An extension of a convexity theorem of the generalized numerical range associated with

For any , let be the following subset of :

We show that if , then is always convex. When , it is an ellipsoid, probably degenerate. The convexity result is best possible in the sense that if we have defined similarly, then there are examples which fail to be convex when and .

The set is also symmetric about the origin for all , and contains the origin when . Equivalent statements of this result are given. The convexity result for is similar to Au-Yeung and Tsing's extension of Westwick's convexity result for .

     

Some characterizations of

We show that a function on the unit disk extends continuously to , the maximal ideal space of iff it is uniformly continuous (in the hyperbolic metric) and close to constant on the complementary components of some Carleson contour.

     

Support cones and convexity of sets in

We discuss several metric characterizations of convexity of sets in non-smooth finite-dimensional Banach spaces. We describe a setting in which convexity is equivalent to the rotation-invariance of various properties, including almost convexity, radial continuity of the metric projection, and Chebyshevity. One of the tools used is a generalization of norm-smoothness which involves support cones of the unit ball.

     

Polynomial continuity on

A mapping between Banach spaces is said to be polynomially continuous if its restriction to any bounded set is uniformly continuous for the weak polynomial topology. A Banach space has property (RP) if given two bounded sequences , we have that for every polynomial on whenever for every polynomial on ; i.e., the restriction of every polynomial on to each bounded set is uniformly sequentially continuous for the weak polynomial topology. We show that property (RP) does not imply that every scalar valued polynomial on must be polynomially continuous.

     

Zero divisors and

Let be a discrete group, the group ring of over and the Lebesgue space of with respect to Haar measure. It is known that if is torsion free elementary amenable, and , then . We will give a sufficient condition for this to be true when , and in the case we will give sufficient conditions for this to be false when .

     

Weak compactness in

We characterize weak compactness and weak conditional compactness of subsets of in terms of regular methods of summability. We also study when these results still hold using only convergence in the sense of Cesàro.

     

On polarized surfaces

Let be a smooth projective surface over and an ample Cartier divisor on . If the Kodaira dimension or , the author proved , where . If , then the author studied with . In this paper, we study the polarized surface with , , and .

     

On the Diophantine equation

If and are positive integers with and , then the equation of the title possesses at most one solution in positive integers and , with the possible exceptions of satisfying , and . The proof of this result relies on a variety of diophantine approximation techniques including those of rational approximation to hypergeometric functions, the theory of linear forms in logarithms and recent computational methods related to lattice-basis reduction. Additionally, we compare and contrast a number of these last mentioned techniques.

     

On the recursive sequence

We show that every positive solution of the equation

where , converges to a period two solution.

     

The

In this paper we analyze the localization of , the fiber of the double suspension map , with respect to . If four cells at the bottom of , the th extended power spectrum of the Moore spectrum, are collapsed to a point, then one obtains a spectrum . Let be the James-Hopf map followed by the collapse map. Then we show that the secondary suspension map has a lifting to the fiber of and this lifting is shown to be a -periodic equivalence, hence an -equivalence.

     

On the diophantine equation

One of the purposes of this note is to correct the proof of a recent result of Y. Guo & M. Le on the equation . Moreover, we prove that the diophantine equation , , , , , gcd, , has only finitely many solutions, all of which satisfying .

     

On the eigenvalues of the

We study the nonlinear eigenvalue problem

where , is a bounded smooth domain in . We prove that the first and the second variational eigenvalues of (1) are continuous functions of . Moreover, we obtain the asymptotic behavior of the first eigenvalue as and .

     

-norm uniform distribution

In this paper, the -norm uniform distribution, which is a generalization of the uniform distribution studied by Cambanis, Huang, and Simon (1981), is defined for any . Then its marginal distributions and order statistics are studied.

     

On the suspension order of

It is shown that the suspension order of the -fold cartesian product of real projective -space is less than or equal to the suspension order of the -fold symmetric product of and greater than or equal to , where and satisfy and . In particular has suspension order , and for fixed the suspension orders of the spaces are unbounded while their stable suspension orders are constant and equal to .

     

Calibrated thin

Let be a compact metric space, and let be a calibrated thin -ideal. Then is . This solves an open problem, which was posed by Kechris, Louveau and Woodin. Using our result we obtain a new proof of Kaufman's theorem concerning -sets and -sets.

     

Gaps in

For a partial order , let denote the statement that for every -increasing -sequence there is a -decreasing -sequence on top of such that is an -gap in . The main result of this paper is that . It is also shown, as a corollary, that but .

     

without sharps

We show that the supremum of the lengths of prewellorderings of the reals can be , with inaccessible to reals, assuming only the consistency of an inaccessible.