The tangential k-Cauchy-Fueter type operator and Penrose type integral formula on the generalized complex Heisenberg group

The tangential k-Cauchy-Fueter operator and k-CF functions are counterparts of the tangential Cauchy–Riemann operator and CR functions on the Heisenberg group in the theory of several complex variables,respectively.In this paper,we introduce a Lie group that the Heisenberg group can be imbedded into and call it generalized complex Heisenberg.We investigate quaternionic analysis on the generalized complex Heisenberg.We also give the Penrose integral formula for k-CF functions and construct the tangential k-Cauchy-Fueter complex.

Rational Invariants of the Generalized Classical Groups

In this paper,we give transcendence bases of the rational invariants fields of the generalized classical groups and their subgroups B,N and T,and we also compute the orders of them.Furthermore,we give explicit generators for the rational invariants fields of the Borel subgroup and the Neron-Severi subgroup of the general linear group.

The Invariant Rings of the Generalized Transvection Groups in the Modular Case

In this paper, first we investigate the invariant rings of the finite groups G ≤ GL（n, F;） generated by i-transvections and i-reflections with given invariant subspaces H over a finite field F;in the modular case. Then we are concerned with general groups G;（ω） and G;（ω）;named generalized transvection groups where ωis a k-th root of unity. By constructing quotient group and tensor, we calculate their invariant rings. In the end, we determine the properties of Cohen-Macaulay,Gorenstein, complete intersection, polynomial and Poincare series of these rings.

The Related Properties of Generalized Orthogonal Group in Specific Normed Linear Spaces

Firstly, in the general normed linear space, the concepts of generalized isosceles orthogonal group, generalized Birkhoff orthogonal group, generalized Roberts orthogonal group, strong Birkhoff orthogonal group and generalized orthogonal basis are introduced. Secondly, the conclusion that any two nonzero generalized orthogonal groups must be linearly independent group is proven. And the existence of nonzero generalized orthogonal group and its linear correlation are discussed preliminarily, as well as some related properties of nonempty generalized orthogonal group in specific normed linear space namely the lp space.

ADMISSIBLE WAVELETS ON THE PRODUCT HEISENBERG GROUP H_n

Let H. be the n-direct proded of Heisenbery group H, P the affine group of Hn. Then P ho a natural unitary representation U on L2(Hn). In this paper,the direct sum decomposition of irreducible invariant closed subspaces under unitary representation U for L2(Hn) is given. The restrictins of U on these subspaces are square-integrable. The charactedsation of admissible condition is obtained in ierms of the Fourier transform. By seleting appropriately an orthogonal wavelet basis and the wavelet transform,the authors obtain the orthogonal direct chin decomposinon of function space L2(P,dμl).

Generation of Spin 1 System

In this paper, we shall establish the connection between the group theory and quantum mechanics by showing how the group theory helps us to construct the spin operators. We look to the group generators SU (3). From these generators, new spin 1 operators will be constructed. These operators S-x, S-y and S-z satisfy all the properties of Pauli spin operators S-x, S-y and S-z. We shall discuss the notion of spin squeezing and correlations for pure spin 1 system using our spin operators S-x, S-y and S-z.

Three types of generalized Kadomtsev-Petviashvili equations arising from baroclinic potential vorticity equation

By means of the reductive perturbation method, three types of generalized （2＋l）-dimensional Kadomtsev- Petviashvili （KP） equations are derived from the baroclinic potential vorticity （BPV） equation, including the modified KP （mKP） equation, standard KP equation and cylindrical KP （cKP） equation. Then some solutions of generalized cKP and KP equations with certain conditions are given directly and a relationship between the generalized mKP equation and the mKP equation is established by the symmetry group direct method proposed by Lou et al. From the relationship and the solutions of the mKP equation, some solutions of the generalized mKP equation can be obtained. Furthermore, some approximate solutions of the baroclinic potential vorticity equation are derived from three types of generalized KP equations.

Generalization of Spin Operators

The study shall look to the group of generators SU(4). From these generators, a new group spin operator will be constructed. We will classify these groups into right handed groups and left handed groups. These two groups will satisfy all the properties of Pauli spin operators Sx, Sy and Sz with respect to the frame xyz. The analysis shows that the number of groups spin operators depends on the order of the group. This leads us to construct the theorem which defines the number of the groups spin operators. The analysis also leads to two kinds of frames: left handed frame (LHF) and right handed frame (RHF). The right handed operators will act on the RHF, and left hand operators act on the LHF. The study shall discuss the notion of spin squeezing for pure spin 3/2 system by using our new frames and new spin operators. It will show that our calculation is equivalent to the calculation by using Pauli spin operators.

ON THE SUPERSTABILITY OF THE PEXIDER TYPE GENERALIZED TRIGONOMETRIC FUNCTIONAL EQUATIONS

The aim of this paper is to investigate the superstability problem for the pexiderized trigonometric functional equation∑ v∈Φ∫Kf（xkv（y）k^-1）dwK（k）= Φ g（x）h（y）, x, y ∈ G,where G is any topological group, K is a compact subgroup of G, ωK is the normalized Haar measure of K, Φ is a finite group of K-invariant morphisms of G and f, g, h are continuous complex-valued functions.Consequently, we have generalized the results of stability for d＇Alembert＇s and Wilson＇s equations by R. Badora, J. Baker, B. Bouikhalene, P. Gavruta, S. Kabbaj, Pl. Kannappan, G. H.Kim, J.M. Rassias, A. Roukbi, L. Sz′ekelyhidi, D. Zeglami, etc.

Generating Relations for Grothendieck Groups of Trivial Extension Algebras

In this paper we describe explicitly the generating relations for the Grothendieck groups of trivial extension algebras of tame hereditary algebras.

Coleman Automorphisms of Generalized Dihedral Groups

Let D be a generalized dihedral group and Autcol（D） its Coleman automorphism group. Denote by Outcol（D） the quotient group of Autcol（D） by Inn（D）, where Inn（D） is the inner automorphism group of D. It is proved that either Outcol（D） = i or Outcol（D） is an elementary abelian 2-group whose order is completely determined by the cardinality of π（D）. Furthermore, a necessary and sufficient condition for Outcol（D） = 1 is obtained. In addition, whenever Outcol（D） ≠ 1, it is proved that Autcol（D） is a split extension of Inn（D） by an elementary abelian 2-group for which an explicit description is given.

On Nonsingularity and Group Inverse of Linear Combinations of Generalized and Hypergeneralized Projectors

In this paper, some conditions for the nonsingularity and group inverses of linear combinations of generalized and hypergeneralized projectors are established. Moreover, some formulae for the inverses and group inverses of them are derived. The work of this paper extends some previous results.

GENERALIZED INTUITIONISTIC FUZZY HYBRID CHOQUET AVERAGING OPERATORS

In this study a new hybrid aggregation operator named as the generalized intuitionistic fuzzy hybrid Choquet averaging（GIFHCA） operator is defined.Meantime,some desirable properties are studied, and several important cases are examined.Furthermore,we define the generalized Shapley GIFHCA （GS-GIFHCA） operator,which does not only overall consider the importance of elements and their ordered positions,but also globally reflect the correlations among them and their ordered positions.In order to simplify the complexity of solving a fuzzy measure,we further define the generalizedλ-Shapley GIFHCA（GλS-GIFHCA） operator.