Characterization of Lie higher derivations on triangular algebras

Let $\mathcal{A}$ and $\mathcal{B}$ be unital rings, and $\mathcal{M}$ be an $\left( {\mathcal{A},\mathcal{B}} \right)$ -bimodule, which is faithful as a left $\mathcal{A}$ -module and also as a right $\mathcal{B}$ -module. Let $\mathcal{U} = Tri\left( {\mathcal{A},\mathcal{M},\mathcal{B}} \right)$ be the triangular algebra. In this paper, we give some different characterizations of Lie higher derivations on $\mathcal{U}$ .     

Characterizing centralizers and generalized derivations on triangular algebras by acting on zero product

Let U = Tri(A,M,B) be a triangular ring, where A and B are unital rings, and M is a faithful (A, B)-bimodule. It is shown that an additive map Φ on U is centralized at zero point (i.e., Φ(A)B = AΦ(B) = 0 whenever AB = 0) if and only if it is a centralizer. Let δ : U → U be an additive map. It is also shown that the following four conditions are equivalent: (1) δ is specially generalized derivable at zero point, i.e., δ(AB) = δ(A)B + Aδ(B) Aδ(I)B whenever AB = 0; (2) δ is generalized derivable at zero point, i.e., there exist additive maps τ1 and τ2 on U derivable at zero point such that δ(AB) = δ(A)B + Aτ1 (B) = τ2 (A)B + Aδ(B) whenever AB = 0; (3) δ is a special generalized derivation; (4) δ is a generalized derivation. These results are then applied to nest algebras of Banach space.     

Optical solitons in the parabolic law media with high-order dispersion

The transmission equation of ultrashort optical pulse in the high-order dispersion media with the parabolic law (cubic–quintic) nonlinearity has been studied with the help of the subsidiary ordinary differential equation expansion method. As a result, the optical solitons and triangular periodic solutions are obtained, and the conditions for exact solutions to exist are also given.     

A process analytical approach for quality control of dapivirine in HIV preventive vaginal rings by Raman spectroscopy

An estimated 34 million people are living with human immunodeficiency virus (HIV) worldwide. A polymeric drug delivery device for women has been developed to help reduce male to female vaginal transmission of HIV‐1. The device is an intravaginal slow‐release ring containing the antiretroviral drug dapivirine as the active pharmaceutical ingredient (API). In all pharmaceutical delivery systems, API content is a critical control point. Conventional quantification of the API in vaginal rings is a stepwise, destructive and time‐consuming process involving solvent extraction and high pressure liquid chromatography. This study investigates the potential of Raman spectroscopy for fast and non‐destructive quantification of dapivirine in intravaginal rings. Wide‐area illumination measurements were carried out on rotating rings using a custom built ring spinner. Customized reference rings of known concentrations were used to build calibration models, and the models were verified by rings from production using the established method as reference analysis. Bi‐ and multivariate calibration methods were applied. Models based on band ratios and partial least squares (PLS) regression models performed similarly well resulting in low model and prediction errors. Next to an alternative reference analysis for quality assurance, Raman can also be used as a production process performance analysis and optimization tool due to its non‐destructive nature and speed of analysis. Using measurements on connected spots over the entire ring circumference, the within ring variation in API was determined. ANOVA showed that there was a statistical difference in API distribution over the rings, information that can be of use in process optimization, for example. However, Tukey's honestly significant difference (HSD) test proved that no regions of the ring were out of specifications, indicating a stable production system. Copyright © 2014 John Wiley & Sons, Ltd.     

Entropy of spin clusters with frustrated geometry

Geometrically frustrated clusters of Ising spins of different shapes on a triangular lattice are studied by exact enumeration. The focus is laid on the ground-state energy and residual entropy behaviors as functions of the cluster shape and size, as well as the spin value. Depending on the cluster shape, the residual entropy density in approach to the thermodynamic limit can either vanish or remain finite and the dependence can be decreasing, increasing or non-monotonic. Nevertheless, the relative entropies normalized by the respective thermodynamic limit values turn out to be little sensitive to the spin value. Attention is drawn to magnetocaloric properties of systems of selected cluster shapes.     

On generalized Jordan 1-derivation in rings

Let n ⩾ 1 be a fixed integer and let R be an (n + 1)!-torsion free 1-ring with identity element e. If F, d:R → R are two additive mappings satisfying F(xⁿ⁺¹) = F(x)(x¹)ⁿ + xd(x)(x¹)ⁿ⁻¹ + x²d(x)(x¹)ⁿ⁻²+ ⋯ +xⁿd(x) for all x ∈ R, then d is a Jordan 1-derivation and F is a generalized Jordan 1-derivation on R.     

$GP$-$V^{'}$-环的非奇异性与正则性

In this paper,we introduce a non-trivial generalization of ZI-rings-quasi ZI-rings.A ring R is called a quasi ZI-ring,if for any non-zero elements a,b ∈ R,ab = 0 implies that there exists a positive integer n such that an = 0 and anRbn = 0.The non-singularity and regularity of quasi ZI,GP-Vˊ-rings are studied.Some new characterizations of strong regular rings are obtained.These effectively extend some known results.     

Finiteness results concerning algebraic power series

We construct an explicit filtration of the ring of algebraic power series by constructible sets, measuring the complexity of these series. As an example of use of this, we give a bound on the dimension of the set of algebraic power series of bounded complexity lying on an algebraic variety defined over the field of power series.     

Ext groups in the category of bimodules over a simple Leibniz algebra

In this article, we generalize Loday and Pirashvili's [11] computation of the Ext-category of Leibniz bimodules for a simple Lie algebra to the case of a simple (non Lie) Leibniz algebra. Most of the arguments generalize easily, while the main new ingredient is the Feldvoss-Wagemann's cohomology vanishing theorem for semi-simple Leibniz algebras.     

Noetherian rings of low global dimension and syzygetic prime ideals

Let R be a Noetherian ring. We prove that R has global dimension at most two if, and only if, every prime ideal of R is of linear type. Similarly, we show that R has global dimension at most three if, and only if, every prime ideal of R is syzygetic. As a consequence, we derive a characterization of these rings using the André-Quillen homology.