Covers ℘ for Abstract Regular Polytopes \mathcal {Q} such that \mathcal{Q}=\mathcal{P}/\mathbf{Z}_{p}^{k}

One key problem in the theory of abstract polytopes is the so-called amalgamation problem. In its most general form, this is the problem of characterising the polytopes with given facets  $\mathcal {K}$ and vertex figures ?. The first step in solving it for particular  $\mathcal{K}$ and ? is to find the universal such polytope, which covers all the others. This article explains a construction that may be attempted on an arbitrary polytope ?, which often yields an infinite family of finite polytopes covering ? and sharing its facets and vertex figures. The existence of such an infinite family proves that the universal polytope is infinite; alternatively, the construction can produce an explicit example of an infinite polytope of the desired type. An algorithm for attempting the construction is explained, along with sufficient conditions for it to work. The construction is applied to a few  $\mathcal{K}$ and ? for which it was previously not known whether or not the universal polytope was infinite, or for which only a finite number of finite polytopes was previously known. It is conjectured that the construction is quite broadly applicable.     

与群PSL2(\mathcal{R})相关的交叉积{\mathcal{R}(\mathcal{A}},α)的一点注记

在上半复平面$\mathbb{H}$上给定双曲测度$dxdy/y^{2}$, 群$G={\rm PSL}_{2}(\mathbb{R})$ 在$\mathbb{H}$上的分式线性作用导出了$G$在Hilbert空间$L^{2}(\mathbb{H}, dxdy/y^{2})$上的酉表示$\alpha$. 证明了交叉积 $\mathcal{R}(\mathcal{A}, \alpha)$是$\mathrm{I}$型von Neumann代数, 其中$\mathcal{A}= \{M_{f}:f\in L^{\infty}(\mathbb{H},dxdy/y^{2} )\}$. 具体地, 交叉积代数$\mathcal{R}(\mathcal{A}, \alpha)$与von Neumann代数$\mathcal{B}(L^{2}(P, \nu))\overline{\otimes}\mathcal{L}_{K}$是*-同构的, 其中$\mathcal{L}_{K}$是$G$中子群 $K$的左正则表示生成的群von Neumann代数.     

Z∗$$ {\mathcal{Z}}^{\ast } $$-Semilocal Modules and the Proper Class ℛS$$ \mathrm{\mathcal{R}}\mathcal{S} $$

Ukrainian Mathematical Journal - Over an arbitrary ring, a module M is said to be $$ {\mathcal{Z}}^{\ast } $$-semilocal if every submodule U of M has a $$ {\mathcal{Z}}^{\ast } $$ -supplement V in...     

Discreteness criteria for Möbius groups acting on $$\mathcal{O}\mathcal{L}_p $$

In this paper, three new discreteness criteria for Möbius groups acting on\(\bar R^{n * } \) are obtained; they are generalizations of known results using the information of two-generator subgroups.     

The Quasiapproximate $(\mathcal{U} + \mathcal{K})$ –Invariants of Essentially Normal Operators

For a class of essentially normal operators, we characterize their norm closures of –orbits. Moreover, we introduce a notion of the quasiapproximate – equivalence of essentially normal operators and determine completely the quasiapproximate –invariants. Finally, we give the canonical forms of essentially normal operators under this quasiapproximate –equivalence.     

Cartan-Dieudonné Theorem for {\mathcal {A}}-Modules

Like the classical Cartan-Dieudonné theorem, the sheaf-theoretic version shows that A{\mathcal {A}}-isometries on a convenient A{\mathcal {A}}-module E{\mathcal {E}} of rank n can be decomposed in at most n orthogonal symmetries (reflections) with respect to non-isotropic hyperplanes. However, the coefficient sheaf of \mathbb C{\mathbb {C}}-algebras A{\mathcal {A}} is assumed to be a PID \mathbb C{\mathbb {C}}-algebra sheaf and, if (E,f){(\mathcal {E},\phi)} is a pairing with f{\phi} a non-degenerate A{\mathcal {A}}-bilinear morphism, we assume that E{\mathcal {E}} has nowhere-zero (local) isotropic sections; but, for Riemannian sheaves of A{\mathcal {A}}-modules, this is not necessarily required.     

