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.     

Approach Theory in a Category: A Study of Compactness and Hausdorff Separation

In a topological construct endowed with a proper -factorization system and a concrete functor , we study -compactness and -Hausdorff separation, where is a class of “closed morphisms” in the sense of Clementino et al. (A functional approach to general topology. In: Categorical Foundations. Encyclopedia of Mathematics and Its Applications, vol. 97, pp. 103–163. Cambridge University Press, Cambridge, 2004), determined by Λ. In particular, we point out under which conditions on Λ, the notion of -compactness of an object of coincides with 0-compactness of the image in Prap. Our results will be illustrated by some examples: except for some well-known ones, like b-compactness of a topological space, we also capture some compactness notions that were not considered before in the literature. In particular, we obtain a generalization of b-compactness to the setting of approach spaces. This notion is shown to play an important role in the study of uniformizability. The author is research assistant at the Fund of Scientific Research Vlaanderen (FWO).     

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.     

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]).      

Graph von Neumann Algebras

In this paper, we will define a graph von Neumann algebra over a fixed von Neumann algebra M, where G is a countable directed graph, by a crossed product algebra = M × _α , where is the graph groupoid of G and α is the graph-representation. After defining a certain conditional expectation from onto its M-diagonal subalgebra we can see that this crossed product algebra is *-isomorphic to an amalgamated free product where = vN(M × _α where is the subset of consisting of all reduced words in {e, e ^–1} and M × _α is a W ^*-subalgebra of as a new graph von Neumann algebra induced by a graph G _e . Also, we will show that, as a Banach space, a graph von Neumann algebra is isomorphic to a Banach space ⊕ where is a certain subset of the set E(G)* of all words in the edge set E(G) of G. The author really appreciates to Prof F. Radulescu and Prof P. Jorgensen for the valuable discussion and kind advice. Also, he appreciates all supports from St. Ambrose Univ.. In particular, he thanks to Prof T. Anderson and Prof V. Vega for the useful conversations and suggestions.     

Weak Quantum Enveloping Algebras of Borcherds Superalgebras

In present paper we define a new kind of weak quantized enveloping algebra of Borcherds superalgebras . It is a noncommutative and noncocommutative weak graded Hopf algebra. Using localizing with some Ore set, we obtain a different kind of quantized enveloping algebras of Borcherds superalgebras . It has a homomorphic image which is isomorphic to the usual quantum enveloping algebra of . Moreover, is isomorphic to a direct sum of and an other algebra as algebras. The author is sponsored by ZJNSF No. Y607136.     

On Some Properties of Randomly Indexed Sequences of Random Elements

Let be a probability space with a nonatomic measure P and let (S,ρ) be a separable complete metric space. Let {N _n ,n≥1} be an arbitrary sequence of positive-integer valued random variables. Let {F _k ,k≥1} be a family of probability laws and let X be some random element defined on and taking values in (S,ρ). In this paper we present necessary and sufficient conditions under which one can construct an array of random elements {X _n,k ,n,k≥1} defined on the same probability space and taking values in (S,ρ), and such that , and moreover as  n→∞. Furthermore, we consider the speed of convergence to X as n→∞.      

Realization of the mapping class group by homeomorphisms

In this paper, we show that the mapping class group of a closed surface can not be geometrically realized as a group of homeomorphisms of that surface. More precisely, let denote the standard projection of the group of homeomorphisms to the mapping class group of a closed surface M of genus g>5. We show that there is no homomorphism , such that is the identity. This answers a question by Thurston (see [11]). Mathematics Subject Classification (2000)  Primary 20H10, 37F30     

Riesz and Poisson-Jensen Representation Formulas for a Class of Ultraparabolic Operators on Lie Groups

The aim of this paper is to give some representation formulas of Riesz and Poisson-Jensen type for super-solutions to a class of hypoelliptic ultraparabolic operators on a homogeneous Lie group . Our results complete the ones obtained in Cinti (Math Scand 100:1–21, 2007). We also provide a suitable theory for -Green functions and for -Green potentials of Radon measures. The proofs mostly rely on the use of appropriate techniques relevant to the Potential Theory for . Investigation supported by University of Bologna. Funds for selected research topics.     

Dimension of the Torelli group for Out(F<Subscript>n</Subscript>)

Let be the kernel of the natural map Out(F_n)→GL_n(ℤ). We use combinatorial Morse theory to prove that has an Eilenberg–MacLane space which is (2n-4)-dimensional and that is not finitely generated (n≥3). In particular, this shows that the cohomological dimension of is equal to 2n-4 and recovers the result of Krstić–McCool that is not finitely presented. We also give a new proof of the fact, due to Magnus, that is finitely generated.     

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 .     

Moduli spaces of quadratic complexes and their singular surfaces

We construct the coarse moduli space of quadratic line complexes with a fixed Segre symbol σ as well as the moduli space of the corresponding singular surfaces. We show that the map associating to a quadratic line complex its singular surface induces a morphism . Finally we deduce that the varieties of cosingular quadratic line complexes are almost always curves.      

Special Quasigraded Lie Algebras and Integrable Hamiltonian Systems

We construct a family of special quasigraded Lie algebras of functions of one complex variables with values in finite-dimensional Lie algebra , labeled by the special 2-cocycles F on . The main property of the constructed Lie algebras is that they admit Kostant-Adler-Symes scheme. Using them we obtain new integrable finite-dimensional Hamiltonian systems and new hierarchies of soliton equations.     

