For each positive integer n it is possible to partition the Euclidean plane into n (disjoint) congruent connected sets [1], but if n > 2, it is impossible to partition the plane into n congruent continuumwise connected sets such that some one of the sets can be translated onto another one [2]. This paper is concerned with the possibility of partitioning the plane into three congruent sets without any topological restrictions whatever.

Metric axioms have been given in [3] for space euclidean geometry. If we replace the "similarity axiom" by the "congruence axiom", where congruence is defined to be a similarity of ratio one, the resulting structure is absolute geometry. In order to show this we choose a suitable definition for absolute geometry. The P a s c h system of axioms, given in an improved formulation by H. S. M. Coxeter in [4], is particularly suitable; the primitive notions are points, betweenness relation, and congruence relation. We can verify that every axiom for the absolute geometry in [4] in a theorem in [3] where the similarity axiom has been replaced by the congruence axiom. The only case for which it is not obvious is axiom 15.15 in [4] which says that if ABC and A' B' C' are two triangles with BC ≡ B'C' CA ≡ C'A1, AB ≡ A ' B ', while D and D' are two further points such that [B, C, D] and [B', C' D'] and BD ≡ B' D', then AD ≡ A' D'. In that case we first prove that if two triangles ABC and A'B C are such that AB/A'B' ≡ BC/B'C' ≡ CA/C'A' ≡ 1 then they are congruent; a proof of this, independent of the similarity axiom, can be found in [2]. The proof of 15. 15 in [4] is then obvious. As every axiom in the weakened structure of [3] is a theorem of absolute geometry we have a definition for this geometry.

In this paper we prove that if {sm, n(x, y)} is the sequence of partial sums of the Fourier series of a function f(x, y), which is periodic in each variable and of bounded variation in the sense of Hardy-Krause in the period rectangle, then {sm, n(x, y)} converges uniformly to f(x, y) in any closed region D in which this function is continuous at every point. This result is then used to prove that the regular Hausdorff means of the Fourier series of such a function also converge uniformly in such a region.

Consider the subset containing those functions for which

One never attempts to visualize ; it is just a compact blur in the infinite-dimensional space . Nevertheless, we want to establish that it shares with several other sets an odd but rather remarkable "geometric" property: it is overwhelmingly concentrated around a single element. This element we call the nucleus of .

The sequence to sequence quasi - Hausdorff transformations were defined by Hardy [1] 1 1. 19 p. 277 as follows. For a given sequence {μn} (n ≥ 0) of real or complex numbers, define the operator Δ by for k > l. {tm} (m ≥ 0) is called the sequence to sequence quasi-Hausdorff transform by means of {μn} (or, in short, the [QH, μn] transform) of {sn} (n ≥ 0) if if , provided that the sums on the right-hand side converge for all m ≥ 0. Ramanujan in [11] and [12] has defined the series to series quasi-Hausdorff transformation s and has proved necessary and sufficient conditions for the regularity of the two kinds of transformations.

Let be a non-decreasing sequence of non-negative numbers, and let U∘=0. Then we have

Yang [3] proved the following integral inequality:

If y(x) is absolutely continuous on a≤x≤X, with y(a) = 0, then

Let

be a polynomial of degree n with real and non-negative zeros x1 ≤ x2 ≤ … xn. The zeros xj. will be said to have extent 1 if

Let ξ1 ≤ ξ2 ≤ … ξn-1 be the zeros of the derived polynomial p'(x). The zeros ξ1, ξ2, …, ξn-1 are real and non-negative, and moreover their extent can be at most equal to the extent of the zeros x1, x2, …, xn. The two can indeed be equal. For if the extent of the zeros xj. is 1 and 1 is a multiple zero of p'(x) then ξn-1 = 1. However it is not quite clear how small ξn-1 = 1 can be if Xn = 1. The extent ξn-1 = 1. of the zeros of p'(x) is less than 1 only if 1 is not a multiple zero of p(x). So let us suppose that p(x) has a simple zero at x = 1. Consequently xn-1 is the largest zero of p(x)/(x-l) or equivalently the largest zero of p(x) in 0 < x < 1 and it follows by Rolle's theorem that p'(x) n as a zero in the interval (xn-1,1). Thus the extent ξn-1 of the zeros of p'(x) is greater than xn-1 and it remains to see how small it can be.

In this paper, we have obtained the conditions that a sub-manifold of a locally product Riemannian manifold may also be locally product. Generalizations of the equations of Gauss, Mainardi-Codazzi and Ricci have been obtained for the locally product sub-manifold of the locally product Riemannian manifold.

It might be thought that, as far as the oscillation of solutions is concerned, the behaviour of

and

would be the same as long as t - α(t) → ∞ as t→∞. To motivate the theorem presented in this note, we show first that this is not the case. Consider the above equation with α(t) = 3t/4, a(t) = l/2t2 i.e.

