Search

A monoid presented by a single relation has elements ≠ 1 of finite order if and only if the presentation is 〈 X ; p m p ′ = p n p ′〉, where p is a primitive word in the free monoid on X and p ′ is a left factor of p . The word problem and the divisibility problems are shown to be solvable. Also necessary and sufficient conditions for residual finiteness are given.

For a fixed integer m≥4, we find the number of elements x in a complete residue system modulo m(m−1)(m−2) which simultaneously satisfy the three divisibility conditions (m−2) | (x−2), (m−1)(m−2) | (x−1)(x−2), and m(m−1)(m−2)| x(x−1)(x−2).

This paper shows that a module which admits determinants has additional structure beyond the known properties of being a finitely generated projective module with localizations at all maximal ideals having constant rank. In particular the rank of a maximal free direct factor is connected to divisibility properties of a generator for the free top exterior power, and the complement of a maximal free direct factor is studied and shown to satisfy a condition which is easily expressible in terms of a Koszal Cohomology. As a side result we settle a question of H. Flanders involving the relation between generators of exterior powers of a module and the exterior rank of the module.

In a very interesting recent article [1] W. A. Broomhead described an investigation carried out by staff and pupils at Tonbridge School of the patterns which result when the numbers in Pascal’s triangle are reduced modulo m. For the case when m equals a prime number, p, the pattern formed by the zeros in the reduced triangle (corresponding to binomial coefficients divisible p) was completely described and the following result (stated by G. Gilbart-Smith) was proved:In the (n + l)th row of Pascal’s triangle, there are

It is the purpose of this article to extend and to clarify the public goods approach to the study of alliances. In particular, the paper examines the nature of defense as a pure public good and draws the conclusion that some defense expenditure may be best classified as an impure public good due to the presence of divisibility and exclusion properties. The traditional pure public good model of alliances is analyzed in a more general framework in order to introduce more fully resource scarcity and to demonstrate the symbiotic properties of the military alliance. Two models of increasing generality recast the analysis so that defense is an impure public good. Both optimal membership size and production efficiency are studied in the impure public good model. The paper concludes with a rationale for world peace organizations based on the economic theory of clubs.

If D is an integral domain with quotient field K, the group of divisibility G(D) of D is the partially ordered group of non-zero principal fractional ideals with aD ≦ bD if and only if aD contains bD. If K* denotes the multiplicative group of K and U(D) the group of units of D, then G(D) is order isomorphic to K*/U(D), where aU(D) ≦ bU(D) if and only if b/a ∊ D.

LetA be an augmentedK-algebra; defineT:A →A ⊗kkA byT(a)=1⊗a −a⊗1,a ∈A. We prove, under some conditions, thatg is in the subalgebraK[f] ofA generated byf if and only ifT(g) is in the principal ideal generated byT(f) inA⊗kkA. WhenA=K[[X]],T(f) is a multiple ofT(X) if and only iff belongs to the ringL obtained by localizingK[X] at (X).

A topological semiring (S, + , ·) is a nonempty Hausdorff space S on which are defined continuous and associative operations, termed addition (+) and multiplication (·), such that the multiplication distributes over addition from left and right. The additive semigroup (S, +) need not be commutative.We prove that the set A of additively divisible elements of a compact semiring S is a two-sided multiplicative ideal, containing the set E[+] of additive idempotents, with the property that (A.S) ∪ (S.A) ⊂ E[+].

Let a, < n2 < . . . be a sequence of integers denoted by A. Put A(x) = xCniCs 1. If no ai divides any other then A is called a p&m&e sequence. It is well known and easy to see that, for a primitive sequence, max A(z) = [3(x+ l)]. B esicovitch (1) constructed a primitive sequence of positive upper density and Behrend and Erd6s (1) proved that every primitive sequence has lower density 0. Davenport and Erdijs (1) proved that if A has positive upper logarithmic density then there is an infinite subsequence (ni,)j,r,. of A such that CZ,~ j aij+l. Erdijs (2) proved that’, if we assume that no ai divides the product of two others, then$

For any prime p, the sequence of Catalan numbers a n = 1 n 2n−2 n−1 is divided by the an prime to p into blocks Bk(k > 0) of an divisible by p. The lengths and positions of the Bk are determined. Additional results are obtained on prime power divisibility of Catalan numbers.

