Let $x$ and $y$ be two random variables with continuous cumulative distribution functions $f$ and $g$. A statistic $U$ depending on the relative ranks of the $x$'s and $y$'s is proposed for testing the hypothesis $f = g$. Wilcoxon proposed an equivalent test in the Biometrics Bulletin, December, 1945, but gave only a few points of the distribution of his statistic. Under the hypothesis $f = g$ the probability of obtaining a given $U$ in a sample of $n x's$ and $m y's$ is the solution of a certain recurrence relation involving $n$ and $m$. Using this recurrence relation tables have been computed giving the probability of $U$ for samples up to $n = m = 8$. At this point the distribution is almost normal. From the recurrence relation explicit expressions for the mean, variance, and fourth moment are obtained. The 2rth moment is shown to have a certain form which enabled us to prove that the limit distribution is normal if $m, n$ go to infinity in any arbitrary manner. The test is shown to be consistent with respect to the class of alternatives $f(x) > g(x)$ for every $x$.

Let S={a1≥a2≥…≥am} be a sequence of real numbers. If ai≥0, aj≤0 then the replacement of ai, aj by ai+aj will be called a merger. The set S can be transformed into a set S1 of s≤m real numbers by a sequence of mergers if and only if∑1saj⩾∑1maj⩾∑m−s+1maj

Let A be the incidence matrix of a block design constructed from a relative difference set. Let rp be the rank mod p of A where p is a prime. In this paper we find inequalities for rp and determine rp completely in some cases, in particular when A is the incidence matrix of the hyperplanes of a projective or Euclidean geometry. An inequality for the p-rank of arbitrary balanced incomplete block designs is also obtained.

The following theorems are proved:o(1)Let A⊕B=A∪B∪(A+B). If G is a finite Abelian group and A1+…+Ak subsets of G with |A1|+…+|Ak|≥|G| then either A1⊕…⊕Ak=G or 0∈A2+…+Ak. For k=2 this statement is true for any group.(3)Let a1, …, ap+k−1 be a sequence of p+k−1 integers. Then it is possible to select k distinct indices i1, …, ik such that ai1+…+aik≡0(mod p).

According to Schuetzenberger [5] we also know, that if v is even then k-X is a square. The incidence matrix of a (bib) with parameters (4N1, 2N -1, N-1) can be used to construct Hadamard matrices of order 4N. To do this one replaces the zeros of the incidence matrix by -1 and borders the resulting matrix by a row and a column of l's. Other Hadamard matrices may be obtained directly (without bordering) from (bib) designs with parameters 4N2, 2N2 N, N2 N. Hadamard matrices have been used in the construction of binary codes [4], and there is no reason why other (bib) designs should not prove advantageous especially in asymmetric channels. It seems also reasonable to expect that codes constructed from groups and especially from Abelian groups will be relatively easy to implement. Apart from its intrinsic interest as a problem in combinatorial analysis, therefore, the construction of difference sets and the question of their existence for certain parameter combinations is of interest in the theory of error correcting codes. We shall restrict ourselves here principally to Abelian groups because not much is known about difference sets in non-Abelian groups. The difference sets with k =v and with k = v-I are called trivial and will not be considered here. It is easy to see that the complement of every difference set is a difference set so that we may always assume k

Multipliers are useful in constructing difference sets as well as in impossibility proofs. The existence of a fixed set facilitates their application, so that the results of this paper should prove to be useful in numerical work concerning difference sets. Let G be a group which we shall write multiplicatively. For any set A of elements of G, we define (1)

The apparent degree of soiling of a coloured object is a matter of subjective assessment, but is, as we have been able to show with a group of observers, closely paralleled by the physical quantity Hs, the diminution (%) in the brightness coefficient of the surface–Hs = YO−YS/YO×100A study of soiled coloured starch has indicated the possibility of relating Hs and hence degree of apparent soiling to the physical concentration of soil and the reflectance spectrum of the original surface. It is possible to compute from the spectrum a characteristic empirical quantity, the Sim value, which ranks the coloured material in order of relative apparent unsoilability in a series of materials, similar in structure but differing in colour. The derivation of SMR values for coloured starch involves no optical hypothesis, and the generalisation to fibrous materials depends only on the essential optical similarities of the systems. Full practical details, with tabulated data, are given for the determination of the SME values of coloured fibres. The conclusions are discussed with special reference to the behaviour of carpets made from viscose rayon staple.

H. B. Mann and T. H. Morton, Discuss. Faraday Soc., 1954, 16, 75 DOI: 10.1039/DF9541600075