A well known theorem by S. Banach states that a contractive function f: X → X on a complete metric space X has a fixed point, and that this fixed point is unique. This result has a partial extension to multi-functions: every contractive compact-valued multi-function on a complete metric space has a fixed point (see Definition 1 and Theorem 1 below). But simple examples show that this fixed point is no longer unique. We investigate some questions concerned with the properties of the fixed point set Φ of a contractive multi-function φ. Is, e.g., Φ connected if φ is connected-valued? Is Φ convex if φ is convex-valued? The answer is yes if X is the real line (§2), but examples in §3 and §4 show that in general the answer is no.

In this paper we provide new proofs of some interesting results of Firey [2] on isoperimetric ratios of Reuleaux polygons. Recall that a Reuleaux polygon is a plane convex set of constant width whose boundary consists of a finite (odd) number of circular arcs. Equivalently, it is the intersection of a finite number of suitably chosen congruent discs. For more details, see [1, p. 128].

If a Reuleaux polygon has n sides (arcs) of positive length (where n is odd and ≥ 3), we will refer to it as a Reuleaux n-gon, or sometimes just as an n-gon. If all of the sides are equal, it is termed a regular n-gon.

The kernels of the transforms in the class that we shall treat satisfy

1.1

where Re ak = ak Re ck = ck , 0 ≤ ak/ck and Σa-2k (see also [1], [2], [3], [6], [7] and [8]).

Let f(τ) be a complex valued function, defined and analytic in the upper half of the complex τ plane (τ=x+iy, y > 0), such that f(τ+λ) = f(τ) where λ is real and f(-1/τ) = γ(-iτ)k f(τ), k being a complex number. The function (—iτ)k is defined as e k log(-iτ) where log(—iτ) has the real value when — iτ is positive and γ is a complex number with absolute value 1. Such functions have been studied by E. Hecke [4] who calls them functions with signature (λ, k, γ). We further assume that f(τ) = O(|y| -c ) as y tends to zero uniformly for all x, c being a positive real number. It then follows that f(τ) has a Fourier expansion of the type f(τ) = a 0 + Σ a n exp(2πinτ/λ) (n = 1,2,…), the series being convergent absolutely in the upper half plane.

In this work we consider the equation

1

where K(x, y) is singular in the sense that it does not properly belong to L 2 and f(x) is an arbitrary L 2 function.

A Lebesgue measurable function K(x, y) of two variables, having real values on [0.1] × [0.1] is called a singular normal kernel of

1. (i)

   There exists approximating kernels Km(x, y) satisfying

2. (ii)

3. (iii)

4. (iv)

For statistical inference on the unknown parameter θ of a probability distribution with a known form of its density function f(x | θ), the late Sir R. A. Fisher suggests that inference should be based on the observed value of a sufficient statistic when it exists. In the absence of a sufficient statistic, fiducial probability statements about θ can still be made, according to Fisher, if we can find an ancillary statistic so that "the most likely estimate could be made exhaustive by means of the ancillary values" [1, p. 138].

A selfmapping f of a metric space (X, d) is nonexpansive (ε-nonexpansive) if d(f(x), f(y)) ≤ d(x, y) for all x, y ∊ X (respectively if d(x, y) < ε). In [1], M. Edelstein proved that a nonexpansive mapping f of En admits a fixed point provided the f-closure of En (i.e. the set of all points which are cluster points of {fn (x)} for some x) is nonempty. R. D. Holmes [2] considered commutative semigroups of selfmappings of a metric space and obtained fixed point theorems for such semigroups under certain contractivity conditions.

In this note we apply a theorem of T. Mitchell on fixed points of contraction maps to a particular set of contraction maps of metric spaces. Also, part of Mitchell's theorem is extended to permit application, under suitable conditions, when the space involved is not compact.

We employ the following terminology due to Mitchell [1]. Let S be a semigroup; m(S) the space of all bounded real functions on S with sup norm; and X a subset of m(S). For all s ∊ S, the left translation ls {right translation rs } of m(s) by s is given by (lsf)(s') = f(ss') {(rsf)(s')=f(s's)} where f ∊ m(S) and s' ∊ S.

Let A be a commutative complex Banach algebra with identity e. Gleason [1] (cf. also Kahane and Żelazko [2]) has given the following characterization of maximal ideals in A.

Theorem. A subspace X ⊂ A of codimension one is a maximal ideal in A if and only if it consists of non-invertible elements.

The proofs given by Gleason and by Kahane and Żelazko are both based on the use of Hadamard's factorization theorem for entire functions. In this note we show that this can be avoided by using elementary properties of analytic functions.

