Kummer subfields of tame division algebras over Henselian fields

By generalizing the method used by Tignol and Amitsur in [J.-P. Tignol, S.A. Amitsur, Kummer subfields of Malcev-Neumann division algebras, Israel Journal of Math. 50 (1985), 114-144], we determine necessary and sufficient conditions for an arbitrary central division algebra D over a Henselian valued field E to have Kummer subfields when the characteristic of the residue field of E does not divide the degree of D. We prove also that if D is a semiramified division algebra of degree n [resp., of prime power degree p^r] over E such that does not divide n and [resp., and p³ divides ], then D is non-cyclic [resp., D is not an elementary abelian crossed product].     

Strong continuity of generalized Feynman-Kac semigroups: Necessary and sufficient conditions

Let (E,D(E)) be a strongly local, quasi-regular symmetric Dirichlet form on L²(E;m) and ((Xt)_t?0,(Px)_x∈E) the diffusion process associated with (E,D(E)). For u∈De(E), u has a quasi-continuous version and has Fukushima's decomposition: , where is the martingale part and is the zero energy part. In this paper, we study the strong continuity of the generalized Feynman-Kac semigroup defined by , t?0. Two necessary and sufficient conditions for to be strongly continuous are obtained by considering the quadratic form (Qu,Db(E)), where Qu(f,f):=E(f,f)+E(u,f²) for f∈Db(E), and the energy measure μ〈u〉 of u, respectively. An example is also given to show that is strongly continuous when μ〈u〉 is not a measure of the Kato class but of the Hardy class with the constant (cf. Definition 4.5).     

Strong Mori and Noetherian properties of integer-valued polynomial rings

Let D be a domain with quotient field K and let Int(D) be the ring of integer-valued polynomials {f∈K[X]|f(D)⊆D}. We give conditions on D so that the ring Int(D) is a Strong Mori domain. In particular, we give a complete characterization in the case that the conductor is nonzero, where D′ is the integral closure of D. We also show that when D is quasilocal with or D is Noetherian, Int(D) is a Strong Mori domain if and only if Int(D) is Noetherian.     

Chen-Ruan cohomology of some moduli spaces of parabolic vector bundles

Let (X,D) be an ?-pointed compact Riemann surface of genus at least two. For each point x∈D, fix parabolic weights such that . Fix a holomorphic line bundle ξ over X of degree one. Let PMξ denote the moduli space of stable parabolic vector bundles, of rank two and determinant ξ, with parabolic structure over D and parabolic weights . The group of order two line bundles over X acts on PMξ by the rule E_∗⊗L?E_∗⊗L. We compute the Chen-Ruan cohomology ring of the corresponding orbifold.     

Affine Demazure modules and T-fixed point subschemes in the affine Grassmannian

Let G be a simple algebraic group defined over C and T be a maximal torus of G. For a dominant coweight λ of G, the T-fixed point subscheme of the Schubert variety in the affine Grassmannian GrG is a finite scheme. We prove that for all such λ if G is of type A or D and for many of them if G is of type E, there is a natural isomorphism between the dual of the level one affine Demazure module corresponding to λ and the ring of functions (twisted by certain line bundle on GrG) of . We use this fact to give a geometrical proof of the Frenkel-Kac-Segal isomorphism between basic representations of affine algebras of A,D,E type and lattice vertex algebras.     

Broadcasting algorithms in radio networks with unknown topology

In this paper we present new randomized and deterministic algorithms for the classical problem of broadcasting in radio networks with unknown topology. We consider directed n-node radio networks with specified eccentricity D (maximum distance from the source node to any other node). Bar-Yehuda et al. presented an algorithm that for any n-node radio network with eccentricity D completes the broadcasting in time, with high probability. This result is almost optimal, since as it has been shown by Kushilevitz and Mansour and Alon et al., every randomized algorithm requires Ω(Dlog(n/D)+log²n) expected time to complete broadcasting.Our first main result closes the gap between the lower and upper bound: we describe an optimal randomized broadcasting algorithm whose running time complexity is , with high probability. In particular, we obtain a randomized algorithm that completes broadcasting in any n-node radio network in time , with high probability.The main source of our improvement is a better “selecting sequence” used by the algorithm that brings some stronger property and improves the broadcasting time. Two types of “selecting sequences” are considered: randomized and deterministic ones. The algorithm with a randomized sequence is easier (more intuitive) to analyze but both randomized and deterministic sequences give algorithms of the same asymptotic complexity.Next, we demonstrate how to apply our approach to deterministic broadcasting, and describe a deterministic oblivious algorithm that completes broadcasting in time , which improves upon best known algorithms in this case. The fastest previously known algorithm had the broadcasting time of , it was non-oblivious and significantly more complicated; our algorithm can be seen as a natural extension of our randomized algorithm. In this part of the paper we assume that each node knows the eccentricity D.Finally, we show how our randomized broadcasting algorithm can be used to improve the randomized complexity of the gossiping problem.     

Direct summands of direct sums of modules whose endomorphism rings have two maximal right ideals

Let M₁,…,M_n be right modules over a ring R. Suppose that the endomorphism ring of each module M_i has at most two maximal right ideals. Is it true that every direct summand of M₁⊕?⊕M_n is a direct sum of modules whose endomorphism rings also have at most two maximal right ideals? We show that the answer is negative in general, but affirmative under further hypotheses. The endomorphism ring of uniserial modules, that is, the modules whose lattice of submodules is linearly ordered under inclusion, always has at most two maximal right ideals, and Pavel P?íhoda showed in 2004 that the answer to our question is affirmative for direct sums of finitely many uniserial modules.     

