Some Generalized Normal Subgroups and p-nilpotency of Finite Groups

In this paper, the influence of s-semipermutable, c~#-normal, subnormally embedded and ss-quasinormal subgroups on the p-nilpotency of finite groups is investigated and some recent results are generalized.

循环的次正规子群为正规的群

称循环的次正规子群正规的群为CT群 ,本文对CT群进行了研究 。

SUBNORMAL SOLUTIONS OF DIFFERENTIAL EQUATIONS WITH PERIODIC COEFFICIENTS

In this article, we apply the concept of hyper-order to higher order linear differential equations with periodic coefficients, investigate the existence and the form of its subnormal solution, and estimate the growth of all other solutions, and answer the question raised by Gundersen and Steinbart for more general periodic differential equations.

Simulation Study on the Self-Sustained Oscillations in DC Driven Glow Discharges at Atmospheric Pressure Under Different Gas Gaps

In this paper, a one-dimensional plasma fluid model is employed to study the self- sustained oscillations in DC-driven helium glow discharges at atmospheric pressure under different gas gaps. Our simulation results indicate that a harmonic current oscillation with tiny amplitude always occur at the onset of instability and transits into a relaxation one as the conductivity of the semiconductor is decreased. It is found that the dynamics of the oscillations are dependent on the gas gaps. The discharge can only exhibit a simple oscillation with unique amplitude and frequency at smaller gas gaps （〈2 mm） while it can exhibit a more complex oscillation with several different amplitudes and frequencies at larger gas gaps （〉2 mm）. The discharge modes in these current oscillations have also been analyzed.

2D DC Subnormal Glow Discharge in Argon

A two-dimensional time-dependent fluid model was developed and used to describe a DC subnormal glow discharge in argon with Cartesian geometry. This configuration allows us to take into account the transverse expansion of the discharge. A hydrodynamic fluid model used in this paper is based on the moments of the Boltzmann transport equation. The resultant set of governing equations consists of continuity equations （fluxes and densities） for electrons and ions, an energy equation for electrons, and Poisson＇s equation. Simulation results are presented for the densities of charged particles, the electric voltage, the electric field, and the electron temperature of the discharge. The results were compared with those obtained in the literature.

On the Representation of Some Subnormal Operators

Let T be pure subnormal operator. In this paper necessary and sufficiert conditions that T=N+K are given,where N is normal, K is quasinormal and NK=KN.

Near Subnormality of Weighted Shifts and the Answer to the Hilbert Space Problem 160

In this paper, necessary and sufficient conditions are obtained for unilateral weighted shifts to be near subnormal. As an application of the main results, many answers to the Hilbert space problem 160 are presented at the end of the paper.

A Generalization of Cuccia＇s Theorem

For a finite group G, let S（G） be the set of minimal subgroups of odd order,which are complemented in G. It is proved that if every minimal subgroup X of odd orderof G which does not belong to S（G）, CG（X） is either subnormal or abnormal in G. Then Gsolvable.

New Constructions of Nullnorms on Bounded Lattices

We propose two more general methods to construct nullnorms on bounded lattices. By some illustrative examples, we demonstrate that the new method differ from the existing approaches.

A NOTE ON NORMAL ELEMENTARY OPERATORS

Let（T1,T2）and （S1*,S2*） be double commuting subnormal operator pairs,X and K be bounded linear operators with T1KS1+T2KS2=0. Then （1）||T1XS1+T2XS2+K||≥||K||and （2）||T1XS1+T2XS2+K1|22=||T1XS1+T2XS2 ||22+||K||22if K is Hilbert-Schmidt operator,Let T and S* be dominant operator and subnormal operator, respectively, and K is Hilbert-Schmidt operator, then （3）||TX-XS+K||22=||TX-XS||22+||K||22if TK+KS;and （4）||TXS-X+K||22=||TXS-X||22+||K||22 if TKS=K. These generalize the results of （1）and （3）.