Constructions of p-ary Quadratic Bent Functions

Bent functions have many applications in the fields of coding theory, communications and cryptography. This paper studies the constructions of bent functions having the form for odd n and for even n, over the finite field of odd characteristic p, where . Based on the irreducibility of some polynomials on , we focus on characterizing the bent functions for n=p ^v q ^r and n=2p ^v q ^r , where is an odd prime and p a primitive root modulo q ². Moreover, the enumerations of those functions are also considered. Partially supported by the NSF of China under Grants No. 60603012 and No. 60573053.     

Primitive Ideals and Automorphisms of Quantum Matrices

Let be a field and q be a nonzero element of that is not a root of unity. We give a criterion for 〈0〉 to be a primitive ideal of the algebra of quantum matrices. Next, we describe all height one primes of ; these two problems are actually interlinked since it turns out that 〈0〉 is a primitive ideal of whenever has only finitely many height one primes. Finally, we compute the automorphism group of in the case where m ≠ n. In order to do this, we first study the action of this group on the prime spectrum of . Then, by using the preferred basis of and PBW bases, we prove that the automorphism group of is isomorphic to the torus when m ≠ n and (m,n) ≠ (1, 3),(3, 1). This research was supported by a Marie Curie Intra-European Fellowship within the 6th European Community Framework Programme and by Leverhulme Research Interchange Grant F/00158/X.     

Computing Minimum-Volume Enclosing Axis-Aligned Ellipsoids

Given a set of points and ε>0, we propose and analyze an algorithm for the problem of computing a (1+ε)-approximation to the minimum-volume axis-aligned ellipsoid enclosing . We establish that our algorithm is polynomial for fixed ε. In addition, the algorithm returns a small core set , whose size is independent of the number of points m, with the property that the minimum-volume axis-aligned ellipsoid enclosing is a good approximation of the minimum-volume axis-aligned ellipsoid enclosing . Our computational results indicate that the algorithm exhibits significantly better performance than the theoretical worst-case complexity estimate. This work was supported in part by the National Science Foundation through CAREER Grants CCF-0643593 and DMI-0237415.     

Semicrossed products of simple C*-algebras

Let and be C*-dynamical systems and assume that is a separable simple C*-algebra and that α and β are *-automorphisms. Then the semicrossed products and are isometrically isomorphic if and only if the dynamical systems and are outer conjugate. K. R. Davidson was partially supported by an NSERC grant. E. G. Katsoulis was partially supported by a summer grant from ECU     

Kaplansky classes and derived categories

We put a monoidal model category structure on the category of chain complexes of quasi-coherent sheaves over a quasi-compact and semi-separated scheme X. The approach generalizes and simplifies the method used by the author in (Trans Am Math Soc 356(8) 3369–3390, 2004) and (Trans Am Math Soc 358(7), 2855–2874, 2006) to build monoidal model structures on the category of chain complexes of modules over a ring and chain complexes of sheaves over a ringed space. Indeed, much of the paper is dedicated to showing that in any Grothendieck category , any nice enough class of objects induces a model structure on the category Ch() of chain complexes. The main technical requirement on is the existence of a regular cardinal κ such that every object satisfies the following property: Each κ-generated subobject of F is contained in another κ-generated subobject S for which . Such a class is called a Kaplansky class. Kaplansky classes first appeared in Enochs and López-Ramos (Rend Sem Mat Univ Padova 107, 67–79, 2002) in the context of modules over a ring R. We study in detail the connection between Kaplansky classes and model categories. We also find simple conditions to put on which will guarantee that our model structure is monoidal. We will see that in several categories the class of flat objects form such Kaplansky classes, and hence induce monoidal model structures on the associated chain complex categories. We will also see that in any Grothendieck category , the class of all objects is a Kaplansky class which induces the usual (non-monoidal) injective model structure on Ch().     

Boundary Limits for Bounded Quasiregular Mappings

In this paper we establish results on the existence of nontangential limits for weighted -harmonic functions in the weighted Sobolev space , for some q>1 and w in the Muckenhoupt A _q class, where is the unit ball in . These results generalize the ones in Sect. 3 of Koskela et al., Trans. Am. Math. Soc. 348(2), 755–766, 1996, where the weight was identically equal to one. Weighted -harmonic functions are weak solutions of the partial differential equation

Optimisation of Measures on a Hyperfinite Adapted Probability Space

The minimisation problem for a functional is considered, where is an ℝ ⁿ -valued stochastic process, defined on some filtered probability space , and P is an admissible probability measure in the sense that it obeys (1) some uniform equivalence condition with respect to the given measure ℙ on Γ, and (2) a finite number (possibly zero) of arbitrarily given other conditions that require the expectation (with respect to P) of some continuous bounded function φ of , for t ₁,…,t _k ∈[0,1], to lie within some closed set. We assume that u can be formulated through finite compositions of conditional expectations and bounded continuous functions. Under the assumption of |φ| being uniformly bounded from below and some condition on the dimension of , the existence of a solution on hyperfinite adapted probability spaces, as well as its minimality among admissible measures on any other adapted probability space, is proven. Also, a coarseness result for the Loeb operation is established. The main result of this paper, however, is a “standard result”: It does not include any reference to nonstandard analysis and can be perfectly understood without any familiarity with nonstandard analysis.