Regular differential forms on stratified spaces

Given a stratified space W, we present for each perversity a complex of differential forms which are defined on any stratum, are “regular” with respect to a fixed family of trivializations of W and whose cohomology is dual to the intersection homology of W for the complementary perversity . Lavoro svolto nell'ambito del GNSAGA del CNR con contributo MURST fondi 40%     

Real interpolation of spaces of differential forms

In this paper, we study interpolation of Hilbert spaces of differential forms using the real method of interpolation. We show that the scale of fractional order Sobolev spaces of differential l-forms in H ^s with exterior derivative in H ^s can be obtained by real interpolation. Our proof heavily relies on the recent discovery of smoothed Poincaré lifting for differential forms [M. Costabel and A. McIntosh, On Bogovskii and regularized Poincare integral operators for de Rham complexes on Lipschitz domains, Math. Z. 265(2): 297–320, 2010]. They enable the construction of universal extension operators for Sobolev spaces of differential forms, which, in turns, pave the way for a Fourier transform based proof of equivalences of K-functionals.     

Dual spaces of local Morrey-type spaces

In this paper we show that associated spaces and dual spaces of the local Morrey-type spaces are so called complementary local Morrey-type spaces. Our method is based on an application of multidimensional reverse Hardy inequalities.     

Normal forms for rank two linear irregular differential equations and moduli spaces

Periodica Mathematica Hungarica - We provide a unique normal form for rank two irregular connections on the Riemann sphere. In fact, we provide a birational model where we introduce apparent...     

Stability of Hodge decompositions in finite element spaces of differential forms in arbitrary dimension

We elaborate on the interpretation of some mixed finite element spaces in terms of differential forms. In particular we define regularization operators which, combined with the standard interpolators, enable us to prove discrete Poincaré–Friedrichs inequalities and discrete Rellich compactness for finite element spaces of differential forms of arbitrary degree on compact manifolds of arbitrary dimension.     

Dual affine-metrically connected spaces

In this paper we study the dual geometry of a normalized affinely connected space A _n,n . In particular, we consider the dual affine-metrically connected spaces \(\mathop M\limits^p _{n,n} \) induced by a nondegenerate normalization of an affine-metrically connected space M _n,n .     

Finite-dimensional period spaces for the spaces of cusp forms

It is shown that each bounded linear operator on a separable Hilbert space which generates a finite type I von Neumann algebra has, up to unitary equivalence, a unique representation as a direct integral of inflations of mutually unitarily inequivalent irreducible matrices. This leads to a simplification of the so-called central decomposition and the multiplicity theory for such operators.     

Killing forms on symmetric spaces

Killing forms on Riemannian manifolds are differential forms whose covariant derivative is totally skew-symmetric. We show that a compact simply connected symmetric space carries a non-parallel Killing p-form (p?2) if and only if it isometric to a Riemannian product Sk×N, where Sk is a round sphere and k>p.     

Positive forms on Banach spaces

The first representation theorem establishes a correspondence between positive, self-adjoint operators and closed, positive forms on Hilbert spaces. The aim of this paper is to show that some of the results remain true if the underlying space is a reflexive Banach space. In particular, the construction of the Friedrichs extension and the form sum of positive operators can be carried over to this case. This revised version was published online in June 2006 with corrections to the Cover Date.     

Normal forms in matrix spaces

For the groupGL(m, C)xGL(n, C) acting on the space ofmxn matrices over C, we introduce a class of subgroups which we call admissible. We suggest an algorithm to reduce an arbitrary matrix to a normal form with respect to an action of any admissible group. This algorithm covers various classification problems, including the wild problem of bringing a pair of matrices to normal form by simultaneous similarity. The classical left, right, two-sided and similarity transformations turns out to be admissible. However, the stabilizers of known normal forms (Smith's, Jordan's), generally speaking, are not admissible, and this obstructs inductive steps of our algorithm. This is the reason that we introduce modified normal forms for classical actions.Partially supported by Israel Science Foundation     

Decomposition of spaces of modular forms

Structural theorems for spaces of modular forms with respect to congruence subgroups are proved. The Dedekind η-function plays an important role in our study.     

Finite element differential forms

A differential form is a field which assigns to each point of a domain an alternating multilinear form on its tangent space. The exterior derivative operation, which maps differential forms to differential forms of the next higher order, unifies the basic first order differential operators of calculus, and is a building block for a great variety of differential equations. When discretizing such differential equations by finite element methods, stable discretization depends on the development of spaces of finite element differential forms. As revealed recently through the finite element exterior calculus, for each order of differential form, there are two natural families of finite element subspaces associated to a simplicial triangulation. In the case of forms of order zero, which are simply functions, these two families reduce to one, which is simply the well-known family of Lagrange finite element subspaces of the first order Sobolev space. For forms of degree 1 and of degree n − 1 (where n is the space dimension), we obtain two natural families of finite element subspaces, unifying many of the known mixed finite element spaces developed over the last decades. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)