On a sheaf-theoretic version of the Witt’s decomposition theorem. A Lagrangian perspective

The approach to a counterpart, in Abstract Geometric Algebra, that is, Geometric Algebra via sheaves of modules, of the classical Witt’s decomposition theoremis based on the axiomatization of the classical context, which however leads to the formulation of a specific subcategory of the category of sheaves of modules: the full subcategory of convenient sheaves of modules. Convenient sheaves of modules turn out, by the very essence of the matter at hand, to be of further importance as far as the setting of results leading to the sheaf-theoretic aspect of several forms of the Witt’s theorem is concerned. Further versions of the Witt’s theorem are still to be treated elsewhere.      

The Averaged Robbins – Monro Method for Linear Problems in a Banach Space

We consider a recursive method of Robbins–Monro type to solve the linear problem Ax=V in a Banach space. The bounded linear operator A and the vector V are assumed to be observable with some noise only. According to Polyak and Ruppert we use gains converging to zero slower than 1/n and take the average of the iterates as an estimator for the solution of the linear problem. Under weak conditions on the noise processes almost sure and distributional invariance principles are shown.     

Maximal L p –L q Regularity for Parabolic Partial Differential Equations on Manifolds with Cylindrical Ends

We give a short, simple proof of maximal L^p–L^q regularity for linear parabolic evolution equations on manifolds with cylindrical ends by making use of pseudodifferential parametrices and the concept of -boundedness for the resolvent.      

The G-Fredholm Property of the \bar{\partial} -Neumann Problem

Let G be a unimodular Lie group, X a compact manifold with boundary, and M be the total space of a principal bundle G→M→X so that M is also a strongly pseudoconvex complex manifold. In this work, we show that if G acts by holomorphic transformations in M, then the Laplacian on M has the following properties: The kernel of □ restricted to the forms Λ ^p,q with q>0 is a closed, G-invariant subspace in L ²(M,Λ ^p,q ) of finite G-dimension. Secondly, we show that if q>0, then the image of □ contains a closed, G-invariant subspace of finite G-codimension in L ²(M,Λ ^p,q ). These two properties taken together amount to saying that □ is a G-Fredholm operator. It is a corollary of the first property mentioned that the reduced L ²-Dolbeault cohomology spaces of M are finite G-dimensional for q>0. The boundary Laplacian □ _b has similar properties.      

On the size of the Jacobians of curves over finite fields

Abdract  Given a smooth curve of genus g ≥ 1 which admits a smooth projective embedding of dimension m over the ground field of q elements, we obtain the asymptotic formula q ^g+o(g) for the size of set of the -rational points on its Jacobian in the case when m and q are bounded and g → ∞. We also obtain a similar result for curves of bounded gonality. For example, this applies to the Jacobian of a hyperelliptic curve of genus g → ∞.      

The geometric invariants $${\Omega^{n}}$$ of a product of groups

We compute the geometric invariants of a product G × H of groups in terms of and . This gives a sufficient condition in terms of and for a normal subgroup of G × H with abelian quotient to be of type F _n . We give an example involving the direct product of the Baumslag–Solitar group BS_1,2 with itself.      

On unitary representability of topological groups

We prove that the additive group (E*, τ _k (E)) of an -Banach space E, with the topology τ _k (E) of uniform convergence on compact subsets of E, is topologically isomorphic to a subgroup of the unitary group of some Hilbert space (is unitarily representable). This is the same as proving that the topological group (E*, τ _k (E)) is uniformly homeomorphic to a subset of for some κ. As an immediate consequence, preduals of commutative von Neumann algebras or duals of commutative C*-algebras are unitarily representable in the topology of uniform convergence on compact subsets. The unitary representability of free locally convex spaces (and thus of free Abelian topological groups) on compact spaces, follows as well. The above facts cannot be extended to noncommutative von Neumann algebras or general Schwartz spaces. Research partially supported by Spanish Ministry of Science, grant MTM2008-04599/MTM. The foundations of this paper were laid during the author’s stay at the University of Ottawa supported by a Generalitat Valenciana grant CTESPP/2004/086.     

Dirichlet Problem at Infinity for $\mathcal A$-Harmonic Functions

We study the Dirichlet problem at infinity for -harmonic functions on a Cartan–Hadamard manifold M and give a sufficient condition for a point at infinity x ₀∈M(∞) to be -regular. This condition is local in the sense that it only involves sectional curvatures of M in a set U∩M, where U is an arbitrary neighborhood of x ₀ in the cone topology. The results apply to the Laplacian and p-Laplacian, 1<p<∞, as special cases.      

CR-Admissible $$\mathbb{{Z}}_{2}$$-gradations and CR-symmetries

Complementing the results of (Lotta and Nacinovich, Adv. Math. 191(1): 114–146, 2005), we show that the minimal orbit M of a real form G of a semisimple complex Lie group in a flag manifold is CR-symmetric (see (Kaup and Zaitsev Adv. Math. 149(2):145–181, 2000)) if and only if the corresponding CR algebra admits a gradation compatible with the CR structure.      

From light tails to heavy tails through multiplier

Let X and Y be two independent nonnegative random variables, of which X has a distribution belonging to the class or for some γ ≥ 0 and Y is unbounded. We study how their product XY inherits the tail behavior of X. Under some mild technical assumptions we prove that the distribution of XY belongs to the class or accordingly. Hence, the multiplier Y builds a bridge between light tails and heavy tails.      

