An ordered plane [1, p. 177 ff.] is a set of undefined entities called points, together with an undefined relation of intermediacy, satisfying the following axioms: (The symbol [ABC] means ″B is between A and C″.)

By the celebrated theorem of Jordan [3] and Hölder [2], there is associated with each finite group G a family of distinct simple groups Hi. such that every composition series of G has ni factor groups isomorphic to Hi and no others. We denote the collection of pairs (Hi, ni) by CF(G). Conversely, given k pairs (Hi, ni), we may construct by an easy direct product procedure a group G with CF(G) = { (Hi, ni) | i =1,…, k}. The composition factors, of course, do not in general determine the group.

We call a semigroup S transitive if S is isomorphic to a semigroup T of transformations of some set M into itself, where T acts on M transitively, that is in such a manner that for all x, y ∊ M we have Xπ = y for some transformation π∊T. In [4] the author showed that S is transitive if and only if there exists a right congruence σ (i.e., an equivalence relation for which a σ b always implies ac σ bc for all c ∊ S) on S, satisfying:

1. (1)There exists a left identity modulo σ, that is an element e such that ea σ a for all a ∊ S .

2. (2)Each σ-class meets each right ideal, or, equivalently, for all a, b ∊ S we have ac σ b for some c ∊ S .

3. (3)The relation σ contains ( i. e. , is less fine than) no left congruence except the identity relation (in which each class consists of a single element).

Let A: a1 < a2 < … < ak be a set of non-negative integers We call the corresponding polynomial the characteristic polynomial, or briefly, the c-polynomial of A. Any polynomial of such a form we call a c-polynomial and any factorization of a c-polynomial into others of the kind we call a c-factorization. If a c-polynomial cannot be factored in this way we call it c-irreducible. In this note we will determine all c-factorizations of the polynomial 1 + x + x2 + … + xn-1, and will find under what circumstances the c-irreducible factors are also irreducible in the usual sense, i.e., irreducible over the field of rationals.

In this note the arithmetic functions L(n) and w(n) denote respectively the number and product of the distinct prime divisors of the integer n ≥ 1, and L(l) = 0, w(l) = 1. An arithmetic function f is called multiplicative if f(1) = 1 and f(mn) = f(m)f(n) whenever (m, n) = 1. It is known ([1], [3], [4]) that every multiplicative function f satisfies the identity

1.1

In this paper we shall say “E has the (F, G) (extension) property” to mean the following: F is a subspace of the real normed linear space G, E is a real normed linear space, and any bounded linear mapping F→E has a linear extension G→E with the same bound (equivalently, every linear mapping F→E of bound 1 has a linear extension G→E with bound 1).

Brouwer's celebrated fixed point theorem states that every map of a closed n-cell into itself has a fixed point.

A similar theorem is here proved for coincidences between a pair of maps (f, g): In→In, where I denotes a closed n-cell (i.e. a homeomorph of the n-ball) and a coincidence is a point x∊In for which f(x)=g(x). That two arbitrary maps (f, g): In→In need not have a coincidence is shown by the pair f:In→y0, g:In→y1, where y0, y1∊In and y0≠y1. More generally, one can immediately construct a map g so that (f, g) is coincidence free if f is not surjective.

If

1.1

and

1.2

a ≥ 0, k=1, 2, 3, …; where is the Laguerre polynomial of order v, defined by

then we shall show that under certain conditions

1.3

Following the inversion theory, two representation theorems are given. The proofs of these theorems follow easily along the lines of Widder [4, Ch. VII] and are therefore omitted.

In [3] Rubel proved that if h(n) is an arithmetic function such that , L finite, then where μ(n) is the Mobius function. This result was extended to functions other than μ(n) in [4]; however, (as first pointed out to the author by Benjamin Volk), the order condition imposed there is unnecessary; in fact, utilizing the result of [3], the following slightly more general theorem has an almost trivial proof.

The following proof of Caratheodory's Theorem, while not essentially new, seems to be natural and therefore of interest.

LEMMA 1. Let P denote a supporting hyperplane of the convex polytope K. Then P∩K is a convex polytope whose vertices are vertices of K.

We prove the following

THEOREM 1. Let f1(z), f2(z) be entire functions of exponential type τ1, τ2; respectively. Suppose that for certain constants K1, K2,

on the real line.

Considering Cauchy's functional equation

f(z1+z2)=f(z1)+ f(z2),

where f(z) is an entire function of z, we have the following functional equation:

(1) |f(x+iy)|=|f(x)+f(iy)|,

where x and y are real.

The solution of a linear non-homogeneous differential equation whose non-homogeneous term is of the form tkeαt can be obtained by what is usually called the method of undetermined coefficients. The application of this method may be justified in several different ways (see for example [1, pp. 114–117], [2, pp. 94–99], [3]).

We shall consider the analogous problem for a system of differential equations. It turns out that we can solve this problem using only elementary techniques of linear algebra. The solution has essentially the same form as in the case of a single equation, but may contain terms which would not be expected and may lack terms which would be expected in a straightforward extension of the theory to systems. Our method of obtaining the solution is constructive, in the sense that while our results give only the form of the solution, the solution itself may be found by substitution of this form into the system of differential equations.

In an n-dimensional manifold Vn, coordinates xi for i = 1, …, n, let each curve x(t) for t0≤t≤t1 of class C1 define a corresponding λ by means of the integral equation

1

If Fn is the distribution function of a distribution n with moments up to order n equal to those of the standard normal distribution, then from Kendall and Stuart [1, p.87],

Let n be an arbitrary but fixed positive integer. Let Tn be the set of all monotone - increasing n-tuples of positive integers:

1

Define

2

In this note we prove that ϕ is a 1–1 mapping from Tn onto {1, 2, 3,…}.

In [2, p. 511] Sanderson has shown that if every Large left ideal of a ring R with identity contains a regular element, and if the regular elements in R satisfy Ore's condition, then the complete (Utumi's) ring of quotients coincides with the classical ring of quotients. He conjectured that the above conditions are also necessary. The following is a counter example.

The Cauchy-Peano existence theorem (1) does not allow us to decide from the form of a given system of equations whether or not its solution can be continued for the infinite interval -∞ < t < ∞. Several sufficient conditions for such a continuation were given by A. Wintner in (2). The main result of his paper is the following theorem which is not proven.

In his paper ″Indexed systems of neighborhoods for general topological spaces″ (Amer. Math. Monthly 68, (1961), 886–893), A.S. Davis defined a hierarchy of what he called regularity axioms. The R1 -axiom is independent of both T0 and T1, but is strictly weaker than T2. In this note, we propose to study the properties of the spaces satisfying the R1 -axiom. In particular, we will show that in many well-known results, the hypothesis can be weakened from T2 to R1, which is part of our motivation in studying R1 -spaces.

C. Goffman asserts that "… in a metric space X a set S is compact if and only if it is closed and totally bounded." [1] and "… every totally bounded sequence in a metric space has convergent subsequence." [2].

The statements (incidentally, equivalent to each other) are both wrong, as the following counter-example shows. Take the set of all reals in the open interval (0, 1) with the usual metric. This space is closed and totally bounded, but not compact.