Data-sparse approximation to the operator-valued functions of elliptic operator

In previous papers the arithmetic of hierarchical matrices has been described, which allows us to compute the inverse, for instance, of finite element stiffness matrices discretising an elliptic operator The required computing time is up to logarithmic factors linear in the dimension of the matrix. In particular, this technique can be used for the computation of the discrete analogue of a resolvent

In the present paper, we consider various operator functions, the operator exponential negative fractional powers , the cosine operator function and, finally, the solution operator of the Lyapunov equation. Using the Dunford-Cauchy representation, we get integrals which can be discretised by a quadrature formula which involves the resolvents mentioned above. We give error estimates which are partly exponentially, partly polynomially decreasing.

     

Metrically generated theories

Many examples are known of natural functors describing the transition from categories of generalized metric spaces to the ``metrizable" objects in some given topological construct . If preserves initial morphisms and if is initially dense in , then we say that is -metrically generated. Our main theorem proves that is -metrically generated if and only if can be isomorphically described as a concretely coreflective subconstruct of a model category with objects sets structured by collections of generalized metrics in and natural morphisms. This theorem allows for a unifying treatment of many well-known and varied theories. Moreover, via suitable comparison functors, the various relationships between these theories are studied.

     

Spaces of type and the Hankel convolution

In this paper we introduce new function spaces that are denoted by , -1/2$"> and and that are spaces of type where the Hankel convolution and the Hankel transformation are defined. The spaces will play the same role in the Hankel setting that the spaces play in the theory of Fourier transformation.     

Fundamental groups of moduli and the Grothendieck-Teichmüller group

Let denote the moduli space of Riemann spheres with ordered marked points. In this article we define the group of quasi-special symmetric outer automorphisms of the algebraic fundamental group for all to be the group of outer automorphisms respecting the conjugacy classes of the inertia subgroups of and commuting with the group of outer automorphisms of obtained by permuting the marked points. Our main result states that is isomorphic to the Grothendieck-Teichmüller group for all .

     

The -invariant and Gorensteinness of graded rings associated to filtrations of ideals in regular local rings

Let be a regular local ring and let be a filtration of ideals in such that is a Noetherian ring with . Let and let be the -invariant of . Then the theorem says that is a principal ideal and for all if and only if is a Gorenstein ring and . Hence , if is a Gorenstein ring, but the ideal is not principal.

     

Boundary relations and their Weyl families

The concepts of boundary relations and the corresponding Weyl families are introduced. Let be a closed symmetric linear operator or, more generally, a closed symmetric relation in a Hilbert space , let be an auxiliary Hilbert space, let

and let be defined analogously. A unitary relation from the Krein space to the Krein space is called a boundary relation for the adjoint if . The corresponding Weyl family is defined as the family of images of the defect subspaces , , under . Here need not be surjective and is even allowed to be multi-valued. While this leads to fruitful connections between certain classes of holomorphic families of linear relations on the complex Hilbert space and the class of unitary relations , it also generalizes the notion of so-called boundary value space and essentially extends the applicability of abstract boundary mappings in the connection of boundary value problems. Moreover, these new notions yield, for instance, the following realization theorem: every -valued maximal dissipative (for ) holomorphic family of linear relations is the Weyl family of a boundary relation, which is unique up to unitary equivalence if certain minimality conditions are satisfied. Further connections between analytic and spectral theoretical properties of Weyl families and geometric properties of boundary relations are investigated, and some applications are given.

     

An ultrafilter with property

Let be a -supercompact cardinal. We show that carries a normal ultrafilter with a property introduced by Menas. With it we give a transparent proof of Kamo's theorem that carries a normal ultrafilter with the partition property.

     

An order characterization of commutativity for -algebras

In this paper, we investigate the problem of when a -algebra is commutative through operator-monotonic increasing functions. The principal result is that the function is operator-monotonic increasing on a -algebra if and only if is commutative. Therefore, -algebra is commutative if and only if in for all positive elements in .

     

Analytic contractions, nontangential limits, and the index of invariant subspaces

Let be a Hilbert space of analytic functions on the open unit disc such that the operator of multiplication with the identity function defines a contraction operator. In terms of the reproducing kernel for we will characterize the largest set such that for each , the meromorphic function has nontangential limits a.e. on . We will see that the question of whether or not has linear Lebesgue measure 0 is related to questions concerning the invariant subspace structure of .

We further associate with a second set , which is defined in terms of the norm on . For example, has the property that for all if and only if has linear Lebesgue measure 0.

