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There are two well known constructions of the real numbers from the rationals—namely the Dedekind cuts method in which a real number is defined as a class of rationals, and the Cantor–Cauchy completion method in which a real number is defined as an equivalence class of Cauchy sequences of rational numbers. In this article we give a new construction of the reals from the rationals in which a real number is defined as an equivalence class of sets of natural numbers.

We find the maximum number of orthogonal skewsymmetric anticommuting integer matrices of order n for each natural number n and relate this to finding free direct summands of certain generic projective modules. While studying composition of quadratic forms, Hurwitz [4] and Radon [6] considered families of orthogonal matrices {A1,3L , A,} satisfying the conditions (1) Ai= -At, i= 1, ,s (2) AiAj = -AjAi, i $ j. DEFINITION. (1) A family of orthogonal matrices satisfying (1) and (2) above will be called a Hurwitz-Radon (H-R)family. If n is a positive integer and n=2ab, b odd, then we write a=4c+d where O0d

An interesting way to find the smallest prime factor of a given natural number. A computer program for this procedure is supplied.

giving a way to calculate as many of the B's as desired. Formula (1.2), stated in tha way or in the form (1.3) or in various other forms, has been known a long time. Thus, in 1748 Euler gave (1.3) (obscurely expressed) in section 76 of his Introductio [4], [5]. Adams and Hippisley [1] gave the formula in the form (1.2) (their formula 6.361). Formula (1.3) is given as formula (0.314) in any of the several editions (Russian original, German, or English) of Ryshik and Gradstein [16], where, however, p is restricted to be a natural number. Thinking to remove this restriction of Ryshik and Gradstein, von Holdt [17] published still another derivation using properties of double sums to establish that (1.2) holds true for rational real p. His derivation avoids use of differentiation of the series in (1.1). We mention this because differentiation of formal power series affords a quick proof of (1.2) and has been used before. The basic recurrence relation is not as widely known as it should be, and has been rediscovered repeatedly. Thus Barrucand [2] found (1.2) again and made applications of it. Cappellucci [18] found it in the form (1.3) but attributed it to Hansted [19]. Hindenburg [12], in his treatise on the multinomial theorem (p. 291) gave (1.3) in a perfectly obscure notation. Many other references could be cited. Actually, the recurrence is implicit in still another class of formulas widespread in the literature, stemming from the early work of Hindenburg's student Rothe [15]. I myself have written a number of papers, e.g., [6]-[11] having to do with a formula of Rothe and its consequences for combinatorics, special functions and number theory. What we shall do here is to derive the formulas again and put them in a

Fibonacci groups are the groupswhere r is a natural number. The groups F(2, 8) and F(2, 10) are shown to he infinite, thus leaving F(2, 9) as the only Fibonacci group whose finiteness or infiniteness has not been determined.

It is shown that there exists a sequence of natural numbers {nk} which does not belong to the class B2 and which cannot be decomposed into a finite number of lacunary sequences such that: a) if the series\(\sum\nolimits_{k^{ = - \infty } }^\infty {c_k e^{in_k x} }\) converges on a set of positive measure, then the series consisting of the squares of the coefficients converges; b) for each set E of positive measure we can remove from the system\(\left\{ {e^{in_k x} } \right\}_{k = - \infty }^\infty\) a finite number of terms with the result that what is left is a Bessel system in L2(E); and c) if the series\(\sum\nolimits_{k^{ = - \infty } }^\infty {c_k e^{in_k x} }\) converges to zero on a set of positive measure, then each coefficient is zero.

The principal result of this paper establishes the validity of a conjecture by Graham and Rothschild. This states that, if the natural numbers are divided into two classes, then there is a sequence drawn from one of those classes such that all finite sums of distinct members of that sequence remain in the same class.

We consider in this paper cones of generalized absolutely monotone functions with respect to an infinite sequence ofC ∞ [a, b] functions {u i } i = 0/∞ such that for alln, n=0,1,2,...{u 0, u1,...,un} constitutes an Extended Tchebycheff System on [a,b]. We find a necessary and sufficient condition for the functionsu i, i=0,1,2,... to generate all the extreme rays in this cone. We conclude by constructing a cone of generalized absolutely monotone functions where theu i′s do not generate all of its extreme rays.

