For every normed space , we note its closed unit ball and unit sphere by and , respectively. Let and be normed spaces such that is Lipschitz homeomorphic to , and is Lipschitz homeomorphic to .
We prove that the following are equivalent:
1. is Lipschitz homeomorphic to .
2. is Lipschitz homeomorphic to .
3. is Lipschitz homeomorphic to .
This result holds also in the uniform category, except (2 or 3) 1 which is known to be false.
Let and be finite groups and let be a hilbertian field. We show that if has a generic extension over and satisfies the arithmetic lifting property over , then the wreath product of and also satisfies the arithmetic lifting property over . Moreover, if the orders of and are relatively prime and is abelian, then any extension of by (which is necessarily a semidirect product) has the arithmetic lifting property.
A variety is a class of Banach algebras , for which there exists a family of laws such that is precisely the class of all Banach algebras which satisfies all of the laws (i.e. for all , . We say that is an -variety if all of the laws are homogeneous. A semivariety is a class of Banach algebras , for which there exists a family of homogeneous laws such that is precisely the class of all Banach algebras , for which there exists 0$"> such that for all homogeneous polynomials , , where . However, there is no variety between the variety of all -algebras and the variety of all -algebras, which can be defined by homogeneous laws alone. So the theory of semivarieties and the theory of varieties differ significantly. In this paper we shall construct uncountable chains and antichains of semivarieties which are not varieties.
We consider the problem of minimizing the energy of the maps from the annulus to such that is equal to for , and to , for , where is a fixed angle.
We prove that the minimum is attained at a unique harmonic map which is a planar map if , while it is not planar in the case \pi^2$">.
Moreover, we show that tends to as , where minimizes the energy of the maps from to , with the boundary condition , .
For an element of a commutative complex Banach algebra we investigate the following property: every complete norm on making the multiplication by from to itself continuous is equivalent to .
We consider constant symmetric tensors on , , and we study the problem of finding metrics conformal to the pseudo-Euclidean metric such that . We show that such tensors are determined by the diagonal elements and we obtain explicitly the metrics . As a consequence of these results we get solutions globally defined on for the equation Moreover, we show that for certain unbounded functions defined on , there are metrics conformal to the pseudo-Euclidean metric with scalar curvature .
A -algebra is said to have the FS-property if the set of all self-adjoint elements in has a dense subset of elements with finite spectrum. We shall show that this property is not stable under taking the minimal -tensor products even in case of separable nuclear -algebras.
We consider a class of compact spaces for which the space of probability Radon measures on has countable tightness in the topology. We show that that class contains those compact zero-dimensional spaces for which is weakly Lindelöf, and, under MA + CH, all compact spaces with having property (C) of Corson.
Let be an -dimensional normal projective variety with only Gorenstein, terminal, -factorial singularities. Let be an ample line bundle on . Let denote the nef value of . The classification of via the nef value morphism is given for the situations when satisfies or .
The author discusses the semilinear parabolic equation with . Under suitable assumptions on and , he proves that, if with , then the solutions are global, while if with 1$">, then the solutions blow up in a finite time, where is a positive solution of , with .
We show that a family of functions meromorphic in some domain is normal, if for all the derivative omits the value and if the values that can take at the zeros of satisfy certain restrictions. As an application we obtain a new proof of a theorem of Langley which classifies the functions meromorphic in the plane such that and have no zeros.
Given a topological system on a -compact Hausdorff space and its factor we show the existence of a largest topological factor containing such that for each -invariant measure , . When a relative variational principle holds, .
Let be a generalized frame in a separable Hilbert space indexed by a measure space , and assume its analysing operator is surjective. It is shown that is essentially discrete; that is, the corresponding index measure space can be decomposed into atoms such that is isometrically isomorphic to the weighted space of all sequences of complex numbers with , where This provides a new proof for the redundancy of the windowed Fourier transform as well as any wavelet family in .
Let be a self-similar probability measure on satisfying where 0$"> and Let be the Fourier transform of A necessary and sufficient condition for to approach zero at infinity is given. In particular, if and for then 0$"> if and only if is a PV-number and is not a factor of . This generalizes the corresponding theorem of Erdös and Salem for the case