$$ \mathcal{I}_g $$-normal and $$ \mathcal{I}_g $$-regular spaces

$ \mathcal{I}_g $ \mathcal{I}_g -normal and $ \mathcal{I}_g $ \mathcal{I}_g -regular spaces are introduced and various characterizations and properties are given. Characterizations of normal, mildly normal, g-normal, regular and almost regular spaces are also given.     

<Emphasis Type="Italic">g</Emphasis>-closed sets in ideal topological spaces

Characterizations and properties of $ \mathcal{I}_g $ \mathcal{I}_g -closed sets and $ \mathcal{I}_g $ \mathcal{I}_g -open sets are given. A characterization of normal spaces is given in terms of $ \mathcal{I}_g $ \mathcal{I}_g -open sets. Also, it is established that an $ \mathcal{I}_g $ \mathcal{I}_g -closed subset of an $ \mathcal{I} $ \mathcal{I} -compact space is $ \mathcal{I} $ \mathcal{I} -compact.     

A$\mathcal{T}$-algebras and extensions of A$\mathbb{T}$-algebras

Lin and Su classified A$ \mathcal{T} $ \mathcal{T} -algebras of real rank zero. This class includes all A$ \mathbb{T} $ \mathbb{T} -algebras of real rank zero as well as many C*-algebras which are not stably finite. An A$ \mathcal{T} $ \mathcal{T} -algebra often becomes an extension of an A$ \mathbb{T} $ \mathbb{T} -algebra by an AF-algebra. In this paper, we show that there is an essential extension of an A$ \mathbb{T} $ \mathbb{T} -algebra by an AF-algebra which is not an A$ \mathcal{T} $ \mathcal{T} -algebra. We describe a characterization of an extension E of an A$ \mathbb{T} $ \mathbb{T} -algebra by an AF-algebra if E is an A$ \mathcal{T} $ \mathcal{T} -algebra.     

Finite rank operators in Jacobson radical \mathcal{R}_{\mathcal{N} \otimes \mathcal{M}}

In this paper we investigate finite rank operators in the Jacobson radical of Alg( ), where are nests. Based on the concrete characterizations of rank one operators in Alg( ) and , we obtain that each finite rank operator in can be written as a finite sum of rank one operators in and the weak closure of equals Alg( ) if and only if at least one of is continuous.     

Super quasiperiodic wave solutions and asymptotic analysis for $$ \mathcal{N} = 1 $$ supersymmetric KdV-type equations

Based on a general multidimensional Riemann theta function and the super Hirota bilinear form, we extend the Hirota method to construct explicit super quasiperiodic (multiperiodic) wave solutions of $ \mathcal{N} = 1 $ \mathcal{N} = 1 supersymmetric KdV-type equations in superspace. We show that the supersymmetric KdV equation does not have an N-periodic wave solution with arbitrary parameters for N ≥ 2. In addition, an interesting influencing band occurs among the super quasiperiodic waves under the presence of a Grassmann variable. We also observe that the super quasiperiodic waves are symmetric about this band but collapse along with it. We present a limit procedure for analyzing the asymptotic properties of the super quasiperiodic waves and rigorously show that the super periodic wave solutions tend to super soliton solutions under some “small amplitude” limits.     

On $$ \mathfrak{F}_h $$-normal subgroups of finite groups

Let G be a finite group, and let $ \mathfrak{F} $ \mathfrak{F} be a formation of finite groups. We say that a subgroup H of G is $ \mathfrak{F}_h $ \mathfrak{F}_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 $ \mathfrak{F} $ \mathfrak{F} -hypercenter $ Z_\infty ^\mathfrak{F} $ Z_\infty ^\mathfrak{F} (G/H _G ) of G/H _G . In this paper, we obtain some results about the $ \mathfrak{F}_h $ \mathfrak{F}_h -normal subgroups and then use them to study the structure of finite groups.     