0. Summary and notation. In this paper a special type of co-ordinal, called rich, is studied. Basic properties of rich co-ordinals are proved in ?1. In ?2 rich co-ordinals are seen to be the co-ordinals occurring in paths which are addition isomorphic to initial segments of the classical ordinals. The results of ?1 are applied to obtain information about and examples of such paths. In the next section the order types of rich co-ordinals with a given field, X, is seen to be determined essentially by the finite divisors of X. RETs satisfying various divisibility conditions are constructed in ?4. These RETs and the results of ?3 determine the collection of sets of ordinals which are order types of rich co-ordinals with a fixed field.

1. THERE is an increasing tendency in modern economic theory towards greater rigour both in the formulation of its assumptions and in the standards of proof which those assumptions imply. This tendency, far from being 'too abstract', is to be commended on several grounds, one of which is that it is rigorous, and we can therefore have greater confidence that our conclusions really do follow from our hypotheses. An equally important reason for welcoming this approach is that we are often able to dispense with unnecessarily unrealistic economic assumptions, such as perfect competition and divisibility, and unnecessarily restrictive mathematical conditions, such as differentiability, in the proof of the desired results. Thus this method of attack leads at once both to more precise theory and to greater realism. To say this is certainly not to favour this method to the exclusion of all others: nor is it to endorse its indiscriminate application, for a devotion to rigour for its own sake can lead to mere formalism, even in mathematics itself. Economic content is, as always, the chief criterion by which to judge the results reached by this method. Its main disadvantage is that the level of abstraction on which it operates is at present often difficult for economists to understand. This unfamiliarity may be expected to lessen as time goes on, and in any case does not constitute a valid objection against its employment. This paper, as a small example of this method, presents the theory of revealed preference in reasonably rigorous terms. As a result of a careful formulation of the usual assumptions, a number of unresolved problems in this field can easily be cleared up. In particular, the vexed problem of integrability can be assigned its proper, minor, place.

The values of the first factor H 1 ( m ) {H_1}(m) of the class number of the m m th cyclotomic field are tabulated for 120 composite m m ’s larger than 100. A conjecture concerning a divisibility property of H 1 ( m ) {H_1}(m) is stated.

The procedure described here employs face-to-face intersection analysis to determine whether two or more polyhedral objects intersect. As means to minimize the number of pairs of faces which should be examined for face-to-face intersection analysis, a solution box approach, mutual divisibility, and visibility of two faces are considered. Intersection detection between two faces is done by the determination of their parity mode.

In a partially ordered abelian groupGG, two elementsaaandbbarepseudo-disjointifa,b≧0a,b \geqq 0and either one is zero, or both are strictly positive and eachoo-ideal which is maximal with respect to not containingaacontainsbb, and vice versa.GGis apseudo lattice-groupif every element ofGGcan be written as a difference of pseudo-disjoint elements. We prove the following theorem: supposeGGis an abelian pseudo lattice-group; if there is anx>0x > 0and a finite set of pairwise pseudo-disjoint elementsx1,x2,⋯,xk{x_1},{x_2}, \cdots ,{x_k}all of which exceedxx, and in addition this set is maximal with respect to the above properties, thenGGis not a group of divisibility. The main consequence of this result is that every so-called “vv-group”V(Λ,Rλ)V(\Lambda ,{R_\lambda })for a given partially ordered setΛ\Lambda, and whereRλ{R_\lambda }is a subgroup of the additive reals in their usual order, is a group of divisibility only ifΛ\Lambdais a root system, and henceV(Λ,Rλ)V(\Lambda ,{R_\lambda })is a lattice-ordered group. We do give examples of pseudo lattice-groups which are not lattice-groups, and yet are groups of divisibility. Finally, we compute for each integral domainDDwhose group of divisibility is a lattice-group, the group of divisibility of the polynomial ringD[x]D[x]in one variable.