A note on the crossed product $$ \mathcal{R}(\mathcal{A},\alpha ) $$ associated with PSL2(ℝ)

Given the hyperbolic measure dxdy/y ² on the upper half plane ℍ, the rational actions of PSL₂(ℝ) on ℍ induces a continuous unitary representation α of this group on the Hilbert space L ²(ℍ, dxdy/y ²). Supposing that = {M _f : f ∈ L ^∞ (ℍ, dxdy/y ²)}, we show that the crossed product is of type I. In fact, the crossed product is *-isomorphic to the von Neumann algebra , where is the abelian group von Neumann algebra generated by the left regular representation of K. This work was supported by the Youth Foundation of Sichuan Education Department of China (Grant No. 2003B017)     

关于$\mathcal{F}$-S-可补子群

设$\mathcal{F}$是一个群类. 群$G$的子群$H$称为在$G$中$\mathcal{F}$-S-可补的，如果存在$G$的一个子群$K$,使得$G=HK$且$K/K\cap{H_G}\in\mathcal{F}$, 其中$H_G=\bigcap_{g\in G}H^g$是包含在$H$中的$G$的最大正规子群．本文利用子群的$\mathcal{F}$-S-可补性, 给出了有限群的可解性, 超可解性和幂零性的一些新的刻画. 应用这些结果, 我们可以得到一系列推论, 其中包括有关已知的著名结果.     

The variety generated by {\mathbb {A}(\mathcal {T})}– two counterexamples

We show that \({\mathcal {V}(\mathbb {A}(\mathcal {T}))}\) does not have definable principal subcongruences or bounded Maltsev depth. When the Turing machine \({\mathcal {T}}\) halts, \({\mathcal {V}(\mathbb {A}(\mathcal {T}))}\) is an example of a finitely generated semilattice based (and hence congruence \({\wedge}\)-semidistributive) variety with only finitely many subdirectly irreducible members, all finite. This is the first known example of a variety with these properties that does not have definable principal subcongruences or bounded Maltsev depth.     

Subdirectly irreducible \mathcal{RGC} -commutative right H-semigroups

A semigroup S is said to be ℛ-commutative if, for all elements a,b∈S, there is an element x∈S ¹ such that ab=bax. A semigroup S is called a generalized conditionally commutative (briefly, -commutative) semigroup if it satisfies the identity aba ²=a ² ba. An ℛ-commutative and -commutative semigroup is called an -commutative semigroup. A semigroup S is said to be a right H-semigroup if every right congruence of S is a congruence of S. In this paper we characterize the subdirectly irreducible semigroups in the class of -commutative right H-semigroups. Research supported by the Hungarian NFSR grant No T029525.     

A note on κ-uniform points of G^{\mathcal{LUC}}

Let G be a Hausdorff, non-compact, locally compact topological group. We show that, for every infinite cardinal number κ with κ≤κ(G), the set of all κ-uniform points of $G^{\mathcal{LUC}}$ is a closed, two-sided ideal of $G^{\mathcal{LUC}}$ .     

Classes of semigroups modulo Green’s relation \mathcal{H}

Inverse semigroups and orthodox semigroups are either defined in terms of inverses, or in terms of the set of idempotents E(S). In this article, we study analogs of these semigroups defined in terms of inverses modulo Green’s relation \(\mathcal{H}\) , or in terms of the set of completely regular elements H(S). Results are obtained both for the regular and the non-regular cases. We then study the interplays between these new classes of semigroups, as well as with various known classes notably of inverse, orthodox, E-solid and cryptic semigroups.     

On $${\mathcal {I}}_{

Tóth  János T.  Filip  Ferdinánd  Bukor  József  Zsilinszky  László 《Periodica Mathematica Hungarica》2021,82(2):125-135

A sequence space representation of L. Schwartz’ space {{\mathcal{O}_{C}}}

We derive a representation of the isomorphic spaces ${\mathcal{O}_{C}}$ of very slowly increasing functions and ${\mathcal{O}_{M}'}$ of very rapidly decreasing distributions as a completed topological tensor product of sequence spaces. In order to describe this completed topological tensor product as a space of double sequences, we construct a representation as an inductive limit of vector valued sequence spaces. Moreover we compare the representations of ${\mathcal{O}_{C}}$ and ${\mathcal{O}_{M}}$ .     

