Hom-Lie algebras were introduced by J. Hartwig, D. Larsson, and S. Silvestrov as a generalized Lie algebra. When studying the homology and cohomology theory of Hom-Lie algebras, the authors find that the lowdimensional cohomology theory of Hom-Lie algebras is not well studied because of the Hom-Jacobi identity. In this paper, the authors compute the first and second cohomology groups of the q-deformed Heisenberg-Virasoro algebra of Hom-type, which will be useful to build the low-dimensional cohomology theory of Hom-Lie algebras.

Two kinds of eigentime identity for asymmetric finite Markov chains are proved both in the ergodic case and the transient case.

In this article, we apply the localization techniques to right (?)-serial coalgebras and obtain some interesting results. In particular, we give a characterization of right (?)-serial coalgebras by means of its ‘local structure’, which is the localized right (?)-serial coalgebras, and we get a main result—the periodicity theorem.

Let G be a finite group, and let ? be a formation of finite groups. We say that a subgroup H of G is ?h-normal in G if there exists a normal subgroup T of G such that HT is a normal Hall subgroup of G and (H ∩ T )H_G/H_G is contained in the ?-hypercenter Z∞?(G/HG) of G/H_G. In this paper, we obtain some results about the ?h-normal subgroups and then use them to study the structure of finite groups.

It is an interesting topic to determine the structure of a finite group with a given number of elements of maximal order. In this article, we classify finite groups with 24 elements of maximal order.

In this paper, we first analyze the structure of a finite nonsolvable group in which every cyclic subgroup of order 2 and 4 of every second maximal subgroup is an NE-subgroup. Next, we prove that a finite group G is solvable if every nonnilpotent subgroup of G is a PE-group.

Suppose that cause-effect relationships between variables can be described by a causal network with a linear structural equation model. Kuroki and Miyakawa proposed a graphical criterion for selecting covariates to identify the effect of a conditional plan with one control variable [J. Roy. Statist. Soc. Ser. B, 2003, 65: 209-222]. In this paper, we study a particular type of conditional plan with more than one control variable and propose a graphical criterion for selecting covariates to identify the effect of a conditional plan of the studied type.

In this paper, we use the general quantization method by Drinfel’d twists to quantize the Schr?dinger-Virasoro Lie algebra whose Lie bialgebra structures were recently discovered by Han-Li-Su. We give two different kinds of Drinfel’d twists, which are then used to construct the corresponding Hopf algebraic structures. Our results extend the class of examples of noncommutative and noncocommutative Hopf algebras.

A t-(v, k, 1) directed design (or simply a t-(v, k, 1)DD) is a pair (S, ?), where S is a v-set and ? is a collection of k-tuples (called blocks) of S, such that every t-tuple of S belongs to a unique block. The t-(v, k, 1)DD is called resolvable if ? can be partitioned into some parallel classes, so that each parallel class is a partition of S. It is proved that a resolvable 3-(v, 4, 1)DD exists if and only if v ≡ 0 (mod 4).

We derive the gradient estimates and Harnack inequalities for positive solutions of the diffusion equation u_t = Δu^m on Riemannian manifolds. Then, we prove a Liouville type theorem.

J. Wei recently proposed a concept of ?s-modules which is another generalization of ?-modules besides ?n-modules [J. Algebra, 2005, 291: 312-324]. In this paper, we consider the co-?s-modules and give some characterizations and properties. It is found that the class of co-?s-modules contains co-selfsmall injective cogenerators. The relations between co-?s-modules and co-?n-modules are also considered.

Let k be a field and E(n) be the 2ⁿ⁺¹-dimensional pointed Hopf algebra over k constructed by Beattie, D?sc?lescu and Grünenfelder [J. Algebra, 2000, 225: 743-770]. E(n) is a triangular Hopf algebra with a family of triangular structures R_M parameterized by symmetric matrices M in M_n(k). In this paper, we study the Azumaya algebras in the braided monoidal category E(n)?RM and obtain the structure theorems for Azumaya algebras in the category E(n)?RM, where M is any symmetric n × n matrix over k.

The aim of this paper is to study the boundedness of the windowed-Kontorovich-Lebedev transforms. For this purpose, we first define the translation associated to the Kontorovich-Lebedev transform and a generalized convolution product, then obtain some harmonic analysis results. We present a sufficient and necessary condition for the boundedness of the windowed-Kontorovich-Lebedev transform. Finally, we define the corresponding Weyl operator, and study the boundedness and compactedness of the Weyl operator with symbols in L^q (q ∈ [1, 2]) acting on L^p.

In this paper, we characterize the nilpotency and supersolvability of a finite group G by assuming some subgroups of prime power order have the semi cover-avoiding property in G. Some earlier results are generalized.

In this paper, we establish an exact asymptotic formula for the finite-time ruin probability of a nonstandard compound renewal risk model with constant force of interest in which claims arrive in groups, their sizes in one group are identically distributed but negatively dependent, and the inter-arrival times between groups are negatively dependent too.