Cofiniteness with respect to a Serre subcategory

Let Φ be a system of ideals in a commutative Noetherian ring R, and let be a Serre subcategory of R-modules. We set $$ H_\Phi ^i ( \cdot , \cdot ) = \mathop {\lim }\limits_{\overrightarrow {\mathfrak{b} \in \Phi } } Ext_R^i (R/\mathfrak{b}| \otimes R \cdot , \cdot ). $$ . Suppose that a is an ideal of R, and M and N are two R-modules such that M is finitely generated and N ∈ . It is shown that if the functor $ D_\Phi ( \cdot ) = \mathop {\lim }\limits_{\overrightarrow {\mathfrak{b} \in \Phi } } Hom_R (\mathfrak{b}, \cdot ) $ is exact, then, for any $ \mathfrak{b} \in \Phi ,Ext_R^j (R/\mathfrak{b},H_\Phi ^i (M,N)) $ ∈ for all i, j ≥ 0. It is also proved that if there is a nonnegative integer t such that $ H_\mathfrak{a}^i (M,N) $ ∈ for all i < t, then $ Hom_R (R/\mathfrak{a},H_\mathfrak{a}^t (M,N)) $ ∈ , provided that is contained in the class of weakly LaskerianR-modules. Finally, it is shown that if L is an R-module and t is the infimum of the integers i such that $ H_\mathfrak{a}^i (L) $ ∈ , then $ Ext_R^j (R/\mathfrak{a},H_\mathfrak{a}^t (M,L)) $ ∈ if and only if $ Ext_R^j (R/\mathfrak{a},Hom_R (M,H_\mathfrak{a}^t (L))) $ ∈ for all j ≥ 0.     

Martindale rings and H-module algebras with invariant characteristic polynomials

Under study is the category of the possibly noncommutative H-module algebras that are mapped homomorphically onto commutative algebras. The H-equivariant Martindale ring of quotients Q _H (A) is shown to be a finite-dimensional Frobenius algebra over the subfield of invariant elements Q _H (A) ^H and also the classical ring of quotients for A. We introduce a full subcategory of such that the algebras in are integral over its subalgebras of invariants and construct a functor ?? , which is left adjoined to the inclusion ?? .     

Continuous embeddings of Besov-Morrey function spaces

We study embeddings of spaces of Besov-Morrey type, M Bp1,q1s1,r1(Rd ) → M Bp2 ,q2s2 ,r2 (R d ), and obtain necessary and sufficient conditions for this. Moreover, we can also characterise the special weighted situation Bp1 ,r1s1 (R d , w) → M Bp2 ,q2s2 ,r2 (Rd ) for a Muckenhoupt A ∞ weight w, with wα(x) = |x|α , α -d1, as a typical example.     

Convolution functions of several variables with generalized bounded variation

Let L ¹ be the class of all complex-valued functions, with period 2π in each variable, in the space , where $\mathbb{T} = [0,2\pi )$ is the one-dimensional torus. Here, it is observed that L ¹ * E ? E for E = Lip(p; α ₁, α ₂, ..., α _N ) over , for , for , and for in the sense of Vitali as well as Hardy.     

Coefficient characterizations and sections for some univalent functions

Let (α) denote the class of locally univalent normalized analytic functions f in the unit disk |z| < 1 satisfying the condition $Re\left( {1 + \frac{{zf''(z)}} {{f'(z)}}} \right) < 1 + \frac{\alpha } {2}for|z| < 1 $ and for some 0 < α ≤ 1. We firstly prove sharp coefficient bounds for the moduli of the Taylor coefficients a _n of f ∈ (α). Secondly, we determine the sharp bound for the Fekete-Szegö functional for functions in (α) with complex parameter λ. Thirdly, we present a convolution characterization for functions f belonging to (α) and as a consequence we obtain a number of sufficient coefficient conditions for f to belong to (α). Finally, we discuss the close-to-convexity and starlikeness of partial sums of f ∈ (α). In particular, each partial sum s _n (z) of f ∈ (1) is starlike in the disk |z| ≤ 1/2 for n ≥ 11. Moreover, for f ∈ (1), we also have Re(s′ _n (z)) > 0 in |z| ≤ 1/2 for n ≥ 11.     

Special framed Morse functions on surfaces

Let M be a smooth closed orientable surface. Let F be the space of Morse functions on M and $\mathbb{F}^1$ be the space of framed Morse functions both endowed with the C ^??-topology. The space $\mathbb{F}^0$ of special framed Morse functions is defined. We prove that the inclusion mapping is a homotopy equivalence. In the case when at least x(M) + 1 critical points of each function of F are marked, the homotopy equivalences and are proved, where is the complex of framed Morse functions, is the universal moduli space of framed Morse functions, is the group of self-diffeomorphisms of M homotopic to the identity.     

On the class of groups with pronormal hall π-subgroups

Given a set π of prime numbers, we define the class of all finite groups in which Hall π-subgroups exist and are pronormal by analogy with the Hall classes , , and . We study whether is closed under the main class-theoretic closure operations. In particular, we establish that is a saturated formation.     

Maximal function on the dual of Laguerre hypergroup

In this paper, we are interested in the Laguerre hypergroup $\mathbb{K} = [0,\infty ) \times \mathbb{R}$ which is the fundamental manifold of the radial function space for the Heisenberg group. So, we consider the generalized shift operator generated by the dual of the Laguerre hypergroup which can be topologically identified with the so-called Heisenberg fan, the subset of ?²: $$\bigcup\limits_{j \in \mathbb{N}} {\left\{ {(\lambda ,\mu ) \in \mathbb{R}^2 :\mu = \left| \lambda \right|(2j + \alpha + 1),\lambda \ne 0} \right\} \cup \left\{ {(0,\mu ) \in \mathbb{R}^2 :\mu \geqslant 0} \right\}} ,$$ by means of which the maximal function is investigated. For 1 < p ?? ??, the L _p ( )-boundedness and weak L ₁( )-boundedness result for the maximal function is obtained.     

