In following a suggestion of S. Chowla to apply a method of C. Hooley [3] to obtain an asymptotic formula for the sum ∑ r(n)r(n+a), where r(n) denotes the number of representations of n≤x

n as the sum of two squares and is positive integer, we have had to obtain non-trivial estimates for the error term in the asymptotic expansion of

1

Let K be a field and Ka its algebraic closure. A valuation a ring A of K is called henselian, if there is only one valuation ring C of Ka which lies over A (i.e. such that C ∩ K = A) or, equivalently, if Hensel's Lemma is valid for K, A (see [5], F). In the following, we shall consider only rank one valuation rings.

A Room design of order 2n, where n is a positive integer, is an arrangement of 2n objects in a square array of side 2n - 1, such that each of the (2n - 1)2 cells of the array is either empty or contains exactly two distinct objects; each of the 2n objects appears exactly once in each row and column; and each (unordered) pair of objects occurs in exactly one cell. A Room design of order 2n is said to be cyclic if the entries in the (i + l) th row are obtained by moving the entries in the i th row one column to the right (with entries in the (2n - l)th column being moved to the first column), and increasing the entries in each occupied cell by l(mod 2n - 1), except that the digit 0 remains unchanged.

A path of maximum length in a graph G is referred to as a detour path of G and the length of such a path is called the detour number of G. It is not surprising that the study of detour paths is closely associated with the problem of investigating hamiltonian paths in graphs. Evidently few results have been obtained in this area, although Ore [3] has shown that any two detour paths intersect. It is the purpose of this article to further investigate these concepts. In particular, we obtain bounds for several graph theoretic parameters in terms of the detour number and also present formulae for the detour numbers of several important classes of graphs.

In 1879 Kempe [5] presented what has become the most famous of all incorrect proofs of the Four Colour Conjecture, but even though his proof was erroneous his method has become quite useful. In 1890 Heawood [4] was able to modify Kempe's method to establish the Five Colour Theorem for planar graphs. In this article we show that other modifications of Kempe's method can be made which enable one to establish more results about planar graphs. By this process we obtain upper bounds for several parameters which involve partitioning the point set of a graph. In particular, we show that the point set of any planar graph can be partitioned into four or less subsets such that the subgraph induced by each subset is either disconnected or trivial (consists of a single point). We also show that the point set of any planar graph can be partitioned into three or less subsets such that the subgraph induced by each subset contains no cycles.

The theorem of the title asserts that every non-degenerate continuum (that is, every compact connected Hausdorff space containing more than one point) contains at least two non-cutpoints. This is a fundamental result in set - theoretic topology and several standard proofs, each varying from the others to some extent, have been published. (See, for example, [1], [4] and [5]). The author has presented a less standard proof in [3] where the non-cutpoint existence theorem was obtained as a corollary to a result on partially ordered spaces. In this note a refinement of that argument is offered which seems to the author to be the simplest proof extant. To facilitate its exposition, the notion of a weak partially ordered space is introduced and the cutpoint partial order of connected spaces is reviewed.

Meir and Sharma [1] have suggested a modification of Romberg quadrature using Newton-Cotes and, in particular, Simpson sums in place of trapezoidal sums. By comparing the error term with that obtained by Bulirsch [3] for trapezoidal sums, they concluded that the use of Simpson sums would lead to an improvement of the results. The procedure adopted by Meir and Sharma [1] permits them to obtain an expression for the error in the numerical quadrature. However, for the purpose of numerical computation, this procedure appears to be less suitable. In section 3, we give an alternative formulation which would enable us to carry out the computation, using Simpson sums, in the same wasy as is done in the case of Romberg quadrature with trapezoidal sums. Some numerical results are discussed in section 4.

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 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.

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.

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.

Let (xn) be a sequence of real numbers. (xn) corresponds to a number Lim xn called the Banach limit of (xn) satisfying the following conditions:

1. (1) Lim (axn + byn) = a Lim xn + b Lim yn

2. (2) If xn ≥ 0 for every n, then Lim xn ≥ 0

3. (3) Lim xn+1 = Lim xn

4. (4) If xn = 1 for every n, then Lim xn = 1

The existence of such limits is proved by Banach [1].

This paper gives alternative straightforward and simpler proofs of some of the results of Laurent's [10], and Likes' [11], [13]. The derivation of the results is simplified by using the theory of Dirichlet's multiple integral and the transformation used to derive this multiple integral. Some applications of Dirichlet's transformation to order statistic theory from gamma, and normal populations, have been already given by Kabe [7].

It is known that if K is a field, then the ring of formal power series in one or more variables, with coefficients in K, is Henselian at its maximal ideal. In this note we show that if R is a ring (commutative and with identity element) which is Henselian at the maximal ideals M1, M2, … then R[[x]] - the ring of formal power series in x with coefficients from R - is also Henselian at the maximal ideals M1 ⋅ R[[x]] + x⋅ R[[x]], etc.

Let f(x) be a polynomial of degree d ≥ 2 defined over the finite field kq with q = pn elements. Let

If f*(x, y) has no irreducible factor over kq which is absolutely irreducible, f is called an exceptional polynomial [1]. Davenport and Lewis have noted that when d is small compared with p, a permutation (substitution) polynomial is necessarily an exceptional polynomial. It is the purpose of this paper to prove the converse; that is, we will show the existence of a constant a(d), depending only on d, such that if f(x.) is an exceptional polynomial over kq, where p ≥ a(d), then f(x) is a permutation polynomial.

J. A. Erdös proved recently [1] that every singular matrix over a field F is a product of idempotent matrices. He gave two proofs, one valid for matrices which are similar to triangular matrices and the other valid in general. We shall give a simple geometric proof of the above result. Instead of matrices we use linear operators. Moreover we get an explicit factorization in terms of projectors (idempotent operators).

Let A be an m×n (0, l )-matrix. Let C1, C2, …, Cn denote its columns. A sequence of distinct columns is said to form a chain if the inner product of and (for 1 ≤ t ≤ k-l) is at least one. k-1 is called the length of the chain and this chain is said to connect are said to be connected. As can be easily seen, connectedness is an equivalence relation on the set of columns. A matrix is called connected if all its columns belong to the same equivalence class. If Ci and Cj belong to the same equivalence class, then s(Ci, Cj) will denote the length of the shortest chain between Ci and Cj We define the distance between any two columns Ci and Cj to be denoted by d(Ci, Cj), in the following manner.

J.H. van Lint has recently shown [1] that if A(n, m) denotes the n number of representations of m in the form , where εk = 0 or 1 for -n ≤ k ≤ n then

(1)

Using this result, the fact that A(n, m) is a non-increasing function of |m|, and a simple recurrence relation for A(n, m) we derive the following extension of (1):

(2)

where [0 (n)] is any integral valued function m(n) = 0(n).

The nth power Gn of a connected graph G is the graph with the same point set as G and where two points u and v are adjacent in Gn if and only if the distance between u and v in G is at most n. The graph G2 is called the square of G while G3 is referred to as the cube of G.