Angular derivatives at boundary fixed points for self-maps of the disk

Let be a one-to-one analytic function of the unit disk into itself, with . The origin is an attracting fixed point for , if is not a rotation. In addition, there can be fixed points on where has a finite angular derivative. These boundary fixed points must be repelling (abbreviated b.r.f.p.). The Koenigs function of is a one-to-one analytic function defined on such that , where . If is the first iterate of that does have b.r.f.p., we compute the Hardy number of , , in terms of the smallest angular derivative of at its b.r.f.p.. In the case when no iterate of has b.r.f.p., then , and vice versa. This has applications to composition operators, since is a formal eigenfunction of the operator . When acts on , by a result of C. Cowen and B. MacCluer, the spectrum of is determined by and the essential spectral radius of , . Also, by a result of P. Bourdon and J. Shapiro, and our earlier work, can be computed in terms of . Hence, our result implies that the spectrum of is determined by the derivative of at the fixed point and the angular derivatives at b.r.f.p. of or some iterate of .

     

Subaveraging estimate for

Let be a smooth real hypersurface of and a compact submanifold of . We generalize a result of A. Boggess and R. Dwilewicz giving, under some geometric conditions on and , an estimate of the submeanvalue on of any function on a neighbourhood of , by the norm of on a neighbourhood of in .

     

Eigenvalues of the form valued Laplacian for Riemannian submersions

Let be a Riemannian submersion of closed manifolds. Let be an eigen -form of the Laplacian on with eigenvalue which pulls back to an eigen -form of the Laplacian on with eigenvalue . We are interested in when the eigenvalue can change. We show that , so the eigenvalue can only increase; and we give some examples where , so the eigenvalue changes. If the horizontal distribution is integrable and if is simply connected, then , so the eigenvalue does not change.

     

Analytic subgroups of the reals

We prove that every analytic proper subgroup of the reals can be covered by an null set. We also construct a proper Borel subgroup of the reals that cannot be covered by countably many sets such that is nowhere dense for every

     

Enveloping algebras of lie color algebras: primeness versus graded-primeness

Let be a finite abelian group and let be a, possibly restricted, -graded Lie color algebra. Then the enveloping algebra is also -graded, and we consider the question of whether being graded-prime implies that it is prime. The first section of this paper is devoted to the special case of Lie superalgebras over a field of characteristic . Specifically, we show that if and if has a unique minimal graded-prime ideal, then this ideal is necessarily prime. As will be apparent, the latter result follows quickly from the existence of an anti-automorphism of whose square is the automorphism of the enveloping algebra associated with its -grading. The second section, which is independent of the first, studies more general Lie color algebras and shows that if is graded-prime and if most homogeneous components of are infinite dimensional over , then is prime. Here we use -methods to study the grading on the extended centroid of . In particular, if is generated by the infinite support of , then we prove that is homogeneous.

     

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 .

     

A uniqueness theorem for harmonic functions

The main result of this note is the following theorem: Theorem 1. Let be a half ball in and . Assume that is in and harmonic in , and that for every positive integer there exists a constant such that

Then .

First we prove it for , and then we show by induction that it holds for all .

     

Uniqueness for an overdetermined boundary value problem for the p-Laplacian

For set , and let be a measure with compact support. Suppose, for , there are functions and (bounded) domains , both containing the support of with the property that in (weakly) and in the complement of . If in addition is convex, then and .

     

The property and the reals

The property may be generalized by using filters on in a very natural way. We analyze the necessary requirements for a space to have property for a filter . We construct special filters for which has the property, in particular a P-point and a Q-point.

     

Renormalized oscillation theory for Dirac operators

Oscillation theory for one-dimensional Dirac operators with separated boundary conditions is investigated. Our main theorem reads: If and if solve the Dirac equation , (in the weak sense) and respectively satisfy the boundary condition on the left/right, then the dimension of the spectral projection equals the number of zeros of the Wronskian of and . As an application we establish finiteness of the number of eigenvalues in essential spectral gaps of perturbed periodic Dirac operators.

     

On certain character sums over

Let be the finite field with elements and let denote the ring of polynomials in one variable with coefficients in . Let be a monic polynomial irreducible in . We obtain a bound for the least degree of a monic polynomial irreducible in ( odd) which is a quadratic non-residue modulo . We also find a bound for the least degree of a monic polynomial irreducible in which is a primitive root modulo .

     

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 .

     

On asymmetry of topological centers of the second duals of Banach algebras

Let be a Banach algebra with a bounded approximate identity and let and be the left and right topological centers of . It is shown that i) is not sufficient for ; ii) the inclusion is not sufficient for ; iii) is not sufficient for to be weakly sequentially complete. These results answer three questions of Anthony To-Ming Lau and Ali Ülger.

     

There is a paracompact Q-set space in ZFC

We construct a paracompact space such that every subset of is an -set, yet is not -discrete. We will construct our space not to have a -diagonal, which answers questions of A.V. Arhangel'skii and D. Shakhmatov on cleavable spaces.

     

Derived lengths and character degrees

Let be a finite solvable group. Assume that the degree graph of has exactly two connected components that do not contain . Suppose that one of these connected components contains the subset , where and are coprime when . Then the derived length of is less than or equal to .

     

A note on Greenberg's conjecture and the abc conjecture

For any totally real number field and any prime number , Greenberg's conjecture for asserts that the Iwasawa invariants and are both zero. For a fixed real abelian field , we prove that the conjecture is ``affirmative' for infinitely many (which split in if we assume the abc conjecture for .

     

On scrambled sets and a theorem of Kuratowski on independent sets

The measure of scrambled sets of interval self-maps was studied by many authors, including Smítal, Misiurewicz, Bruckner and Hu, and Xiong and Yang. In this note, first we introduce the notion of ``-chaos" which is related to chaos in the sense of Li-Yorke, and we prove a general theorem which is an improvement of a theorem of Kuratowski on independent sets. Second, we apply the result to scrambled sets of higher dimensional cases. In particular, we show that if a map of the unit -cube is -chaotic on , then for any there is a map such that and are topologically conjugate, and has a scrambled set which has Lebesgue measure 1, and hence if , then there is a homeomorphism with a scrambled set satisfying that is an -set in and has Lebesgue measure 1.

     

A note on norm attaining functionals

We are concerned in this paper with the density of functionals which do not attain their norms in Banach spaces. Some earlier results given for separable spaces are extended to the nonseparable case. We obtain that a Banach space is reflexive if and only if it satisfies any of the following properties: (i) admits a norm with the Mazur Intersection Property and the set of all norm attaining functionals of contains an open set, (ii) the set of all norm one elements of contains a (relative) weak* open set of the unit sphere, (iii) has and contains a (relative) weak open set of the unit sphere, (iv) is , has and contains a (relative) weak open set of the unit sphere. Finally, if is separable, then is reflexive if and only if contains a (relative) weak open set of the unit sphere.

     

A pointwise spectrum and representation of operators

For a self-adjoint operator commuting with an increasing family of projections we study the multifunction an open set of the topology containing , where is the spectrum of on . Let be the measure of maximal spectral type. We study the condition that is essentially a singleton, is not a singleton. We show that if is the density topology and if satisfies the density theorem, in particular if it is absolutely continuous with respect to the Lebesgue measure, then this condition is equivalent to the fact that is a Borel function of . If is the usual topology then the condition is equivalent to the fact that is approched in norm by step functions , where the set of intervals covers the set where is a singleton.