Narrow Results

We study the primeness of noncommutative polynomials on prime rings. Let $R$ be a prime ring with extended centroid $C$, $\rho$ a right ideal of $R$, $f(X_1,\dots ,X_t)$ a noncommutative polynomial over $C$, which is not a polynomial identity (PI) for $\rho$, and $a,b\in R\setminus \{0\}$. Then $af(x_1,\dots ,x_t)b=0$ for all $x_1,\dots ,x_t\in \rho$ if and only if one of the following holds: (i) $a\rho =0$; (ii) $\rho C=eRC$ for some idempotent $e\in RC$ and $b\in \rho C$ such that either $f(X_1,\dots ,X_t)X_{t+1}$ is a PI for $\rho$ or $f(X_1,\dots ,X_t)$ is central-valued on $eRCe$ and $ab=0$. We then apply the result to higher commutators of right ideals. Some results of the paper are also studied from the view of point of the notion of $X$-primeness of rings.

This paper focuses on a comparative study of a reduced order linear quadratic regulator control (ROLQR) and conventional variable frequency hysteresis controller (HC) for power factor correction, and their performances are compared. A prototype of front-end AC–DC converter followed by DC–DC Cuk converter controlled by a dSPACE signal processor was set up for 60 watts. Experimental results are presented to validate the simulation results. Simulation and experimental results reveal that ROLQR control can achieve good output voltage regulation and also provide improved robustness in shaping the input current in the presence of load variations when compared to conventional HC.

This paper proposes the idea of mathematical modeling of the buck converter. The modified approach is achieved through the relay feedback (RF) method and the changes are introduced on the converter output with a soft start. Mathematical modeling of the buck converter is achieved by the state-space model (SSM) and converter’s transfer function (TF) to attain stability. By iteration, the proportional-integral (PI) parameters are tuned by a controller that is incorporated in the RF. The output voltage changes are introduced by maintaining the relay-tuned PI control. The concept requires small tuning times which is obtained at a reduced cost. The effectiveness of buck converter using RF is validated in simulation and hardware.

Let R be a prime ring that is not commutative and such that R ≇ M₂(GF(2)), let D, G be two generalized derivations of R, and let m, n be two fixed positive integers. Then D(x^m)xⁿ = xⁿG(x^m) for all x ∈ R iff the following two conditions hold: (1) There exists w ∈ Q, the symmetric Martindale quotient ring of R, such that D(x) = xw and G(x) = wx for all x ∈ R; (2) either w ∈ C, or x^m and xⁿ are C-dependent for all x ∈ R. We also consider the situation for the semiprime case.

Let R be a prime ring, which is not commutative, with involution * and with Q_ms(R) the maximal symmetric ring of quotients of R. An additive map δ : R → R is called a Jordan *-derivation if δ(x²) = δ(x)x* + xδ(x) for all x ∈ R. A Jordan *-derivation of R is called X-inner if it is of the form x ↦ xa - ax* for x ∈ R, where a ∈ Q_ms(R). We prove that any Jordan *-derivation of R is X-inner if char R ≠ 2 or deg(S(R)) > 4, where S(R) := {x ∈ R|x* = x}.

Let R be a prime ring with extended centroid C. We prove that an additive map from R into RC + C can be characterized in terms of left and right b-generalized derivations if it has a generalized derivation expansion. As a consequence, a generalization of the Noether–Skolem theorem is proved among other things: A linear map from a finite-dimensional central simple algebra into itself is an elementary operator if it has a generalized derivation expansion.

Let $R$ be an associative ring. Given a positive integer $n\ge 2$, for $a_1,\dots ,a_n\in R$ we define ${[a_1,\dots ,a_n]}_n:=a_1{a}_2\cdots a_n-a_n{a}_{n-1}\cdots a_1$, the $n$-generalized commutator of $a_1,\dots ,a_n$. By an $n$-generalized Lie ideal of $R$ (at the $(r+1)$th position with $r\ge 0$) we mean an additive subgroup $A$ of $R$ satisfying ${[x_1,\dots ,x_r,a,y_1,\dots ,y_s]}_n\in A$ for all $x_i,y_j\in R$ and all $a\in A$, where $r+s=n-1$. In the paper, we study $n$-generalized commutators of rings and prove that if $R$ is a noncommutative prime ring and $n\ge 3$, then every nonzero $n$-generalized Lie ideal of $R$ contains a nonzero ideal. Therefore, if $R$ is a noncommutative simple ring, then $R={[R,\dots ,R]}_n$. This extends a classical result due to Herstein [Generalized commutators in rings, Portugal. Math. 13 (1954) 137–139]. Some generalizations and related questions on $n$-generalized commutators and their relationship with noncommutative polynomials are also discussed.

Polymer layers adsorbed to a surface or in a confined environment often change their mechanical properties. There is even the possibility of solidification of the confined layer. To judge the stiffness of such a layer, we used the Hertz model to calculate the Young's modulus of the polymer layer in the confinement of AFM experiments with silicon nitride tip with a radius of curvature of R ≈ 50 nm and a glass sphere attached to the cantilever R = 5 μm. Since there is no visible indentation of the layer in the AFM experiments, the layer is either penetrated very easily, or the indentation is too small to be seen in a force curve. The latter would be the case for a polymer layer with a Young's modulus above 4 × 10⁸Pa in case of an experiment with a silicon nitride tip and 4 × 10⁵Pa in case of a glass sphere.

Let R be a prime ring and n > 1 be a fixed positive integer. If g is a nonzero generalized derivation of R such that g(x)ⁿ=g(x) for all x ∈ R, then R is commutative except when R is a subring of the 2 × 2 matrix ring over a field. Moreover, we generalize the result to the case g(f(x_i))ⁿ = g(f(x_i)) for all x₁, x₂, …, x_t∈ R, where f(X_i) is a multilinear polynomial.