Invariants of 2 × 2 matrices, irreducible {\hbox{SL}(2, \mathbb{C})} characters and the Magnus trace map

We obtain an explicit characterization of the stable points of the action of on the cartesian product G  ^{× n} by simultaneous conjugation on each factor in terms of the corresponding invariant functions. From this, a simple criterion for the irreducibility of representations of finitely generated groups into G is derived. We also obtain analogous results for the action of on the vector space of n-tuples of 2 × 2 complex matrices. For a free group F _n of rank n, we show how to generically reconstruct the 2 ^n-2 conjugacy classes of representations F _n → G from their values under the map considered in Magnus [Math. Zeit. 170, 91–103 (1980)], defined by certain 3n − 3 traces of words of length one and two.      

Inverse Conductivity Problem on Riemann Surfaces

An electrical potential U on a bordered Riemann surface X with conductivity function σ>0 satisfies equation d(σ d ^c U)=0. The problem of effective reconstruction of σ from electrical currents measurements (Dirichlet-to-Neumann mapping) on the boundary: U| _bX ↦ σ d ^c U| _bX is studied. We extend to the case of Riemann surfaces the reconstruction scheme given, firstly, by R. Novikov (Funkc. Anal. Ego Priloz. 22:11–22, 2008) for simply connected X. We apply for this new kernels for on the affine algebraic Riemann surfaces constructed in Henkin (, 2008).      

Schur–Weyl duality for higher levels

We extend Schur–Weyl duality to an arbitrary level l ≥ 1, level one recovering the classical duality between the symmetric and general linear groups. In general, the symmetric group is replaced by the degenerate cyclotomic Hecke algebra over parametrized by a dominant weight of level l for the root system of type A_∞. As an application, we prove that the degenerate analogue of the quasi-hereditary cover of the cyclotomic Hecke algebra constructed by Dipper, James and Mathas is Morita equivalent to certain blocks of parabolic category for the general linear Lie algebra.      

The \bar \partial -Neumann operator and commutators of the Bergman projection and multiplication operators

We prove that compactness of the canonical solution operator to restricted to (0, 1)-forms with holomorphic coefficients is equivalent to compactness of the commutator defined on the whole L _(0,1)²(Ω), where is the multiplication by and is the orthogonal projection of L _(0,1)²(Ω) to the subspace of (0, 1) forms with holomorphic coefficients. Further we derive a formula for the -Neumann operator restricted to (0, 1) forms with holomorphic coefficients expressed by commutators of the Bergman projection and the multiplications operators by z and . Partially supported by the FWF grant P19147-N13.     

On the Intertwining of \partial{\mathcal{D}}-Isometries

Let be a strictly pseudoconvex bounded domain in with C ² boundary . If a subnormal m-tuple T of Hilbert space operators has the spectral measure of its minimal normal extension N supported on , then T is referred to as a -isometry. Using some non-trivial approximation theorems in the theory of several complex variables, we establish a commutant lifting theorem for those -isometries whose (joint) Taylor spectra are contained in a special superdomain Ω of . Further, we provide a function-theoretic characterization of those subnormal tuples whose Taylor spectra are contained in Ω and that are quasisimilar to a certain (fixed) -isometry T (of which the multiplication tuple on the Hardy space of the unit ball in is a rather special example). Submitted: September 9, 2007. Revised: October 10, 2007. Accepted: October 24, 2007.     

Analytic properties in the spectrum of certain Banach algebras

We show a sufficient condition for a domain in to be a H ^∞-domain of holomorphy. Furthermore if a domain has the Gleason property at a point and the projection of the n − 1th order generalized Shilov boundary does not coincide with Ω then is schlicht. We also give two examples of pseudoconvex domains in which the spectrum is non-schlicht and satisfy several other interesting properties.      

On Attractors for Multivalued Semigroups Defined by Generalized Semiflows

In this paper we extend results from Semigroup Theory on existence and characterization of attractors in order to include multivalued semigroups T(t) defined by generalized semiflows . In particular we show that, if is continuous, possesses a Lyapunov function, and has a global attractor which is maximal compact invariant, then  =  W ^u (Z()), where Z() is the stationary solutions set and W ^u (Z()) is the unstable set of Z(). We introduce the -attractor concept which does not enjoy any uniformity on time of attraction and we prove, under suitable conditions, that the global -attractor is the set of asymptotic states described by Z(). Jacson Simsen is supported by CAPES-Brazil.     

The rectifiability threshold in abelian groups

For any abelian group G and integer t ≥ 2 we determine precisely the smallest possible size of a non-t-rectifiable subset of G. Specifically, assuming that G is not torsion-free, denote by p the smallest order of a non-zero element of G. We show that if a subset S ⊆ G satisfies |S| ≤ ⌌log _t p⌍, then S is t-isomorphic (in the sense of Freiman) to a set of integers; on the other hand, we present an example of a subset S ⊆ G with |S| = ⌌log _t p⌍ + 1 which is not t-isomorphic to a set of integers.     

The cubic Segre variety in PG(5, 2)

The Segre variety in PG(5, 2) is a 21-set of points which is shown to have a cubic equation Q(x) = 0. If T(x, y, z) denotes the alternating trilinear form obtained by completely polarizing the cubic polynomial Q, then the associate U ^# of an r-flat is defined to be

Fundamentals for symplectic $$ \mathcal{A} $$-modules. Affine Darboux theorem

In his [9–11], the first author shows that the sheaf-theoreti-cally based Abstract Differential Geometry incorporates and generalizes classical differential geometry. Here, we undertake to explore the implications of Abstract Differential Geometry to classical symplectic geometry. The full investigation will be presented elsewhere.