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.     

Groupoid and inverse semigroup presentations of ultragraph C *-algebras

Inspired by the work of Paterson on C ^* -algebras of directed graphs, we show how to associate a groupoid to an ultragraph in such a way that the C ^*-algebra of is canonically isomorphic to Tomforde’s C ^*-algebra . The groupoid is built from an inverse semigroup naturally associated to . A.E. Marrero was supported by grants from the National Science Foundation and the Sloan Foundation and by a GAANN Fellowship. Many of the results here are taken from this author’s dissertation [7]. P.S. Muhly was supported by a grant from the National Science Foundation (DMS-0355443).     

Relations between different types of spectra and spectral characterizations

In the study of the asymptotic behaviour of solutions of differential-difference equations the -spectrum has been useful, where and implies Fourier transform , with given , φ∈L ^∞(ℝ,X), X a Banach space, (half)line. Here we study and related concepts, give relations between them, especially weak Laplace half-line spectrum of φ, and thus ⊂ classical Beurling spectrum = Carleman spectrum =  ; also  = Beurling spectrum of “φ modulo ” (Chill-Fasangova). If satisfies a Loomis type condition (L _U ), then countable and uniformly continuous ∈U are shown to imply ; here (L _U ) usually means , indefinite integral Pf of f in U imply Pf in (the Bohl-Bohr theorem for = almost periodic functions, U=bounded functions). This spectral characterization and other results are extended to unbounded functions via mean classes , ℳ ^m U ((2.1) below) and even to distributions, generalizing various recent results for uniformly continuous bounded φ. Furthermore for solutions of convolution systems S*φ=b with in some we show . With these above results, one gets generalizations of earlier results on the asymptotic behaviour of solutions of neutral integro-differential-difference systems. Also many examples and special cases are discussed.     

Geometry of the mapping class groups I: Boundary amenability

We construct a geometric model for the mapping class group of a non-exceptional oriented surface S of genus g with k punctures and use it to show that the action of on the compact metrizable Hausdorff space of complete geodesic laminations for S is topologically amenable. As a consequence, the Novikov higher signature conjecture holds for every subgroup of .     

On congruences on \mathcal{T} -regular semigroups

A nontrivial regular semigroup S with zero in which every interval of idempotents is a finite chain is said to be -regular. The structure of these semigroups is described in terms of trees of completely 0-simple semigroups. For S in this form, we study congruences which we express in terms of congruence aggregates. We determine the inclusion relation, meet and join of congruences, their kernel and trace, and the ends of the intervals which form their classes. We characterize those S for which the kernel relation on the congruence lattice is a congruence, and those for which the operators and are homomorphisms.     

Vive la différence III

We show that, consistently, there is an ultrafilter on ω such that if N _n^ℓ = (P _n^ℓ ∪ Q _n^ℓ, P _n^ℓ, Q _n^ℓ, R _n^ℓ) (for ℓ = 1, 2, n < ω), P _n^ℓ ∪ Q _n^ℓ ⊆ ω, and are models of the canonical theory t ^ind of the strong independence property, then every isomorphism from onto is a product isomorphism. The first version of this work done in 93; First typed: May 1993. This research was partially supported by the United States-Israel Binational Science Foundation. Publication 509     

On pairs of commuting nilpotent matrices

Let B be a nilpotent matrix and suppose that its Jordan canonical form is determined by a partition λ. Then it is known that its nilpotent commutator is an irreducible variety and that there is a unique partition μ such that the intersection of the orbit of nilpotent matrices corresponding to μ with is dense in . We prove that map given by is an idempotent map. This answers a question of Basili and Iarrobino [9] and gives a partial answer to a question of Panyushev [18]. In the proof, we use the fact that for a generic matrix the algebra generated by A and B is a Gorenstein algebra. Thus, a generic pair of commuting nilpotent matrices generates a Gorenstein algebra. We also describe in terms of λ if has at most two parts.     

Bounded universal functions for sequences of holomorphic self-maps of the disk

We give several characterizations of those sequences of holomorphic self-maps {φ _n } _n≥1 of the unit disk for which there exists a function F in the unit ball of H ^∞ such that the orbit {F∘φ _n :n∈ℕ} is locally uniformly dense in . Such a function F is said to be a -universal function. One of our conditions is stated in terms of the hyperbolic derivatives of the functions φ _n . As a consequence we will see that if φ _n is the nth iterate of a map φ of into , then {φ _n } _n≥1 admits a -universal function if and only if φ is a parabolic or hyperbolic automorphism of . We show that whenever there exists a -universal function, then this function can be chosen to be a Blaschke product. Further, if there is a -universal function, we show that there exist uniformly closed subspaces consisting entirely of universal functions.     

