Commutators with idempotent values on multilinear polynomials in prime rings

Let R be a prime ring of characteristic different from 2, C its extended centroid, d a nonzero derivation of R, f(x ₁, . . . , x _n ) a multilinear polynomial over C, ρ a nonzero right ideal of R and m > 1 a fixed integer such that

$$\qquad \left ([d(f(r_{1},\ldots ,r_{n})),f(r_{1},\ldots ,r_{n})]\right )^{m}=[d(f(r_{1},\ldots ,r_{n})),f(r_{1},\ldots ,r_{n})] $$

for all r ₁, . . . , r _n ∈ρ. Then either [f(x ₁,…,x _n ),x _n+1]x _n+2 is an identity for ρ or d(ρ)ρ = 0.

     

Cocentralizing and vanishing derivations on multilinear polynomials in prime rings

Let R be a prime ring of characteristic different from 2 and extended centroid C and let f(x1,..., x _n ) be a multilinear polynomial over C not central-valued on R, while δ is a nonzero derivation of R. Suppose that d and g are derivations of R such that

$\delta (d(f(r_1 , \ldots ,r_n ))f(r_1 , \ldots ,r_n ) - f(r_1 , \ldots ,r_n )g(f(r_1 , \ldots ,r_n ))) = 0$

for all r1,..., r _n ∈ R. Then d and g are both inner derivations on R and one of the following holds: (1) d = g = 0; (2) d = ?g and f(x ₁,..., x _n )² is central-valued on R.

     

Power-Commuting Generalized Skew Derivations in Prime Rings

Let R be a non-commutative prime ring of characteristic different from 2 with extended centroid C, F ≠ 0 a generalized skew derivation of R, and n ≥ 1 such that [F(x), x] ⁿ  = 0, for all x ∈ R. Then there exists an element λ ∈ C such that F(x) = λx, for all x ∈ R.     

Drift perturbation of subordinate Brownian motions with Gaussian component

Let d ≥ 1 and Z be a subordinate Brownian motion on R~d with infinitesimal generator ? + ψ(?),where ψ is the Laplace exponent of a one-dimensional non-decreasing L′evy process(called subordinator). We establish the existence and uniqueness of fundamental solution(also called heat kernel) pb(t, x, y) for non-local operator L~b= ? + ψ(?) + b ?, where Rb is an Rd-valued function in Kato class K_(d,1). We show that p~b(t, x, y)is jointly continuous and derive its sharp two-sided estimates. The kernel pb(t, x, y) determines a conservative Feller process X. We further show that the law of X is the unique solution of the martingale problem for(L~b, C_c~∞(R~d)) and X is a weak solution of Xt = X0+ Zt + integral from n=0 to t(b(Xs)ds, t ≥ 0).Moreover, we prove that the above stochastic differential equation has a unique weak solution.     

Engel type identities with derivations for right ideals in prime rings

Let R be a prime ring of char R≠2, d a non-zero derivation of R and ρ a non-zero right ideal of R such that [[d(x),d(y)]_n [y,x]_m] = 0 for all x,y ∈ ρ or [[d(x),d(y)]_n d[y,x]_m] = 0 for all x,y ∈ ρ, n, m ≥ 0 are fixed integers. If [ρ,ρ]ρ ≠ 0, then d(ρ)ρ = 0.     

The Fixed Points of Contractions of <Emphasis Type="Italic">f</Emphasis>-Quasimetric Spaces

The recent articles of Arutyunov and Greshnov extend the Banach and Hadler Fixed-Point Theorems and the Arutyunov Coincidence-Point Theorem to the mappings of (q₁, q₂)-quasimetric spaces. This article addresses similar questions for f-quasimetric spaces.Given a function f: R ₊² → R₊ with f(r₁, r₂) → 0 as (r₁, r₂) → (0, 0), an f-quasimetric space is a nonempty set X with a possibly asymmetric distance function ρ: X² → R+ satisfying the f-triangle inequality: ρ(x, z) ≤ f(ρ(x, y), ρ(y, z)) for x, y, z ∈ X. We extend the Banach Contraction Mapping Principle, as well as Krasnoselskii’s and Browder’s Theorems on generalized contractions, to mappings of f-quasimetric spaces.     