The realization of hyperelliptic curves through endomorphisms of Kronecker modules

Let K be an algebraically closed field and A the Kronecker algebra over K. A general problem is to study the endomorphism algebras of A-modules M that are extensions of finite-dimensional, torsion-free, rank-one A-modules P, by infinite-dimensional, torsion-free, rank-one A-modules N. Such endomorphism algebras can be studied by means of a quadratic polynomial f(Y) in one variable Y over the rational function field K(X). We call this f(Y) the regulator of the extension. We prove that if the regulator has non-zero discriminant, then is a Noetherian, commutative K-algebra. We also prove that, subject to a regulator with non-zero discriminant, is affine over K if and only if End N is affine, in which case is the coordinate ring of a hyperelliptic curve.     

Sheaf coalgebras and duality

In this paper we construct an adjoint pair of functors between the category of sheaves on a smooth manifold M and the category of coalgebras over the ring of smooth functions with compact support on M. We show that the sheaf coalgebra associated to a sheaf E on M determines the sheaf E uniquely up to an isomorphism, and that the adjoint functors restrict to an equivalence between sheaves on M and sheaf coalgebras over .     

Inducing regulation of any digraphs

For a given structure D (digraph, multidigraph, or pseudodigraph) and an integer r large enough, a smallest inducing r-regularization of D is constructed. This regularization is an r-regular superstructure of the smallest possible order with bounded arc multiplicity, and containing D as an induced substructure. The sharp upper bound on the number, ρ, of necessary new vertices among such superstructures for n-vertex general digraphs D is determined, ρ being called the inducing regulation number of D. For being the maximum among semi-degrees in D, simple n-vertex digraphs D with largest possible ρ are characterized if either or (where the case is not a trivial subcase of ).     

On spaces which are D, linearly D and transitively D

In this note, we comment on D-spaces, linearly D-spaces and transitively D-spaces. We show that every meta-Lindelöf space is transitively D. If X is a weak -refinable TD-scattered space, then X is transitively D, where TD is the class of all transitively D-spaces. If X is a weak -refinable -scattered space, then X is a D-space, where is the class of all D-spaces, and hence every weak -refinable (or submetacompact) scattered space is a D-space. This gives a positive answer to a question mentioned by Martínez and Soukup. In the last part of this note, we show that if X is a weak -refinable space then X is linearly D.     

Crossed-products by finite index endomorphisms and KMS states

Given a unital -algebra A, an injective endomorphism preserving the unit, and a conditional expectation E from A to the range of α we consider the crossed-product of A by α relative to the transfer operator L=α⁻¹E. When E is of index-finite type we show that there exists a conditional expectation G from the crossed-product to A which is unique under certain hypothesis. We define a “gauge action” on the crossed-product algebra in terms of a central positive element h and study its KMS states. The main result is: if h>1 and E(ab)=E(ba) for all a,b∈A (e.g. when A is commutative) then the KMS_β states are precisely those of the form ψ=φ°G, where φ is a trace on A satisfying the identity

     

On Bernardi's integral operator and the Briot-Bouquet differential subordination

Let A,B,D,E∈[−1,1]. Conditions on A,B,D and E are determined so that

     

Note on the rank of quadratic twists of Mordell equations

Let E be the elliptic curve given by a Mordell equation y²=x³−A where A∈Z. Michael Stoll found a precise formula for the size of a Selmer group of E for certain values of A. For D∈Z, let ED denote the quadratic twistDy²=x³−A. We use Stoll's formula to show that for a positive square-free integer A≡1 or and for a nonnegative integer k, we can compute a lower bound for the proportion of square-free integers D up to X such that rankED(Q)?2k. We also compute an upper bound for a certain average rank of quadratic twists of E.     

Integral generators for the cohomology ring of moduli spaces of sheaves over Poisson surfaces

Let M be a smooth and compact moduli space of stable coherent sheaves on a projective surface S with an effective (or trivial) anti-canonical line bundle. We find generators for the cohomology ring of M, with integral coefficients. When S is simply connected and a universal sheaf E exists over S×M, then its class [E] admits a Künneth decomposition as a class in the tensor product of the topological K-rings. The generators are the Chern classes of the Künneth factors of [E] in . The general case is similar.     

Periodic polynomial of trace maps

Let σ be an endomorphism of the free group on two generators and Φσ the trace map associated with σ. A polynomial P is said to be periodic for σ if, for some positive integer n, it is invariant under , i.e., . In this note we study the structure of the ring of periodic polynomials for σ.     

Isochronous centers of Lienard type equations and applications

In this work we study Eq. (E) with a center at 0 and investigate conditions of its isochronicity. When f and g are analytic (not necessary odd) a necessary and sufficient condition for the isochronicity of 0 is given. This approach allows us to present an algorithm for obtained conditions for a point of (E) to be an isochronous center. In particular, we find again by another way the isochrones of the quadratic Loud systems (LD,F). We also classify a 5-parameters family of reversible cubic systems with isochronous centers.