Annals Of Mathematics Research Articles

William W. Boone. The word problem. Annals of mathematics, vol. 70 (1959), pp. 207–265.

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Anil Nerode. Extensions to isols. Annals of mathematics, second series, vol. 73 (1961), pp. 362–403.

Anil Nerode. Extensions to isols. Annals of mathematics, second series, vol. 73 (1961), pp. 362–403.

William W. Boone. An analysis of Turing's “The word problem in semi-groups with cancellation.”Annals of mathematics, ser. 2 vol. 67 (1958), pp. 195–202.

William W. Boone. An analysis of Turing's “The word problem in semi-groups with cancellation.”Annals of mathematics, ser. 2 vol. 67 (1958), pp. 195–202.

Michael O. Rabin. Recursive unsolvability of group theoretic problems. Annals of mathematics, ser. 2 vol. 67 (1958), pp. 172–194.

Michael O. Rabin. Recursive unsolvability of group theoretic problems. Annals of mathematics, ser. 2 vol. 67 (1958), pp. 172–194.

Clifford Spector. On degrees of recursive unsolvability. Annals of mathematics, ser. 2 vol. 64 (1956), pp. 581–592.

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A. Seidenberg. A new decision method for elementary algebra. Annals of mathematics, ser. 2 vol. 60 (1954), pp. 365–374.

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S. C. Kleene and Emil L. Post. The upper semi-lattice of degrees of recursive unsolvability. Annals of mathematics, ser. 2 vol. 59 (1954), pp. 379–407.

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A. M. Turing. The word problem in semi-groups with cancellation. Annals of mathematics, ser. 2 vol. 52 (1950), pp. 491–505.

A. M. Turing. The word problem in semi-groups with cancellation. Annals of mathematics, ser. 2 vol. 52 (1950), pp. 491–505.

An inequality in the theory of arithmetic on algebraic varieties

The inequality which is proved here was established under different conditions in a paper entitled ‘Periodic points on an algebraic variety', which is to appear in the Annals of Mathematics. It is hoped that the new formulation will be more useful for applications. The present investigations use the rather sophisticated techniques of the ‘Theory of distributions’ which was developed by Weil (1). As the content of this theory may be unfamiliar to the reader, Weil's results are described at some length, though of course, his proofs are not given. The one result which we need and which is not given explicitly by Weil, is proved in detail.

On the Converse of Bianchi's Permutability Theorem

Bianchi's permutability theorem, can be stated under the following form.' Let two W-congruences have one focal sheet in common, let the other focal sheets be respectively S' and S, then there are oo 'surfaces which are together with S' and S the focal sheets of two other W-congruences. The conviction that it is possible to state a converse of Bianchi's permutability theorem, has led G. Fubini,2 to consider this problem in a paper in the Annals of Mathematics, but without giving it a complete solution. Because of the role played by this theorem in metric and projective differential geometry, and in Fubini's theory of stratifiable-congruences of lines, a complete proof of the converse of this theorem, should be of interest. The converse of Bianchi's permutability theorem, which we prove here, can be stated as follows: If co' surfaces M' are congruence-transforms of two surfaces Mo and M2, all the congruences (M'Mo) and (M'M2) are W-congruences. For proving this theorem, we shall use Cartan's3 method of the moving reference system. Let the coordinates of the four points Mi(i = 0, 1, 2, 3) be functions of two parameters u, v, which generate, as u, v, vary, four surfaces, such that the surfaces Mo and M2 are congruence-transforms2 of the surfaces Ml and M3. Then the tangent planes to the surfaces Mi(i = 0, 2) and M,(p = 1, 3) pass respectively through Ml, M3 and Mo, M2. As a moving reference system, we consider the tetrahedron depending on the parameters u, v and formed by the points Mo, 1112 and by two arbitrary points of the M' i. e Ml and M3. Therefore Mj = Ml + pjM3. Consequently3 we have:

J. C. C. McKinsey and Alfred Tarski. On closed elements in closure algebras. Annals of mathematics, ser. 2 vol. 47 (1946), pp. 122–162.

J. C. C. McKinsey and Alfred Tarski. On closed elements in closure algebras. Annals of mathematics, ser. 2 vol. 47 (1946), pp. 122–162.

J. C. C. McKinsey and Alfred Tarski. The algebra of topology. Annals of mathematics, ser. 2 vol. 45 (1944), pp. 141–191.

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M. H. A. Newman. On theories with a combinatorial definition of “equivalence.”Annals of mathematics, ser. 2 vol. 43 (1942), pp. 223–243.