Let (X, , μ) and (Y, , ν) be two σ-finite measure spaces. An operator T, defined by h = Tf, which maps functions on X into functions on Y is called quasilinear if T(f+g) is uniquely defined whenever Tf and Tg are defined, and if

1.1

where is independent of f and g. If the operator T is called sublinear.

A lower bound for a family of Ramsey numbers is derived using a geometrical argument.

A (simple, undirected) graph G is vertex transitive if for any two vertices of G there is an automorphism of G that maps one to the other. Similarly, G is edge transitive if for any two edges [a, b] and [c, d] of G there is an automorphism of G such that {c, d} = {f(a), f(b)}. A 1-path of G is an ordered pair (a, b) of (distinct) vertices a and b of G, such that a and b are joined by an edge. G is 1-transitive if for any two 1-paths (a, b) and (c, d) of G there is an automorphism f of G such that c = f(a) and d = f(b). A graph is regular of valency d if each of its vertices is incident with exactly d of its edges.

By an L-algebra we mean a power-associative nonassociative algebra (not necessarily finite-dimensional) over a field F in which every subalgebra generated by a single element is a left ideal. An H-algebra is a power-associative algebra in which every subalgebra is an ideal. The H-algebras were characterized by D. L. Outcalt in [2]. Let Sα be the semigroup with cardinality α such that if x, y ∊ S α then xy = y. Consider the algebra over a field F with basis S α. Such an algebra is an L-algebra that is not an H-algebra unless S α contains only one element. In this paper we will prove that an algebra A over a field F with char. ≠ 2 is an L-algebra if and only if it is either an H-algebra or has a basis S α where α is the dimension of A.

In this note we define two concepts which can be thought of as a generalization of noetherian concepts.

The main result is as follows (Corollary A): If R is a ring whose countably generated (left) ideals are (left) principal, then R is a (left) principal ideal ring.

This result if obtained, more generally, for any (left) R-module and any regular cardinal ℵα (Corollary 1); a cardinal ℵα is regular whenever W(ℵα) = {ordinals γ | card γ < ℵα} has no cofinal subset of cardinality less than ℵα.

A result proved by Favard for scalar-valued almost-periodic functions has an immediate extension to Banach space valued functions (see [2] and [3] for explicit details).

The result says that integrals of almost-periodic functions whose 'spectrum' is at positive distance from 0 are again almost-periodic. Our aim here is to indicate a more general formulation of this result using strongly almost-periodic oneparameter groups of operators in Banach spaces.

Whitman [2] solved the word problem for lattices by giving an explicit construction of the free lattice, FL(X), on a given set of generators X.

The solution is the following:

For x, y ∊ X, and a, b, c, d ∊ FL(X),

(W1)

(W2)

(W3)

(W4)

where [p, q] = {x; p ≤ x ≤ q}.

The purpose of this note is to give a simple nonconstructive proof that the condition (W4) must hold in every projective (hence every free) lattice. Jonsson [1] has shown that in every equational class of lattices (Wl), (W2), and (W3) hold. Therefore the combination of these results gives a complete nonconstructive solution to the word problem for lattices.

Using the theory of algebraic numbers, Mordell [1] has shown that the Diophantine equation

1

possesses only two solutions in positive integers; these are given by n = 2, m = 1, and n = 14, m = 5. We are interested in positive integer solutions to the generalized equation

2

and in this paper we prove for several choices of k and l that (2) has no solutions, in other cases the only solutions are given, and numerical evidence for all values of k and l for which max (k, l) ≤ 15 is also exhibited.

In [3] D. B. Fuks defined a duality of functors in the category of weak homotopy types. In general this duality is more difficult to work with than the duality of functors of the category of pointed Kelley spaces [2]. It happens however that all so-called strong functors [2] F of induce functors of , and if we denote the duality operators of and by and D respectively, then there are many cases where .

Let (Ω, , P) be a probability space, and A 1, A 2… be a sequence of members of . The classical Borel-Cantelli problem is to determine the probability that infinitely many events A k occur. The classical results may be found in Feller [2, p. 188]; while related work may be found in Spitzer [3, p. 317], and Dawson and Sankoff [1]. The latter works are generalizations of the Borel-Cantelli lemmas, taken in different directions.

In this paper, necessary and sufficient conditions will be given for infinitely many events A k to occur, with probability 1. A lower bound for the probability that only finitely many A k occur, is developed. In addition, necessary and sufficient conditions that only finitely many A k occur, with probability 1, are given.