Let be a field and . There exist a differential graded -module and various approximations to a differential on one of which gives a non-trivial deformation, another is obstructed, and another is unobstructed at order . The analogous problem in the category of -algebras in characteristic zero remains a long-standing open question.
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If is a smoothly bounded multiply-connected domain in the complex plane and where we show that is compact if and only if its Berezin transform vanishes at the boundary.
Let be the disk algebra. In this paper we address the following question: Under what conditions on the points do there exist operators such that
and , , for every ? Here the convergence is understood in the sense of norm in . Our first result shows that if satisfy Carleson condition, then there exists a function such that , . This is a non-trivial generalization of results of Somorjai (1980) and Partington (1997). It also provides a partial converse to a result of Totik (1984). The second result of this paper shows that if are required to be projections, then for any choice of the operators do not converge to the identity operator. This theorem generalizes the famous theorem of Faber and implies that the disk algebra does not have an interpolating basis.
We show that if is an -regular set in for which the triple integral of the Menger curvature is finite and if , then almost all of can be covered with countably many curves. We give an example to show that this is false for .
In this paper we consider the long time behavior of solutions of the initial value problem for semi-linear wave equations of the form
Here 0.$">
We prove that if m \ge 1,$"> then for any 0$"> there are choices of initial data from the energy space with initial energy such that the solution blows up in finite time. If we replace by , where is a sufficiently slowly decreasing function, an analogous result holds.
A commutative Banach algebra is said to have the property if the following holds: Let be a closed subspace of finite codimension such that, for every , the Gelfand transform has at least distinct zeros in , the maximal ideal space of . Then there exists a subset of of cardinality such that vanishes on , the set of common zeros of . In this paper we show that if is compact and nowhere dense, then , the uniform closure of the space of rational functions with poles off , has the property for all . We also investigate the property for the algebra of real continuous functions on a compact Hausdorff space.
Let be a self-similar probability measure on satisfying where 0$"> 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
Suppose is not a non-trivial linear combination of with non-negative coefficients. Then we describe the sector such that every interior integer point of the sector is a linear combination of over , but infinitely many points on each of its hyperfaces are not. For the result reduces to a formula of Sylvester corresponding to Frobenius' Coin-changing Problem in the case of coins of two denominations.
The following dichotomy is established for any pair , of hereditary families of finite subsets of : Given , an infinite subset of , there exists an infinite subset of so that either , or , where denotes the set of all finite subsets of .
ABSTRACT. We construct the Cremona transformations of satisfying the following property: there exist such that the image of all straight lines through are straight lines through . We characterise these transformations, and for all non-negative integer we give a formula for the dimension of the set of those whose degree is .
Let be an open set and let denote the class of real analytic functions on . It is proved that for every surjective linear partial differential operator and every family depending holomorphically on there is a solution family depending on in the same way such that The result is a consequence of a characterization of Fréchet spaces such that the class of ``weakly' real analytic -valued functions coincides with the analogous class defined via Taylor series. An example shows that the analogous assertions need not be valid if is replaced by another set.
where is a continuous function. We show that a necessary and sufficient condition on for this problem to have positive solutions which are arbitrarily large at is that be less than 1 on a sequence of points in which tends to .