A noncommutative symmetric system over the Grossman-Larson Hopf algebra of labeled rooted trees

In this paper, we construct explicitly a noncommutative symmetric ( CS) system over the Grossman-Larson Hopf algebra of labeled rooted trees. By the universal property of the CS system formed by the generating functions of certain noncommutative symmetric functions, we obtain a specialization of noncommutative symmetric functions by labeled rooted trees. Taking the graded duals, we also get a graded Hopf algebra homomorphism from the Connes-Kreimer Hopf algebra of labeled rooted forests to the Hopf algebra of quasi-symmetric functions. A connection of the coefficients of the third generating function of the constructed CS system with the order polynomials of rooted trees is also given and proved.     

Admissible Galois Structures and Coverings in Regular Mal’cev Categories

Given a regular Gumm category such that any regular epimorphism is effective for descent, we prove that any Birkhoff subcategory in gives rise to an admissible Galois structure. This result allows one to consider some new applications of the categorical Galois theory in the context of topological algebras. Given a regular Mal’cev category , we first characterize the coverings of the Galois structure induced by the subcategory of the abelian objects in . Then we consider as a subcategory of the category of the equivalence relations in , and we characterize the coverings of the corresponding Galois structure . By composing the Galois structures and we obtain the Galois structure induced by as a subcategory of . We give the characterization of the -coverings in terms of the coverings of and .     

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.     

On cyclotomic schemes over finite near-fields

We introduce a concept of cyclotomic association scheme over a finite near-field . It is proved that any isomorphism of two such nontrivial schemes is induced by a suitable element of the group AGL(V), where V is the linear space associated with . A sufficient condition on a cyclotomic scheme that guarantee the inclusion where is a finite field with elements, is given. I. Ponomarenko partially supported by RFFI, grants 03-01-00349, NSH-2251.2003.1.     

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.     

Reduction of exterior differential systems with infinite dimensional symmetry groups

A symmetry-based method for constructing solutions to systems of differential equations founded on the reduction of exterior differential systems invariant under the action of an infinite dimensional pseudogroup is proposed. One can associate to any system of differential equations Δ=0 with a symmetry group an exterior differential system invariant under so that solutions of Δ=0 correspond to integral manifolds . The -invariant exterior differential system gives rise to a reduced system specified on a cross section to the pseudogroup orbits, and it is shown that solutions to Δ=0 can be reconstructed from integral manifolds by solving an equation of generalized Lie type for the jets of pseudogroup transformations. In particular, as opposed to the classical method of symmetry reduction, every solution to the system of differential equations can, under some mild regularity assumptions, be constructed by the present algorithm. AMS subject classification (2000)  58A15, 58A20, 58H05, 58J70     

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).     

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.     

Recursively enumerable sets of polynomials over a finite field are Diophantine

We construct a Diophantine interpretation of over . Using this together with a previous result that every recursively enumerable (r.e.) relation over is Diophantine over , we will prove that every r.e. relation over is Diophantine over . We will also look at recursive infinite base fields , algebraic over . It turns out that the Diophantine relations over are exactly the relations which are r.e. for every recursive presentation.     

Rigidity of hyperstable complexes

A module J over a ring is said to be hyperstable when . Over a module M for which Ext we show that the projective n-stems for which is hyperstable constitute a single homotopy type. Received: 17 November 2006     

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.     

Positive elimination in valued fields

Let K be an algebraically closed field with a valuation ring or a real closed field with a convex valuation ring . We show that the projection of a basic (see “Introduction”) subset of to K ⁿ is again basic.     

Symmetric groups and expander graphs

We construct explicit generating sets S _n and of the alternating and the symmetric groups, which turn the Cayley graphs and into a family of bounded degree expanders for all n.     

Norm-preserving extensions of functionals and denting points of convex sets

Let W and Z be Banach spaces, and let and be closed subspaces. Let be a subspace of , the Banach space of bounded linear operators from W* to Z**, containing . We describe, for and , all norm-preserving extensions of to the space in terms of convergence of convex combinations. We also characterize denting points of bounded convex subsets of Banach spaces in similar terms. Various applications are presented. Supported by Estonian Science Foundation Grant 5704.     

Schur–Weyl Reciprocity between the Quantum Superalgebra and the Iwahori–Hecke Algebra

In this paper, we establish Schur–Weyl reciprocity between the quantum general super Lie algebra and the Iwahori–Hecke algebra . We introduce the sign -permutation representation of on the tensor space of dimensional -graded -vector space . This action commutes with that of derived from the vector representation on . Those two subalgebras of satisfy Schur–Weyl reciprocity. As special cases, we obtain the super case (), and the quantum case (). Hence this result includes both the super case and the quantum case, and unifies those two important cases.Presented by A. Verschoren.     

Representation theory of $\mathcal{W}$-algebras

We study the representation theory of the -algebra associated with a simple Lie algebra at level k. We show that the “-” reduction functor is exact and sends an irreducible module to zero or an irreducible module at any level k∈ℂ. Moreover, we show that the character of each irreducible highest weight representation of is completely determined by that of the corresponding irreducible highest weight representation of affine Lie algebra of . As a consequence we complete (for the “-” reduction) the proof of the conjecture of E. Frenkel, V. Kac and M. Wakimoto on the existence and the construction of the modular invariant representations of -algebras. Mathematics Subject Classification (1991)  17B68, 81R10     

Reduced Minimum Modulus Preserving in Banach Space

Let be the algebra of all bounded linear operators on a complex Banach space X and γ(T) be the reduced minimum modulus of operator . In this work, we prove that if , is a surjective linear map such that is an invertible operator, then , for every , if and only if, either there exist two bijective isometries and such that for every , or there exist two bijective isometries and such that for every . This generalizes for a Banach space the Mbekhta’s theorem [12].      

Homotopy Method for a General Multiobjective Programming Problem

In this paper, we study a class of vector minimization problems on a complete metric space X which is identified with the corresponding complete metric space of objective functions . We do not impose any compactness assumption on X. We show that, for most (in the sense of Baire category) functions , the corresponding vector optimization problem has a solution.