Phase portraits for quadratic homogeneous polynomial vector fields on $$ \mathbb{S}^2 $$

Let X be a homogeneous polynomial vector field of degree 2 on $ \mathbb{S}^2 $ \mathbb{S}^2 . We show that if X has at least a non-hyperbolic singularity, then it has no limit cycles. We give necessary and sufficient conditions for determining if a singularity of X on $ \mathbb{S}^2 $ \mathbb{S}^2 is a center and we characterize the global phase portrait of X modulo limit cycles. We also study the Hopf bifurcation of X and we reduce the 16 ^th Hilbert’s problem restricted to this class of polynomial vector fields to the study of two particular families. Moreover, we present two criteria for studying the nonexistence of periodic orbits for homogeneous polynomial vector fields on $ \mathbb{S}^2 $ \mathbb{S}^2 of degree n.     

Distortio n theorems o n the Lie ball $$ \mathcal{R}_{IV} $$ ( n) i n ℂ n

In this paper, we introduce the subfamilies H _m ($ \mathcal{R}_{IV} $ \mathcal{R}_{IV} (n)) of holomorphic mappings defined on the Lie ball $ \mathcal{R}_{IV} $ \mathcal{R}_{IV} (n) which reduce to the family of holomorphic mappings and the family of locally biholomorphic mappings when m = 1 and m → +∞, respectively. Various distortion theorems for holomophic mappings H _m ($ \mathcal{R}_{IV} $ \mathcal{R}_{IV} (n)) are established. The distortion theorems coincide with Liu and Minda’s as the special case of the unit disk. When m = 1 and m → +∞, the distortion theorems reduce to the results obtained by Gong for $ \mathcal{R}_{IV} $ \mathcal{R}_{IV} (n), respectively. Moreover, our method is different. As an application, the bounds for Bloch constants of H _m ($ \mathcal{R}_{IV} $ \mathcal{R}_{IV} (n)) are given.     

On the 2 k-th power mean of $$ \frac{{L'}} {L}(1,\chi ) $$ with the weight of Gauss sums

The main purpose of this paper is to study the hybrid mean value of $ \frac{{L'}} {L}(1,\chi ) $ \frac{{L'}} {L}(1,\chi ) and Gauss sums by using the estimates for trigonometric sums as well as the analytic method. An asymptotic formula for the hybrid mean value $ \sum\limits_{\chi \ne \chi _0 } {|\tau (\chi )||\frac{{L'}} {L}(1,\chi )|^{2k} } $ \sum\limits_{\chi \ne \chi _0 } {|\tau (\chi )||\frac{{L'}} {L}(1,\chi )|^{2k} } of $ \frac{{L'}} {L} $ \frac{{L'}} {L} and Gauss sums will be proved using analytic methods and estimates for trigonometric sums.     

Fourier-Feynman transforms of unbounded functionals on abstract Wiener space

Huffman, Park and Skoug established several results involving Fourier-Feynman transform and convolution for functionals in a Banach algebra S on the classical Wiener space. Chang, Kim and Yoo extended these results to abstract Wiener space for a more generalized Fresnel class $ \mathcal{F}_{\mathcal{A}_1 ,\mathcal{A}_2 } $ \mathcal{F}_{\mathcal{A}_1 ,\mathcal{A}_2 } _A1,A2 than the Fresnel class $ \mathcal{F} $ \mathcal{F} (B)which corresponds to the Banach algebra S. In this paper we study Fourier-Feynman transform, convolution and first variation of unbounded functionals on abstract Wiener space having the form

