On Additive Maps and Commutativity in Rings

We study additive maps which are skew-commuting or skew-centralizing on appropriate subsets of a ring R; and we investigate commutativity in prime and semiprime rings admitting a nonzero derivation d such that [d(x),d(y)] = 0 for all x,y in some nonzero one-sided ideal. This paper has two main parts. The first, motivated by a recent result of Bre?ar [3] on triviality of skew-commuting additive maps on prime rings, is a study of additive maps which are skew-commuting or skew-centralizing on subsets of certain rings. The second continues a study, begun years ago by Herstein [7], of prime and semiprime rings R admitting a nonzero derivation d such that d(x)d(y) ? d(y)d(x) = 0 for all x, y in a suitably chosen subset of R.     

On one sided ideals of a semiprime ring with generalized derivations

Let R be a ring with center Z(R). An additive mapping ${F : R \longrightarrow R}$ is said to be a generalized derivation on R if there exists a derivation ${d : R \longrightarrow R}$ such that F(xy) = F(x)y + xd(y), for all ${x, y \in R}$ (the map d is called the derivation associated with F). Let R be a semiprime ring and U be a nonzero left ideal of R. In the present note we prove that if R admits a generalized derivation F, d is the derivation associated with F such that d(U) ≠ (0) then R contains some nonzero central ideal, if one of the following conditions holds: (1) R is 2-torsion free and ${F(xy) \in Z(R)}$ , for all ${x, y \in U}$ , unless F(U)U = UF(U) = Ud(U) = (0); (2) ${F(xy) \mp yx \in Z(R)}$ , for all ${x,y \in U}$ ; (3) ${F(xy) \mp [x,y] \in Z(R)}$ , for all ${x,y \in U}$ ; (4) F ≠ 0 and F([x,y]) = 0, for all ${x, y \in U}$ , unless Ud(U) = (0); (5) F ≠ 0 and ${F([x, y]) \in Z(R)}$ , for all ${x, y \in U}$ , unless either d(Z(R))U = (0) or Ud(U) = (0)n.     

On almost-quasi-commutative rings

A ring R is called almost-quasi-commutative if for each x, y ∈ R there exist nonzero relatively prime integers j = j(x, y) and k = k(x, y) and a non-negative integer n = n(x, y) such that jxy = k(yx) ⁿ . We establish some general properties of such rings, study commutativity of almost-quasi-commutative R, and consider several examples.     

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.     

Equilibria of a nonautonomous Lotka-Volterra model with shelter for the prey

For the nonautonomous Lotka-Volterra model

$\dot x = \alpha (t)(x - M^{ - 1} x^2 - K^{ - 1} (x - \phi (x,y))y),\dot y = \beta (t)y(L^{ - 1} (x - \phi (x,y)) - 1),$

where some part φ(x, y) of the prey population is out of reach of the predator, we obtain sufficient conditions for the existence of a positive asymptotically stable equilibrium in the domain of admissible values of the variables x and y. We consider the cases in which φ(x, y) = m, φ(x, y) = mx, and φ(x, y) = my.

     

Derivations with engel conditions in prime and semiprime rings

Let R be a prime ring, I a nonzero ideal of R, d a derivation of R and m, n fixed positive integers. (i) If (d[x, y]) ^m = [x, y] _n for all x, y ∈ I, then R is commutative. (ii) If Char R ≠ 2 and [d(x), d(y)] _m = [x, y] ⁿ for all x, y ∈ I, then R is commutative. Moreover, we also examine the case when R is a semiprime ring.     

Skew Derivations of Prime Rings

Given a prime ring R, a skew g-derivation for g : R → R is an additive map f : R → R such that f(xy) = f(x)g(y) + xf(y) = f(x)y + g(x)f(y) and f(g(x)) = g(f(x)) for all x, y ∈ R. We generalize some properties of prime rings with derivations to the class of prime rings with skew derivations.     

