On generalized k-syzygy modules

In this paper, we first introduce the notion of generalized k-syzygy modules, and then give an equivalent characterization that the class of generalized k-syzygy modules coincides with that ofω-k-torsionfree modules. We further study the extension closure of the category consisting of generalized k-syzygy modules. Some known results are obtained as corollaries.     

The generalized Rédei-matrix

Let F be a number field with odd class number, and let E be a quadratic extension of F. Our main aim is to prove that the 4-rank of the class group C(E) of E is equal to m − 1 − rank R _E/F , where m is the number of primes of F ramifying in E, R _E/F is the generalized Rédei-matrix of local Hilbert symbols with coefficients in and the rank is the rank over . We determine the generalized Rédei-matrices R _E/F explicitly for biquadratic number fields E. The research is partly supported by NNSF of China (No. 10371054, No. 10771100) and the Morningside Center of Mathematics in Beijing (MCM).     

Classes of generalized ∗-modules

We study the class, STAR , of all ?-modules by means of the classes, S_λ, of all ?λ-modules, λ> 0 being a cardinal. Since STAR, equals the intersection of the decreasing chain S_λ, λ > 0, our approach ‘from above’ complements the usual approach ‘from below’ consisting in the study of quasi-progenerators and tilting modules. We present relations between categorical properties of ?-;modules and those of ?-modules. Answering a question of Menini, we use the solution of the Artin's problem to show that the chain Sλ,λ > No, is strictly decreasing in general.     

Pillai’s generalized function

The distribution of values of Pillai’s function over the ring of Gauss integers $ \mathbb{Z}\left[ i \right] $ is studied. The asymptotic formulas for the summators $ {\varSigma_{{N\left( \alpha \right)\leq x}}}\frac{{g\left( \alpha \right)}}{{{N^{\alpha }}\left( \alpha \right)}} $ , where $ a\in \mathbb{R} $ , are found.     

A fiducial argument for generalized p-value

This paper provides a general method for constructing generalized p-value via the fiducial inference.Furthermore,the power properties of the generalized test are discussed.As illustrations, the two-parameter exponential distribution and unbalanced two-fold nested design are researched.It is shown that the resulting generalized p-values are of good frequency property.     

Effective stability for generalized Hamiltonian systems

An effective stability result for generalized Hamiltonian systems is obtained by applying the simultaneous approximation technique due to Lochak. Among these systems, dimensions of action variables and angle variables might be distinct.     

On generalized Rubel’s equation

We solve generalized the generalized Rubel equation on the space of analytic functions in domains.     

Multivariate generalized Ornstein–Uhlenbeck processes

De Haan and Karandikar (1989) [7] introduced generalized Ornstein–Uhlenbeck processes as one-dimensional processes _(Vt)t≥0${\left(V_t\right)}_{t\ge 0}$ which are basically characterized by the fact that for each h>0$h>0$ the equidistantly sampled process _(Vnh)n∈N0${\left(V_{nh}\right)}_{n\in {\mathbb{N}}_0}$ satisfies the random recurrence equation _Vnh=_{A(n−1)h,nhV(n−1)h}+_B(n−1)h,nh$V_{nh}=A_{(n-1)h,nh}{V}_{(n-1)h}+B_{(n-1)h,nh}$, n∈N$n\in \mathbb{N}$, where _{(A(n−1)h,nh,B(n−1)h,nh)n∈N}${(A_{(n-1)h,nh},B_{(n-1)h,nh})}_{n\in \mathbb{N}}$ is an i.i.d. sequence with positive _A0,h$A_{0,h}$ for each h>0$h>0$. We generalize this concept to a multivariate setting and use it to define multivariate generalized Ornstein–Uhlenbeck (MGOU) processes which occur to be characterized by a starting random variable and some Lévy process (X,Y)$(X,Y)$ in R^m×m×R^m${\mathbb{R}}^{m\times m}\times {\mathbb{R}}^m$. The stochastic differential equation an MGOU process satisfies is also derived. We further study invariant subspaces and irreducibility of the models generated by MGOU processes and use this to give necessary and sufficient conditions for the existence of strictly stationary MGOU processes under some extra conditions.     

μ-Separations in generalized topological spaces

This paper takes some investigations on generalized topological spaces with some μ- separations. Some characterizations of μTi-spaces for i = 0,1,2,3,4, μTD-spaces and μR0-spaces are obtained and some relations among these spaces are established.     

Lie bialgebras of generalized Witt type

In this paper, all Lie bialgebra structures on the Lie algebras of generalized Witt type are considered. It is proved that, for any Lie algebra W of generalized Witt type, all Lie bialgebras on W are the coboundary triangular Lie bialgebras. As a by-product, it is also proved that the first cohomology group H1(W, W(?)W) is trivial.     

ζ μ -sets in generalized topological spaces

We define ζ _μ -sets, (ζ,μ)-closed sets and generalized ζ _μ -sets in a generalized topological space and investigate properties of several low separation axioms of generalized topologies constructed by the families of these sets. Characterizations of some properties of (ζ,μ)-R ₀ and (ζ,μ)-R ₁ generalized topological spaces will be given.     

Fuzzy generalized Bi-ideals of Γ-semigroups

In this paper, we introduce the concept of fuzzy generalized bi-ideal of a ??-semigroup, which is an extension of the concept of a fuzzy bi-ideal of a ??-semigroup and characterize regular ??-semigroups in terms of fuzzy generalized biideals.     

Counting generalized Jenkins–Strebel differentials

We study the combinatorial geometry of “lattice” Jenkins–Strebel differentials with simple zeroes and simple poles on $\mathbb{C }\!\mathrm{P }^1$ and of the corresponding counting functions. Developing the results of Kontsevich (Commun Math Phys 147:1–23, 1992) we evaluate the leading term of the symmetric polynomial counting the number of such “lattice” Jenkins–Strebel differentials having all zeroes on a single singular layer. This allows us to express the number of general “lattice” Jenkins–Strebel differentials as an appropriate weighted sum over decorated trees. The problem of counting Jenkins–Strebel differentials is equivalent to the problem of counting pillowcase covers, which serve as integer points in appropriate local coordinates on strata of moduli spaces of meromorphic quadratic differentials. This allows us to relate our counting problem to calculations of volumes of these strata . A very explicit expression for the volume of any stratum of meromorphic quadratic differentials recently obtained by the authors (Athreya et al. 2012) leads to an interesting combinatorial identity for our sums over trees.     

Factorization in generalized Calogero–Moser spaces

Using a recent construction of Bezrukavnikov and Etingof, [R. Bezrukavnikov, P. Etingof, Induction and restriction functors for rational Cherednik algebras, arXiv: 0803.3639], we prove that there is a factorization of the Etingof–Ginzburg sheaf on the generalized Calogero–Moser space associated to a complex reflection group. In the case $W=S_n$, this confirms a conjecture of Etingof and Ginzburg, [P. Etingof, V. Ginzburg, Symplectic reflection algebras, Calogero–Moser space, and deformed Harish-Chandra homomorphisms, Invent. Math 147 (2002) 243–348].     

A comment on generalized αβ-closed sets

In this note, we will show that the notions of generalized alpha b-closed (gαb-closed) sets and b-closed set are equivalent.