Dominant representations and a markaracter table for a group of finite order

Summary The concept of markaracter is proposed to discuss marks and characters for a group of finite order on a common basis. Thus, we consider a non-redundant set of dominant subgroups and a non-redundant set of dominant representations (SDR), where coset representations concerning cyclic subgroups are named dominant representations (DRs). The numbers of fixed points corresponding to each DR are collected to form a row vecter called a dominant markaracter (mark-character). Such dominant markaracters for the SDR are collected as a markaracter table. The markaracter table is related to a subdominant markaracter table of its subgroup so that the corresponding row of the former table is constructed from the latter. The data of the markaracter table are in turn used to construct a character table of the group, after each character is regarded as a markaracter and transformed into a multiplicity vector. The concept of orbit index is proposed to classify multiplicity vectors; thus, the orbit index of each DR is proved to be equal to one, while that corresonding to an irreducible representation is equal to zero.     

Equivariant Mapping Class Group and Orbit Braid Group

Motivated by the work of Birman about the relationship between mapping class groups and braid groups, the authors discuss the relationship between the orbit braid group and the equivariant mapping class group on the closed surface M with a free and proper group action in this paper. Their construction is based on the exact sequence given by the fibration F₀^GM → F(M/G, n). The conclusion is closely connected with the braid group of the quotient space. Comparing with the situation without the group action, there is a big difference when the quotient space is T².     

Domains in complex surfaces with non-compact automorphism groups

Let X be an arbitrary complex surface and D a domain in X that has a non-compact group of holomorphic automorphisms. A characterization of those domains D that admit a smooth, weakly pseudoconvex, finite type boundary orbit accumulation point is obtained.     

Movement of Intransitive Permutation Groups Having Maximum Degree

Let G be a permutation group on a set Ω with no fixed points in,and m be a positive integer.Then the movement of G is defined as move(G):=sup Γ {|Γg\Γ| | g ∈ G}.It was shown by Praeger that if move(G) = m,then |Ω| 3m + t-1,where t is the number of G-orbits on.In this paper,all intransitive permutation groups with degree 3m+t-1 which have maximum bound are classified.Indeed,a positive answer to her question that whether the upper bound |Ω| = 3m + t-1 for |Ω| is sharp for every t > 1 is given.     

Strength of convergence in the orbit space of a transformation group

Let (G,X) be a second-countable transformation group with G acting freely on X. It is shown that measure-theoretic accumulation of the action and topological strength of convergence in the orbit space X/G provide equivalent ways of quantifying the extent of nonproperness of the action. These notions are linked via the representation theory of the transformation-group C^∗-algebra C₀(X)?G.     

Homoclinic Bifurcation of Orbit Flip with Resonant Principal Eigenvalues

Codimension-3 bifurcations of an orbit-flip homoclinic orbit with resonant principal eigenvalues are studied for a four-dimensional system. The existence, number, co-existence and noncoexistence of 1-homoclinic orbit, 1-periodic orbit, 2n-homoclinic orbit and 2n-periodic orbit are obtained. The bifurcation surfaces and existence regions are also given.     

The uniqueness of limit cycles for Liénard system

In monographs [Theory of Limit Cycles, 1984] and [Qualitative Theory of Differential Equations, 1985], eleven propositions by several mathematicians are listed on the uniqueness of limit cycles for equations of type (I), (II), and (III) of the quadratic ordinary differential systems. In this paper, we first point out that all these propositions were not completely proved since the equations under consideration do not satisfy the conditions of the theorems used to guarantee the uniqueness of limit cycles. Then we give a new set of theorems that guarantee the uniqueness of limit cycles for the Liénard systems, which not only can be applied to complete the proof of the propositions mentioned above but generalize many other uniqueness theorems as well. The conditions in these uniqueness theorems, which are independent and were obtained by different methods, can be combined into one improved general theorem that is easy to apply. Thus many of the most frequently used theorems on the uniqueness of limit cycles are corollaries of the results in this paper.     

On equivalence of generalized multi-valued contractions and Nadler's fixed point theorem

We consider two generalizations of Nadler's theorem, one proved by Mizoguchi and Takahashi in response to the Reich conjecture and another theorem proved by Kaneko. We show that due to the additional conditions of these theorems the given multi-valued map reduces to a multi-valued contraction map. We prove this result by showing that the orbit of the multi-valued map is bounded under the contractive conditions of the two generalizations.     

Double quantization of CP type orbits by generalized Verma modules

It is known that symmetric orbits in g^* for any simple Lie algebra g are equipped with a Poisson pencil generated by the Kirillov-Kostant-Souriau bracket and the reduced Sklyanin bracket associated to the “canonical” R-matrix. We realize quantization of the Poisson pencil CPⁿ type orbits (i.e. orbits in sl(n + 1)^* whose real compact form is CPⁿ) by means of q-deformed Verma modules.     

Pointwise Recurrent Dynamical Systems with Pseudo-orbit Tracing Property

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