Inverse Sturm-Liouville problems with homogeneous delay

We study the inverse Sturm-Liouville problem with a delay $$- y''(x) + q(x)y(\alpha x), q \in L_2 [0,\pi ], \alpha \in (0,1],$$ and the boundary conditions y(0) = y(π) = 0.     

On Heuvers’ Logarithmic Functional Equation

?(x + y) - ?(x) - ?(y) = ?(x ^?1 + y ^?l) are identical to those of the Cauchy equation ?(xy) = ?(x) + ?(y) when ? is a function from the positive real numbers into the reals. In the present article, we prove this equivalence for functions mapping the set of nonzero elements of a field (excluding ?₂) .     

Another logarithmic functional equation

Summary. Let f : ]0,￥[? \Bbb R f :\,]0,\infty[\to \Bbb R be a real valued function on the set of positive reals. The functional equations¶¶f(x + y) - f(x) - f(y) = f(x^-1 + y^-1) f(x + y) - f(x) - f(y) = f(x^{-1} + y^{-1}) ¶and¶f(xy) = f(x) + f(y) f(xy) = f(x) + f(y) ¶are equivalent to each other.     

A fourth-order difference method for elliptic equations with nonlinear first derivative terms

We present a nine-point fourth-order finite difference method for the nonlinear second-order elliptic differential equation Au_xx + Bu_yy = f(x, y, u, u_x, u_y) on a rectangular region R subject to Dirichlet boundary conditions u(x, y) = g(x, y) on ?R. We establish, under appropriate conditions O(h⁴)-convergence of the finite difference scheme. Numerical examples are given to illustrate the method and its fourth-order convergence.     

A method of feasible directions using function approximations,with applications to min max problems

This paper presents a demonstrably convergent method of feasible directions for solving the problem min{φ(ξ)| gⁱ(ξ)?0i=1,2,…,m}, which approximates, adaptively, both φ(x) and ▽φ(x). These approximations are necessitated by the fact that in certain problems, such as when $\text{φ(x) =}\text{max}\text{{f(x, y) ¦ y ? Ω}{}_{\mathrm{y}}\text{}}$, a precise evaluation of φ(x) and ▽φ(x) is extremely costly. The adaptive procedure progressively refines the precision of the approximations as an optimum is approached and as a result should be much more efficient than fixed precision algorithms.It is outlined how this new algorithm can be used for solving problems of the form $\text{min}{}_{\mathrm{y}}\text{? Ω}{}_{\mathrm{x}}\text{max}{}_{\mathrm{y}}\text{? Ω}{}_{\mathrm{y}}\text{f(x, y)}$ under the assumption that Ωm_ξ={x|gⁱ(x)?0, j=1,…,s} ∩$\text{R}$ⁿ, Ω_y={y|ζ_i(y)?0, i-1,…,t} ∩ $\text{R}$^m, with f, g^j, ζⁱ continuously differentiable, f(x, ·) concave, ζⁱ convex for $\text{i = 1,…, t,}\text{and}\text{Ω}{}_{\mathrm{x}}\text{, Ω}{}_{\mathrm{y}}$ compact.     

Constructive existence and uniqueness for some nonlinear two-point boundary value problems

Constructive existence and uniqueness theorems are presented for the problem $\text{y″ = ?(x, y), y(0) = y}{}_0\text{, y(1) = y}{}_1$. Applications to several problems are also given including one in which the boundary values are y′(0) = y′₀, y(1) = y₁.     

Commutativity of rings with derivations

I. N. Herstein [10] proved that a prime ring of characteristic not two with a nonzero derivation d satisfying d(x)d(y) = d(y)d(x) for all x, y must be commutative, and H. E. Bell and M. N. Daif [8] showed that a prime ring of arbitrary characteristic with nonzero derivation d satisfying d(xy) = d(yx) for all x, y in some nonzero ideal must also be commutative. For semiprime rings, we show that an inner derivation satisfying the condition of Bell and Daif on a nonzero ideal must be zero on that ideal, and for rings with identity, we generalize all three results to conditions on derivations of powers and powers of derivations. For example, let R be a prime ring with identity and nonzero derivation d, and let m and n be positive integers such that, when charR is finite, m ∨ n < charR. If d(x ^m y ⁿ ) = d(y ⁿ x ^m ) for all x, y ∈ R, then R is commutative. If, in addition, charR≠ 2 and the identity is in the image of an ideal I under d, then d(x) ^m d(y) ⁿ = d(y) ⁿ d(x) ^m for all x, y ∈ I also implies that R is commutative.     

