E. Hewitt [1] used the Daniell approach to define a real-valued measure function on a σ-algebra of the real line. He began by defining an arbitrary non-negative linear functional I on L∞ ∞(R), (the space of all complex-valued continuous functions on the real line R which vanish off some compact subset of R).

In this note real valued functions, defined on a linear sublattice S of a linear lattice R and satisfying the two order conditions (M1) and (M2), are studied from the point of view of the existence and uniqueness of extensions to R. The paper is partly expository and supplements and extends §3 of [4] where S was assumed to be an ℓ-ideal.

Let f(t) be integrable L(-π, π) and periodic with period 2π, and let

1.1

be its Fourier series. The series

1.2

obtained by the term by term differentiation of the series (1. 1), is called the Derived Fourier series of (1. 1). The series conjugate to (1. 2),

1.3

is called the Derived conjugate series of the Fourier series of f.

In a paper by LaSalle [l] on linear time optimal control the following lemma is proved:

Let Ω be the set of all r-dimensional vector functions U(τ) measurable on [ 0, t] with |ui(τ)≦1. Let Ωo be the subset of functions uo(τ) with |uoi(τ) = 1. Let Y(τ) be any (n × r ) matrix function in L1([ 0, t]).

Recently B. Palczewski and W. Pawelski [l] have given some sufficient conditions of uniqueness for the solutions of the Darboux problem for equations of the form:

1

The criteria given there for equations of the form (1) are natural generalizations of the criteria given by Krasnosielski and Krein [2] in the corresponding ordinary differential case. The purpose of the present note is to give a further generalization of the above result and two other uniqueness conditions for the solutions of the Darboux problem of the same form.

The Cantor function C [2; p. 213], which appears in analysis as a simple example of a continuous increasing function which is not absolutely continuous, has the following properties:

1. (i) C is defined on [0,1], with C(0) = 0, C (l) = l;

2. (ii) C is continuous and non-decreasing on [0,1];

3. (iii) C is constant on each interval contiguous to the perfect Cantor set P;

4. (iv) C fails to be constant on any open interval containing points of P;

5. (v) The set of points at which C is non-differentiable is non-denumerable.

Classically, there has been, for obvious reasons, an intimate relation between the concepts "rings of quotients" and "divisible modules". Recently, however, their generalizations have appeared to diverge.

For example, Hattori ([9]) and Levy ([15]) have generalized the concept of "divisibility" as follows: Hattori (respectively Levy) defines a left R-module M over a ring R to be divisible if and only if Ext1R(R/I, M)=0 for every principal left ideal I ⊂ R (respectively, every principal left ideal I ⊂ R which is generated by a regular element of R).

The purpose of this paper is to present a geometric theorem which provides a proof of a fundamental theorem of finite Markov chains.

The theorem, stated in matrix theoretic terms, concerns the asymptotic behaviour of the powers of an n by n stochastic matrix, that is, a matrix of non-negative entries each of whose row sums is 1. The matrix might arise from a repeated physical process which goes from one of n possible states to another at each iteration and whose probability of going to a state depends only on the state it is in at present and not on its more distant history.

We consider a system of differential equations of second order given by

1

(' = d/dt) where P and Q have continuous first partial derivatives with respect to x and y in some open and simply connected set R containing O = (0, 0) which we assume to be the only singular point in R. In fact, let R be the whole plane; for if not then the following discussion can be modified.

Recently A. Haimovici [1] has proved a general fixed point theorem of transformations in metric spaces from which he obtained existence theorems for certain types of ordinary and partial differential equations. However, both the result and the proof are given for a rather special case. One of the purposes of this present note is to put his result on a more concrete basis and give a stronger characterization of the kind of transformations used in [l]. (Theorem 3).

This paper is a continuation of two papers [4], [5] and brings out the solution of the ballot problem in its general form.

In [5], Narayana has considered a generalised occupancy problem which can be viewed as a problem in compositions of integers. In what follows, we use the definitions of [6].

A family of sets is said to possess property if there exists a set such that and F ⊄ B for each In [1], P. Erdos and A. Hajnal ask the following question: Does there exist for every positive integer k a finite family of finite sets satisfying

1. (i) |F|=k for each

1. (ii) | F∩ G| ≤ 1 for each F, , F ≠ G

1. (iii) does not possess property ?

The nth Taylor mean of order r of a sequence {sn} is given by

1.1

where

1.2

Cowling [l] has shown that this method is regular if and only if 0 ≦ r < 1. Since r = 0 corresponds to ordinary convergence, it will be assumed here that 0 < r < 1.

G. de Rham [1] has shown that certain remarkable curves, defined by limits of sequences of inscribed polygons, are solutions of functional equations of the form

If the functions FP (z) (p = 1,2) are contraction mappings of the complex plane into itself, i.e. for fixed r < 1

If (X, S,μ) is an arbitrary complemented measure space and X is σ-finite then (L1)* = L∞ or, more precisely, (L1)* is isometric and isomorphic to L∞ by the correspondence

It is well known that there exist non σ-finite spaces with (L1)* ≥ L∞.

Egoroff' s theorem [1] was extended by Kvačko [3] to functions with values in a separable metric space; and, as is easily seen, this result applies also to separable pseudometric spaces. In the present note we shall use this theorem to obtain some propositions on iterated limits, which, despite their simplicity, seem not yet to be known in the proposed generality.

A semi-lattice (Birkhoff, Lattice Theory, p. 18, Ex. 1) is an algebra <A,.> with a single binary operation satisfying: (1) x = xx, (2) xy = yx, and (3) (xy)z = x(yz). In this note we show that the three identities may be reduced to two but cannot be reduced to one.

It is easy to see that (2), (3) imply (4) (uv)((wx)(yz)) = ((vu)(xw))(zy). Setting w = y = u and x = z = v in (4) and using (1) we get uv = vu. Setting v = u, x = w, and z =y in (4) and using (1) we get u(wy) = (uw)y. And so (1) and (4) imply (2) and (3).

If R is a simple ring with 1 which contains a maximal annihilator right ideal then R is the endomorphism ring of a unital torsion-free module over an integral domain.

We first prove the following:

Let R be a ring with 1. If a ϵ R such that (a)r = {r ϵ R|ar = 0} is a maximal annihilator right ideal then HomR(aR, aR) is an integral domain.

Let F(x) be the continuous distribution function of a random variable X and Fn(x) be the empirical distribution function determined by a random sample X1, …, Xn taken on X. Using the method of Birnbaum and Tingey [1] we are going to derive the exact distributions of the random variables

and and where the indicated sup' s are taken over all x' s such that -∞ < x < xb and xa ≤ x < + ∞ with F(xb) = b, F(xa) = a in the first two cases and over all x' s so that Fn(x) ≤ b and a ≤ Fn(x) in the last two cases.