The {{\mathcal {HS}} = {\mathcal {SH}}} problem for coalgebras

H. P. Gumm and T. Schröder stated a hypothesis that the preservation of preimages by a functor T for which |T1| = 1 is equivalent to the satisfaction of the class equality ${{\mathcal {HS}}(\tt {K}) = {\mathcal {SH}}(\tt {K})}$ for any class K of T-coalgebras. Although we were not able to find a full solution, our paper gives a positive answer to this problem for a very wide class of Set-endofunctors.     

Lattices of topologies of unary algebras of the variety $$ \mathcal{A}_{1,1} $$

We consider the variety of unary algebras 〈A, f, g〉 defined by the identities f(g(x)) = g(f(x)) = x. We describe algebras of this variety, whose topology lattices are modular, distributive, linearly ordered, complemented, or pseudocomplemented.     

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.     

Finite rank operators in Jacobson radical \mathcal{R}_{\mathcal{N} \otimes \mathcal{M}}

In this paper we investigate finite rank operators in the Jacobson radical of Alg( ), where are nests. Based on the concrete characterizations of rank one operators in Alg( ) and , we obtain that each finite rank operator in can be written as a finite sum of rank one operators in and the weak closure of equals Alg( ) if and only if at least one of is continuous.     

Determinantal Resolution of the Universal Subscheme in $$\mathcal{S}{\text{ }} \times {\text{ }}H_{d + 1} $$

We construct a determinantal resolution of singularities for the universal subscheme in and prove that it is isomorphic to the variety of total pairs .     

$${\mathcal{C}}_{0}$$ ( 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     

Note on $$ \mathcal{I}R $$-closed and $$ A_{\mathcal{I}R} $$-sets

We characterize and discuss the properties of $ \mathcal{I}R $ \mathcal{I}R -closed sets and $ A_{\mathcal{I}R} $ A_{\mathcal{I}R} -sets. Also, we give characterizations of weakly $ \mathcal{I} $ \mathcal{I} -locally closed sets and $ \mathcal{I} $ \mathcal{I} -submaximal spaces. A characterization of codense ideals in terms of $ \mathcal{I}R $ \mathcal{I}R -closed sets is also given.     

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 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{F}}_\phi and {\mathcal{F}}_g Classes

In this paper we give a generalization of the Zhao F(p, q, s)-spaces by using operators instead of functions. In this way we unify and simplify several important results about the classic spaces D_p, Q_p{\mathcal{Q}}_{p} ,B^α, etc.     

$$ \mathcal{I}_g $$-normal and $$ \mathcal{I}_g $$-regular spaces

$ \mathcal{I}_g $ \mathcal{I}_g -normal and $ \mathcal{I}_g $ \mathcal{I}_g -regular spaces are introduced and various characterizations and properties are given. Characterizations of normal, mildly normal, g-normal, regular and almost regular spaces are also given.     

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 \({\mathbb {S}}^3\times {\mathbb {S}}^3\).     

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.     

Three-dimensional $$ \mathcal{N} $$=4 superconformal superfield theories

We consider the map of three-dimensional N=4 superfields to the N=3 harmonic superspace. The left and right representations of the N=4 superconformal group are constructed on N=3 analytic superfields. These representations are convenient for describing N=4 superconformal couplings of Abelian gauge superfields to hypermultiplets. We investigate the N=4 invariance in the non-Abelian N=3 Yang-Mills theory.     

Amenability,tubularity, and embeddings into $$\mathcal{R}^{\omega}$$

Suppose M is a tracial von Neumann algebra embeddable into (the ultraproduct of the hyperfinite II₁-factor) and X is an n-tuple of selfadjoint generators for M. Denote by Γ(X; m, k, γ) the microstate space of X of order (m, k ,γ). We say that X is tubular if for any ε >  0 there exist and γ > 0 such that if then there exists a k × k unitary u satisfying for each 1 ≤  i ≤  n. We show that the following conditions are equivalent:

Yangians and $$\mathcal{W} $$ -Algebras

We present a connection between -algebras and Yangians in the case of gl(N) algebras, as well as for twisted Yangians and super-Yangians. We illustrate this connection, which allows constructing an R-matrix for the -algebras and classifying their finite-dimensional irreducible representations, in the framework of the nonlinear Schrödinger equation in 1+1 dimensions.     

The $${\mathbb {Q}}$$ Q -Korselt set of $$\mathrm {pq}$$ pq

Periodica Mathematica Hungarica - Let N be a positive integer, $${\mathbb {A}}$$ be a nonempty subset of $${\mathbb {Q}}$$ and $$\alpha =\dfrac{\alpha _{1}}{\alpha _{2}}\in {\mathbb {A}}{\setminus...     

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 .      

Between quantum Virasoro algebra $$\mathcal{L}_c $$ and generalized Clifford algebras

In this paper we construct the quantum Virasoro algebra generators in terms of operators of the generalized Clifford algebras C_n^k. Precisely, we show that can be embedded into generalized Clifford algebras. Junior Associate at The Abdus Salam ICTP, Trieste, Italy.