Diophantine inequality involving binary forms

Let d ? 3 be an integer, and set r = 2^d?1 + 1 for 3 ? d ? 4, \(\tfrac{{17}}{{32}} \cdot 2^d + 1\) for 5 ? d ? 6, r = d²+d+1 for 7 ? d ? 8, and r = d²+d+2 for d ? 9, respectively. Suppose that Φ _i (x, y) ∈ ?[x, y] (1 ? i ? r) are homogeneous and nondegenerate binary forms of degree d. Suppose further that λ₁, λ₂,..., λ _r are nonzero real numbers with λ₁/λ₂ irrational, and λ₁Φ₁(x₁, y₁) + λ₂Φ₂(x₂, y₂) + · · · + λ _r Φ _r (x _r , y _r ) is indefinite. Then for any given real η and σ with 0 < σ < 2^2?d, it is proved that the inequality

$$\left| {\sum\limits_{i = 1}^r {{\lambda _i}\Phi {}_i\left( {{x_i},{y_i}} \right) + \eta } } \right| < {\left( {\mathop {\max \left\{ {\left| {{x_i}} \right|,\left| {{y_i}} \right|} \right\}}\limits_{1 \leqslant i \leqslant r} } \right)^{ - \sigma }}$$

has infinitely many solutions in integers x₁, x₂,..., x _r , y₁, y₂,..., y _r . This result constitutes an improvement upon that of B. Q. Xue.

     

Some results on the annihilator graph of a commutative ring

Let R be a commutative ring. The annihilator graph of R, denoted by AG(R), is the undirected graph with all nonzero zero-divisors of R as vertex set, and two distinct vertices x and y are adjacent if and only if ann _R (xy) ≠ ann _R (x) ∪ ann _R (y), where for z ∈ R, ann _R (z) = {r ∈ R: rz = 0}. In this paper, we characterize all finite commutative rings R with planar or outerplanar or ring-graph annihilator graphs. We characterize all finite commutative rings R whose annihilator graphs have clique number 1, 2 or 3. Also, we investigate some properties of the annihilator graph under the extension of R to polynomial rings and rings of fractions. For instance, we show that the graphs AG(R) and AG(T(R)) are isomorphic, where T(R) is the total quotient ring of R. Moreover, we investigate some properties of the annihilator graph of the ring of integers modulo n, where n ? 1.     

Centralizers of generalized skew derivations on multilinear polynomials

Let R be a prime ring of characteristic different from 2, let Q be the right Martindale quotient ring of R, and let C be the extended centroid of R. Suppose that G is a nonzero generalized skew derivation of R and f(x ₁,..., x _n ) is a noncentral multilinear polynomial over C with n noncommuting variables. Let f(R) = {f(r ₁,..., r _n ): r _i ∈ R} be the set of all evaluations of f(x ₁,..., x _n ) in R, while A = {[G (f(r ₁,..., r _n )), f(r ₁,..., r _n )]: r _i ∈ R}, and let C _R (A) be the centralizer of A in R; i.e., C _R (A) = {a ∈ R: [a, x] = 0, ? _x ∈ A }. We prove that if A ≠ (0), then C _R (A) = Z(R).     

On boundary value problems for systems of nonlinear generalized ordinary differential equations

A general theorem (principle of a priori boundedness) on solvability of the boundary value problem dx = dA(t) · f(t, x), h(x) = 0 is established, where f: [a, b]×R ⁿ → R ⁿ is a vector-function belonging to the Carathéodory class corresponding to the matrix-function A: [a, b] → R ^n×n with bounded total variation components, and h: BV_s([a, b],R ⁿ ) → R ⁿ is a continuous operator. Basing on the mentioned principle of a priori boundedness, effective criteria are obtained for the solvability of the system under the condition x(t₁(x)) = B(x) · x(t ₂(x))+c ₀, where t _i: BV_s([a, b],R ⁿ ) → [a, b] (i = 1, 2) and B: BV_s([a, b], R ⁿ ) → R ⁿ are continuous operators, and c ₀ ∈ R ⁿ .     

Planar Graded Lattices and the <Emphasis Type="Bold">c</Emphasis><Subscript>1</Subscript>-Median Property

Let L be a lattice of finite length, ξ = (x ₁,…, x _k )∈L ^k , and y ∈ L. The remoteness r(y, ξ) of y from ξ is d(y, x ₁)+?+d(y, x _k ), where d stands for the minimum path length distance in the covering graph of L. Assume, in addition, that L is a graded planar lattice. We prove that whenever r(y, ξ) ≤ r(z, ξ) for all z ∈ L, then y ≤ x ₁∨?∨x _k . In other words, L satisfies the so-called c ₁ -median property.     

Weak- and strong-type inequality for the cone-like maximal operator in variable Lebesgue spaces

The classical Hardy-Littlewood maximal operator is bounded not only on the classical Lebesgue spaces L_p(R^d) (in the case p > 1), but (in the case when 1/p(·) is log-Hölder continuous and p- = inf{p(x): x ∈ R^d > 1) on the variable Lebesgue spaces L_p(·)(R^d), too. Furthermore, the classical Hardy-Littlewood maximal operator is of weak-type (1, 1). In the present note we generalize Besicovitch’s covering theorem for the so-called γ-rectangles. We introduce a general maximal operator M_s^γδ, and with the help of generalized Φ-functions, the strong- and weak-type inequalities will be proved for this maximal operator. Namely, if the exponent function 1/p(·) is log-Hölder continuous and p- ≥ s, where 1 ≤ s ≤ ∞ is arbitrary (or p- ≥ s), then the maximal operator M_s^γδ is bounded on the space L_p(·)(R^d) (or the maximal operator is of weak-type (p(·), p(·))).     

