identity element. For any x, y ∈ R. Recall that R is prime if xRy = 0 implies x = 0 or y = 0, and is semiprime if xRx = 0 implies x = 0. Given an integer n ≥ 2, R is said to be n−torsion free if for x ∈ R, nx = 0 implies x = 0.An additive mapping d : R → R is called a derivation if d(xy) = d(x)y + yd(x) holds for all x, y ∈ R, and it is called a...

Let X be an irreducible smooth complex projective curve of genus g > 2, and let x ∈ X be a fixed point. A framed bundle is a pair (E,φ), where E is a vector bundle over X, of rank r and degree d, and φ : Ex −→ C r is a non–zero homomorphism. There is a notion of (semi)stability for framed bundles depending on a parameter τ > 0, which gives rise to the moduli space of τ– semistable framed bundle...

Let m ≥ 0, n ≥ 0 be fixed integers with m + n 6= 0 and let R be a ring. It is our aim in this paper to investigate additive mapping T : R → R satisfying the relation (m + n)T (x2) = mT (x)x + nxT (x) for all x ∈ R. This research is a continuation of our earlier work ([11]). Throughout, R will represent an associative ring with center Z(R). Given an integer n ≥ 2, a ring R is said to be n−torsio...