On certain functional equations related to Jordan triple (q, f){(\theta, \phi)}-derivations on semiprime rings

The main purpose of this paper is to prove the following result. Let R be a 2-torsion free semiprime ring with symmetric Martindale ring of quotients Q _s and let q{\theta} and f{\phi} be automorphisms of R. Suppose T:R? R{T:R\rightarrow R} is an additive mapping satisfying the relation T(xyx)=T(x)q(y)q(x)-f(x)T(y)q(x)+f(x)f(y)T(x){T(xyx)=T(x)\theta (y)\theta (x)-\phi (x)T(y)\theta (x)+\phi (x)\phi (y)T(x)}, for all pairs x,y ? R{x,y\in R}. In this case T is of the form 2T(x)=qq(x)+f(x)q{2T(x)=q\theta (x)+\phi (x)q}, for all x ? R{x\in R} and some fixed element q ? Q_s{q\in Q_{s}}.     

A new continuation method for complementarity problems with uniformP-functions

The complementarity problem with a nonlinear continuous mappingf from the nonnegative orthantR ₊ ⁿ ofR ⁿ intoR ⁿ can be written as the system of equationsF(x, y) = 0 and(x, y) R ₊ ²ⁿ , whereF denotes the mapping from the nonnegative orthantR ₊ ²ⁿ ofR ²ⁿ intoR ₊ ⁿ × Rⁿ defined byF(x, y) = (x ₁y₁,,x_ny_n, f₁(x) – y₁,, f_n(x) – y_n) for every(x, y) R ₊ ²ⁿ . Under the assumption thatf is a uniformP-function, this paper establishes that the mappingF is a homeomorphism ofR ₊ ²ⁿ ontoR ₊ ⁿ × Rⁿ. This result provides a theoretical basis for a new continuation method of tracing the solution curve of the one parameter family of systems of equationsF(x, y) = tF(x ⁰, y⁰) and(x, y) R ₊ ²ⁿ from an arbitrary initial point(x ⁰, y⁰) R ₊ ²ⁿ witht = 1 until the parametert attains 0. This approach is an extension of the one used in the polynomially bounded algorithm recently given by Kojima, Mizuno and Yoshise for solving linear complementarity problems with positive semi-definite matrices.     

Convex envelopes of products of convex and component-wise concave functions

In this paper, we consider functions of the form f(x,y)=f(x)g(y){\phi(x,y)=f(x)g(y)} over a box, where f(x), x ? \mathbb R{f(x), x\in {\mathbb R}} is a nonnegative monotone convex function with a power or an exponential form, and g(y), y ? \mathbb Rⁿ{g(y), y\in {\mathbb R}^n} is a component-wise concave function which changes sign over the vertices of its domain. We derive closed-form expressions for convex envelopes of various functions in this category. We demonstrate via numerical examples that the proposed envelopes are significantly tighter than popular factorable programming relaxations.     

Bounds for singular integrals associated with a class of hypersurfaces

For the hypersurface Γ=(y,γ(y)), the singular integral operator along Γ is defined by. $$Tf(x,x_n ) = P.V.\int_{\mathbb{R}^n } {, f(x - y,x_n ) - } \gamma (y))_{\left| y \right|^{n - 1} }^{\Omega (v)} dy$$ where Σ is homogeneous of order 0, $ \int_{\Sigma _{n \lambda } } {\Omega (y')dy'} = 0 $ . For a certain class of hypersurfaces, T is shown to be bounded on L^p(Rⁿ) provided Ω∈L _α ¹ (Σ _n?2),P>1.