Relations between the $${\mathcal {}}$$-ultrafilters

Under CH we show the following results:

1. (1)
   There is a discrete ultrafilter which is not a \({\mathcal {Z}}_{0}\)-ultrafilter.
    
2. (2)
   There is a \(\sigma \)-compact ultrafilter which is not a \({\mathcal {Z}}_{0}\)-ultrafilter.
    
3. (3)
   There is a \({\mathcal {J}}_{\omega ^{3}}\)-ultrafilter which is not a \({\mathcal {Z}}_{0}\)-ultrafilter.
    

     

M-$${\mathcal{}}$$-Injective (Flat) and Strongly $${\mathcal{B}}$$B-Injective (Flat) Modules

Let M be a left R-module, \({\mathcal{A}}\)be a family of some submodules of M and \({\mathcal{B}}\)be a family of some left R-modules. In this article, we introduce and characterize \({\mathcal{A}}\)-coherent, \({P\mathcal{A}}\), \({F\mathcal{A}}\), M-\({\mathcal{A}}\)-injective (flat) and strongly \({\mathcal{B}}\)-injective (flat) modules, which are generalizations of coherent, PS, FS, M-injective (flat) and strongly M-injective modules, respectively. We extend some known results to this general structure.     

$${\varvec{}}$$-Jorda homomorphisms ad pseudo $${\varvec{}}$$-Jorda homomorphisms o Baach algebras

Let \(n\in \mathbb {N}\), A and B be Banach algebras and let B be a right A-module. We say that a linear mapping \(\varphi :A\longrightarrow B\) is a pseudo n-Jordan homomorphism if there exists an element \(w\in A\) such that \(\varphi (a^nw)=\varphi (a)^n\cdot w\), for every \(a\in A\) and \(n\ge \) 2. In this paper, among other things, we show that under some conditions if a linear mapping \(\varphi \) is a (pseudo) n-Jordan homomorphism, then it is a (pseudo) \((n + 1)\)-Jordan homomorphism. Additionally, we investigate automatic continuity of surjective pseudo n-Jordan homomorphisms under some conditions.     

On quasi $${\mathcal{H}}$$ -closed subspaces

In this paper some properties concerning the notion of quasi H{\mathcal{H}} -closedness in topological spaces are obtained. Locally quasi H{\mathcal{H}} -closed spaces are studied. Properties of these types of spaces are investigated, when the subsets involved are α-open, semi-open, clopen, or preopen.     

Homological algebra i $${\varvec{}}$$-abelia categories

In this paper, we study the homological theory in n-abelian categories. First, we prove some useful properties of n-abelian categories, such as \((n+2)\times (n+2)\)-lemma, 5-lemma and n-Horseshoes lemma. Secondly, we introduce the notions of right(left) n-derived functors of left(right) n-exact functors, n-(co)resolutions, and n-homological dimensions of n-abelian categories. For an n-exact sequence, we show that the long n-exact sequence theorem holds as a generalization of the classical long exact sequence theorem. As a generalization of \(\textsf {Ext}^*(-,-)\), we study the n-derived functor \(\textsf {nExt}^*(-,-)\) of hom-functor \(\mathrm {Hom}(-,-)\). We give an isomorphism between the abelian group of equivalent classes of m-fold n-extensions \(\textsf {nE}^m(A,B)\) of A, B and \(\textsf {nExt}_{\mathcal A}^m(A,B)\) using n-Baer sum for \(m,n\ge 1\).     

On the Class of Banach Spaces with James Constant $${\sqrt{}}$$ : Part II

The class of two-dimensional normed spaces with James constant \({\sqrt{2}}\) is studied. It is shown that, if the norm is \({\pi/2}\)-rotation invariant, then its James constant is \({\sqrt{2}}\) if and only if the norm is \({\pi/4}\)-rotation invariant. We also present a characterization of \({\pi/4}\)-rotation invariant norms using some properties of certain convex functions on the unit interval, which allow us to easily construct norms the James constant of which are \({\sqrt{2}}\). Moreover, two important examples are given, which show that neither absoluteness, symmetry nor \({\pi/2}\)-rotation invariance can be a characteristic property of the norms with James constant \({\sqrt{2}}\).     

On $${\mathcal{F}}$$ -supplemented subgroups of finite groups

Let G be a finite group and a formation of finite groups. We say that a subgroup H of G is -supplemented in G if there exists a subgroup T of G such that G = TH and is contained in the -hypercenter of G/H _G . In this paper, we use -supplemented subgroups to study the structure of finite groups. A series of previously known results are unified and generalized. Research of the author is supported by a NNSF grant of China (Grant #10771180).     

Compactified Picard stacks over $${\overline{\mathcal M}_g}$$

We study algebraic (Artin) stacks over [`(M)]<sub>g</sub>{\overline{\mathcal M}_{g}} giving a functorial way of compactifying the relative degree d Picard variety for families of stable curves. We also describe for every d the locus of genus g stable curves over which we get Deligne–Mumford stacks strongly representable over[`(M)]_g{\overline{\mathcal M}_{g}} .     