The name is often given to branches of mathematics in which the central concept is an equivalence relation defined by means of certain transformations or A class of objects is given, and it is declared of certain pairs of them that one is obtained from the other by a move; and two objects are regarded as equivalent if, and only if, one is obtainable from the other by a series of moves. For example, in the theory of free groups the objects are words made from an alphabet a, b, , a-, b', and a move is the insertion or removal of a consecutive pair of letters xx-l or x-lx. In combinatorial topology the objects are complexes, and the allowed are an edge by the insertion of a new vertex, or the reverse of this process.' In Church's calculus2 the rules II and III are ''moves of this kind. In many such theories the fall naturally into two classes, which may be called and negative. Thus in the free group the cancelling of a pair of letters may be called a positive move, the insertion negative; in topology the breaking of an edge, in the conversion calculus the application of Rule II (elimination of a X), may be taken as the positive moves. In theories that have this dichotomy it is always important to discover whether there is what may be called a of confluence, namely, whether if A and B are equivalent it follows that there exists a third object, C, derivable both from A and from B by positive only. A closely connected problem is the search for endforms, or normal forms, i.e. objects which admit no positive move. It is obvious that in a theory in which the confluence theorem holds no equivalence class can contain more than one end-form, but there remains the question whether in such a class any random series of positive must terminate at the end-form, or whether infinite series of may also exist. The purpose of this paper is to make a start on a general theory of of moves by obtaining some conditions under which the answers to both the above questions are favorable. The results are essentially about partially-ordered systems, i.e. sets in which there is a transitive relation >, and sufficient conditions are given for every two elements to have a lower bound (i.e. for the set to be directed) if it is known that every two sufficiently near elements have a lower bound. What further conditions are required for the existence of a greatest lower bound is not relevant to the present purpose, and is reserved for a later discussion.

On converse gap theorems

Presented to the Society, February 22, 1941; received by the editors June 4, 1941. (1) In the following the two parts of my paper Untersuchungen iiber Lxicken und Singularitdten von Potenzreihen, Mathematische Zeitschrift, vol. 29 (1929), pp. 549-640 and Annals of Mathematics, (2), vol. 34 (1933), pp. 731-777 will be quoted as LS I and LS II. For Theorem I (Fabry's theorem) see LS I, p. 627, Theorem VIa; for Theorem II (theorem of the present author) see LS II, p. 737, Theorem B. (2) These problems have been stated, with an indication of the proof: I in Comptes Rendus de l'Academie des Sciences, Paris, vol. 208 (1939), pp. 709-711; II in the Bulletin of the American Mathematical Society, vol. 47 (1941), p. 207. A problem related to I was stated by G. Szego, Acta Litterarum ac Scientiarum Szeged, vol. 1 (1923), p. 73.

B. O. Koopman. Intuitive probabilities and sequences. Annals of mathematics, ser. 2 vol. 42 (1941), pp. 169–187.

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A Property of Integral Functions of Order Less Than Two with Real Roots

P. Erdos and T. Grilinwakl in their paper On polynomials with real roots, in Annals of Mathematics, Vol. 40, No. 3, July 1939, p. 537, have proved the following: Let f(x) be a real polynomial with real roots having consecutive single roots at x = a,, x as , and let P be the point of intersection of the tangents to the curve Y = f(x) at (a,, 0), (as , 0), and let M(a,as) be maximum of f(x) in (a,as).

B. O. Koopman. The axioms and algebra of intuitive probability. Annals of mathematics, ser. 2 vol. 41 (1940), pp. 269–292.

B. O. Koopman. The axioms and algebra of intuitive probability. Annals of mathematics, ser. 2 vol. 41 (1940), pp. 269–292.

B. A. Bernstein. Postulates for Boolean algebra involving the operation of complete disjunction. Annals of mathematics, 2 s. vol. 37 (1936), pp. 317–325.

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Linkages illustrating the cubic transformation of elliptic functions

The connection between poristic polygons and closed linkages, originally pointed out by Prof. Arnold Emch in the 'Annals of Mathematics,' Series 2, vol. 1 (1900), and further developed in the 'Proceedings of the London Mathematical Society,' Series 2, vol. 11 (1911), affords a means of associating linkages with the subject of the transformation of elliptic functions. Darboux was the first to show systematically how all the variable angles and lines of a three-bar linkage can be expressed in terms of a single elliptic parameter. To him also is due the idea of combining a series of identical three-bar linkages into a closed system; the closure being possible in virtue of the real periodicity of the elliptic functions expressing the variable lines and angles.

On the Locus of the Foci of a System of Similar Conics through three Points

1. The problem has been proposed by Steiner* of finding the envelope of a system of similar conics circumscribed about a given triangle, and of finding the loci of the centres and foci of the conics of the system. He states that the envelope is a curve of the fourth order having three double points, and gives some of its properties. The problem has been treated by P. H. Schoute in a paper entitled Application de la transformation par droites symétriques à un problème de Steiner.† In this paper the author discusses the problem of the envelope in detail by a geometrical method, and gives the order of the locus of centres, of the locus of foci, and of the locus of vertices, and the class of the envelope of asymptotes, of the envelope of axes, and of the envelope of directrices. But in every case (except that of the locus of centres), he gives the degree or class twice as great as it should be. In a paper published in the Annals of Mathematics‡ I have found explicit equations for the locus of centres and the envelope of asymptotes, showing that each asymptote envelopes a three-eusped hypocycloid; and in a paper in the Transactions of the American Mathematical Society,§ I have obtained an equation for the envelope of the axes, showing that each axis also envelopes a three-cusped hypocycloid. I have also obtained the equation of the envelope of the directrices (a curve of the fourth class) and the equation of the locus of the vertices (a curve of the eighth degree); but the results have not been published. In the solution of a problem in the Educational Times (proposed by himself), Cayley* proved that the locus of the foci of the parabolas which pass through three fixed points is a unicursal quintic passing through the two circular points at infinity, by showing that the co-ordinates of the focus of any such parabola may be explicitly expressed as rational functions of a parameter of the fifth degree. As the locus of the foci of a system of similar conies passing through three fixed points is not in general a unicursal curve, it is hardly likely that Cayley's method could be extended to the general case.