Variations on ω-boundedness

Let be a property (or, equivalently, a class) of topological spaces. A space X is called -bounded if every subspace of X with (or in) has compact closure. Thus, countable-bounded has been known as ω-bounded and (σ-compact)-bounded as strongly ω-bounded. In this paper we present a systematic study of the interrelations of these two known “boundedness” concepts with -boundedness where is one of the further countability properties weakly Lindelöf, Lindelöf, hereditarily Lindelöf, and ccc.     

Associative n-Tuple algebras

In the paper,we study algebras having n bilinearmultiplication operations : A×A → A, s = 1, …, n, such that (a b) c = a (b c), s, r = 1,..., n, a, b, c ∈ A. The radical of such an algebra is defined as the intersection of the annihilators of irreducible A-modules, and it is proved that the radical coincides with the intersection of the maximal right ideals each of which is s-regular for some operation . This implies that the quotient algebra by the radical is semisimple. If an n-tuple algebra is Artinian, then the radical is nilpotent, and the semisimple Artinian n-tuple algebra is the direct sum of two-sided ideals each of which is a simple algebra. Moreover, in terms of sandwich algebras, we describe a finite-dimensional n-tuple algebra A, over an algebraically closed field, which is a simple A-module.     

Trace series on

Let K be a number field which contains the absolute Galois group of K. Let be the completion of relative to the spectral norm . A trace series and a trace function are associated to an element . A measure preserving function H:G{inK}→[0,1] is constructed and a Fourier type theory is considered for elements of .     

Characterization of compact quantum group

Given a C^＊-algebra A and a comultiplication Ф on A, we show that the pair （A, Ф） is a compact quantum group if and only if the associated multiplier Hopf ^＊-algebra （A, ФA） is a compact Hopf ^＊-algebra.     

Topological spaces compact with respect to a set of filters

If is a family of filters over some set I, a topological space X is sequencewise -compact if for every I-indexed sequence of elements of X there is such that the sequence has an F-limit point. Countable compactness, sequential compactness, initial κ-compactness, [λ; µ]-compactness, the Menger and Rothberger properties can all be expressed in terms of sequencewise -compactness for appropriate choices of . We show that sequencewise -compactness is preserved under taking products if and only if there is a filter such that sequencewise -compactness is equivalent to F-compactness. If this is the case, and there exists a sequencewise -compact T ₁ topological space with more than one point, then F is necessarily an ultrafilter. The particular case of sequential compactness is analyzed in detail.     

On pointwise estimate for Gamma operators

For linear combinations of Gamma operators, if 0＜a＜2r, 1/2-1/2r≤λ≤l, or 0≤λ＜1/2-1/2r(r≥2),0＜a＜r+10＜a<(r+1)/1-λ, we obtain an equivalent theorem with ωuλρ(f ,t) instead of ωrλφ(f,t), where ωuφ(f,t) is theDitzian-Totik moduli of smoothness.     

Variational method for precise asymptotic formulas for nonlinear eigenvalue problems

We consider the nonlinear eigenvalue problem motivated by the perturbed elliptic sine-Gordon equation where p >1 is a constant and λ ∈ R is an eigenvalue parameter. Our aim is to clarify the asymptotic relationship between L ^p+1-norm, L ^∞-norm of the solutions and λ when these norms are large. To this end, we consider an associated variational problem with this equation on a manifold new parameter), and obtain a solution pair . We establish the precise asymptotic formulas for .     

Toral rank conjecture for moment-angle complexes

We consider an operation K ? L(K) on the set of simplicial complexes, which we call the “doubling operation.” This combinatorial operation was recently introduced in toric topology in an unpublished paper of Bahri, Bendersky, Cohen and Gitler on generalized moment-angle complexes (also known as K-powers). The main property of the doubling operation is that the moment-angle complex can be identified with the real moment-angle complex for the double L(K). By way of application, we prove the toral rank conjecture for the spaces by providing a lower bound for the rank of the cohomology ring of the real moment-angle complexes . This paper can be viewed as a continuation of the author’s previous paper, where the doubling operation for polytopes was used to prove the toral rank conjecture for moment-angle manifolds.     

On minimal non- -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.     

Upper estimates for the approximation numbers of the generalized Laplace transform

A Laplace-transform-type operator acting in the Lebesgue spaces of real functions on the half-axis is considered. Sufficient conditions under which belongs to some Schattentype classes are found. Upper asymptotic estimates for the approximation numbers of are obtained.     

Peierls substitution and the Maslov operator method

We consider a periodic Schrödinger operator in a constant magnetic field with vector potential A(x). A version of adiabatic approximation for quantum mechanical equations with rapidly varying electric potentials and weak magnetic fields is the Peierls substitution which, in appropriate dimensionless variables, permits writing the pseudodifferential equation for the new auxiliary function: , where is the corresponding energy level of some auxiliary Schrödinger operator, assumed to be nondegenerate, and µ is a small parameter. In the present paper, we use V. P. Maslov’s operator method to show that, in the case of a constant magnetic field, such a reduction in any perturbation order leads to the equation with the operator represented as a function depending only on the operators of kinetic momenta $ \hat P_j = - i\mu \partial _{x_j } + A_j \left( x \right) $ .