Distances Between <Emphasis Type="Italic">σ</Emphasis>-Fields on a Probability Space

For a probability space (Ω,ℱ,P) and two sub-σ-fields we consider two natural distances: and . We investigate basic properties of these distances. In particular we show that if a distance (ρ or ) from ℬ to is small then there exists Z∈ℱ with small P(Z), such that for every B∈ℬ there exists such that B∖Z and A∖Z differ by a set of probability zero. This improves results of Neveu (Ann. Math. Stat. 43(4):1369–1371, [1972]), Jajte and Paszkiewicz (Probab. Math. Stat. 19(1):181–201, [1999]).      

Wedge Products and Cotensor Coalgebras in Monoidal Categories

The construction of the cotensor coalgebra for an “abelian monoidal” category which is also cocomplete, complete and AB5, was performed in Ardizzoni et al. (Comm Algebra 35(1):25–70, 2007). It was also proved that this coalgebra satisfies a meaningful universal property which resembles the classical one. Here the lack of the coradical filtration for a coalgebra E in is filled by considering a direct limit of a filtration consisting of wedge products of a subcoalgebra D of E. The main aim of this paper is to characterize hereditary coalgebras , where D is a coseparable coalgebra in , by means of a cotensor coalgebra: more precisely, we prove that, under suitable assumptions, is hereditary if and only if it is formally smooth if and only if it is the cotensor coalgebra if and only if it is a cotensor coalgebra , where N is a certain D-bicomodule in . Because of our choice, even when we apply our results in the category of vector spaces, new results are obtained. This paper was written while A. Ardizzoni was member of G.N.S.A.G.A. with partial financial support from Mi.U.R.     

<Emphasis Type="Italic">F</Emphasis>-isocristaux surconvergents et surcohérence différentielle

Résumé Soient un anneau de valuation discrète complet d’inégales caractéristiques, de corps résiduel parfait k, un -schéma formel propre et lisse, T un diviseur de la fibre spéciale P de , U l’ouvert de P complémentaire de T, Y un sous-k-schéma fermé lisse de U. Nous prouvons que la catégorie des F-isocristaux surconvergents sur Y est équivalente à celle des F-isocristaux surcohérents sur Y (voir [Car, 6.2.1 et 6.4.3.a)]). Plus généralement, nous établissons par recollement une telle équivalence pour tout k-schéma séparé lisse Y. Nous vérifions de plus que les F-complexes de -modules à cohomologie bornée et -surcohérente se dévissent en F-isocristaux surconvergents.     

The extension of the operation in the Stone-Čech compactification of semigroups

For a large class of locally compact semitopological semigroups S, the Stone-Čech compactification β S is a semigroup compactification if and only if S is either discrete or countably compact. Furthermore, for this class of semigroups which are neither discrete nor countably compact, the quotient contains a linear isometric copy of ℓ _∞. These results improve theorems by Baker and Butcher and by Dzinotyiweyi.     

Generalized Continuous Posets and a New Cartesian Closed Category

With every subset selection for posets, there is associated a certain ideal completion . As shown by Erné, such completions help to extend classical results on domains and similar structures in the absence of the required joins. Some results about –predistributive or –precontinuous posets and –continuous functions are summarized and supplemented. In particular, several central results on function spaces in domain theory are extended to the setting of productive closed subset selections. The category FSBP, in which objects are finitely separated and upper bounded posets and arrows are continuous functions between them, is shown to be cartesian closed. This research is supported by the National Natural Science Foundation of China, 10471035.     

Equivariant vector bundles on Drinfeld’s upper half space

Let be Drinfeld’s upper half space over a finite extension K of ℚ _p . We construct for every GL _d+1-equivariant vector bundle on ℙ ^d _K , a GL _d+1(K)-equivariant filtration by closed subspaces on the K-Fréchet . This gives rise by duality to a filtration by locally analytic GL _d+1(K)-representations on the strong dual . The graded pieces of this filtration are locally analytic induced representations from locally algebraic ones with respect to maximal parabolic subgroups. This paper generalizes the cases of the canonical bundle due to Schneider and Teitelbaum [ST1] and that of the structure sheaf by Pohlkamp [P].     