On Finite Groups that are the Product of Two Subnormal Supersoluble Subgroups

Denote the class of finite groups that are the product of two normal supersoluble subgroups and the class of groups that are the product of two subnormal supersoluble subgroups by B_(1)and B_(2),respectively.In this paper,a characterisation of groups in B_(1)or in B_(2)is given.By applying this new characterisation,some new properties of B_(1)(B_(2))and new proofs of many known results about B_(1)or B_(2)are obtained.Further,closure properties of B_(1)and B_(2)are discussed.

On subnormal solutions of second order linear periodic differential equations

In this paper, we investigate the existence and the form of subnormal solution for a class of second order periodic linear differential equations, estimate the growth properties of all solutions, and answer the question raised by Gundersen and Steinbart.

On quasi-similarity of subnormal operators

For a compact subset K in the complex plane, let Rat(K) denote the set of the rational functions with poles off K. Given a finite positive measure with support contained in K, let R2(K,v) denote the closure of Rat(K) in L2(v) and let Sv denote the operator of multiplication by the independent variable z on R2(K, v), that is, Svf = zf for every f∈R2(K, v). SupposeΩis a bounded open subset in the complex plane whose complement has finitely many components and suppose Rat(Ω) is dense in the Hardy space H2(Ω). Letσdenote a harmonic measure forΩ. In this work, we characterize all subnormal operators quasi-similar to Sσ, the operators of the multiplication by z on R2(Ω,σ). We show that for a given v supported onΩ, Sv is quasi-similar to Sσif and only if v/■Ω■σ and log(dv/dσ)∈L1(σ). Our result extends a well-known result of Clary on the unit disk.

Numbers of Subnormal Solutions for Higher Order Periodic Differential Equations

In this paper, we estimate the number of subnormal solutions for higher order linear periodic differential equations, and estimate the growth of subnormal solutions and all other solutions. We also give a representation of subnormal solutions of a class of higher order linear periodic differential equations.

The influence of X-semipermutability of subgroups on the structure of finite groups

Let A be a subgroup of a group G and X be a nonempty subset of G. A is said to be X-semipermutable in G if A has a supplement T in G such that A is X-permutable with every subgroup of T. In this paper, we investigate further the influence of X-semipermutability of some subgroups on the structure of finite groups. Some new criteria for a group G to be supersoluble or p-nilpotent are obtained.

On minimal non-J N J-groups

A finite group G is called an J N J-group if every proper subgroup H of G is either subnormal in G or self-normalizing. We determinate the structure of non-J N J-groups in which all proper subgroups are J N J- groups.

F-subnormal and F-subabnormal chains in finite groups

The class F pc of all finite groups G is defined such that X≤G implies that there exists an F subnormal subgroup S of G containing X such that X is F subabnormal in S, where F is a saturated formation, closed under taken subgroups. Groups in F pc are characterized by F projectors and F covering subgroups.

Extension respect to complement space

Let H K be a pair of Hilbert spaces and i be inclusion from H into K If i is acontraction, then we denote H K In this case, P=ii~* is a positive operator in theHilbert space K,0≤P≤I, and P=i~*i positive operator in the Hilbert space H, 0≤P≤I and0∈σ_p(P). L. de Branges proved that in this case there is a unique complement space L ofH in K such that L K and for x∈H, v∈L,it holds that (x+y,x+y)_k≤(x,x)_H+ (y,y)_LAnd for:z∈K there is unique decomposition z=x+y for which the above equality holds.

Weakly k-hyponormal and polynomially hyponormal commuting operator pairs

We introduce the notion of weak k-hyponormality and polynomial hyponormality for commuting operator pairs on a Hilbert space and investigate their relationship with k-hyponormality and subnormality.We provide examples of 2-variable weighted shifts which are weakly 1-hyponormal but not hyponormal.By relating the weak k-hyponormality and k-hyponormality of a commuting operator pair to positivity of restriction of some linear functionals to corresponding cones of functions,we prove that there is an operator pair that is polynomially hyponormal but not 2-hyponormal,generalizing Curto and Putinar’s result(1991,1993)to the two-variable case.

Finite Groups of Spencer Height ≤ 3

In this paper, we describe the finite groups G in which every maximal chain M3 〈 M2 〈 M1 〈 M0 = G contains a proper subnormal subgroup of G.