On principal eigenvalues for boundary value problems with indefinite weight and Robin boundary conditions

We investigate the existence of principal eigenvalues (i.e., eigenvalues corresponding to positive eigenfunctions) for the boundary value problem on ; on , where is a bounded region in , is an indefinite weight function and may be positive, negative or zero.

     

-identities on associative algebras

Let be an algebra over a field and a finite group of automorphisms and anti-automorphisms of . We prove that if satisfies an essential -polynomial identity of degree , then the -codimensions of are exponentially bounded and satisfies a polynomial identity whose degree is bounded by an explicit function of . As a consequence we show that if is an algebra with involution satisfying a -polynomial identity of degree , then the -codimensions of are exponentially bounded; this gives a new proof of a theorem of Amitsur stating that in this case must satisfy a polynomial identity and we can now give an upper bound on the degree of this identity.

     

Sharper changes in topologies

Let be a Polish group, a Polish topology on a space , acting continuously on , with -invariant and in the Borel algebra generated by . Then there is a larger Polish topology on so that is open with respect to , still acts continuously on , and has a basis consisting of sets that are of the same Borel rank as relative to .

     

A characterization of the Clifford torus

In this paper, we prove that an -dimensional closed minimal hypersurface with Ricci curvature of a unit sphere is isometric to a Clifford torus if , where is the squared norm of the second fundamental form of .

     

On almost representations of groups

We say that a group belongs to the class if every nonunit quotient group of has an element of order two.

Let be a Hilbert space and let be its group of unitary operators. Suppose that groups and belong to the class and the order of is more than two. Then the free product has the following property. For any there exists a mapping satisfying the following conditions :

1)

2) for any representation the relation

holds.

     

Relative Brauer groups of discrete valued fields

Let be a non-trivial finite Galois extension of a field . In this paper we investigate the role that valuation-theoretic properties of play in determining the non-triviality of the relative Brauer group, , of over . In particular, we show that when is finitely generated of transcendence degree 1 over a -adic field and is a prime dividing , then the following conditions are equivalent: (i) the -primary component, , is non-trivial, (ii) is infinite, and (iii) there exists a valuation of trivial on such that divides the order of the decomposition group of at .

     

A Dauns-Hofmann theorem for TAF-algebras

Let be a TAF-algebra, the centre of the ideal lattice of , and the space of meet-irreducible elements of , equipped with the hull-kernel topology. It is shown that is a compact, locally compact, second countable, -space, that is an algebraic lattice isomorphic to the lattice of open subsets of , and that is isomorphic to the algebra of continuous, complex functions on . If is semisimple, then is isomorphic to the algebra of continuous, complex functions on , the primitive ideal space of . If is strongly maximal, then the sum of two closed ideals of is closed.

     

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

     

Normalizers of the congruence subgroups of the Hecke group

Let cos and let be the Hecke group associated to . In this article, we show that for a prime ideal in , the congruence subgroups of are self-normalized in .

     

Tensor products of subnormal operators

We shall use a -algebra approach to study operators of the form where is subnormal and is normal. We shall determine the spectral properties for these operators, and find the minimal normal extension and the dual operator. We also give a necessary condition for to contain a compact operator and a sufficient condition for the algebraic equivalence of and .

We also consider the existence of a homomorphism satisfying . We shall characterize the operators such that exists for every operator .

The problem of when is unitarily equivalent to is considered. Complete results are given when and are positive operators with finite multiplicity functions and has compact self-commutator. Some examples are also given.

     

Some corollaries of Frobenius' normal -complement theorem

For a prime divisor of the order of a finite group , we present the set of -subgroups generating . In particular, we present the set of primary subgroups of generating the last member of the lower central series of . The proof is based on the Frobenius Normal -Complement Theorem and basic properties of minimal nonnilpotent groups. Let be a group and a group-theoretic property inherited by subgroups and epimorphic images such that all minimal non--subgroups (-subgroups) of are not nilpotent. Then (see the lemma), if is generated by all -subgroups of it follows that is a -group.

     

Group rings whose symmetric elements are Lie nilpotent

Let be the group ring of a group over a field , with characteristic different from . Let denote the natural involution on sending each group element to its inverse. Denote by the set of symmetric elements with respect to this involution. A paper of Giambruno and Sehgal showed that provided has no -elements, if is Lie nilpotent, then so is . In this paper, we determine when is Lie nilpotent, if does contain -elements.

     

Existence of unliftable modules

Let be a commutative Noetherian local ring, and let where is a non-zerodivisor of contained in . Then a finitely generated -module is said to lift to if there exists a finitely generated -module such that is -regular and . In this paper we give a general construction of finitely generated -modules of finite projective dimension over which fail to lift to provided and the depth of is at least 2.

     

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 .

     

Sets of range uniqueness for classes of continuous functions

Diamond, Pomerance and Rubel (1981) proved that there are subsets of the complex plane such that for any two entire functions and if , then . Baraducci and Dikranjan showed in 1993 that the continuum hypothesis (CH) implies the existence of a similar set for the class of continuous nowhere constant functions from to , while it follows from the results of Burke and Ciesielski (1997) and Ciesielski and Shelah that the existence of such a set is not provable in ZFC. In this paper we will show that for several well-behaved subclasses of , including the class of differentiable functions and the class of absolutely continuous functions, a set with the above property can be constructed in ZFC. We will also prove the existence of a set with the dual property that for any if , then .

     

Projections from

Let and be two infinite dimensional real Banach spaces. The following question is classical and long-standing. Are the following properties equivalent?

a) There exists a projection from the space of continuous linear operators onto the space of compact linear operators.

b) .

The answer is positive in certain cases, in particular if or has an unconditional basis. It seems that there are few results in the direction of a general solution. For example, suppose that and are reflexive and that or has the approximation property. Then, if , there is no projection of norm 1, from onto . In this paper, one obtains, in particular, the following result:

Theorem. Let be a real Banach space which is reflexive (resp. with a separable dual), of infinite dimension, and such that has the approximation property. Let be a real scalar with . Then can be equivalently renormed such that, for any projection from onto , one has . One gives also various results with two spaces 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 , , .

     

Reflection and uniqueness theorems for harmonic functions

Suppose that is harmonic on an open half-ball in such that the origin 0 is the centre of the flat part of the boundary . If has non-negative lower limit at each point of and tends to 0 sufficiently rapidly on the normal to at 0, then has a harmonic continuation by reflection across . Under somewhat stronger hypotheses, the conclusion is that . These results strengthen recent theorems of Baouendi and Rothschild. While the flat boundary set can be replaced by a spherical surface, it cannot in general be replaced by a smooth -dimensional manifold.

     

Densely hereditarily hypercyclic sequences and large hypercyclic manifolds

We prove in this paper that if is a hereditarily hypercyclic sequence of continuous linear mappings between two topological vector spaces and , where is metrizable, then there is an infinite-dimensional linear submanifold of such that each non-zero vector of is hypercyclic for . If, in addition, is metrizable and separable and is densely hereditarily hypercyclic, then can be chosen dense.

     

Gradient estimates for positive solutions of the Laplacian with drift

Let be a complete Riemannian manifold of dimension without boundary and with Ricci curvature bounded below by where If is a vector field such that and on for some nonnegative constants and then we show that any positive solution of the equation satisfies the estimate

on , for all In particular, for the case when this estimate is advantageous for small values of and when it recovers the celebrated Liouville theorem of Yau (Comm. Pure Appl. Math. 28 (1975), 201-228).