A generalization of Mathieu subspaces to modules of associative algebras

We first propose a generalization of the notion of Mathieu subspaces of associative algebras $ \mathcal{A} $ \mathcal{A} , which was introduced recently in [Zhao W., Generalizations of the image conjecture and the Mathieu conjecture, J. Pure Appl. Algebra, 2010, 214(7), 1200–1216] and [Zhao W., Mathieu subspaces of associative algebras], to $ \mathcal{A} $ \mathcal{A} -modules $ \mathcal{M} $ \mathcal{M} . The newly introduced notion in a certain sense also generalizes the notion of submodules. Related with this new notion, we also introduce the sets σ(N) and τ(N) of stable elements and quasi-stable elements, respectively, for all R-subspaces N of $ \mathcal{A} $ \mathcal{A} -modules $ \mathcal{M} $ \mathcal{M} , where R is the base ring of $ \mathcal{A} $ \mathcal{A} . We then prove some general properties of the sets σ(N) and τ(N). Furthermore, examples from certain modules of the quasi-stable algebras [Zhao W., Mathieu subspaces of associative algebras], matrix algebras over fields and polynomial algebras are also studied.     

On the shift semigroup on the Hardy space of Dirichlet series

We develop a Wold decomposition for the shift semigroup on the Hardy space $ \mathcal{H}^2 $ \mathcal{H}^2 of square summable Dirichlet series convergent in the half-plane $ \Re (s) > 1/2 $ \Re (s) > 1/2 . As an application we have that a shift invariant subspace of $ \mathcal{H}^2 $ \mathcal{H}^2 is unitarily equivalent to $ \mathcal{H}^2 $ \mathcal{H}^2 if and only if it has the form $ \phi \mathcal{H}^2 $ \phi \mathcal{H}^2 for some $ \mathcal{H}^2 $ \mathcal{H}^2 -inner function φ.     

Explicit sum-product theorems for large subsets of $$ \mathbb{F} $$p

In this note, we use ‘classical’ methods to obtain sum-product theorems for subsets A⊂$ \mathbb{F} $ \mathbb{F} _p .     

On a semilattice of numberings

We study some properties of a $ \mathfrak{c} $ \mathfrak{c} -universal semilattice $ \mathfrak{A} $ \mathfrak{A} with the cardinality of the continuum, i.e., of an upper semilattice of m-degrees. In particular, it is shown that the quotient semilattice of such a semilattice modulo any countable ideal will be also $ \mathfrak{c} $ \mathfrak{c} -universal. In addition, there exists an isomorphism $ \mathfrak{A} $ \mathfrak{A} such that $ {\mathfrak{A} \mathord{\left/ {\vphantom {\mathfrak{A} {\iota \left( \mathfrak{A} \right)}}} \right. \kern-\nulldelimiterspace} {\iota \left( \mathfrak{A} \right)}} $ {\mathfrak{A} \mathord{\left/ {\vphantom {\mathfrak{A} {\iota \left( \mathfrak{A} \right)}}} \right. \kern-\nulldelimiterspace} {\iota \left( \mathfrak{A} \right)}} will be also $ \mathfrak{c} $ \mathfrak{c} -universal. Furthermore, a property of the group of its automorphisms is obtained. To study properties of this semilattice, the technique and methods of admissible sets are used. More exactly, it is shown that the semilattice of mΣ-degrees $ L_{m\Sigma }^{\mathbb{H}\mathbb{F}\left( S \right)} $ L_{m\Sigma }^{\mathbb{H}\mathbb{F}\left( S \right)} on the hereditarily finite superstructure $ \mathbb{H}\mathbb{F} $ \mathbb{H}\mathbb{F} (S) over a countable set S will be a $ \mathfrak{c} $ \mathfrak{c} -universal semilattice with the cardinality of the continuum.     

