Let α be an open arc on the unit circle

and for z=reiθ, 0 ≤ r < 1, let

(1)

The function ω(z; α) is called the harmonic measure of the arc α with respect to the unit disc, (Nevanlinna 2); it is harmonic and bounded in the unit disc and possesses (Fatou) boundary values 1 and 0 at interior points of α and the complementary arc β respectively.

J. Mycielski has conveyed to me the following problem by P. Erdös. Let {tn} be a sequence of natural numbers such that

(a) Does there exist an irrational number η such that the sequence {tnη (mod 1)} is not dense in [0,1)?

A partially ordered space is an ordered pair (X, ≤) where X is a compact metric space and ≤ is a partial ordering on X such that ≤ is a closed subset of the Cartesian product X×X. ≤ is said to be a closed partial order on X.

We are concerned with the nth order differential equation y(n) = (x, y, y′, …,y(n-1)), where it is assumed throughout that f is continuous on [α,β) × Rn, α < β≤∞, and that solutions of initial value problems are unique and exist on [α, β). The definition of the first conjugate point function η1(t) for linear homogeneous equations is extended to this nonlinear case. Our main concern is what properties of this conjugacy function are valid in the nonlinear case.

It is well-known that the join-irreducible elements J(L) and the meet-irreducible elements M(L) of a lattice L of finite length play a central role in its arithmetic and, especially, in the case that L is distributive. In [3] it was shown that the quotient set Q(L) = {b/a | a ∊ J(L), b ∊ M(L), a ≤ b} plays a somewhat analogous role in the study of the sublattices of L. Indeed, in a lattice L of finite length, if S is a sublattice of L then S = L — ∪b/a∊A [a, b] for some A ⊆ Q(L). Furthermore, the converse actually characterizes finite distributive lattices [3].

A well-known theorem which goes back to R. E. Johnson [4], asserts that if R is a ring then Q(R), its maximal ring of quotients is regular (in the sense of v. Neumann) if and only if the singular ideal of R vanishes. In the theory of semigroups a natural question is therefore the following: Do there exist properties which characterize those semigroups whose maximal semigroups of quotients are regular? Partial answers to this question have been given in [3], [7] and [8]. In this paper we completely solve the commutative case, i.e. we give necessary and sufficient conditions for a commutative semigroup S in order that Q(S), the maximal semigroup of quotients, is regular. These conditions reflect very closely the property of being semiprime, which in the theory of commutative rings characterizes those rings which have a regular ring of quotients.

For each Hausdorff space X, let FX be an Hausdorff extension of X. The existence of the largest subcategory of HAUS on which F s a functor and an epi-reflection is investigated.

In [4] Carroll and the author have treated the following problem

(1)

where Λ is a closed densely defined self-adjoint operator in a separable Hilbert space H with (Λu, u) ≥ c ‖u‖2, c > 0, Λ-1 ∊ L(H) (L(E, F) is the space of continuous linear maps from E to F; in particular, L(H) = L(H, H)), a(t) > 0 for t > 0 a(0) = 0 and S(t), R(t), B(t) ∈ L(H).

Inspired by a question of J. B. Miller, L. Fejes Tóth asked for a catalogue of those subsets of Euclidean n-space which are homothetic (similar and similarly situated) to their intersection with a suitably translated copy of themselves. For example, a triangle is homothetic to its intersection with an arbitrarily translated replica provided only that the intersection has non-void interior. In 3-space a cube is homothetic to its intersection with a replica translated part way along any body diagonal. With these two preliminary examples for motivation, let us make a definition.

We introduce almost polar-dense lattices and prove that the generalized interval topology of an almost polar-dense, modular lattice is equivalent to its interval topology. Furthermore, for totally ordered sets, the converse holds: if the generalized interval topology is the interval topology, then the set is almost polar-dense.

For α>0 the Riemann-Liouville Integral J(α) is given for suitable functions g by

1

For a variety of function spaces (e.g., C[0, 1] or Lp(0, 1) with p≥1) this defines a C0 semigroup which has been extensively studied (cf., e.g., [3]).

All rings are associative. A ring T is said to be radical over a subring R if for every t ∈ T, there exists a natural number n(t) such that tn(t) ∈ R.

In [1] Faith showed that if T is radical over R and T is primitive, then R is primitive. We might then ask if the same is true if prime is substituted for primitive.

Nous donnons, dans ce qui suit, la preuve de la proposition suivante.

The set of projections in the BAER*-semigroup of hemimorphisms on an orthomodular lattice L can be partially ordered such that the subset of closed projections becomes an orthocomplemented lattice isomorphic to the underlying lattice L. The set of closed projections is identical with the set of Sasaki-projections on L (Foulis [2]). Another interesting class of (in general nonclosed) projections, first investigated by Janowitz [4], are the symmetric closure operators. They map onto orthomodular sublattices where Sasaki-projections map onto segments of the lattice L.

Let X be a random variable having the extreme value density of the form

(1)

where r is assumed to be a positive Lebesgue measurable function of x and the function q is defined by

for all θ in Ω = (0, ∞). It is further assumed that q(θ) approaches zero as θ → ∞.

In this note we are concerned with estimating parametric functions g(θ) of the form [1/q(θ)]α, α any real number. The loss function is assumed to be squared error and the estimators are assumed to be functions of a single observation X.

We consider here the nonlinear differential equation

(1.1)

where x ∊l= [a, ∞). We will make the following assumptions

1. (A) f is continuous on [a, ∞) × Rn,

2. (B) solutions of initial value problems (I.V.P.'s) are unique and extend throughout [a, ∞).

3. (C)

In this note we give an elementary proof of the following.

Let be an integer. n≥1 Then, every positive even integer less than or equal to n(n2—1)/3 can be expressed as a sum ofn squares of integers from the set {0, 1, 2, …, n - 1}.

Theorem. Letbe a function holomorphic in the disk, wherep is a natural number andIfthen then f(z) assumes every complex value infinitely often in every sector.

The purpose of this note is to prove the above result. To do this, we first observe that from the condition a<∞, we can easily show that the derivative f′(z) satisfying

In this note, we prove analogues of the classical maximum modulus theorem and Schwarz lemma, for sequence spaces. We begin by stating these two results in a convenient way; that is for the unit disk and functions of bound one.

It follows from [3] that all morphisms of free closed categories on finite discrete categories are components of natural or “generalized” natural transformations, and from [8] that all hom-sets of such categories are finite. The purpose of this paper is to show that neither statement remains true if the categories are also assumed to be cartesian.