{\boldsymbol(}\boldsymbol{\mathcal{Z}}_{\bf 1}, \boldsymbol{\mathcal{Z}}_{\bf 2}{\boldsymbol)}-Complete Partially Ordered Sets and Their Representations by \boldsymbol{\mathcal{Q}}-Spaces

The present paper proposes a general theory for $\left( \mathcal{Z}_{1}, \mathcal{Z}_{2}\right) $ -complete partially ordered sets (alias $\mathcal{Z} _{1}$ -join complete and $\mathcal{Z}_{2}$ -meet complete partially ordered sets) and their Stone-like representations. It is shown that for suitably chosen subset selections $\mathcal{Z}_{i}$ (i?=?1,...,4) and $\mathcal{Q} =\left( \mathcal{Z}_{1},\mathcal{Z}_{2},\mathcal{Z}_{3},\mathcal{Z} _{4}\right) $ , the category $\mathcal{Q}$ P of $\left( \mathcal{Z}_{1},\mathcal{Z}_{2}\right) $ -complete partially ordered sets and $\left( \mathcal{Z}_{3},\mathcal{Z}_{4}\right) $ -continuous (alias $\mathcal{ Z}_{3}$ -join preserving and $\mathcal{Z}_{4}$ -meet preserving) functions forms a useful categorical framework for various order-theoretical constructs, and has a close connection with the category $\mathcal{Q}$ S of $\mathcal{Q}$ -spaces which are generalizations of topological spaces involving subset selections. In particular, this connection turns into a dual equivalence between the full subcategory $ \mathcal{Q}$ P _s of $\mathcal{Q}$ P of all $\mathcal{Q}$ -spatial objects and the full subcategory $\mathcal{Q}$ S _s of $\mathcal{Q}$ S of all $\mathcal{Q}$ -sober objects. Here $\mathcal{Q}$ -spatiality and $\mathcal{Q}$ -sobriety extend usual notions of spatiality of locales and sobriety of topological spaces to the present approach, and their relations to $\mathcal{Z}$ -compact generation and $\mathcal{Z}$ -sobriety have also been pointed out in this paper.     

\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.      

Nonzero Infinitesimal Blocks of Category $\boldsymbol{\mathcal{O}}_{\textbf{\emph{S}}}$

Category \(\mathcal{O}\) is a nice category of modules for a finite dimensional semisimple Lie algebra \(\mathfrak{g}\) It was first introduced by Bernstein–Gelfand–Gelfand in 1976. In the early 1980’s, Rocha–Caridi introduced parabolic category \(\mathcal{O}_S\), where S is a subset of simple roots. Category \(\mathcal{O}_S\) is a generalization of ordinary category O. These are highest weight categories that decompose into certain subcategories, called infinitesimal blocks. An infinitesimal block contains at most finitely many simple modules, and some contain only the zero module. The representation type of the infinitesimal blocks of category \(\mathcal{O}_S\) has been studied by Futorny–Nakano–Pollack, Brüstle–König–Mazorchuk, and Boe–Nakano. The representation type of the singular blocks of category \(\mathcal{O}_S\) is still generally unknown, though Boe–Nakano classified certain of these. Understanding when a singular block is zero is an important step to understanding the singular blocks in general. In this work, we will answer this question. It is given in terms of nilpotent orbits of \(\mathfrak{g}\).     

{\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.     

Convergence of polyharmonic splines on semi-regular grids \mathbb{Z}{{\boldsymbol{\times}} \boldsymbol{a}} {\mathbb{Z}^{\it\boldsymbol{n}}} for {\boldsymbol{a}\rightarrow\mathbf {0}}

Let p, n ∈ ? with 2p ≥ n + 2, and let I _a be a polyharmonic spline of order p on the grid ? × a? ⁿ which satisfies the interpolating conditions $I_{a}\left( j,am\right) =d_{j}\left( am\right) $ for j ∈ ?, m ∈ ? ⁿ where the functions d _j : ? ⁿ → ? and the parameter a > 0 are given. Let $B_{s}\left( \mathbb{R}^{n}\right) $ be the set of all integrable functions f : ? ⁿ → ? such that the integral $$ \left\| f\right\| _{s}:=\int_{\mathbb{R}^{n}}\left| \widehat{f}\left( \xi\right) \right| \left( 1+\left| \xi\right| ^{s}\right) d\xi $$ is finite. The main result states that for given $\mathbb{\sigma}\geq0$ there exists a constant c>0 such that whenever $d_{j}\in B_{2p}\left( \mathbb{R}^{n}\right) \cap C\left( \mathbb{R}^{n}\right) ,$ j ∈ ?, satisfy $\left\| d_{j}\right\| _{2p}\leq D\cdot\left( 1+\left| j\right| ^{\mathbb{\sigma}}\right) $ for all j ∈ ? there exists a polyspline S : ? ⁿ⁺¹ → ? of order p on strips such that $$ \left| S\left( t,y\right) -I_{a}\left( t,y\right) \right| \leq a^{2p-1}c\cdot D\cdot\left( 1+\left| t\right| ^{\mathbb{\sigma}}\right) $$ for all y ∈ ? ⁿ , t ∈ ? and all 0 < a ≤ 1.     

