(1) The symbols p and q stand for prime numbers and throughout the paper we assume that p is fixed and contained in {3, 5, 7}. Let L be an algebraic number field (i.e., L is a finite extension of Q). Then prime divisors of L dividing p (resp. q) are denoted by (resp. ). The completion of L with respect to is denoted by . Let S be a finite set of prime numbers, and let M/L be a Galois extension with abelian Galois group of exponent p.

Definition. M/L is said to be little ramified outside S if for primes q ∉ S and all one has

with k ∊ N and . Here ζp is a pth root of unity, u1, …, uk are elements in and is the normed valuation belonging to . In particular M/L is unramified at all divisors of primes q ∉ S ∪ {p}.

A sequence, a1 < a2 < a3 < …, of positive integers is called lacunary if the difference sequence dn = an+l — an tends to infinity as n → ∞.

In several recent papers we have made use of these sequences in analysis and combinatorics. In [6] we show that the class of all sets which are either finite or the range of a lacunary sequence is “full” in the sense that if (tk) is a real sequence and for each then (tk) is an l1 sequence, that is,

In [3] the class of all finite unions of sets of is shown to consist of exactly those sets of integers, A, whose characteristic sequence, χA, is in the well known summability space bs + c0. More recently, in [1], we study lacunary sequences in connection with the conjecture of P.

To shed light on the following unsolved problem, several authors have considered related problems. The problem is that of finding commuting approximants to pairs of asymptotically commuting self-adjoint matrices:

Suppose that Hn and Kn are self-adjoint matrices of dimension m(n), with ║Hn║, ║Kn║ ≦ 1, which commute asymptotically in the sense that

Must there then exist commuting self-adjoint matrices H′n and K′n for which

One may alter the conditions imposed on Hn and Kn, for example, by requiring Hn to be normal and Kn to be self-adjoint, and ask whether commuting approximants H′n and K′n can be found satisfying the same conditions. Some of these related problems have been solved. This paper will examine their solutions from a K-theoretic point of view, illustrating the difficulty inherent in modifying them to work for the original problem.

In recent years there has been a growing interest in the existence and asymptotic behavior of entire solutions for second order nonlinear elliptic equations. By an entire solution we mean a solution of the elliptic equation under consideration which is guaranteed to exist in the whole Euclidean N-space RN, N ≧ 2. For standard results on the subject the reader is referred to the papers [2-7, 9-21].

The study of entire solutions, which at an early stage was restricted to simple equations of the form Δu + f(x, u) = 0, x ∊ RN, Δ being the N-dimensional Laplacian, has now been extended and generalized to elliptic equations of the type

A

where

Nonsmooth analysis has provided important new mathematical tools for the study of problems in optimization and other areas of analysis [1, 2, 6-12, 28]. The basic building blocks of this subject are local approximations to sets called tangent cones.

Definition 1.1. Let E be a real, locally convex, Hausdorff topological vector space (abbreviated l.c.s.). A tangent cone (on E) is a mapping A:2E × E → 2E such that A(C, x) is a (possibly empty) cone for all nonempty C in 2E and x in E.

In the sequel, we will say that a tangent cone has a certain property (e.g. “A is closed” or “A is convex“) if A(C, x) has that property for all non-empty sets C and all x in C. (If A(C, x) is empty, it will be counted as having the property trivially.)

Let Ω be a bounded open set in Rn. An immediate consequence of the maximum principle is that if s is a function continuous on and subharmonic on Ω, then

(1)

Of course (1) is no longer true if Ω is not bounded. For example in C ∼ R2 consider the functions

However, if we restrict the growth of s, then (1) may still hold even if the open set Ω is no longer bounded and such is the theme of Phragmèn-Lindelöf type theorems. If we assume even more, namely, that s is upper-bounded, then we can again infer (1) for unbounded open sets Ω. We shall return to this point later.

In the present note, we wish to prove (1) for an arbitrary subharmonic function s on an open subset Ω of Rn. In particular, we do not assume that s is bounded or even of restricted growth. Rather, we impose restrictions on the (possibly unbounded) set Ω.

Much of [2] was devoted to studying pairs of subrings A ⊂ B of a field with the property that A and B have the same prime ideals. In this paper, we continue that investigation, but we no longer assume that A and B are comparable. Interestingly, most of the results of [2] carry over to this more general context. Besides such extensions of [2], additional motivation for the more general context comes from the need to explicate some naturally occurring examples (see Examples 2.5, 3.6, and 4.3).

Section 2 begins by showing that we may reduce to the case in which R is a quasilocal domain with nonzero maximal ideal M and quotient field K. Proposition 2.3 establishes that the set C(R) of all subrings A of K with Spec(A) = Spec(R) forms a complete semilattice.