On the relation of Voevodsky’s algebraic cobordism to Quillen’s <Emphasis Type="Italic">K</Emphasis>-theory

Quillen’s algebraic K-theory is reconstructed via Voevodsky’s algebraic cobordism. More precisely, for a ground field k the algebraic cobordism P¹-spectrum MGL of Voevodsky is considered as a commutative P¹-ring spectrum. Setting we regard the bigraded theory MGL ^p,q as just a graded theory. There is a unique ring morphism which sends the class [X]_MGL of a smooth projective k-variety X to the Euler characteristic of the structure sheaf . Our main result states that there is a canonical grade preserving isomorphism of ring cohomology theories

A Conditional 0–1 Law for the Symmetric <Emphasis Type="Italic">σ</Emphasis>-field

Let (Ω,ℬ,P) be a probability space, a sub-σ-field, and μ a regular conditional distribution for P given . For various, classically interesting, choices of (including tail and symmetric), we prove the following 0–1 law: There is a set such that P(A ₀)=1 and μ(ω)(A)∈{0,1} for all and ω∈A ₀. If ℬ is countably generated (and certain regular conditional distributions exist), the result applies whatever P is.      

Hermite Subdivision Schemes and Taylor Polynomials

We propose a general study of the convergence of a Hermite subdivision scheme ℋ of degree d>0 in dimension 1. This is done by linking Hermite subdivision schemes and Taylor polynomials and by associating a so-called Taylor subdivision (vector) scheme . The main point of investigation is a spectral condition. If the subdivision scheme of the finite differences of is contractive, then is C ⁰ and ℋ is C ^d . We apply this result to two families of Hermite subdivision schemes. The first one is interpolatory; the second one is a kind of corner cutting. Both of them use the Tchakalov-Obreshkov interpolation polynomial.      

Nested set complexes of Dowling lattices and complexes of Dowling trees

Given a finite group G and a natural number n, we study the structure of the complex of nested sets of the associated Dowling lattice (Proc. Internat. Sympos., 1971, pp. 101–115) and of its subposet of the G-symmetric partitions which was recently introduced by Hultman (, 2006), together with the complex of G-symmetric phylogenetic trees . Hultman shows that the complexes and are homotopy equivalent and Cohen–Macaulay, and determines the rank of their top homology. An application of the theory of building sets and nested set complexes by Feichtner and Kozlov (Selecta Math. (N.S.) 10, 37–60, 2004) shows that in fact is subdivided by the order complex of . We introduce the complex of Dowling trees and prove that it is subdivided by the order complex of . Application of a theorem of Feichtner and Sturmfels (Port. Math. (N.S.) 62, 437–468, 2005) shows that, as a simplicial complex, is in fact isomorphic to the Bergman complex of the associated Dowling geometry. Topologically, we prove that is obtained from by successive coning over certain subcomplexes. It is well known that is shellable, and of the same dimension as . We explicitly and independently calculate how many homology spheres are added in passing from to . Comparison with work of Gottlieb and Wachs (Adv. Appl. Math. 24(4), 301–336, 2000) shows that is intimely related to the representation theory of the top homology of . Research partially supported by the Swiss National Science Foundation, project PP002-106403/1.     

On the Number of Topological Types Occurring in a Parameterized Family of Arrangements

Let be an o-minimal structure over ℝ, a closed definable set, and

Region-Fault Tolerant Geometric Spanners

We introduce the concept of region-fault tolerant spanners for planar point sets and prove the existence of region-fault tolerant spanners of small size. For a geometric graph on a point set P and a region F, we define to be what remains of after the vertices and edges of intersecting F have been removed. A  -fault tolerant t-spanner is a geometric graph  on P such that for any convex region F, the graph is a t-spanner for , where is the complete geometric graph on P. We prove that any set P of n points admits a -fault tolerant (1+ε)-spanner of size for any constant ε>0; if adding Steiner points is allowed, then the size of the spanner reduces to  , and for several special cases, we show how to obtain region-fault tolerant spanners of size without using Steiner points. We also consider fault-tolerant geodesic t -spanners: this is a variant where, for any disk D, the distance in between any two points u,v∈P∖D is at most t times the geodesic distance between u and v in ℝ²∖D. We prove that for any P, we can add Steiner points to obtain a fault-tolerant geodesic (1+ε)-spanner of size  . M.A. Abam was supported by the Netherlands’ Organisation for Scientific Research (NWO) under project no. 612.065.307 and by the MADALGO Center for Massive Data Algorithmics, a Center of the Danish National Research Foundation. M. de Berg was supported by the Netherlands’ Organisation for Scientific Research (NWO) under project no. 639.023.301. M. Farshi was supported by Ministry of Science, Research and Technology of I.R. Iran. NICTA is funded by the Australian Government as represented by the Department of Broadband, Communications and the Digital Economy and the Australian Research Council through the ICT Centre of Excellence program.