Complete positivity of elementary operators

In this paper, we prove that if is an -dimensional subspace of , then is -reflexive, where denotes the greatest integer not larger than . By the result, we show that if is an elementary operator on a -algebra , then is completely positive if and only if is -positive.

     

An extremal problem for trigonometric polynomials

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

     

On the product of two generalized derivations

Two elements and in a ring determine a generalized derivation on by setting for any in . We characterize when the product is a generalized derivation in the cases when the ring is the algebra of all bounded operators on a Banach space , and when is a -algebra . We use these characterizations to compute the commutant of the range of .

     

A Liouville-type theorem on halfspaces for the Kohn Laplacian

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 .

     

A factorization theorem for the derivative of a function in

We show that a function is the derivative of a function in the Hardy space of the unit disk for if and only if where and . Here, can be chosen to be non-vanishing, , and . As an application, we characterize positive measures on the unit disk such that the operator is bounded from the tent space to , where .

     

Periodic solutions of a periodic delay predator-prey system

The existence of a positive periodic solution for

is established, where , , , , are positive periodic continuous functions with period , and , are periodic continuous functions with period .

     

Large orbits in actions of nilpotent groups

If a nontrivial nilpotent group acts faithfully and coprimely on a group , it is shown that some element of has a small centralizer in and hence lies in a large orbit. Specifically, there exists such that , where is the smallest prime divisor of .

     

On the topology of isoparametric hypersurfaces with four distinct principal curvatures

Let be the pair of multiplicities of an isoparametric hypersurface in the unit sphere with four distinct principal curvatures -w.r.g., we assume that . In the present paper we prove that, in the case 4B2 of U. Abresch (Math. Ann. 264 (1983), 283-302) (i.e., where ), must be either 2 or 4. As a by-product, we prove that the focal manifold of an isoparametric hypersurface is homeomorphic to a bundle over if one of the following conditions holds: (1) and or ; (2) and . This generalizes partial results of Wang (1988) about the topology of Clifford type examples. Consequently, the hypersurface is homeomorphic to an iterated sphere bundle under the above condition.

     

The growth theorem of convex mappings on the unit ball in

Let be an arbitrary norm on . Let be a normalized biholomorphic convex mapping on the unit ball in with respect to the norm . We will give an upper bound of the growth of .

     

An extension of H. Cartan's theorem

In this article we prove that if , , is a bounded pseudoconvex domain with real analytic boundary, then for each , there exists a fixed open neighborhood of and an open neighborhood of in such that any can be extended holomorphically to , and that the action defined by

is real analytic in joint variables. This extends H. Cartan's theorem beyond the boundary. Some applications are also discussed here.

     

Fixed point iteration for pseudocontractive maps

Let be a compact convex subset of a real Hilbert space, ; a continuous pseudocontractive map. Let and be real sequences in [0,1] satisfying appropriate conditions. For arbitrary define the sequence iteratively by where are arbitrary sequences in . Then, converges strongly to a fixed point of . A related result deals with the convergence of to a fixed point of when is Lipschitz and pseudocontractive. Our theorems also hold for the slightly more general class of continuous hemicontractive nonlinear maps.

     

On covering multiplicity

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

     

Smith equivalence of representations for finite perfect groups

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 , , .

     

On real quadratic function fields of Chowla type with ideal class number one

Let be the finite field with elements, (2), , where is a square-free polynomial in with and . In this paper several equivalent conditions for the ideal class number to be one are presented and all such quadratic function fields with are determined.

     

Oscillating global continua of positive solutions of semilinear elliptic problems

Let be a bounded domain in , , with boundary , and consider the semilinear elliptic boundary value problem

where is a uniformly elliptic operator on , , is strictly positive in , and the function is continuously differentiable, with , . A well known result of Rabinowitz shows that an unbounded continuum of positive solutions of this problem bifurcates from the principal eigenvalue of the linear problem. We show that under certain oscillation conditions on the nonlinearity , this continuum oscillates about , in a certain sense, as it approaches infinity. Hence, in particular, the equation has infinitely many positive solutions for each in an open interval containing .

     

Degenerations for modules over representation-finite algebras

Let be a representation-finite algebra. We show that a finite dimensional -module degenerates to another -module if and only if the inequalities hold for all -modules . We prove also that if for any indecomposable -module , then any degeneration of -modules is given by a chain of short exact sequences.

     

Further extension of the Heinz-Kato-Furuta inequality

Let be a bounded operator on a Hilbert space and positive definite operators. Kato has shown that if and for all , then where are operator monotone functions defined on such that . Furuta has shown that Let be any continuous operator monotone functions, and set for We will show that is well defined and Moreover, we will extend this result for unbounded closed operators densely defined on

     

An uncertainty principle for Hankel transforms

There exists a generalized Hankel transform of order on , which is based on the eigenfunctions of the Dunkl operator

For this transform coincides with the usual Fourier transform on . In this paper the operator replaces the usual first derivative in order to obtain a sharp uncertainty principle for generalized Hankel transforms on . It generalizes the classical Weyl-Heisenberg uncertainty principle for the position and momentum operators on ; moreover, it implies a Weyl-Heisenberg inequality for the classical Hankel transform of arbitrary order on

     

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 .

     

Extremal points of a functional on the set of convex functions

We investigate the extremal points of a functional , for a convex or concave function . The admissible functions are convex themselves and satisfy a condition . We show that the extremal points are exactly and if these functions are convex and coincide on the boundary . No explicit regularity condition is imposed on , , or . Subsequently we discuss a number of extensions, such as the case when or are non-convex or do not coincide on the boundary, when the function also depends on , etc.