Let R be a ring, K a field, and n a natural number. We will be concerned with the following type of questions: (a) Classify the representations of R in (hO, (or n dimensional representations). (b) Classify n dimensional representations up to the natural equivalence. If ~bt, ~b2 : R ~ (K), we say ~bx is equivalent to ~b 2 if there is a K automorphism 7 of (K), such that 7q~1 = 4)2. (c) Classify equivalence classes of irreducible and semisimple representations. For the sake of simplicity we will restrict ourselves mainly to the case that K is algebraically closed and R is a finitely generated K algebra. In fact, for most of this paper, we will assume that R = K{xa, ..., x,) is a free algebra; at the end we will deduce from the results in this ease the more general theorems for not necessarily free algebras. Before describing the theorems that we will obtain, let us digress in order to motivate the use that we will make of the theory of rings with polynomial identities as a tool for attacking the problems stated before. Let us recall, therefore, some of the basic structural results of this theory and interpret them in terms of representation theory.

This paper presents a simple algorithm for finding the number of restricted k-partitions of a natural number n. The unrestricted k-partitions of n are expressed as the sum of these restricted k-partitions, called inadmissible, and the admissible k-partitions. The simplicity of the algorithm is striking, though all the implications are unclear.

For a graph G, the Ramsey number r(G) is the smallest natural number p such that given a graph H with p points. H or [email protected]? contains a G. An (n,j)-linear forest L is the disjoint union of nontrivial paths, j of which have an odd number of points, and such that the union has n points. If L is an (n,j)-linear forest, then we show that .

In his paper, 'Constructivity, Consistency and Natural Languages" Tucker seeks to define both the notion of a constructive procedure and the notion of a non-constructive procedure. 'Constructive procedures are defined as procedures which can, in some sense, be carried out whereas non-constructive procedures are those which, while they can (in some sense) be specified, cannot be carried out' (p. I52). Under this definition of constructive procedure Tucker claims that some procedures that have been regarded in the past as non-constructive are constructive according to the new definition (p. I53). It is the purpose of this discussion note to show that this claim is ill-founded. His first example is the diagonal procedure. He claims that this procedure has been regarded as non-constructive in the past but is in fact constructive. He supports his claim by giving an interpretation of the term 'indenumerable set'. 'Taken in the referential sense, the expression 'indenumerable set' means an actual infinity which is greater than an actual denumerable infinity' (p. I56). But there is another interpretation open to us, namely, 'a non-referential interpretation in which it means a set which contains an indenumerable element; where by an indenumerable element is meant an element which differs systematically from each element in an unending series whose generative recipe is given' (p. I56). It is difficult to make sense of this interpretation. What is needed is a clarification of the terms 'indenumerable element' and the term on which it depends, 'generative recipe'. From a later exposition that Tucker gives (p. I59) it appears that D is an indenumerable element of S if D belongs to S and is different from each element of S given by some law determining an initial element and the successor of any element. Under this definition it will turn out that many sets thought of as denumerable will be indenumerable. (Perhaps both denumerable and indenumerable, but Tucker does not define denumerable.) Even the set of natural numbers would become indenumerable. For the generative recipe-initial element 3, successor of an element x, x + i-will give two indenumerable elements, i and 2. i and 2 belong to the set of natural numbers and yet differ from each of the elements given by the generative recipe. That the set of natural numbers is non-denumerable is an absurd consequence and makes nonsense of the distinction initially brought in by Cantor. It might be said that my example ignores the fact that there is a generative recipe for the natural numbers and if I had taken this recipe then I should not have succeeded in obtaining an indenumerable element. In general this would mean that there is only one proper generative recipe