Let S be a commutative semigroup. For a, b ES, we say that a divides b (or b is a multiple of a), and write a|Ilb, if either a=b or ax= b for some xE S. We say that a properly divides b if a b and b does not divide a. We call S archimedean if for all a, b ES, a divides some power of b. It is known that every commutative semigroup is uniquely expressible as a semilattice of archimedean semigroups (Clifford and Preston [2, Theorem 4.13, p. 132], which is an easy consequence of Tamura and Kimura [7]). Those commutative archimedean semigroups which are naturally totally ordered (that is, those in which the divisibility relation is a total order) have been studied by Clifford [1 ] and other authors (see Fuchs [3, Chapter 11] for references). Tamura [5], [6] has begun the study of those which are naturally partly ordered. The purpose of the present paper is to determine all those commutative archimedean semigroups which satisfy the following three conditions: (1) There is no infinite sequence of elements in which each term properly divides the one preceding it. (2) If a b and b a, then a = b. (3) If aIb and a|c, then either bIc or c|b. Condition (1) is essentially the maximal condition on principal ideals. It is, of course, satisfied whenever S is finite. Condition (2) states that S is naturally partly ordered. It can be shown that a commutative archimedean semigroup satisfies (2) if and only if it either contains no idempotent or contains a zero element; however, we shall not need to use this fact. The effect of (3) is to assert that every set of elements having a common divisor is naturally totally ordered. Thus (3) generalizes natural total ordering.

There's magic in the number 1,001, which is as old as the tales of the Arabian Nights. A little-used algorithm makes use of that magic to reveal divisibility by those mystic primes, 7, 11, and 13, almost as easily as rubbing a magic lantern. Unlike Aladdin, we shall probably not be content to use the scheme unless we know why it works. The key to understanding depends basically on the fact that 7 × 11 × 13 = 1,001 and on a little elementary number theory.

Problem . To find a proceeding by which the divisibility of a proposed integer N by 7 or 13, or by both 7 and 13, may be determined through the same rule. Solution . We can designate the number N by abcd . . . mn , so that ( a ) be the first or highest, and ( n ) the last or lowest digit in it, therefore we may put N = abcde . . . mn .

M { UCH has been written during The last thirty years or so on the subject of the basic ingredients of Anaxagoras' universe,' but comparatively little attention has been paid to the cosmogonical process by which Anaxagoras believed the universe to have developed out of those ingredients into its present state. That process is my subject, but before dealing with any part of Anaxagoras' physical theories it always behoves a writer to explain how he hiimself understands Anaxagoras' first priniciples. There is, we are told, a constant, infinite, and infinitely divisible amount of matter in the universe (Frags. 1-7, Diels-Kranz). Since all things are infinitely divisible, they cannot be composed of a finite number of Empedoclean type elements, for any such theory entails a limit to divisibility (cf. Arist. de Caelo 305 a I). It would seem to follow that if there is an infinite number of substances in the phenomenal world there must be an infinite number of primary substances. Parmenides had already shown that the Real must be unchanging; but an infinite number of homogeneous, unchanging, entities (such as Melissus hinted at, DK 3o B 8) could only explain an unchanging universe. Consequently Anaxagoras' entities are not homogeneous. As he reiterates, e 7rocVTL tvTo (0pop svearLV. Frag. 4 and the phrase 67MaMCEp pte of the universe. It is over the precise meaning of ncxv-d and 7caNxT0 that scholars especially disagree. I agree, following the ancient commentators, with Bailey, Cherniss, Raven, and Mathewson, that Tannery, Burnet, and Cornford were wrong in thinking that the 'portions' that are in everything are just the traditional 'opposites' or opposite 'quality-things.' I, too, take the view that Anaxagoras meant to include anything that his predecessors had thought of as primary, including the Hot and the Cold (Frag. 8), the Wet and the Dry (cf. Anaximander), the Bright and the

In scientific literature the evaporation of musk has a considerable interest. Almost every writer on the divisibility of matter cites it as an instance of the extremely minute division of which matter is capable, our sense of smell enabling us to detect a more minute quantity of this form of matter than can be detected of any other kind of matter by any of the modern refined methods, such as the spectroscope, or chemical processes. It is possible we may shortly have to modify this statement, as extremely small quantities of some kinds of matter can be detected by their radio-activity; but, so far as I know, no reliable numerical values have been obtained in this direction.

