The maximal ideal space of with respect to the Hadamard product

It is shown that the space of all regular maximal ideals in the Banach algebra with respect to the Hadamard product is isomorphic to The multiplicative functionals are exactly the evaluations at the -th Taylor coefficient. It is a consequence that for a given function in and for a function holomorphic in a neighborhood of with and for all the function is in

     

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.

     

The distribution of solutions of the congruence

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 .

     

New facts

We use ``iterated square sequences' to show that there is an -definable partition such that if is an inner model not containing :

(a)

For some is stationary.

(b)

For each there is a generic extension of in which does not exist and is non-stationary.

This result is then applied to show that if is an inner model without , then some sentence not true in can be forced over .

     

The connected stable rank of the purely infinite simple -algebras

Suppose that is a unital purely infinite simple -algebra. If the class [1] of the unit 1 in has torsion, then ; if [1] is torsion-free in , then . If is a non-unital purely infinite simple -algebra, then .

     

Rules and reals

A ``-rule" is a sequence of pairwise disjoint sets , each of cardinality and subsets . A subset (a ``real') follows a rule if for infinitely many , .

Two obvious cardinal invariants arise from this definition: the least number of reals needed to follow all -rules, , and the least number of -rules with no real that follows all of them, .

Call a bounded rule if is a -rule for some . Let be the least cardinality of a set of bounded rules with no real following all rules in the set.

We prove the following: and for all . However, in the Laver model, .

An application of is in Section 3: we show that below one can find proper extensions of dense independent families which preserve a pre-assigned group of automorphisms. The original motivation for discovering rules was an attempt to construct a maximal homogeneous family over . The consistency of such a family is still open.

     

The holomorphic extension of -CR functions on tube submanifolds

We consider the set of CR functions on a connected tube submanifold of satisfying a uniform bound on the -norm in the tube direction. We show that all such CR functions holomorphically extend to functions on the convex hull of the tube (). The -norm of the extension is shown to be the same as the uniform -norm in the tube direction of the CR function.

     

A description of Hilbert -modules in which all closed submodules are orthogonally closed

Let , be -algebras and a full Hilbert --bimodule such that every closed right submodule is orthogonally closed, i.e., . Then there are families of Hilbert spaces , such that and are isomorphic to -direct sums , resp. , and is isomorphic to the outer direct sum .

     

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 .

     

Common fixed points of commuting holomorphic maps in the unit ball of

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.

     

The location of the zeros of the higher order derivatives of a polynomial

Let be a complex polynomial of degree having zeros in a disk . We deal with the problem of finding the smallest concentric disk containing zeros of . We obtain some estimates on the radius of this disk in general as well as in the special case, where zeros in are isolated from the other zeros of . We indicate an application to the root-finding algorithms.

     

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 .

     

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 .

     

Finite time blow-up for the inhomogeneous equation

We consider the inhomogeneous equation

where , and , and give criteria on , and which determine whether for all and all the solution blows up in finite time or whether for and sufficiently small, the solution exists for all time.

     

Behaviour of holomorphic automorphisms on equicontinuous subsets of the space

Consider a compact Hausdorff topological space , a -triple and , the -triple of all continuous -valued functions with the pointwise operations and the norm of the supremum. Let be the group of all holomorphic automorphisms of the unit ball of that map every equicontinuous subset lying strictly inside into another such a set. The real Banach-Lie group and its Lie algebra are investigated. The identity connected component of is identified when has the strong Banach-Stone property. This extends to the infinite dimensional setting a well known result concerning the case .

     

On -cobordisms of spherical space forms

Given a manifold of dimension at least 4 whose universal covering is homeomorphic to a sphere, the main result states that a compact manifold is isomorphic to a cylinder if and only if is homotopy equivalent to this cylinder and the boundary is isomorphic to two copies of ; this holds in the smooth, PL and topological categories. The result yields a classification of smooth, finite group actions on homotopy spheres (in dimensions ) with exactly two singular points.

     

A characterization of

This work is devoted to the relationship between topological properties of a space and those of (= the space of continuous real-valued functions on , with the topology of pointwise convergence). The emphasis is on -compactness of and on location of in . In particular, -compact cosmic spaces are characterized in this way.

     

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.

     

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 .

     

Nonoscillation theorems for second order nonlinear differential equations

We prove nonoscillation theorems for the second order Emden-Fowler equation (E): , , where and . It is shown that when is nondecreasing for any and is bounded above, then (E) is nonoscillatory. This improves a well-known result of Belohorec in the sublinear case, i.e. when and .