In connection with the translation equation

(T)

three differential equations arise together with a differential initial condition. They are satisfied by the differentiable solutions of (T) and of the initial condition

(I)

These equations are attributed in [9] to E. Jabotinsky who seems to have been the first who treated these equations in connection with the theory of analytic iteration (see [6], cf. [7, 8], but see also [1, 2, 3]).

Gronau (see [9]) asked whether, conversely, it is true that all solutions of each of these “Jabotinsky differential equations”, possibly with some further initial conditions added, are also solutions of the translation equation. In this paper we give counter examples but also partial positive answers to these questions. First we show how one obtains the Jabotinsky equations from (T) and (I).

An H-relation, as introduced by Rossa and Tangeman [4], is a relation σ on the class of associative rings with their subrings satisfying the following conditions:

(1) IσR implies that I is a subring of R;

(2) if IσR and f is a homomorphism of R, then (If)σ(Rf);

(3) if IσR and J is an ideal of R, then (I∩J)σJ.

Puczylowski [3] imposes also the condition

(4) if J is an ideal of R, then JσR.

A further condition satisfied by many familiar H-relations is the following:

(5) if f is a homomorphism from a ring R onto a ring S and BσS, then there exists a subring A of R such that AσR and Af = B.

Let X be a complete, separable metric space, and a family of probability measures on the Borel subsets of X. We say that obeys the large deviation principle (LDP) with a rate function I( · ) if there exists a function I( · ) from X into [0, ∞] satisfying:

• (i) 0 ≦ I(x) ≦ ∞ for all x ∊ X,

• (ii) I( · ) is lower semicontinuous,

• (iii) for each 1 < ∞ the set {x:I(x) ≦ 1} is compact set in X,

• (iv) for each closed set C ⊂ X

  

• (v) for each open set U ⊂ X

  

It is easy to see that if A is a Borel set such that

then

where A0 and Ā are respectively the interior and the closure of the Borel set A.

By a compactum, we mean a compact metric space. A continuum is a connected compactum. A curve is a 1-dimensional continuum. Let X be a continuum and let C(X) be the hyperspace of (nonempty) subcontinua of X, C(X) is metrized with the Hausdorff metric (e.g., see [12] or [18]). One of the most convenient tools in order to study the structure of C(X) is a monotone map ω:C(X) → [0, ω(X)] defined by H. Whitney [25]. A map ω:C(X) → [0, ω(X)] is said to be a Whitney map for C(X) provided that

The continua {ω−1} (0 < t < ω(X)) are called the Whitney continua of X. We may think of the map ω as measuring the size of a continuum. Note that ω−1(0) is homeomorphic to X and ω−1(ω(X)) = {X}. Naturally, we are interested in the structures of ω−1(t)(0 < t < ω(X)). In [14], J. Krasinkiewicz proved that if X is a circle-like continuum and ω is any Whitney map for C(X), then for any 0 < t < ω(X)ω−1(t) is shape equivalent to X, i.e., Sh ω−1(t) = Sh X (e.g., see [1] or [17]). In [8], we proved the following: If one of the conditions (i) and (ii) is satisfied, then the shape morphism

which is defined in [7] and [8], is a shape equivalence.

Let Δ = {z│ │z│ < 1} be the unit disk and f an analytic function in Δ. The Dirichlet integral DΔ(f) of f on Δ is defined by

and we denote by AD(Δ) the space of all functions f analytic on Δ for which DΔ(f) < ∞. We denote by BMOA(Δ) the space of analytic functions f in Δ for which

and by VMOA(Δ) the space of those analytic functions f in BMOA(Δ) satisfying the condition

Throughout R is a noetherian Witt ring. The basic example is the Witt ring WF of a field F of characteristic not 2 and finite. We study the structure of (noetherian) Witt rings which are also Gorenstein rings (i.e., have a finite injective resolution). The underlying motivation is the elementary type conjecture. The Gorenstein Witt rings of elementary type are group ring extensions of Witt rings of local type. We thus wish to compare the two classes of Witt rings: Gorenstein and group ring over local type. We show the two classes enjoy many of the same properties and are, in several cases, equal. However we cannot decide if the two classes are always equal.

In the first section we consider formally real Witt rings R (equivalently, dim R = 1). Here the total quotient ring of R is R-injective if and only if R is reduced. Further, R is Gorenstein if and only if R is a group ring over Z. This result appears to be somewhat deep.

The complex maximal theorem of Hardy and Little-wood states:

(Mp). For 0 < p < ∞, there exists a positive constant Cp such that if f is holomorphic in the unit disc U of the complex plane then

where

The corresponding statement to the limiting case p = 0 can be stated as follows:

(M0) There exists a positive constant C0 such that if f is holomorphic in U

where log+t = max(log t, 0).