The least possible jump for a degree of unsolvability a is its join a ∪ 0′ with 0′. Friedberg [1] showed that each degree b ≥ 0′ is the jump of a degree a realizing least possible jump (i.e., satisfying the equation a′ = a ∪ 0′). Sacks (cf. Stillwell [8]) showed that most (in the sense of Lebesgue measure) degrees realize least possible jump. Nevertheless, degrees not realizing least possible jump are easily found (e.g., any degree b ≥ 0′) even among the degrees <0′ (cf. Shoenfield [5]) and the recursively enumerable (r.e.) degrees (cf. Sacks [3]).A degree is called minimal if it is minimal in the natural partial ordering of degrees excluding least element 0. The existence of minimal degrees <0” was first shown by Spector [7]; Sacks [3] succeeded in replacing 0” by 0′ using a priority argument. Yates [9] asked whether all minimal degrees <0′ realize least possible jump after showing that some do by exhibiting minimal degrees below each r.e. degree. Cooper [2] subsequently showed that each degree b > 0′ is the jump of a minimal degree which, as corollary to his method of proof, realizes least possible jump. We show with the aid of a simple combinatorial device applied to a minimal degree construction in the manner of Spector [7] that not all minimal degrees realize least possible jump. We have observed in conjunction with S. B. Cooper and R. Epstein that the new combinatorial device may also be applied to minimal degree constructions in the manner of Sacks [3], Shoenfield [6] or [4] in order to construct minimal degrees <0′ not realizing least possible jump. This answers Yates' question in the negative. Yates [10], however, has been able to draw this as an immediate corollary of the weaker result by carrying out the proof in his new system of prioric games.

The prediction that the ordinal property of natural number symbols (using these symbols to represent the terms in an ordered progression) is more easily learned than the cardinal property of natural number symbols (using these symbols to represent the manyness of collections) was examined in this experiment. Preschoolers who evidenced no proficiency with either the ordinal or cardinal properties of natural number symbols were trained to acquire these properties via simple feedback. Both properties proved to be trainable. The most important findings were that the ordinal property was much easier to train than the cardinal property, ordinal training effects were more durable across a 1-week interval than cardinal training effects, and ordinal training appeared to transfer better than cardinal training.

In the nineteenth century there grew and came slowly to maturity two very different points of view which were not only ways of thinking about mathematics but also, at least partly, two different ways of thinking in mathematics. The first one, to which the name of set-theoretical reductionism seems to be appropriate, went through many stages. One of them centers around the name of Cauchy and consist essentially in what may be called a set theoretical analysis of the main concepts of concrete real and complex analysis. The second stage has a well-established historiographical name, viz. the arithmetisation of analysis. It culminated in the Dedekindian and the Cantorian set-theoretical analysis of the real number concept. At the third stage we have on the one hand a straightforward continuation of the preceding stage as far as set-theoretical analysis of traditional finite number concepts is concerned. This continuation is the well-known lo gistic foundation of natural numbers. On the other hand we have the construction, along with its set-theoretical foundation, of the transfinite number theories.

Let A and S be subsets of the natural numbers. Let A′(n) be the number of partitions of n where each part appears exactly m times for some m ϵ A. Let S(n) be the number of partitions of n into parts which are elements of S.

Giuseppe Peano's development of the real number system from his postulates for the natural numbers and some of his views on definitions in mathematics are presented in order to clarify his concept of number. They show that his use of the axiomatic method was intended to make mathematical theory clearer, more precise, and easier to learn. They further reveal some of his reasons for not accepting the contemporary “philosophies” of logicism and formalism, thus showing that he never tried to found mathematics on anything beyond our experience of the material world.

LET US look at the natural or counting numbers. Some of them, such as the number 7891, contain a 9 as a digit in their decimal (base 10) repre sentation, while others, such as 4515, do not. If we had a basket containing all the natural numbers and we reached in and pulled out, without looking, a single natural number, what would the chances be that our pick would have no 9 for any of its digits? Consider the following comments

After a review of the work of Lawvere and Tierney, it is shown that every topos may be exactly embedded in a product of topoi each with 1 as a generator, and near-exactly embedded in a power of the category of sets. Several metatheorems are then derived. Natural numbers objects are shown to be characterized by exactness properties, which yield the fact that some topoi can not be exactly embedded in powers of the category of sets, indeed that the “arithmetic” arising from a topos dominates the exactness theory. Finally, several, necessarily non-elementary, conditions are shown to imply exact embedding in powers of the category of sets.

In this work we construct an open set V in the space of coefficients An of the equations dw/dz = Pn(z, w)/Qn(z, w) such that on the solutions of an arbitrary equation αV there exist a countable number of homotopically distinct limit cycles. Also, for each natural number N we construct an open set VN⊂An such that an arbitrary equation αVN has a Petrovskiĭ-Landis genus which exceeds N. Bibliography: 9 items.