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.     

$${\mathcal{C}}_{0}$$ ( 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     

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 .      

$${\mathcal{M}}$$ -decomposability and symmetric unimodal densities in one dimension

In this paper, we introduce the notion of -decomposability of probability density functions in one dimension. Using -decomposability, we derive an inequality that applies to all symmetric unimodal densities. Our inequality involves only the standard deviation of the densities concerned. The concept of -decomposability can be used as a non-parametric criterion for mode-finding and cluster analysis.     

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.     

Monotone hulls for {\mathcal {N}}\cap {\mathcal {M}}

Using the method of decisive creatures [see Kellner and Shelah (J Symb Log 74:73–104, 2009)] we show the consistency of “there is no increasing \(\omega _2\) –chain of Borel sets and \(\mathrm{non}({\mathcal N})= \mathrm{non}({\mathcal M})=\mathrm{non}({\mathcal N}\cap {\mathcal M})=\omega _2=2^\omega \) ”. Hence, consistently, there are no monotone Borel hulls for the ideal \({\mathcal M}\cap {\mathcal N}\) . This answers Balcerzak and Filipczak (Math Log Q 57:186–193, 2011 [Questions 23, 24]). Next we use finite support iteration of ccc forcing notions to show that there may be monotone Borel hulls for the ideals \({\mathcal M},{\mathcal N}\) even if they are not generated by towers.     

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.     

Analytic Hypoellipticity for a New Class of Sums of Squares of Vector Fields in $${\mathcal {R}}^3$$

This paper is devoted to a substantial generalization of previous work on the analytic hypoellipticity of sums of squares \(P=\sum _1^4X^2_j\) of real vector fields with real analytic coefficient in three variables. For p(x, y) quasi-homogeneous in (x, y), consider the vector fields

$$\begin{aligned} X_1 = \frac{\partial }{\partial x}, \quad X_2=-\frac{\partial }{\partial y} + p(x,y)\frac{\partial }{\partial t}, \quad X_3=x^{n_1}\frac{\partial }{\partial t}, \quad X_4=y^{n_2}\frac{\partial }{\partial t}, \end{aligned}$$

\( n_1, n_2 \ne 0\). We show that the operator

$$\begin{aligned} P=\sum _1^4 X_j^2, \end{aligned}$$

well known to be \(C^\infty \)-hypoelliptic, is actually analytic hypoelliptic near the origin in \({\mathcal {R}}^3\).

     

Subseries of $$ \mathrm{\mathcal{I}}$$-convergent series

We say that an ideal \( \mathrm{\mathcal{I}}\) has property (T) if for every \( \mathrm{\mathcal{I}}\)-convergent series \( {\sum}_{n=1}^{\infty }{x}_n \), there exists a set A ∈ \( \mathrm{\mathcal{I}}\) such that ∑_n?∈??\Ax _n converges in the usual sense. The main aim of this paper is to focus on several different classes of ideals, such as summable ideals, F _σ ideals, and matrix summability ideals, and to show that they do not have the mentioned property.     

The Stieltjes condition and multidimensional $${\mathcal {K}}$$ K -moment problems

Archiv der Mathematik - We prove a solvability theorem for the Stieltjes problem on $$\mathbb {R}^d$$ which is based on the multivariate Stieltjes condition $$\sum _{n=1}^\infty...     

Scaling of conformal blocks and generalized theta functions over $$\overline{\mathcal {M}}_{g,n}$$

By way of intersection theory on \(\overline{\mathcal {M}}_{g,n}\), we show that geometric interpretations for conformal blocks, as sections of ample line bundles over projective varieties, do not have to hold at points on the boundary. We show such a translation would imply certain recursion relations for first Chern classes of these bundles. While recursions can fail, geometric interpretations are shown to hold under certain conditions.     

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.     

$$L^p$$Lp-Boundedness and $$L^p$$Lp-Nuclearity of Multilinear Pseudo-differential Operators on $${\mathbb {Z}}^n$$Zn and the Torus $${\mathbb {T}}^n$$Tn

In this article, we begin a systematic study of the boundedness and the nuclearity properties of multilinear periodic pseudo-differential operators and multilinear discrete pseudo-differential operators on \(L^p\)-spaces. First, we prove analogues of known multilinear Fourier multipliers theorems (proved by Coifman and Meyer, Grafakos, Tomita, Torres, Kenig, Stein, Fujita, Tao, etc.) in the context of periodic and discrete multilinear pseudo-differential operators. For this, we use the periodic analysis of pseudo-differential operators developed by Ruzhansky and Turunen. Later, we investigate the s-nuclearity, \(0<s \le 1,\) of periodic and discrete pseudo-differential operators. To accomplish this, we classify those s-nuclear multilinear integral operators on arbitrary Lebesgue spaces defined on \(\sigma \)-finite measures spaces. We also study similar properties for periodic Fourier integral operators. Finally, we present some applications of our study to deduce the periodic Kato–Ponce inequality and to examine the s-nuclearity of multilinear Bessel potentials as well as the s-nuclearity of periodic Fourier integral operators admitting suitable types of singularities.

     

Rigidity of unary algebras and its application to the $${\mathcal {HS} = \mathcal {SH}}$$ problem

H. P. Gumm and T. Schröder stated a conjecture that the preservation of preimages by a functor T for which |T1| = 1 is equivalent to the satisfaction of the class equality \({{\mathcal {HS}}({\sf K}) = {\mathcal {SH}}({\sf K})}\) for any class K of T-coalgebras. Although T. Brengos and V. Trnková gave a positive answer to this problem for a wide class of Set-endofunctors, they were unable to find the full solution. Using a construction of a rigid unary algebra we prove \({{\mathcal {HS}} \neq {\mathcal {SH}}}\) for a class of Set-endofunctors not preserving non-empty preimages; these functors have not been considered previously.     

\boldsymbol{\mathcal{Q}}-*-Categories

We consider the theory of categories enriched in an involutive quantaloid : the -*-categories. After giving an introduction to involutive quantaloids and nuclei, we use matrices with entries in to define -*-categories. Then we examine the relations between two kinds of morphisms between them, the functors and the *-maps, to provide a basis to study completeness properties. These results are used to provide a definition of pseudo-presheaves, presheaves and sheaves on involutive quantaloids in order to get a generalization of presheaves and sheaves on sites. Finally a characterization of these sheaves in terms of covers and compatible families is presented.      

Fixed points in {\mathcal{M}}-fuzzy metric spaces

In this paper, we give some common fixed point theorems for five mappings satisfying some conditions in -fuzzy metric spaces.     

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.     

The $${\mathcal {F}}$$ F -transitivity and recurrence of translation semigroups on complex sectors

Semigroup Forum - In this paper, we study the $${\mathcal {F}}$$ -transitive behaviour of the translation semigroups on complex sectors, where $${\mathcal {F}}$$ is a Furstenberg family of the...