Consider the graph Mn, where n = 2r ≥ 6, consisting of a polygonof length n and all n/2 chords joining opposite pairs of vertices. Thisgraph has 2r vertices which we denote by 1, 2, 3,..., 2r, and the 3r(undirected) edges

We call Mn the n-ladder, defined thus far only for n even. Thethree smallest n-ladders with n even are shown in Figure 1.

We use the terminology of [1]. Let D be a strongly connected digraph on npoints and containing m lines, and let A = A(D) be the correspondingadjacency matrix, so that A is an n x n 0-1 matrix containing m unitelements. We recall that A and D are said to be p-cyclic if p is thegreatest common divisor of the lengths of all directed cycles of D. Clearly,the larger the value of p, the smaller the value of m must be; in this notewe make the latter and related statements precise.

To each tournament Tn with n nodes n there corresponds theautomorphism group G(Tn) consisting n of all dominance preservingpermutations of the set of nodes. In a recent paper [3], Myron Goldberg andJ. W. Moon consider the maximum order g(n) which the group of a tournamentwith n nodes may have. Among other results they prove that

1

2

Throughout this paper (X, d) will be a metric space with metric d, and h ahomeomorphism of X onto itself. For any real number r > 0, and p ∊ X,U(p, r) will denote the open r - sphere about p. Any point p ∊ X is calledregular [3] if for any given ∊ > 0 there exists a δ > 0 such that d(p,y) <δ implies d(hn(p), hn(y)) < ∊ for allintegers n, where hn denotes the iterates of h for n > 0, of h-1 for n < 0, and h0 is the identity. Any pointof X which is not a regular point i s called an irregular point. Let I(h)denote the set of all the irregular points of X and R(h) = X-I(h). Lim infand Lim sup are defined as in [4].

Let Ⓗ be an abstract space and for every positive integer n let Fn, θ(x, y), θ ∈ Ⓗ, be a family of distribution n, 0 function s ofrandom variables (Xn, Yn)θ, θ ∈ Ⓗ. Forevery θ ∈ Ⓗ, Eθg (Xn, Yn) will denote theexpected value of the function g of (Xn, Yn )θ. The following proposition is proved.

Let μ∗ be an outer measure on (X, S) with σ- algebra S and let μ* be the inner measure induced by μ∗. A set M is ameasurable cover of a set A ⊆ X if A ⊆ M, M is measurable, and μ∗(M-A) = 0. We assume that every subset of X has a measurable cover; thisholds, for example, if μ∗ is the outer measure induced by ameasure which is σ- finite on X [2, theorem C, p. 50].

A vector measure (countable additive set function with values in a Banachspace) on a field may be extended to a vector measure on the generated σ-field, under certain hypotheses. For example, the extension is establishedfor the bounded variation case [2, 5, 8], and there are more generalconditions under which the extension exists [ 1 ]. The above results have ashypotheses fairly strong boundedness conditions on the n o rm of the measureto be extended. In this paper we prove an extension theorem of the same typewith a restriction on the range, supposing further that the measure ismerely bounded. In fact a vector measure on a σ- field is bounded (III. 4. 5of [3]) but it is conceivable that a vector measure on a field could beunbounded.

In 1955, Suranyi and P. Turán [8] considered the problem of existence anduniqueness of interpolatory polynomials of degree ≤ 2n-1 when their valuesand second derivatives are prescribed on n given nodes. Around this kind ofinterpolation - aptly termed (0, 2) interpolation - considerable literaturehas grown up since then. For more complete bibliography on this subject werefer to J. Balazs [3], Later we considered [10] the problem of modified (0,2) interpolation when 2 the abscissas are the zeros of (1-x2) Tn(x), where Tn(x) is the Tchebycheff polynomial ofthe first kind (Tn(x) = cos n θ, x = cos θ).

In 1955, J. Suranyi and P. Turán [8] introduced the nomenclature (0,2)-interpolation for the problem of finding polynomials of degree ≤ 2n-1whose values and second derivatives are prescribed in certain given nodes.In a series of papers ([2], [3], [8]) Professor Turán and his associatesdiscussed the problems of existence, uniqueness, explicit representation andconvergence of such interpolatory 2 polynomials when the nodes are the zerosof (1-x2) P′n- 1(x), Pn- 1(x) being theLegendre polynomial of degree n - 1.

In 1940, I. Niven [2] proved that the gaussian integer z = x + iy is the sumof two squares of gaussian integers if, and only if, y is even and not bothof 1/2x and 1/2y are rational odd integers. In this note we calculate thetotal number g2(z) of representations of z in this form.

1

where a, b, c, d are rational integers, if and only if

2

Let d denote a fixed integer > 1 and let GF(q) denote the finite field ofq = pn elements. We consider q fixed ≥ A(d), where A(d) is a(large) constant depending only on d. Let

1

where each aiεGF(q). Let nr(r = 2, 3, …, d) denote thenumber of solutions in GF(q) of

for which x1, x2, …, xr are alldifferent.

Let p denote a prime number and let κp denote the finite field ofp elements. Let f(x) ∊ κp[x] be of fixed degree d ≥ 2. We supposethat p is also fixed, large compared with d, say, p ≥ p0(d). ByV(f) we denote the number of distinct values of f(x), x ∊ κp. Wecall f maximal if V(f) = p and quasi-maximal if V(f) = p + O(1). Clearly amaximal polynomial is quasi-maximal but it is not known under whatconditions the converse holds.

Let R be a topological (Hausdorff) ring such that for each a ∊ R, aR and Raare closed subsets of R. We will prove that if the set of non - trivialright (left) zero divisors of R is a non-empty set and the set of all right(left) zero divisors of R is a compact subset of R, then R is a compactring. This theorem has an interesting corollary. Namely, if R is a discretering with a finite number of non - trivial left or right zero divisors thenR is a finite ring (Refer [1]).

Hall and Puckett [2] have shown that an arc - preserving function defined ona locally connected continuum having no local separating points is ahomeomorphism if its total image is not an arc or point. This note showsthat their results can be extended to non-compact manifolds.

In this note we study dependence in matroids as an exercise in combinationalalgebra. Because the work seems to have little connection with graph theorywe will not use Tutte′s approach (1) which uses dual concepts. To define amatroid we use Edmunds (2).