Aspects of the Segre variety $${\mathcal{S}_{1,1,1}(2)}$$

We consider various aspects of the Segre variety \({\mathcal{S}:=\mathcal{S} _{1,1,1}(2)}\) in PG(7, 2), whose stabilizer group \({\mathcal{G}_{\mathcal{S}}<{\rm GL}(8,2)}\) has the structure \({\mathcal{N}\rtimes{\rm Sym}(3),}\) where \({\mathcal{N} :={\rm GL}(2,2)\times{\rm GL}(2,2)\times{\rm GL} (2,2).}\) In particular we prove that \({\mathcal{S}}\) determines a distinguished Z ₃-subgroup \({\mathcal{Z}<{\rm GL}(8,2)}\) such that \({A\mathcal{Z}A^{-1}=\mathcal{Z},}\) for all \({A\in\mathcal{G}_{\mathcal{S}},}\) and in consequence \({\mathcal{S}}\) determines a \({\mathcal{G}_{\mathcal{S}}}\)-invariant spread of 85 lines in PG(7, 2). Furthermore we see that Segre varieties \({\mathcal{S}_{1,1,1}(2)}\) in PG(7, 2) come along in triplets \({\{\mathcal{S},\mathcal{S}^{\prime},\mathcal{S}^{\prime\prime}\}}\) which share the same distinguished Z ₃-subgroup \({\mathcal{Z}<{\rm GL}(8,2).}\) We conclude by determining all fifteen \({\mathcal{G}_{\mathcal{S}}}\)-invariant polynomial functions on PG(7, 2) which have degree < 8, and their relation to the five \({\mathcal{G}_{\mathcal{S}}}\)-orbits of points in PG(7, 2).     

$\mathcal{P}$-kernel normal systems for $\mathcal{P}$-inversive semigroups

As a generalization of Preston’s kernel normal systems, P\mathcal{P}-kernel normal systems for P\mathcal{P}-inversive semigroups are introduced, and strongly regular P\mathcal{P}-congruences on P\mathcal{P}-inversive semigroups in terms of their P\mathcal{P}-kernel normal systems are characterized. These results generalize the corresponding results for P\mathcal{P}-regular semigroups and P\mathcal{P}-inversive semigroups.     

One-loop effective action in the $$ \mathcal{N} $$=2 supersymmetric massive Yang-Mills field theory

We consider the =2 supersymmetric massive Yang-Mills field theory formulated in the =2 harmonic superspace. We present various gauge-invariant forms of writing the mass term in the action (in particular, using the Stueckelberg superfield), which result in dual formulations of the theory. We develop a gaugeinvariant and explicitly supersymmetric scheme of the loop expansion of the superfield effective action beyond the mass shell. In the framework of this scheme, we calculate gauge-invariant and explicitly =2 supersymmetric one-loop counterterms including new counterterms depending on the Stueckelberg superfield. We analyze the component structure of one of these counterterms. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 157, No. 1, pp. 22–40, October, 2008.     

On ideals and quotients of A \mathcal{T} -algebras

Some results on A -algebras are given. We study the problem when ideals, quotients and hereditary subalgebras of A -algebras are A -algebras or A -algebras, and give a necessary and sufficient condition of a hereditary subalgebra of an A -algebra being an A -algebra.     

On a bipartition problem of Bollobás and Scott

The bipartite density of a graph G is max {|E(H)|/|E(G)|: H is a bipartite subgraph of G}. It is NP-hard to determine the bipartite density of any triangle-free cubic graph. A biased maximum bipartite subgraph of a graph G is a bipartite subgraph of G with the maximum number of edges such that one of its partite sets is independent in G. Let $ \mathcal{H} $ \mathcal{H} denote the collection of all connected cubic graphs which have bipartite density $ \tfrac{4} {5} $ \tfrac{4} {5} and contain biased maximum bipartite subgraphs. Bollobás and Scott asked which cubic graphs belong to $ \mathcal{H} $ \mathcal{H} . This same problem was also proposed by Malle in 1982. We show that any graph in $ \mathcal{H} $ \mathcal{H} can be reduced, through a sequence of three types of operations, to a member of a well characterized class. As a consequence, we give an algorithm that decides whether a given graph G belongs to $ \mathcal{H} $ \mathcal{H} . Our algorithm runs in polynomial time, provided that G has a constant number of triangles that are not blocks of G and do not share edges with any other triangles in G.