Sylvester [7] proposed the following question in 1893. If a finite set of points in a plane is such that on the line determined by any two points of the set there is always a third point of the set, is the set collinear? Equivalently, given a finite planar set of non-collinear points, does there exist a line containing exactly two of the points?

A surface S of constant width is the boundary of a convex set K of constant width in euclidean 3-dimensional space E3. (See [l] pp. 127–139. )

Our first result concerns the interdependence of five properties which a curve on such a surface may possess. Let S be a surface of constant width D > 0 which satisfies the smoothness condition that it be a 2-dimensional submanifold of E3 of class C2.

The aim of this paper is to collect some interesting unsolved problems in the topology of compact complex manifolds which are scattered in literature; needless to remark, some of them are known to specialists. Besides, the readermay consult the well-known list of problems by F. Hirzebruch [ 5 ] which has stimulated remarkable progress in this direction. But the topology of complex manifolds has always baffled workers in this field and is still far from satisfactory. Of course, there is remarkable progress in the case of homogeneous compact complex manifolds. We start with the following result.

We study free and injective Lie modules by investigating the relationship between Lie modules and (associative) modules. An important role is played by the universal enveloping ring of a Lie ring [4]. If L is an arbitrary Lie ring and W(L) its universal enveloping ring, we show that the category of Lie L-modules and the category of associative W(L)-module s are isomorphic (section 2). In section 3 we study free Lie modules and show how they may be obtained from free associative modules. A Lie module is free if and only if it is a direct sum of copies of the free Lie module on one generator.

By a (generalized) Schreier extension we mean a group G decomposed into a subinvariant series Gn ↣ Gn-1 ↣ Gn-2 … ↣ G1 ↣ G0 = Gn is anti-invariant in G, i. e. the only subgroup of G which is normal in Gn is the trivial one. ( “ ↣ ” denotes a group monomorphism, i. e. an injection homomorphism. ) As is well known, such groups G can be embedded into the repeated wreath product where Fi ≅ Gi / Gi+1 (cf. [ 2 ], notation of M. Hall [ 1 ], p. 81).

Homological monoids, as first defined by Hilton and Ledermann [ l ], are a generalization of abelian categories. It is known that if is an abelian category, so is here we prove the more general theorem that if is a homological monoid, so is . Our definition differs from that originally given by Hilton and Ledermann by the addition of a uniqueness condition in Axiom 1.

Let A be the matrix algebra of type n × n over a finite algebraic number field F, and V the module of matrices of type n × m over F. V is naturally an A-left module. Given a non-singular symmetric matrix S of type m × m over F, we have a bilinear mapping f of V on A such that f(x, y) = xSy' for elements x and y in V where y' is the transpose of y. In this case, corresponding to the arithmetic of A([l]), the arithmetical theory of V will be discussed to some extent as we establish the arithmetic of quadratic forms over algebraic number fields ([2]). In this note, we shall define a lattice in V with respect to a maximal order in A. and determine its structure (Theorem 1), and after giving a structure of a complement of a lattice (Theorem 2), we shall give a finiteness theorem of class numbers of lattices under some assumption (Theorem 3).

If R is a ring and I is a right ideal of R then I is called faithful if R - I is a faithful right R-module, i.e. if { r ∊ R: Rr⊆ I} = (0). I is called irreducible [ 1 ] provided that if J1 and J2 are right ideals such that J1 ∩ J2 = I, then J1 or J2 = I. Let N(I){ r ∊ R: rI⊆ I} and [ I: a ] = { r ∊ R: ar⊆ I} for a ∊ R. We write (a)r for [(0): a ].

An automorphism Φ of a semigroup S is said to be an inner automorphism if there exists a unit u in S such that

for each a in S.

We write ω for the set of natural numbers (including zero) and for the set of all finite sequences of natural numbers. If n ∊ ω we write . If x is a function which takes its values in cu and whose domain of definition contains then we write for the element (x(0), …, x(n)) of .

Let f(x) be a real-valued function defined on an interval Ia: [ 0, a ]. For each point x in Ia we form the series , where u0 and un+1 = f(un) for n ≥ 0. If the series converges, x will be called a point of convergence; if this series diverges, x will be called a point of divergence. La this note several properties of sets of convergence2 will be obtained.

In this note we consider a system of autonomous differential equations

1.1

where f: En → En is a continuously differentials Le mapping for n ≥ 2. We shall assume that f(0) = 0 and that the origin is locally asymptotically stable.

We consider one-parameter families of distributions which are of the general exponential type, the general series distribution type, or which can be transformed into the exponential type by a one-to-one transformation. In this paper we establish theorems to the effect that such distributions may be characterized by a simple differential equation involving the mean function. It is illustrated that almost all the classical one-parameter families of distributions are characterized by these theorems. Multivariate generalizations are given, and it is also noticed that the functional form of the normalizing factor determines the specific distribution in each general family.

Let be an arbitrary family of continuous complex-valued functions defined on a compact Hausdorff space X. A closed subset B ⊆ X is called a boundary for if every attains its maximum modulus at some point of B. A boundary, B, is said to be minimal if there exists no boundary for properly contained in B. It can be shown that minimal boundaries exist regardless of the algebraic structure which may possess. Under certain conditions on the family , it can be shown that a unique minimal boundary for exists. In particular, this is the case if is a subalgebra or subspace of C(X) where X is compact and Hausdorff (see for example [2]). This unique minimal boundary for an algebra of functions is called the Silov boundary of .

This note is concerned with the extension to locally convex spaces of a theorem of J. Y. Barry [ 1 ]. The basic assumptions are as follows. E is a separated locally convex topological vector space, henceforth assumed to be barreled. E' is its strong dual. For any subset A of E, we denote by w(A) the closure of A in the σ-(E, E')-topology. See [ 2 ] for further information about locally convex spaces. By a projection we shall mean a continuous linear mapping of E into itself which is idempotent.

Planarity in graphs implies relatively small valencies and numbers of edges. In this note we find the maximum sum of valencies and the maximum number of incident edges for a set of n vertices in a planar graph with v vertices. Graphs considered are without multiple edges or loops.