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.     

$\boldsymbol{\mathcal{G}-}$Inhomogeneous Markov Systems of High Order

In the present, we introduce and study the G-\mathcal{G-}inhomogeneous Markov system of high order, which is a more general in many respects stochastic process than the known inhomogeneous Markov system. We define the inhomogeneous superficial razor cut mixture transition distribution model extending for the homogeneous case the idea of the mixture transition model. With the introduction of the appropriate vector stochastic process and the establishment of relationships among them, we study the asymptotic behaviour of the G-\mathcal{G-}inhomogeneous Markov system of high order. In the form of two theorems, the asymptotic behaviour of the inherent G-\mathcal{G-}inhomogeneous Markov chain and the expected and relative expected population structure of the G-\mathcal{G-}inhomogeneous Markov system of high order, are provided under assumptions easily met in practice. Finally, we provide an illustration of the present results in a manpower system.     

On the Intertwining of \partial{\mathcal{D}}-Isometries

Let be a strictly pseudoconvex bounded domain in with C ² boundary . If a subnormal m-tuple T of Hilbert space operators has the spectral measure of its minimal normal extension N supported on , then T is referred to as a -isometry. Using some non-trivial approximation theorems in the theory of several complex variables, we establish a commutant lifting theorem for those -isometries whose (joint) Taylor spectra are contained in a special superdomain Ω of . Further, we provide a function-theoretic characterization of those subnormal tuples whose Taylor spectra are contained in Ω and that are quasisimilar to a certain (fixed) -isometry T (of which the multiplication tuple on the Hardy space of the unit ball in is a rather special example). Submitted: September 9, 2007. Revised: October 10, 2007. Accepted: October 24, 2007.     

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 universal difference variety over \overline{\mathcal{M }}_g

We determine the Kodaira dimension of the Deligne–Mumford compactification \(\overline{\mathfrak{Diff }}_g\) of the universal difference variety over the moduli space of curves.     

On diophantine equations of the form {({\boldsymbol x}-{\boldsymbol a}_{\bf 1})({\boldsymbol x}-{\boldsymbol a}_{\bf 2}) \ldots ({\boldsymbol x}-{\boldsymbol a}_{\boldsymbol k}) + {\boldsymbol r} = {\boldsymbol y}^{ \boldsymbol n}}

Erd?s and Selfridge [3] proved that a product of consecutive integers can never be a perfect power. That is, the equation x(x?+?1)(x?+?2)...(x?+?(m???1))?=?y ⁿ has no solutions in positive integers x,m,n where m, n?>?1 and y?∈?Q. We consider the equation $$ (x-a_1)(x-a_2) \ldots (x-a_k) + r = y^n $$ where 0?≤?a ₁?<?a ₂?<???<?a _k are integers and, with r?∈?Q, n?≥?3 and we prove a finiteness theorem for the number of solutions x in Z, y in Q. Following that, we show that, more interestingly, for every nonzero integer n?>?2 and for any nonzero integer r which is not a perfect n-th power for which the equation admits solutions, k is bounded by an effective bound.     

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 .     

Multilevel interpolation of scattered data using H ${\mathcal{H}}$ -matrices

Numerical Algorithms - Scattered data interpolation can be used to approximate a multivariate function by a linear combination of positive definite radial basis functions (RBFs). In practice, the...     

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.     

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.     

关于${cal I}$-覆盖映射与1-${cal I}$-覆盖映射

In this paper,we introduce the concepts of I-covering mappings and 1-I-covering mappings,discuss the difference between sequence-covering and I-covering mappings by some examples.With those concepts,we get some interesting properties of I-covering(1-I-covering)mappings and some characterizations of I-covering(1-I-covering)and compact mapping images of metric spaces.