n/a

Several writers have considered the relations between the c-functions of an algebraic field and some of its subfields. Thus Artint has, in a particular case, considered the question of the divisibility of the c-function of a field by that of a subfield. In another paperl he has shown how all possible c-relations can be found. Explicit results of a general nature are however not arrived at. Herglotz? has investigated fields formed by the composition of several quadratic fields, thus generalizing a well known result of Dirichlet's.? Pollaczekll has obtained results of a similar kind for Abelian fields with group of type (1, 1). In all the cases cited, use is made of Hecke's** functional equation for the c-function in an arbitrary field. Thus, for example, in Artin's first paper, a c-relation is proved, except for a finite number of factors, by quite elementary methods; employing the functional equation, it is seen to hold in its entirety. Again, Herglotz and Pollaczek deduce discriminantal relationships by means of the functional equation. In the following an explicit relation between c-functions is deduced. The fields considered include as special cases those of Artin (first paper), Herglotz, and Pollaczek. No use is made of the Hecke functional equation; instead a method of an elementary nature is employed. Relations between discriminants also are easily proved by direct means. The first result of interest may be formulated thus: Let K be an (absolute) Galois field of degree m and group Gm. tt We make

Hundred boards are devices that can be used to assist in the learning and reinforcement of such mathematical ideas as multiples, factors, primes, composites, and checks for divisibility. Both the level of the experience and the mathematical generalizations that may be formulated depend on the mathematical maturity of the students. The activities described here have been used in college classes in elementary methods. If prospective teachers can be involved with the types of mathematical experiences that are appropriate in the elementary school, it follows that they will be better able to provide similar types of experiences in their own classroom. A hundred board can be constructed

1. The Thirteenth Century was in but its second year when Fibonacci (or Pisano) proposed a problem on the number of offspring of a pair of rabbits, whose solution led to the sequence of numbers now named after him. There is reason to believe that Fermat derived many of his arithmetic theorems from a knowledge of recurrences and, certainly, his celebrated Last Theorem may be stated as a problem on sequences. Lucast was the first to make any extended researches on sequences, establishing a great many properties of certain second order sequences. Carmichael [1 ]1 in 1920 made the first attack on sequences in general, and established their fundamental property of modular periodicity. This paper undertakes a general survey of the modular properties of linear recurring sequences, beginning from the results of a paper by H. T. Engstrom and two by Morgan Ward. ? No problems on sequences are considered here which are not strictly modular, though questions on divisibility sequencesjj and their remarkable factorization properties are closely related. The mechanism which the author uses for examining the properties of sequences is an isomorphism between the set of all sequences satisfying a fixed recurrence and a polynomial ring of operators. The isomorphism is not with the abstract ring but with a particular realization of it, and this is not especially surprising as a linear sequence is essentially an exponential function. This isomorphism may be derived from the theory of generating functions, and includes the fundamental identity used in Ward [11 ]. In Chapter II the isomorphism is set up and the basic properties of the ring are examined and their interpretation is given for the sequences. For example, the zero divisors of the ring correspond to the sequences which satisfy recurrences of lower order.

The inverse weak algorithm is a property of certain filtered rings which was introduced in [-3] to study divisibility properties in free power series rings over a field. Further significant properties of rings with an inverse weak algorithm were obtained by Bergman in [1], and this clearly makes it desirable to have more examples of rings with an inverse weak algorithm, or better, methods of obtaining such rings. Our aim in this note is to give a simple method of constructing a fairly wide class of rings with an inverse weak algorithm. We recall that any such ring has a filtration (taken negatively for convenience)

i.e.,Br(X), is split by a finite, faithfully flat covering Γ-»X.After proving a divisibility result for Pic(X) under suchcoverings and some preliminary investigation of cohomologyin the topology defined from such coverings, the exact sequencewhich is analogous to that of Chase and Rosenberg is obtained.

