On solutions of real analytic equations

Analyticity of solutions , of systems of real analytic equations , is studied. Sufficient conditions for and power series solutions to be real analytic are given in terms of iterative Jacobian ideals of the analytic ideal generated by . In a special case when the 's are independent of , we prove that if a solution satisfies the condition , then is necessarily real analytic.

     

Weakly coupled bound states in quantum waveguides

We study the eigenvalue spectrum of Dirichlet Laplacians which model quantum waveguides associated with tubular regions outside of a
bounded domain. Intuitively, our principal new result in two dimensions asserts that any domain obtained by adding an arbitrarily small ``bump' to the tube (i.e., , open and connected, outside a bounded region) produces at least one positive eigenvalue below the essential spectrum of the Dirichlet Laplacian . For sufficiently small ( abbreviating Lebesgue measure), we prove uniqueness of the ground state of and derive the ``weak coupling' result using a Birman-Schwinger-type analysis. As a corollary of these results we obtain the following surprising fact: Starting from the tube with Dirichlet boundary conditions at , replace the Dirichlet condition by a Neumann boundary condition on an arbitrarily small segment , , of . If denotes the resulting Laplace operator in , then has a discrete eigenvalue in no matter how small is.

     

Local automorphisms and derivations on

Let be an algebra. A mapping is called a -local automorphism if for every there is an automorphism , depending on and , such that and (no linearity, surjectivity or continuity of is assumed). Let be an infinite-dimensional separable Hilbert space, and let be the algebra of all linear bounded operators on . Then every -local automorphism is an automorphism. An analogous result is obtained for derivations.

     

When is a -block?

Let and be distinct prime numbers and let be a finite group. If is a -block of and is a -block, we study when the set of ordinary irreducible characters in the blocks and coincide.

     

Commensurators of parabolic subgroups of Coxeter groups

Let be a Coxeter system, and let be a subset of . The subgroup of generated by is denoted by and is called a parabolic subgroup. We give the precise definition of the commensurator of a subgroup in a group. In particular, the commensurator of in is the subgroup of in such that has finite index in both and . The subgroup can be decomposed in the form where is finite and all the irreducible components of are infinite. Let be the set of in such that for all . We prove that the commensurator of is . In particular, the commensurator of a parabolic subgroup is a parabolic subgroup, and is its own commensurator if and only if .

     

Coprimeness among irreducible character degrees of finite solvable groups

Given a finite solvable group , we say that has property if every set of distinct irreducible character degrees of is (setwise) relatively prime. Let be the smallest positive integer such that satisfies property . We derive a bound, which is quadratic in , for the total number of irreducible character degrees of . Three exceptional cases occur; examples are constructed which verify the sharpness of the bound in each of these special cases.

     

Note on faithful representations and a local property of Lie groups

Let be any analytic group, let be a maximal toroid of the radical of , and let be a maximal semisimple analytic subgroup of . If is the Lie algebra of , is the radical of , and is the center of , we show that has a faithful representation if and only if (i) , and (ii) has a faithful representation.

     

Point spectrum and mixed spectral types for rank one perturbations

We consider examples of rank one perturbations with a cyclic vector for . We prove that for any bounded measurable set , an interval, there exist so that
eigenvalue agrees with up to sets of Lebesgue measure zero. We also show that there exist examples where has a.c. spectrum for all , and for sets of 's of positive Lebesgue measure, also has point spectrum in , and for a set of 's of positive Lebesgue measure, also has singular continuous spectrum in .

     

A Strict Version of the Non-commutative Urysohn Lemma

Given a pair , of -commuting, hereditary -subalgebras of a unital -algebra , such that is -unital and , there is an element in , with , such that is strictly positive in and is strictly positive in in . Moreover, is strictly positive in in .

     

An improved estimate for the highest Lyapunov exponent in the method of freezing

Let and be the eigenvalues of the matrix . The main result of the Method of Freezing states that if , and , then

for the highest exponent of the system, where

The previous best known value and the substantially smaller values of are reduced to the still smaller value.

     

A sequential property of and a covering property of Hurewicz

has the monotonic sequence selection property if there is for each , and for every sequence where for each is a sequence converging pointwise monotonically to , a sequence such that for each is a term of , and converges pointwise to . We prove a theorem which implies for metric spaces that has the monotonic sequence selection property if, and only if, has a covering property of Hurewicz.

     

Semi-free actions of zero-dimensional compact groups on Menger compacta

Let be the -dimensional universal Menger compactum, a -set in and a metrizable zero-dimensional compact group with the unit. It is proved that there exists a semi-free -action on such that is the fixed point set of every . As a corollary, it follows that each compactum with can be embedded in as the fixed point set of some semi-free -action on .

     

Hypersurfaces in a sphere with constant mean curvature

Let be a closed hypersurface of constant mean curvature immersed in the unit sphere . Denote by the square of the length of its second fundamental form. If , is a small hypersphere in . We also characterize all with .

     

Commutative group algebras of -summable abelian groups

In this note we study the commutative modular and semisimple group rings of -summable abelian -groups, which group class was introduced by R. Linton and Ch. Megibben. It is proved that is -summable if and only if is -summable, provided is an abelian group and is a commutative ring with 1 of prime characteristic , having a trivial nilradical. If is a -summable -group and the group algebras and over a field of characteristic are -isomorphic, then is a -summable -group, too. In particular provided is totally projective of a countable length.

Moreover, when is a first kind field with respect to and is -torsion, is -summable if and only if is a direct sum of cyclic groups.

     

Every local ring is dominated by a one-dimensional local ring

Let be a local (Noetherian) ring. The main result of this paper asserts the existence of a local extension ring of such that (i) dominates , (ii) the residue field of is a finite purely transcendental extension of , (iii) every associated prime of (0) in contracts in to an associated prime of (0), and (iv) . In addition, it is shown that can be obtained so that either is the maximal ideal of or is a localization of a finitely generated -algebra.

     

Open covers and the square bracket partition relation

An open cover of an infinite separable metric space is an -cover of if and for every finite subset of there is a such that . Let be the collection of -covers of . We show that the partition relation holds if, and only if, the partition relation holds.

     

A Characterization of the Leinert property

Let be a discrete group and denote by its left regular representation on . Denote further by the free group on generators and its left regular representation. In this paper we show that a subset of has the Leinert property if and only if for some real positive coefficients the identity

holds. Using the same method we obtain some metric estimates about abstract unitaries satisfying the similar identity

     

Continued-fraction expansions for the Riemann zeta function and polylogarithms

It appears that the only known representations for the Riemann zeta function in terms of continued fractions are those for and 3. Here we give a rapidly converging continued-fraction expansion of for any integer . This is a special case of a more general expansion which we have derived for the polylogarithms of order , , by using the classical Stieltjes technique. Our result is a generalisation of the Lambert-Lagrange continued fraction, since for we arrive at their well-known expansion for . Computation demonstrates rapid convergence. For example, the 11th approximants for all , , give values with an error of less than 10.

     

An extension of the Rabinowitz bifurcation theorem to Lipschitz potential operators in Hilbert spaces

The main result of the paper is an extension of the bifurcation theorem of Rabinowitz to equations with continuous jointly in and of class . We also prove a bifurcation theorem for critical points of the function which is just continuous and changes at an isolated minimum (in ) to isolated maximum when passes, say, zero. The proofs of the theorems, as well as the the theorems themselves, are new, in certain important aspects, even when applied to smooth functions.