It turns out that a.e., by which we mean that has linear Lebesgue measure 0. We will study conditions that imply that a.e.. As one corollary to our results we will show that if dim and if there is a such that for all and all we have , then a.e. and the following four conditions are equivalent:

(1) for some ,

(2) for all , ,

(3) has nonzero Lebesgue measure,

(4) every nonzero invariant subspace of has index 1, i.e., satisfies dim .

     

Tensor products of -weakly closed nest algebra submodules

In this paper we prove that for any unital -weakly closed algebra which is -weakly generated by finite-rank operators in , every -weakly closed -submodule has . In the case of nest algebras, if are nests, we obtain the following -fold tensor product formula:

where each is the -weakly closed Alg -submodule determined by an order homomorphism from into itself.

     

Generating continuous mappings with Lipschitz mappings

If is a metric space, then and denote the semigroups of continuous and Lipschitz mappings, respectively, from to itself. The relative rank of modulo is the least cardinality of any set where generates . For a large class of separable metric spaces we prove that the relative rank of modulo is uncountable. When is the Baire space , this rank is . A large part of the paper emerged from discussions about the necessity of the assumptions imposed on the class of spaces from the aforementioned results.

     

Hyperbolic mean growth of bounded holomorphic functions in the ball

We consider the hyperbolic Hardy class , . It consists of holomorphic in the unit complex ball for which and

where denotes the hyperbolic distance of the unit disc. The hyperbolic version of the Littlewood-Paley type -function and the area function are defined in terms of the invariant gradient of , and membership of is expressed by the property of the functions. As an application, we can characterize the boundedness and the compactness of the composition operator , defined by , from the Bloch space into the Hardy space .

     

Transitive families of projections in factors of type

We introduce a notion of transitive family of subspaces relative to a type factor, and hence a notion of transitive family of projections in such a factor. We show that whenever is a factor of type and is generated by two self-adjoint elements, then contains a transitive family of projections. Finally, we exhibit a free transitive family of projections that generate a factor of type .

     

Flat covers and cotorsion envelopes of sheaves

In this paper we prove that any sheaf of modules over any topological space (in fact, any -module where is a sheaf of rings on the topological space) has a flat cover and a cotorsion envelope. This result is very useful, as we shall explain later in the introduction, in order to compute cohomology, due to the fact that the category of sheaves ( -modules) does not have in general enough projectives.

     

Minimal sufficiency of order statistics in convex models

Let be a convex and dominated statistical model on the measurable space , with minimal sufficient, and let . Then , the -algebra of all permutation invariant sets belonging to the -fold product -algebra , is shown to be minimal sufficient for the corresponding model for independent observations, .

The main technical tool provided and used is a functional analogue of a theorem of Grzegorek (1982) concerning generators of .

     

Critical points of the area functional of a complex closed curve on the manifold of Kähler metrics

We consider a compact complex manifold of dimension that admits Kähler metrics and we assume that is a closed complex curve. We denote by the space of classes of Kähler forms that define Kähler metrics of volume 1 on and define by . We show how the Riemann-Hodge bilinear relations imply that any critical point of is the strict global minimum and we give conditions under which there is such a critical point : A positive multiple of is the Poincaré dual of the homology class of . Applying this to the Abel-Jacobi map of a curve into its Jacobian, , we obtain that the Theta metric minimizes the area of within all Kähler metrics of volume 1 on .

     

Singularities of the hypergeometric system associated with a monomial curve

We compute, using -module restrictions, the slopes of the irregular hypergeometric system associated with a monomial curve. We also study rational solutions and reducibility of such systems.

     

Modified Taylor reproducing formulas and h-p clouds

We study two different approximations of a multivariate function by operators of the form , where is an -reproducing partition of unity and are modified Taylor polynomials of degree expanded at . The first approximation was introduced by Xuli (2003) in the univariate case and generalized for convex domains by Guessab et al. (2005). The second one was introduced by Duarte (1995) and proved in the univariate case. In this paper, we first relax the Guessab's convexity assumption and we prove Duarte's reproduction formula in the multivariate case. Then, we introduce two related reproducing quasi-interpolation operators in Sobolev spaces. A weighted error estimate and Jackson's type inequalities for h-p cloud function spaces are obtained. Last, numerical examples are analyzed to show the approximative power of the method.

     

Enriched Reedy categories

We define the notion of an enriched Reedy category and show that if is a -Reedy category for some symmetric monoidal model category and is a -model category, the category of -functors and -natural transformations from to is again a model category.