Some polynomials associated with the $${\vavec{}}$$-Whitney numbes

In the present article, we study three families of polynomials associated with the r-Whitney numbers of the second kind. They are the r-Dowling polynomials, r-Whitney–Fubini polynomials and the r-Eulerian–Fubini polynomials. Then we derive several combinatorial results by using algebraic arguments (Rota’s method), combinatorial arguments (set partitions) and asymptotic methods.     

On a Class of Central Configurations in the Planar $${\varvec{}}$$-Body Problem

The existence of a central configuration of 2n bodies located on two concentric regular n-gons with the polygons which are homotetic or similar with an angle equal to \(\frac{\pi }{n}\) and the masses on the same polygon, are equal, has proved by Elmabsout (C R Acad Sci 312(5):467–472, 1991). Moreover, the existence of a planar central configuration which consists of 3n bodies, also situated on two regular polygons, the interior n-gon with equal masses and the exterior 2n-gon with masses on the 2n-gon alternating, has shown by author. Following Smale (Invent Math 11:45-64, 1970), we reduce this problem to one, concerning the critical points of some effective-type potential. Using computer assisted methods of proof we show the existence of ten classes of such critical points which corresponds to ten classes of central configurations in the planar six-body problem.     

Some stochastic $${\varvec{}}$$-divergences in information theory

In this paper, we introduce the concept of stochastic HH-divergences based on convex stochastic processes. As an application, we propose some inequalities related to stochastic HH-divergences for convex stochastic processes. Our result extends HH-divergence in the class of f-divergence to the class of convex stochastic processes.     

A compehensive study of $${\vavec{}}$$-Dowling polynomials

After his extensive study of Whitney numbers, Benoumhani introduced Dowling numbers and polynomials as generalizations of the well-known Bell numbers and polynomials. Later, Cheon and Jung gave the r-generalization of these notions. Based on our recent combinatorial interpretation of r-Whitney numbers, in this paper we derive several new properties of r-Dowling polynomials and we present alternative proofs of some previously known ones.     

$${\varvec{\mathcal {}}}$$-minimaxness and local cohomology modules

Let R be a commutative Noetherian ring, \({\mathfrak {a}}\) an ideal of R, M a finitely generated R-module, and \({\mathcal {S}}\) a Serre subcategory of the category of R-modules. We introduce the concept of \({\mathcal {S}}\)-minimax R-modules and the notion of the \({\mathcal {S}}\)-finiteness dimension

$$\begin{aligned} f_{\mathfrak {a}}^{{\mathcal {S}}}(M):=\inf \lbrace f_{\mathfrak {a}R_{\mathfrak {p}}}(M_{\mathfrak {p}}) \vert \mathfrak {p}\in {\text {Supp}}_R(M/ \mathfrak {a}M) \text { and } R/\mathfrak {p}\notin {\mathcal {S}} \rbrace \end{aligned}$$

and we will prove that: (i) If \({\text {H}}_{\mathfrak {a}}^{0}(M), \cdots ,{\text {H}}_{\mathfrak {a}}^{n-1}(M)\) are \({\mathcal {S}}\)-minimax, then the set \(\lbrace \mathfrak {p}\in {\text {Ass}}_R( {\text {H}}_{\mathfrak {a}}^{n}(M)) \vert R/\mathfrak {p}\notin {\mathcal {S}}\rbrace \) is finite. This generalizes the main results of Brodmann–Lashgari (Proc Am Math Soc 128(10):2851–2853, 2000), Quy (Proc Am Math Soc 138:1965–1968, 2010), Bahmanpour–Naghipour (Proc Math Soc 136:2359–2363, 2008), Asadollahi–Naghipour (Commun Algebra 43:953–958, 2015), and Mehrvarz et al. (Commun Algebra 43:4860–4872, 2015). (ii) If \({\mathcal {S}}\) satisfies the condition \(C_{\mathfrak {a}}\), then

$$\begin{aligned} f_{\mathfrak {a}}^{{\mathcal {S}}}(M)= \inf \lbrace i\in {\mathbb {N}}_{0} \vert {\text {H}}_{\mathfrak {a}}^{i}(M) \text { is not } {\mathcal {S}}\hbox {-}minimax\rbrace . \end{aligned}$$

This is a formulation of Faltings’ Local-global principle for the \({\mathcal {S}}\)-minimax local cohomology modules. (iii) \( \sup \lbrace i\in {\mathbb {N}}_{0} \vert {\text {H}}_{\mathfrak {a}}^{i}(M) \text { is not } {\mathcal {S}}\text {-minimax} \rbrace = \sup \lbrace i\in {\mathbb {N}}_{0} \vert {\text {H}}_{\mathfrak {a}}^{i}(M) \text { is not in } {\mathcal {S}} \rbrace \).

