On the $${{\varvec{}}}$$-length of the mutually ermutable roduct of two $${{\varvec{}}}$$-soluble grous

Let a finite group \(G=AB\) be the mutually permutable product of two p-soluble subgroups A and B for some prime p. We give a bound of the p-length of G from the p-lengths of A and B.     

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.     

Null extensions of the bicyclic semigroup {\varvec{\mathcal {B}}} are inflations of {\varvec{\mathcal {B}}}

We prove that a null extension of the bicyclic semigroup is an inflation of it.     

Unirationality of the Hurwitz space $$\varvec{\mathcal {H}_{9,8}}$$

In this paper we prove that the Hurwitz space \(\mathcal {H}_{9,8}\), which parameterizes 8-sheeted covers of \({\mathbb P }^1\) by curves of genus 9, is unirational. Our construction leads to an explicit Macaulay2 code, which will randomly produce a nodal curve of degree 8 of geometric genus 9 with 12 double points and together with a pencil of degree 8.     

$${\varvec{}}$$-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.     

Large $${\varvec{p}}^{\prime }$$-orbits for $${\varvec{p}}$$p-nilpotent linear groups

Let G be a p-nilpotent linear group on a finite vector space V of characteristic p. Suppose that |G||V| is odd. Let P be a Sylow p-subgroup of G. We show that there exist vectors \(v_1\) and \(v_2\) in V such that \(C_G(v_1) \cap C_G(v_2)=P\). A striking conjecture of Malle and Navarro offers a simple global criterion for the nilpotence (in the sense of Broué and Puig) of a p-block of a finite group. Our result implies that this conjecture holds for groups of odd order.     

Properties of singular integral operators $$\varvec{S}_{\varvec{\alpha ,\beta }}$$

For \(\alpha , \beta \in L^{\infty } (S^1),\) the singular integral operator \(S_{\alpha ,\beta }\) on \(L^2 (S^1)\) is defined by \(S_{\alpha ,\beta }f:= \alpha Pf+\beta Qf\), where P denotes the orthogonal projection of \(L^2(S^1)\) onto the Hardy space \(H^2(S^1),\) and Q denotes the orthogonal projection onto \(H^2(S^1)^{\perp }\). In a recent paper, Nakazi and Yamamoto have studied the normality and self-adjointness of \(S_{\alpha ,\beta }\). This work has shown that \(S_{\alpha ,\beta }\) may have analogous properties to that of the Toeplitz operator. In this paper, we study several other properties of \(S_{\alpha ,\beta }\).     

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}} .     

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 \(m,n\ge 1\).     

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.
    

     

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.     

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.     

Hypersurfaces in nearly Kaehler manifold $$\pmb {\mathbb {S}}^{\varvec{3}}\varvec{\times } \pmb {\mathbb {S}}^{\varvec{3}}$$

In this paper, we initiate the study of contact and minimal hypersurfaces in nearly Kaehler manifold \({\mathbb {S}}^3\times {\mathbb {S}}^3\) with a conformal vector field. There are three almost contact metric structures on a hypersurface of \({\mathbb {S}}^3\times {\mathbb {S}}^3\), and we will give some important properties of them. Besides, we study the influence of the conformal vector field on the almost contact metric structures and use it to characterize the hypersurfaces in \({\mathbb {S}}^3\times {\mathbb {S}}^3\).     

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 .      

On Semi-Slant $${\varvec{\xi ^\perp }}$$-Riemannian Submersions

The aim of the present paper is to define and study semi-slant \(\xi ^\perp \)-Riemannian submersions from Sasakian manifolds onto Riemannian manifolds as a generalization of anti-invariant \(\xi ^\perp \)-Riemannian submersions, semi-invariant \(\xi ^\perp \)-Riemannian submersions and slant Riemannian submersions. We obtain characterizations, investigate the geometry of foliations which arise from the definition of this new submersion. After we investigate the geometry of foliations, we obtain necessary and sufficient condition for base manifold to be a locally product manifold and proving new conditions to be totally umbilical and totally geodesicness, respectively. Moreover, some examples of such submersions are mentioned.     

$${\mathcal{C}}_{0}$$ ( 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     

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.     

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.     

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.