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1.
Let be a Polish group, a Polish topology on a space , acting continuously on , with -invariant and in the Borel algebra generated by . Then there is a larger Polish topology on so that is open with respect to , still acts continuously on , and has a basis consisting of sets that are of the same Borel rank as relative to .

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2.
We consider the system of coupled nonlinear Sturm-Liouville boundary value problems where , are real spectral parameters. It will be shown that if the functions and are `generic' then for all integers , there are smooth 2-dimensional manifolds , , of `semi-trivial' solutions of the system which bifurcate from the eigenvalues , , of , , respectively. Furthermore, there are smooth curves , , along which secondary bifurcations take place, giving rise to smooth, 2-dimensional manifolds of `non-trivial' solutions. It is shown that there is a single such manifold, , which `links' the curves , . Nodal properties of solutions on and global properties of are also discussed.

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3.
Using smooth one-fixed-point actions on spheres and a result due to Bob Oliver on the tangent representations at fixed points for smooth group actions on disks, we obtain a similar result for perfect group actions on spheres. For a finite group , we compute a certain subgroup of the representation ring . This allows us to prove that a finite perfect group has a smooth -proper action on a sphere with isolated fixed points at which the tangent representations of are mutually nonisomorphic if and only if contains two or more real conjugacy classes of elements not of prime power order. Moreover, by reducing group theoretical computations to number theory, for an integer and primes , we prove similar results for the group , , or . In particular, has Smith equivalent representations that are not isomorphic if and only if , , .

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4.
We say that a group belongs to the class if every nonunit quotient group of has an element of order two.

Let be a Hilbert space and let be its group of unitary operators. Suppose that groups and belong to the class and the order of is more than two. Then the free product has the following property. For any there exists a mapping satisfying the following conditions :

1)

2) for any representation the relation

holds.

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5.
Let be a system of arithmetic sequences which forms an -cover of (i.e. every integer belongs at least to members of ). In this paper we show the following surprising properties of : (a) For each there exist at least subsets of with such that . (b) If forms a minimal -cover of , then for any there is an such that for every there exists an for which and

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6.
Let be an algebra over a field and a finite group of automorphisms and anti-automorphisms of . We prove that if satisfies an essential -polynomial identity of degree , then the -codimensions of are exponentially bounded and satisfies a polynomial identity whose degree is bounded by an explicit function of . As a consequence we show that if is an algebra with involution satisfying a -polynomial identity of degree , then the -codimensions of are exponentially bounded; this gives a new proof of a theorem of Amitsur stating that in this case must satisfy a polynomial identity and we can now give an upper bound on the degree of this identity.

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7.
We shall use a -algebra approach to study operators of the form where is subnormal and is normal. We shall determine the spectral properties for these operators, and find the minimal normal extension and the dual operator. We also give a necessary condition for to contain a compact operator and a sufficient condition for the algebraic equivalence of and .

We also consider the existence of a homomorphism satisfying . We shall characterize the operators such that exists for every operator .

The problem of when is unitarily equivalent to is considered. Complete results are given when and are positive operators with finite multiplicity functions and has compact self-commutator. Some examples are also given.

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8.
Consider an -superdiffusion on , where is an uniformly elliptic differential operator in , and . The -polar sets for are subsets of which have no intersection with the graph of , and they are related to the removable singularities for a corresponding nonlinear parabolic partial differential equation. Dynkin characterized the -polarity of a general analytic set in term of the Bessel capacity of , and Sheu in term of the restricted Hausdorff dimension. In this paper we study in particular the -polarity of sets of the form , where and are two Borel subsets of and respectively. We establish a relationship between the restricted Hausdorff dimension of and the usual Hausdorff dimensions of and . As an application, we obtain a criterion for -polarity of in terms of the Hausdorff dimensions of and , which also gives an answer to a problem proposed by Dynkin in the 1991 Wald Memorial Lectures.

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9.
For a cube of size , we obtain a lower bound on so that is nonempty, where is the algebraic subset of defined by

a positive integer and an integer not divisible by . For we obtain that is nonempty if , for we obtain that is nonempty if , and for we obtain that is nonempty if . Using the assumption of the Grand Riemann Hypothesis we obtain is nonempty if .

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10.
For let be complex numbers such that is bounded. For define , where . Then the excesses in the sense of Paley and Wiener satisfy .

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11.
Let be a trigonometric polynomial of degree The problem of finding the largest value for in the inequality is studied. We find exactly provided is the conjugate of an even integer and For general we get an interval estimate for where the interval length tends to as tends to

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12.
Fix a smooth projective 3-fold , , with ample, and . Assume the existence of integers with such that is numerically equivalent to . Let be the moduli scheme of -stable rank 2 vector bundles, , on with and . Let be the number ofits irreducible components. Then .

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13.
Let and be the pair of unitary groups over a global field and an irreducible cuspidal representation of which satisfies a certain -function condition. By using a regularized Siegel-Weil formula, we can show that the global theta lifting of in is non-trivial if every local factor of has a local theta lifting (Howe lifting) in .

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14.
Sur les -corps     
Here we give new examples of fields in characteristic whose -invariant and -invariant are different: or . These fields are also -fields.

RSUM. Nous donnons ici de nouveaux exemples de corps en caractéristique dont le -invariant et le -invariant diffèrent. Plus précisément: et ou . Ces corps sont aussi des -corps.

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15.
Erdös and Szemerédi proved that if is a set of positive integers, then there must be at least integers that can be written as the sum or product of two elements of , where is a constant and . Nathanson proved that the result holds for . In this paper it is proved that the result holds for and .

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16.
We consider the balanced weight of a semi-finite weight and a (not necessarily faithful) normal positive functional on a von Neumann algebra , and discuss how the modular operator and the modular conjugation are described under the identification of the standard Hilbert space with , where is the support projection of and .

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17.
Let be the Kohn Laplacian on the Heisenberg group and let be a halfspace of whose boundary is parallel to the center of . In this paper we prove that if is a non-negative -superharmonic function such that

then in .

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18.
We shall prove (a slightly more general version of) the following theorem: let be analytic in the closed unit disk with , and let be a finite Blaschke product. Then there exists a function satisfying: i) analytic in the closed unit disk , ii) , iii) in , such that

satisfies

This completes a recent result of Kühnau for , , where this boundary value problem has a geometrical interpretation, namely that preserves hyperbolic arc length on for suitable
. For these important choices of we also prove that the corresponding functions are uniquely determined by , and that is univalent in . Our work is related to Beurling's and Avhadiev's on conformal mappings solving free boundary value conditions in the unit disk.

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19.
Let be the unit ball of (). We prove that if are holomorphic self-maps of such that , then and have a common fixed point (possibly at the boundary, in the sense of -limits). Furthermore, if and have no fixed points in , then they have the same Wolff point, unless the restrictions of and to the one-dimensional complex affine subset of determined by the Wolff points of and are commuting hyperbolic automorphisms of that subset.

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20.
Let denote the algebra of (bounded linear) operators on the separable complex Hilbert space , and let denote a norm ideal in . For , let the derivation be defined by , and let be defined by . The main result of this paper is to show that if , are contractions, then for every operator such that , then for all .

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