     

Domain decomposition based $${\mathcal H}$$ -LU preconditioning

Hierarchical matrices provide a data-sparse way to approximate fully populated matrices. The two basic steps in the construction of an -matrix are (a) the hierarchical construction of a matrix block partition, and (b) the blockwise approximation of matrix data by low rank matrices. In this paper, we develop a new approach to construct the necessary partition based on domain decomposition. Compared to standard geometric bisection based -matrices, this new approach yields -LU factorizations of finite element stiffness matrices with significantly improved storage and computational complexity requirements. These rigorously proven and numerically verified improvements result from an -matrix block structure which is naturally suited for parallelization and in which large subblocks of the stiffness matrix remain zero in an LU factorization. We provide numerical results in which a domain decomposition based -LU factorization is used as a preconditioner in the iterative solution of the discrete (three-dimensional) convection-diffusion equation. This work was supported in part by the US Department of Energy under Grant No. DE-FG02-04ER25649 and by the National Science Foundation under grant No. DMS-0408950.     

The rational cohomology of $${\overline{\mathcal{M}_4}}$$

We present two approaches to the study of the cohomology of moduli spaces of curves. Together, they allow us to compute the rational cohomology of the moduli space of stable complex curves of genus 4, with its Hodge structure.     

Characterizing cryptogroups with a finite number of $${\mathcal{H}}$$ -classes in each $${\mathcal{D}}$$ -class by their subsemigroups

We construct a family of completely regular semigroups with the property that each completely regular semigroup S with a finite number of -classes in each -class is non-cryptic if and only if S contains an isomorphic image of a member of . Each member F of is an ideal extension of a Rees matrix semigroup J by a cyclic group B with a zero adjoined and the identity of B is the identity of F. Here with I and Λ finite, G is given by generators and relations, and P is given explicitly. Within completely regular semigroups, the cryptic property is equivalent to where is the natural partial order and a if and only if a ² = ab = ba. Hence the above result can be formulated in terms of and .      

Lie n-derivatio ns o n $ {\mathcal {}}$-subspace lattice algebras

Let \(\mathcal {L}\) be a \(\mathcal {J}\)-subspace lattice on a Banach space X over the real or complex field \(\mathbb {F}\) with dimX ≥ 3 and let n ≥ 2 be an integer. Suppose that dimK ≠ 2 for every \(K\in \mathcal {J}{(\mathcal L)}\) and \(L: \text {Alg}\, \mathcal {L}\rightarrow \text {Alg}\,\mathcal {L}\) is a linear map. It is shown that L satisfies \({\sum }_{i=1}^{n}p_{n} (A_{1}, \ldots , A_{i-1}, L(A_{i}), A_{i+1}, \ldots , A_{n})=0\) whenever p _n (A ₁,A ₂,…,A _n ) = 0 for \(A_{1},A_{2},\ldots ,A_{n}\in \text {Alg}\,\mathcal {L}\) if and only if for each \(K\in \mathcal {J}(\mathcal {L})\), there exists a bounded linear operator \(T_{K}\in \mathcal {B}(K)\), a scalar λ _K and a linear functional \(h_{K}: \text {Alg}\,\mathcal {L}\rightarrow \mathbb {F}\) such that L(A)x = (T _K A ? A T _K + λ _K A + h _K (A)I)x for all x ∈ K and all \(A\in \text {Alg}\,\mathcal {L}\). Based on this result, a complete characterization of linear n-Lie derivations on \(\text {Alg}\,\mathcal {L}\) is obtained.     

Minimal non-$${\mathcal {F}}$$ -groups

Let be a saturated formation. We describe minimal non- -, minimal non- -, and minimal non-metabelian groups. Dedicated to L. A. Shemetkov on the occasion of his seventieth birthday.     

$${\mathcal{C}}_{0}$$ ( X)-algebras,stability and strongly self-absorbing $${\mathcal{C}}^{*}$$ -algebras

We study permanence properties of the classes of stable and so-called -stable -algebras, respectively. More precisely, we show that a (X)-algebra A is stable if all its fibres are, provided that the underlying compact metrizable space X has finite covering dimension or that the Cuntz semigroup of A is almost unperforated (a condition which is automatically satisfied for -algebras absorbing the Jiang–Su algebra tensorially). Furthermore, we prove that if is a K ₁-injective strongly self-absorbing -algebra, then A absorbs tensorially if and only if all its fibres do, again provided that X is finite-dimensional. This latter statement generalizes results of Blanchard and Kirchberg. We also show that the condition on the dimension of X cannot be dropped. Along the way, we obtain a useful characterization of when a -algebra with weakly unperforated Cuntz semigroup is stable, which allows us to show that stability passes to extensions of -absorbing -algebras. Research supported by: Deutsche Forschungsgemeinschaft (through the SFB 478), by the EU-Network Quantum Spaces - Noncommutative Geometry (Contract No. HPRN-CT-2002-00280), and by the Center for Advanced Studies in Mathematics at Ben-Gurion University     

Clones with finitely many relative $${\mathcal {R}}$$ -classes

For each clone C{\mathcal {C}} on a set A there is an associated equivalence relation analogous to Green’s R{\mathcal {R}} -relation, which relates two operations on A if and only if each one is a substitution instance of the other using operations from C{\mathcal {C}} . We study the clones for which there are only finitely many relative R{\mathcal {R}} -classes.