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Let a cube of side k in $ {{\mathbf{R}}^n}$ unit cubes. The collection of all affine subspaces of $ {{\mathbf{R}}^n}$ when ordered by inclusion. We explicitly construct a Dilworth partition into chains of .

Although a great deal of work has gone into construction of the irreducible representations of the symmetric group n (and of the general linear group) a simple, intuitive characterization of the symmetry classes is missing. Relying on a systematic distinction between permutations of variables and permutations of places, we provide two such characterizations, showing that elements belonging to any such symmetry class can be described in one of two ways: (i) as the solutions of explicitly given (though not independent) sets of linear equations or (ii) as linear combinations of "simple" elements of a given symmetry class, a simple element being a generalization to an arbitrary symmetry class of the notion of a decomposable skew-symmetric tensor.

The notions of binary string and binary symmetric function are introduced and basic results presented. Boolean algorithms are given for binary addition and multiplication. An analysis of the redundancies involved is straightforward. The examination of carry propagation which arises in the Boolean analysis of functions may lead to a new interpretation of the notion of computational complexity.

The problem of classifying words on four letters into equivalence classes under the operations of conjugacy and reverse complementation is solved by using a surjective mapping to words on two letters of even length. This classification problem for words on four letters arises in the study of DNA sequences. For words on two letters the concept of a lexical word, which is based on a total order relation, allows one to assign a distinguished word to each conjugacy class. The mapping between words on four letters and those on two letters is used to assign a distinguished word to the conjugacy classes of words on four letters. The Möbius inversion formulas are basic to this enumeration problem.

Some statistical tests of randomness are made of the first 88062 binary digits (or equivalent in other bases) of $ \surd n$, $ 2 \leqq n \leqq 15$ square-free) with $ b = 2,4,8,16$ with $ b = 3,5,6,7$. The statistical tests are the $ {\chi ^2}$ in the assigned bases . It seems to be the first statistical study of the digits of any naturally occurring number in bases other than decimal or binary (octal).

A class of recurrence relations is studied in the context of unnormalized arithmetic. The calculational structure of the proposed algorithm is based on error-theoretic considerations, and is designed to yield a measure of error in a natural manner. The algorithm gives consistent results whether the error propagation is stable or unstable.

Calculation in unnormalized arithmetic provides, as has been described elsewhere, an estimate of the error of each quantity, initial, intermediate, and final, in a calculation. In an acceptable computational algorithm, this error estimate should be close to the true probable error. (The probable error is defined as follows: each computed quantity is defined, in the statement of the problem, explicitly or implicitly as some function of the input data, which are however known only up to errors having some distribution. As the input data vary in their joint distribution, the value of this function varies in a distribution whose variance determines the probable error of the result.) Many computational algorithms fail to achieve this goal, because whenever two numbers are combined arithmetically (+, -, ×, or ÷) in a computer, anycorrelation of the errors of the two numbers is perforce ignored; in consequence, the error estimate of a computed result may be greater than, comparable with, or smaller than the probable error, in which case the algorithm is called “conservative”, “faithful”, or “liberal”, respectively.

We give a bijective proof of a classical identity which we have named the cyclotomic identity.

A brief summary is given of the application of the generalized serial test for randomness to the digits of irrational $ \surd n$ where . The results are consistent, except for a few aberrations, with the hypothesis of randomness of the digits.

We present a class of functions gK(w), K≥ 2, for which the recursive sequences wn+ 1= gK(wn) converge to N1/vwith relative error . Newton's method results when K= 2. The coefficients of the gK(w) form a triangle, which is Pascal's for v= 2. In this case, if w1= x1/y1, where x1, y1is the first positive solution of Pell's equation x2− Ny2= 1, then wn+ 1= xn+ 1/yn+ 1is the Knpth or 2Knpth convergent of the continued fraction for , its period pbeing even or odd.