The paper contains a classification of derivations of skew PBW-extensions of rings.

In this paper, we are concerned with an optimal control problem where the system is driven by a fully coupled forward–backward doubly stochastic differential equation. We study the relaxed model for which an optimal solution exists. This is an extension of initial control problem, where admissible controls are measure valued processes. We establish necessary as well as sufficient optimality conditions to the relaxed one.

This paper presents an efficient class of estimators for estimating the population mean of the variate under study in two-phase sampling using information on several auxiliary variates. The expressions for bias and mean square error (MSE) of the proposed class have been obtained using Taylor series method. In addition, the minimum attainable MSE of the proposed class is obtained to the first order of approximation. The proposed class encompasses a wide range of estimators of the sampling literature. Efficiency comparison has been made for demonstrating the performance of the proposed class. An attempt has been made to find optimum sample sizes under a known fixed cost function. Numerical illustrations are given in support of theoretical findings.

The spectrum of a finite group is the set of its element orders, and two groups are said to be isospectral if they have the same spectra. A finite group G is said to be recognizable by spectrum, if every finite group isospectral with G is isomorphic to G. We prove that if S is one of the sporadic simple groups $M^cL$, $M_{12}$, $M_{22}$, He, Suz and $O'N$, then $\mathrm{Aut}(S)$ is recognizable by spectrum. This finishes the proof of the recognizability by spectrum of the automorphism groups of all sporadic simple groups, except $J_2$.

A link tower is a sequence of links with the structure given by removing the last components. Given a link tower, we prove that there is a chain complex consisting of (non-abelian) groups given by the symmetric commutator subgroup of the normal closures in the link group of the meridians excluding the meridian of the last component with the differential induced by removing the last component. Moreover, the homology groups of these naturally constructed chain complexes are isomorphic to the homotopy groups of the manifold M under certain hypothesis. These chain complexes have canonical quotient abelian chain complexes in Minor’s homotopy link groups with their homologies detecting certain differences of the homotopy link groups in the towers.

Recently, Makhnev and Nirova found intersection arrays of distance-regular graphs with $\lambda =2$ and at most 4096 vertices. In the case of primitive graphs of diameter 3 with $\mu = 1$ there corresponding arrays are $\{18,15,9;1,1,10\}$, $\{33,30,8;1,1,30\}$ or $\{39,36,4;1,1,36\}$. In this work, possible orders and subgraphs of fixed points of the hypothetical distance-regular graph with intersection array $\{18,15,9;1,1,10\}$ are studied. In particular, graph with intersection array $\{18,15,9;1,1,10\}$ is not vertex symmetric.