This equation has the non-oscillatory solution y(t) = t1/2 although all solutions of

are oscillatory [1, p. 121].

H. T. Croft [1] has conjectured that among all tetrahedra with fixed total edge length the regular one has the greatest surface area. In this note we prove the following result, which includes this conjecture as a special case

Darling [2] in 1932 gave two types (equations 11 and 18) of transformations connecting generalised hyper geometric functions. The first was studied by Bailey [1] and extended by Sears [4] to a transformation connecting products of basic hyper geometric functions of the type r+1ϕr × r+1ϕr. In a number of papers [6, 7, 8] the author has extended these results to both unilateral and bilateral series with bases q and q1/2. The second type of transformation by Darling for a product 0F1 × 3F2 was extended by Bailey [1] to a transformation between 1F0 × r+1Fr. In the same paper Bailey mentioned the transformation of a 0ϕ1 × 3ϕ2 without proof.

An ellipsoid G is associated uniquely with a positive definite matrix A via

Note that all ellipsoids discussed here are centred at 0. Given G1, and G2 we seek another ellipsoid circumscribed about G1 ∩ G2. It is easy to see that

if and only if x'hx ≤ maxi x'aix for all vectors x.

Given a sequence of r distinct real numbers such that the number of terms of every decreasing subsequence is at most m, then there exists an increasing subsequence of more than n terms, where n is the largest integer less than r/m.

An extremely simple and elegant proof of the theorem was given by A. Seidenberg [2]. This note is intended to point out that a result analogous to the above holds under a more general setting.

The thickness of a graph G is the smallest natural number t such that G is the union of t planar subgraphs. A graph G is t-minimal if its thickness is t and if every proper subgraph of G has thickness < t. (These terms were introduced by Tutte in [3]. In [1, p. 51] Beineke employs the term t-critical instead of t-minimal.) The complete bipartite graph K(m, n) consists of m 'dark1 points, n 'light' points, and the mn lines joining points of different types.

Let f(x) be a polynomial of degree d ≥ 3 with integral coefficients, 'say,

In a previous paper [6] I deduced, from a deep result of Lang and Weil [2], that there is a constant k1(d), depending only on d, such that for all primes p ≥ k1(d), p ⫮ ad, f(x) has a pair of consecutive residues (mod p), that i s, there exists an integer r(0 ≤r ≤ p-1) with the property that

A gap formula for a transform

(1)

is an operator J on f(x)

(2)

Such operators are known for Laplace transform [1; p.91] and [4; pp. 296–299], Stieltjes transform [4; pp. 351–353], a class of convolution transforms [2] and others. Gap formulae for the Weierstrass transform

(3)

and the Weierstrass-Stieltjes transform

(4)

will be proved in the following two theorems.

In an earlier paper [2], we raised the question of determining the minimum span of the kth derivative of a polynomial with real zeros having a given span. More precisely let πn, s denote the class of polynomials , with x1 ≤ x2 ≤ … ≤ xn, and the span σ(P) ≡xn - x1 = 2s (fixed).

This note is concerned with the number C(n) of ordered non-trivial factorizations of an integer n in the special case where n is square free. If F (m) denotes C(p1…Pm) where pi. are distinct primes, it is shown that

and that

The spaces involved in the theory of equations of evolution (that is, the theory of semi-groups) are such that the infinitesimal generators are only densely defined. For the infinitesimal generator to be everywhere defined and smooth (that is, differentiable), one must work with a Frechet space. This is especially important in the non linear theory of Moser (see Moser [3] and Marsden [2]). If the spaces were Banach spaces, the theory would reduce to the classical Picard theory for ordinary differential equations. See Lang [1], for a discussion of the classical theory. In fact in this case the existence theory is both simpler and more comprehensive, because we obtain the important fact that the flow is a diffeomorphism (the solutions depend smoothly, and not merely continuously on the initial data). This has other advantages too, since the theory for smooth flows on Banach manifolds is well developed. (For example, see Marsden [2], theorem. 6.10 for an application we have in mind.)

In graph - theoretic terms a homogeneous p-dendrite, p ≥ 2, is defined as a finite singly-rooted tree in which the root has valency 1 while every other vertex has valency 1 or p. More descriptively, a homogeneous p-dendrite may be imagined to start from its root as the main, or 0th order, branch which proceeds to the first - order branch point where it gives rise top first - order branches. Each of these either terminates at its other end (which is a second-order branch point) or it splits there again into p branches (which are of third order), and so on. The order of the dendrite is the highest order of a branch present in it. For completeness, a 0-th order dendrite is also allowed, this consists of the 0-th order branch alone.