On homogenization for non-self-adjoint locally periodic elliptic operators

In this note we consider the homogenization problem for a matrix locally periodic elliptic operator on R ^d of the form A ^ε = ?divA(x, x/ε)?. The function A is assumed to be Hölder continuous with exponent s ∈ [0, 1] in the “slow” variable and bounded in the “fast” variable. We construct approximations for (A ^ε ? μ)^?1, including one with a corrector, and for (?Δ) ^s/2(A ^ε ? μ)^?1 in the operator norm on L ₂(R ^d ) ⁿ . For s ≠ 0, we also give estimates of the rates of approximation.     

On a functional equation related to derivations in prime rings

In this paper we prove the following result. Let m ≥ 1, n ≥ 1 be fixed integers and let R be a prime ring with m + n + 1 ≤ char(R) or char(R) = 0. Suppose there exists an additive nonzero mapping D : R → R satisfying the relation 2D(x ^n+m+1) = (m + n + 1)(x ^m D(x)x ⁿ + x ⁿ D(x)x ^m ) for all \({x\in R}\). In this case R is commutative and D is a derivation.     

The Folded (2<Emphasis Type="Italic">D</Emphasis> + 1)-cube and Its Uniform Posets

Let Γ denote the folded (2D + 1)-cube with vertex set X and diameter D ≥ 3. Fix x ∈ X. We first define a partial order ≤ on X as follows. For y, z ∈ X let y ≤ z whenever ?(x, y) + ?(y, z) = ?(x, z). Let R (resp. L) denote the raising matrix (resp. lowering matrix) of Γ. Next we show that there exists a certain linear dependency among RL², LRL,L²R and L for each given Q-polynomial structure of Γ. Finally, we determine whether the above linear dependency structure gives this poset a uniform structure or strongly uniform structure.     

Generalized (Jordan) left derivations on rings associated with an element of rings

In this paper, we introduce a new notion of generalized (Jordan) left derivation on rings as follows: let R be a ring, an additive mapping F : R → R is called a generalized (resp. Jordan) left derivation if there exists an element w ∈ R such that F(xy) = xF(y) + yF(x) + yxw (resp. F(x ²) = 2xF(x) + x ² w) for all x, y ∈ R. Then, some related properties and results on generalized (Jordan) left derivation of square closed Lie ideals are obtained.     

On a Functional Equation Related to Jordan Triple Derivations in Prime Rings

A classical result of Herstein asserts that any Jordan derivation on a prime ring with char(R) ≠ 2 is a derivation. It is our aim in this paper to prove the following result, which is in the spirit of Herstein’s theorem. Let R be a prime ring with char(R) = 0 or char(R) > 4, and let D: R → R be an additive mapping satisfying the relation D(x⁴) = D(x)x³ + xD(x²)x + x³D(x) for all x ∈ R. In this case, D is a derivation.     

The Compressed Annihilator Graph of a Commutative Ring

Let R be a commutative ring. In this paper, we introduce and study the compressed annihilator graph of R. The compressed annihilator graph of R is the graph AG_E(R), whose vertices are equivalence classes of zero-divisors of R and two distinct vertices [x] and [y] are adjacent if and only if ann(x)∪ann(y) ? ann(xy). For a reduced ring R, we show that compressed annihilator graph of R is identical to the compressed zero-divisor graph of R if and only if 0 is a 2-absorbing ideal of R. As a consequence, we show that an Artinian ring R is either local or reduced whenever 0 is a 2-absorbing ideal of R.     

Disjoint cliques in claw-free graphs

A graph is said to be claw-free if it does not contain an induced subgraph isomorphic to K_1,3. Let s and k be two integers with 0 ≤ s ≤ k and let G be a claw-free graph of order n. In this paper, we investigate clique partition problems in claw-free graphs. It is proved that if n ≥ 3s+4(k?s) and d(x)+d(y) ≥ n?2s+2k+1 for any pair of non-adjacent vertices x, y of G, then G contains s disjoint K₃s and k ? s disjoint K₄s such that all of them are disjoint. Moreover, the degree condition is sharp in some cases.     

Deviation of elements of a Banach space from a system of subspaces

We prove that if X is a real Banach space, Y ₁ ? Y ₂ ? ... is a sequence of strictly embedded closed linear subspaces of X, and d ₁ ≥ d ₂ ≥ ... is a nonincreasing sequence converging to zero, then there exists an element x ∈ X such that the distance ρ(x, Y _n ) from x to Y _n satisfies the inequalities d _n ≤ ρ(x, Y _n ) ≤ 8d _n for n = 1, 2, ....