On an analogue of Selberg's eigenvalue conjecture for

Let be the homogeneous space associated to the group
. Let where and consider the first nontrivial eigenvalue of the Laplacian on . Using geometric considerations, we prove the inequality . Since the continuous spectrum is represented by the band , our bound on can be viewed as an analogue of Selberg's eigenvalue conjecture for quotients of the hyperbolic half space.

     

On component groups of and degeneracy maps

For an integer and a prime not dividing , we study the kernel of the degeneracy map , where and are the component groups of and , respectively. This is then used to determine the kernel of the degeneracy map when . We also compute the group structure of in some cases.

     

Character degrees and local subgroups of -separable groups

Let be a finite -solvable group for different primes and . Let and be such that . We prove that every of -degree has -degree if and only if and .

     

The exposed points of the set of invariant means on an ideal

Let be a -compact locally compact nondiscrete group and let be a -invariant ideal of . We denote the set of left invariant means on that are zero on (i.e. for all ) by . We show that, when is amenable as a discrete group and the closed -invariant subset of the spectrum of corresponding to is a -set, is very large in the sense that every nonempty -subset of contains a norm discrete copy of , where is the Stone- compactification of the set of positive integers with the discrete topology. In particular, we prove that has no exposed points in this case and every nonempty -subset of the set of left invariant means on contains a norm discrete copy of .

     

Lipschitz images with fractal boundaries and their small surface wrapping

Assume and is a Lipschitz -mapping; and denote the volume and the surface area of . We verify that there exists a figure with , and, of course, , where depends only on the dimension and on . We also give an example when is a square and ; in fact, the boundary of can contain a fractal of Hausdorff dimension exceeding one.

     

A structural result of irreducible inclusions of type

Given an irreducible inclusion of factors with finite index , where is of type , of type , , and are relatively prime positive integers, we will prove that if satisfies a commuting square condition, then its structure can be characterized by using fixed point algebras and crossed products of automorphisms acting on the middle inclusion of factors associated with . Relations between and a certain -kernel on subfactors are also discussed.

     

Sufficient conditions for one domain to contain another in a space of constant curvature

As an application of the analogue of C-S. Chen's kinematic formula in the 3-dimensional space of constant curvature , that is, Euclidean space , -sphere , hyperbolic space (, respectively), we obtain sufficient conditions for one domain to contain another domain in either an Euclidean space , or a -sphere or a hyperbolic space .

     

The maximal normal -group

Let be a prime number and let be an abelian -group. Let be the maximal normal -subgroup of and the maximal -subgroup of its centre. Let be the torsion radical of . Then . The result is new for and 3, and the proof is new and valid for all primes .

     

On uniqueness of -adic meromorphic functions

Let be a complete ultrametric algebraically closed field of characteristic zero, and let be the field of meromorphic functions in . For all set in and for all we denote by the subset of : zero of order After studying unique range sets for entire functions in in a previous article, here we consider a similar problem for meromorphic functions by showing, in particular, that, for every , there exist sets of elements in such that, if have the same poles (counting multiplicities), and satisfy , then . We show how to construct such sets.

     

On the Poincaré series and cardinalities of finite reflection groups

Let be a crystallographic reflection group with length function . We give a short and elementary derivation of the identity , where the product ranges over positive roots , and denotes the sum of the coordinates of with respect to the simple roots. We also prove that in the noncrystallographic case, this identity is valid in the limit ; i.e., .

     

The intersection of three spheres in a sphere and a new application of the Sato-Levine invariant

Take transverse immersions such that (1) is an embedding, (2) and is connected, and (3) . Then we obtain three surface-links = (, ) in , where =(1,2,3), (2,3,1), (3,1,2). We prove that, we have the equality where is the Sato-Levine invariant of , if all are semi-boundary links.

     

Eigenvalue pinching theorems on compact symmetric spaces

We prove two first eigenvalue pinching theorems for Riemannian symmetric spaces (Theorems 1 and 2). As their application, we answer negatively a question raised by Elworthy and Rosenberg, who proposed to show that for every compact simple Lie group with a bi-invariant Riemannian metric on with respect to , being the Killing form of the Lie algebra , the first eigenvalue would satisfy

for all orthonormal bases of tangent spaces of (cf. Corollary 3). This problem arose in an attempt to give a spectral geometric proof that for a Lie group .

     

Zeros of the Zak transform on locally compact Abelian groups

Let be a locally compact abelian group. The notion of Zak transform on extends to . Suppose that is compactly generated and its connected component of the identity is non-compact. Generalizing a classical result for , we then prove that if is such that its Zak transform is continuous on , then has a zero.

     

Singular extensions of the trace and the relative Dixmier property in the type factors

If is an inclusion of type factors with we study the connection between the existence of singular states on which extend the trace on and the Dixmier approximation property in with unitaries in We also prove the existence of singular conditional expectations from certain free product factors onto irreducible hyperfinite subfactors.

     

Glasner sets and polynomials in primes

A set of integers is said to be Glasner if for every infinite subset of the torus and there exists some such that the dilation intersects every integral of length in . In this paper we show that if denotes the th prime integer and is any non-constant polynomial mapping the natural numbers to themselves, then is Glasner. The theorem is proved in a quantitative form and generalizes a result of Alon and Peres (1992).

     

The equivalence of some Bernoulli convolutions to Lebesgue measure

Since the 1930's many authors have studied the distribution of the random series where the signs are chosen independently with probability and . Solomyak recently proved that for almost every the distribution is absolutely continuous with respect to Lebesgue measure. In this paper we prove that is even equivalent to Lebesgue measure for almost all .

     

Algebras of invariant functions on the Shilov boundaries of Siegel domains

Let be a bounded symmetric domain and the Shilov boundary of . Let be the Shilov boundary of the Siegel domain realization of . We consider the case when is the exceptional non-tube type domain of the type . We prove that is not a Gelfand pair and thus resolve an open question of G. Carcano.

     

Eigenvalues of a Sturm-Liouville problem and inequalities of Lyapunov type

We consider the eigenvalue problem in , , where keeps a fixed sign and , and we obtain some lower and upper bounds for in terms of its nonnegative eigenvalues . Two typical results are: (1) if and is not the square of a positive integer; (2) if is the smallest eigenvalue.

     

Spectral multiplier theorem for spaces associated with some Schrödinger operators

Let be the semigroup of linear operators generated by a Schrödinger operator , where is a nonnegative polynomial. We say that is an element of if the maximal function belongs to . A criterion on functions which implies boundedness of the operators on is given.

     

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 .