On the prime model property

Assume is superstable, is a formula over , is countable and is countable and . We investigate models in assuming has the prime model property. We prove some corollaries on the number of models in . We show an example of an -stable and with having exactly 3 models.

     

A topological characterization of linearity for quasi-traces

Let be a -algebra, and let be a (local) quasi-trace on . Then is linear if, and only if, the restriction of to the closed unit ball of is uniformly weakly continuous.

     

Endomorphism rings of completely pure-injective modules

Let be a ring, its injective envelope, and the Jacobson radical of . It is shown that if every finitely generated submodule of embeds in a finitely presented module of projective dimension , then every finitley generated right -module is canonically isomorphic to . This fact, together with a well-known theorem of Osofsky, allows us to prove that if, moreover, is completely pure-injective (a property that holds, for example, when the right pure global dimension of is and hence when is a countable ring), then is semiperfect and is finite-dimensional. We obtain several applications and a characterization of right hereditary right noetherian rings.

     

Isometries of certain operator algebras

To a given basis on an -dimensional Hilbert space , we associate the algebra of all linear operators on having every as an eigenvector. So, is commutative, semisimple, and -dimensional. Given two algebras of this type, and , there is a natural algebraic isomorphism of and . We study the question: When does preserve the operator norm?

     

Differences of vector-valued functions on topological groups

Let be a locally compact group equipped with right Haar measure. The right differences of functions on are defined by for . Let and suppose for some and all . We prove that is a right uniformly continuous function of . If is abelian and the Beurling spectrum does not contain the unit of the dual group , then we show . These results have analogues for functions , where is a separable or reflexive Banach space. Finally, we apply our methods to vector-valued right uniformly continuous differences and to absolutely continuous elements of left Banach -modules.

     

Bounds for the operator norms of some Nörlund matrices

Suppose is a non-increasing sequence of non-negative numbers with , , , and is the lower triangular matrix defined by , , and , . We show that the operator norm of as a linear operator on is no greater than , for ; this generalizes, yet again, Hardy's inequality for sequences, and simplifies and improves, in this special case, more generally applicable results of D. Borwein, Cass, and Kratz. When the tend to a positive limit, the operator norm of on is exactly . We also give some cases when the operator norm of on is less than .

     

Nonresonance problems for differential inclusions in separable Banach spaces

Let be a real separable Banach space. The boundary value problem

is studied on the infinite interval Here, the closed and densely defined linear operator generates an evolution operator The function is measurable in its first variable, upper semicontinuous in its second and has weakly compact and convex values. Either is bounded and is compact for or is compact and is equicontinuous. The mapping is a bounded linear operator and is fixed. The nonresonance problem is solved by using Ma's fixed point theorem along with a recent result of Przeradzki which characterizes the compact sets in

     

When is a -adic power series an endomorphism of a formal group?

If is a noninvertible endomorphism of a formal group, then we have that commutes with an invertible series and is Galois over for all . We shall prove that the converse of this statement is also true.

     

Three counterexamples for a question concerning Green's functions and circular symmetrization

We construct domains in the plane such that if is the Green's function of with pole at zero, while is the symmetric non-increasing rearrangement of for each fixed and is the Green's function of the circular symmetrization , again with pole at zero, then there are positive numbers and such that

whenever . One of our constructions will have simply connected. We also consider the case where the poles of the Green's functions do not lie at the origin. Our work provides a negative answer to a question of Hayman (1967).

     

Almost linearity of -bi-Lipschitz maps between real Banach spaces

Let and be real Banach spaces. A map between and is called an -bi-Lipschitz map if for all . In this note we show that if is an -bi-Lipschitz map with from onto , then is almost linear. We also show that if is a surjective -bi-Lipschitz map with , then there exists a linear isomorphism such that

where as and .

     

A theorem of Briançon-Skoda type for regular local rings containing a field

Let be a regular local ring containing a field. We give a refinement of the Briançon-Skoda theorem showing that if is a minimal reduction of where is -primary, then where and is the largest ideal such that . The proof uses tight closure in characteristic and reduction to characteristic for rings containing the rationals.

     

A note on

There is only one pair of non-real zeros of , and of , in the left half-plane. The Riemann Hypothesis implies that and have no zeros in the strip .

     

A non-homogeneous zero-dimensional is a group

We provide an example of a zero-dimensional (separable metric) absolute Borel set which is not homogeneous, but whose square admits the structure of a topological group. We also construct a zero-dimensional absolute Borel set such that is a homogeneous non-group but is a group. This answers questions of Arhangel'skii and Zhou.

     

Some remarks on the variation of curve length and surface area

Consider the curve , where is absolutely continuous on . Then has finite length, and if is the -neighborhood of in the uniform norm, we compare the length of the shortest path in with the length of . Our main result establishes necessary and sufficient conditions on such that the difference of these quantities is of order as . We also include a result for surfaces.

     

Cohomological detection and regular elements in group cohomology

Suppose that is a compact Lie group or a discrete group of finite virtual cohomological dimension and that is a field of characteristic . Suppose that is a set of elementary abelian -subgroups such that the cohomology is detected on the centralizers of the elements of . Assume also that is closed under conjugation and that is in whenever some subgroup of is in . Then there exists a regular element in the cohomology ring such that the restriction of to an elementary abelian -subgroup is not nilpotent if and only if is in . The converse of the result is a theorem of Lannes, Schwartz and the second author. The results have several implications for the depth and associated primes of the cohomology rings.

     

Lipscomb's universal space is the attractor of an infinite iterated function system

Lipscomb's one-dimensional space on an arbitrary index set is injected into the Tychonoff cube . The image of is shown to be the attractor of an iterated function system indexed by . This system is conjugate, under an injection, with a set of right-shift operators on Baire's space regarded as a code space. This view of extends the fractal nature of initiated in a 1992 joint paper by the author and S. Lipscomb. In addition, we give a new proof that as a subspace of Hilbert's space , the space is complete and hence is closed in .

     

Stability of semigroups commuting with a compact operator

It is proved that if are bounded -semigroups on Banach spaces and , resp., and , are bounded operators with dense ranges such that intertwines with and commutes with , then is strongly stable provided ---the generator of ---does not have eigenvalue on . An analogous result holds for power-bounded operators.

     

On Voiculescu's double commutant theorem

For a separable infinite-dimensional Hilbert space , we consider the full algebra of bounded linear transformations and the unique non-trivial norm-closed two-sided ideal of compact operators . We also consider the quotient -algebra with quotient map

For any -subalgebra of , the relative commutant is given by for all in . It was shown by D. Voiculescu that, for any separable unital -subalgebra of ,

In this note, we exhibit a non-separable unital -subalgebra of for which (VDCT) fails.

     

On the zero sets of certain entire functions

We consider the class of entire functions of the form

where are polynomials and are entire functions. We prove that the zero-set of such an , if infinite, cannot be contained in a ray. But for every region containing the positive ray there is an example of with infinite zero-set which is contained in this region.

     

Representation of continuous functions as sums of Green functions

Let , where is polar and compact and is a domain with Green function . We characterize those subsets of which have the following property: Every positive continuous function on can be written as , where and for each .