The statement (M0) is false as the following example shows.

Suppose throughout that r ≧ 0, α > 0, αq + β > 0 where q is a non-negative integer. Let {sn} be a sequence of real numbers,

The Borel-type summability method (B, α, β) is defined as follows:

The method (B, α, β) is regular [5]; and (B, 1, 1) is the standard Borel exponential method B. For a real sequence {sn} we consider the slowly decreasing-type Tauberian condition

We shall also be concerned with the Cesàro summability method Cp(p > —1), the Valiron method Vα, and the Meyer-König method Sa (0 < a < 1) defined as follows:

The isomorphism problem for modular group algebras of finite p-groups appears to be still far from a solution (see [7] for a survey of the existing results). It is therefore of interest to investigate the problem for special classes of groups.

The groups we consider here are the p-groups of maximal class, which were extensively studied by Blackburn [1]. In this paper we solve the modular isomorphism problem for such groups of order not larger than pp+1, having an abelian maximal subgroup, for odd primes p.

What we in fact do is to generalize methods used by Passman [5] to solve the isomorphism problem for groups of order p4. In Passman's paper the case of groups of maximal class is actually the most difficult one.

Let Ω be a compact complex n + 1-dimensional Hermitian manifold with smooth boundary M. In [2] we proved the following.

THEOREM 1. Suppose satisfies condition Z(q) with 0 ≦ q ≦ n. Let □p,q denote the -Laplacian on (p, q) forms onwhich satisfy the -Neumann boundary conditions. Then as t → 0;,

(0.1)

(If q = n + 1, the -Neumann boundary condition is the Dirichlet boundary condition and the corresponding result is classical.)

Theorem 1 is a version for the -Neumann problem of results initiated by Minakshisundaram and Pleijel [8] for the Laplacian on compact manifolds and extended by McKean and Singer [7] to the Laplacian with Dirichlet or Neumann boundary conditions and by Greiner [5] and Seeley [9] to elliptic boundary value problems on compact manifolds with boundary. McKean and Singer go on to show that the coefficients in the trace expansion are integrals of local geometric invariants.

The most familiar series representation of the Bessel function is

1.1

Jackson [12] gave the following q-analogues:

1.2

1.3

where 0 < q < 1, the q-shifted factorials are defined by

1.4

and the q-gamma function is given by

1.5

By a von Neumann algebra M we mean a weakly closed, self-adjoint algebra of operators on a Hilbert space which contains I, the identity operator. A factor is a von Neumann algebra whose centre consists of scalar multiples of I.

In all that follows ϕ:M → N will be a one to one, *-linear map from the von Neumann factor M onto the von Neumann algebra N such that both ϕ and ϕ−1 preserve commutativity. Our main result states that if M is not of type I2 then where is an isomorphism or an antiisomorphism, c is a non-zero scalar, and λ is a *-linear map from M into ZN, the centre of N.

Our interest in this problem was aroused by several recent results. In [1], Choi, Jafarian, and Radjavi proved that if S is the real linear space of n × n matrices over any algebraically closed field, n ≧ 3, and ψ a linear operator on S which preserves commuting pairs of matrices, then either ψ(S) is commutative or there exists a unitary matrix U such that

for all A in S. They proved an analogous result for the collection of all bounded self-adjoint operators on an infinite dimensional Hilbert space when ψ is one to one. Subsequently, Omladic [7] proved that if ψ:L(X) → L(X) is a bijective linear operator preserving commuting pairs of operators where X is a non-trivial Banach space, then

where U is a bounded invertible operator on X and A′ is the adjoint of A.

B. Sz.-Nagy and C. Foiaş gave the first examples of quasisimilar operators with different spectra [25]. Indeed, quasisimilar operators can even have different spectral radius [19] (see also [15]). Nevertheless, T. B. Hoover has shown in [19] that if T and S are quasisimilar, then σ(T)∩σ(S) ≠ ∅, where σ(R) denotes the spectrum of the (bounded linear) operator R. In [12], the author improved Hoover's result by showing that each component of σ(S) intersects σ(T), and viceversa. This, in turn, was further improved by L. A. Fialkow in [7]. (Actually, Fialkow's results were obtained independently of [12].)

L. A. Fialkow [7] and L. R. Williams [27] independently proved that σe(T)∩σe(S) ≠ ∅, where σe(R) denotes the essential spectrum of R. Several authors have raised the following natural question:

Is it also true that each component of σe(S) intersects σe(T), and viceversa? [7] [21] [26].