Strongly-$\mathcal{I}$-locally closed sets and decompositions of ∗-continuity

We introduce the notion of strongly-I\mathcal{I}-locally closed sets to obtain decompositions of ∗-continuity.     

Infinite time extensions of Kleene’s $${\mathcal{O}}$$

Using infinite time Turing machines we define two successive extensions of Kleene’s O{\mathcal{O}} and characterize both their height and their complexity. Specifically, we first prove that the one extension—which we will call O⁺{\mathcal{O}^{+}}—has height equal to the supremum of the writable ordinals, and that the other extension—which we will call O⁺⁺{\mathcal{O}}^{++}—has height equal to the supremum of the eventually writable ordinals. Next we prove that O⁺{\mathcal{O}^+} is Turing computably isomorphic to the halting problem of infinite time Turing computability, and that O⁺⁺{\mathcal{O}^{++}} is Turing computably isomorphic to the halting problem of eventual computability.     

Embedding FD(ω) into {\mathcal{P}_s} densely

Let be the lattice of degrees of non-empty subsets of 2 ^ω under Medvedev reducibility. Binns and Simpson proved that FD(ω), the free distributive lattice on countably many generators, is lattice-embeddable below any non-zero element in . Cenzer and Hinman proved that is dense, by adapting the Sacks Preservation and Sacks Coding Strategies used in the proof of the density of the c.e. Turing degrees. With a construction that is a modification of the one by Cenzer and Hinman, we improve on the result of Binns and Simpson by showing that for any , we can lattice embed FD(ω) into strictly between and . We also note that, in contrast to the infinite injury in the proof of the Sacks Density Theorem, in our proof all injury is finite, and that this is also true for the proof of Cenzer and Hinman, if a straightforward simplification is made. Thanks to my adviser Peter Cholak for his guidance in my research. I also wish to thank the anonymous referee for helpful comments and suggestions. My research was partially supported by NSF grants DMS-0245167 and RTG-0353748 and a Schmitt Fellowship at the University of Notre Dame.     

On the Hodge–Newton filtration for p-divisible {\mathcal{O}} -modules

The notions Hodge–Newton decomposition and Hodge–Newton filtration for F-crystals are due to Katz and generalize Messing’s result on the existence of the local-étale filtration for p-divisible groups. Recently, some of Katz’s classical results have been generalized by Kottwitz to the context of F-crystals with additional structures and by Moonen to μ-ordinary p-divisible groups. In this paper, we discuss further generalizations to the situation of crystals in characteristic p and of p-divisible groups with additional structure by endomorphisms.     

On Utiyama’s Theme Through “{\mathcal {A}} -Invariance”

The term ?? ${\mathcal {A}}$ -invariance?? refers to the invariance of our results, with respect to the ??arithmetic?? employed, viz. to an appropriate algebra sheaf ${\mathcal {A}}$ . This, combined with the categorical notion of ??adjunction??, particularized here with the homological, in nature Hom-? adjunction, affords the classical perspective of Utiyama, pertaining to the characteristic type of field interactions. Yet, all this, without any ??space-time?? support, in the classical sense of the term (: smooth manifolds), but, just based on the ??functorial?? character of ADG (acronym of ??Abstract Differential Geometry??) and the aforementioned two fundamental principles (:?? ${\mathcal {A}}$ -invariance?? and ??adjunction??).     

Surfaces with $${K^2=2\mathcal{X}-2}$$ and pg ≥ 5

This note describes minimal surfaces S of general type satisfying p _g  ≥ 5 and K ² = 2p _g . For p _g  ≥ 8 the canonical map of such surfaces is generically finite of degree 2 and the bulk of the paper is a complete characterization of such surfaces with non birational canonical map. It turns out that if p _g  ≥ 13, S has always an (unique) genus 2 fibration, whose non 2-connected fibres can be characterized, whilst for p _g  ≤ 12 there are two other classes of such surfaces with non birational canonical map.