In [1] it is shown that for every sequence x = 〈xn : n ∈ ω〉 ∈ Xω Λ there is an isol xω (essentially an immunized product) such thatHere we have used the notation: Λ = the isols, ω = the nonnegative integers, pn is the nth prime rational integer starting with p0 = 2, ∣ denotes divisibility and ∤ its negation. If p is an arbitrary prime, py ∣ x, pz ∣ x, and y < z then py+1 ∣ x. In particular since y ∈ ω is comparable with every element of Λ, the conditions py ∣ x and py+1 ∤ x uniquely determine y. Thus every sequence x ∈ Xωω is uniquely determined by an xω satisfying (1) and consequently may be used as a “code” for that sequence. In Theorem 1 it is shown that (1) does not uniquely determine the values of an arbitrary sequence x ∈ XωΛ, however in Theorem 3 we find a different scheme which does. At the very end of the paper we give some reasons why coding theorems are useful. It should also be mentioned that for a coding theorem to be meaningful it is necessary to restrict the operations by which a sequence can be recaptured from its code. Otherwise a triviality results. Our coding theorem will allow all operations which are first order definable in Λ with respect to addition, multiplication, and exponentiation. We conjecture that the latter operation is really necessary.

In the year 1822, Dr Wollaston published a remarkable paper “on the finite extent of the atmosphere.” Its object is to establish, by observations on the motions of certain of the heavenly bodies, that our atmosphere does not extend into free space, and to deduce from this limitation in its extent, the conclusion, that the air necessarily consists of particles “no longer divisible by repulsion of their parts;” i. e. of true atoms. From this there is the further inference, that, “since the law of definite proportions discovered by chemists, is the same for all kinds of matter, whether solid or fluid, or elastic, if it can be ascertained that any one body consists of particles no longer divisible, we then can scarcely doubt that all other bodies are similarly constituted.” In other words, the existence of a limit to the earth's atmosphere is declared to supply a demonstration of the finite divisibility of matter.

The subject which I am going to discuss in this lecture excels in the richness of its ramifications and in the diversity of its relations to other mathematical topics. I think therefore that it will serve our present purpose better not to at tempt a systematic treatment, beginning with definitions and proceeding to lemmas, theorems, and proofs, but rather to look around and to envisage some outstanding marks scattered in various directions. I hope tha t the intrinsic relationships connecting the problems and theorems which I shall mention will nevertheless remain quite visible. A good deal of the investigations about which I shall report can be subsumed under the heading of analytic number theory, and, more specifically, analytic additive number theory. It would, however, be a misplacement of emphasis if we were to look upon analysis, which here means function theory, only as a tool applied to the investigation of number theory. I t is more the inner harmony of a system which I wish to depict, properties of functions revealing the nature of certain arithmetical facts, and properties of numbers having a bearing on the character of analytic functions. Whereas the multiplicative number theory, which deals with questions of factorization, divisibility, prime numbers, and so on, goes back more than 2000 years to Euclid, the history of additive number theory, in any noteworthy sense, begins with Euler less than 200 years ago. In his famous treatise, Introductio in Analysin Infinitorum (1748), Euler devotes the sixteenth chapter, "De partitione numerorum," to problems of additive number theory. A "partition" is, after Euler, a decomposition of a natural number into summands which are natural numbers, for example, 6 = 1 + 1 + 4 . We can impose various restrictions on the summands ; they may belong to a specified class of numbers, let us say odd numbers, or squares, or cubes, or primes; it may be required that they be all different; or their number may be preassigned. I wish to speak here only about unrestricted partitions. By the way, only the parts are essential, not their arrangement, so that we do not count two decompositions as different if they differ only in the order of the summands; we can therefore take the summands ordered according to their size.

Classically, there has been, for obvious reasons, an intimate relation between the concepts "rings of quotients" and "divisible modules". Recently, however, their generalizations have appeared to diverge.For example, Hattori ([9]) and Levy ([15]) have generalized the concept of "divisibility" as follows: Hattori (respectively Levy) defines a left R-module M over a ring R to be divisible if and only if Ext1R(R/I, M)=0 for every principal left ideal I ⊂ R (respectively, every principal left ideal I ⊂ R which is generated by a regular element of R).

In his paper in Mat. Sb. (N.S.) 61 (103) (1963), Zareckiĭ associated with any binary relation α an ordered pair, (Lα Mα), say, of lattices and showed that α is a left [right] divisor of β if and only if We provide an alternative proof of this result by embedding the category of relations in the category of sets. Our approach provides a unified treatment of several hitherto independent results, and gives new results for the category of partial transformations.

Elementary number theory (divisibility theory) is used to prove that, in the case of repeated sampling, when the k-th generalized sample moment is independent of the sample mean then the sampling is normal if the volume of the sample is large and k is a square-free integer which, if it is even, is not of the form 2(2s +1)).