Search

Let f(m;n) denote the largest integer so that, given any m integers a1 < … < am in [1, 2n], one can always choose f integers b1 < … < bf from [1, n], so that bi + bj = a1 (1 ≤ i ≤ j ≤ f; l ≤ l ≤ m) will never hold. Trivially f(m; n) ≥ n/ (m + 1). In this paper we shall attempt to improve upon this trivial bound by exploiting the possible irregularities of distribution of the sequence among certain congruence classes. One of our main results isprovided m ≥ log n. Related questions and results are also discussed.

The main result of this paper is that the class of congruence lattices of semilattices satisfies no nontrivial lattice identities. It is also shown that the class of subalgebra lattices of semilattices satisfies no nontrivial lattice identities. As a consequence it is shown that if V is a semigroup variety all of whose congruence lattices satisfy some fixed nontrivial lattice identity, then all the members of V are groups with exponent dividing a fixed finite number.

The structure of the sources of a gravitational field in Schwarzchild and Kerr spaces is investigated using the method of geodesic curvature. The curvature is calculated in Schwarzchild space for an Isotropie and time-like congruence and in Kerr space for two Isotropie congruences. An analysis of the curvature is made.

This paper describes a set of algorithms for isolating the real zeros of a univariate polynomial with integer coefficients. The algorithms employ congruence (modular, finite field) arithmetic and are analogous to a set of integer arithmetic algorithms described by the author in a recent paper. The algorithms are analyzed to bound their computing times and these computing times are compared to the computing times of the integer arithmetic algorithms. Some empirical computing times are reported.

By employing an anholonomic description of the governing equations, certain geometric results are obtained for a class of nondissipative magneto-hydrodynamic flows. For the latter, the velocity and magnetic field lines are geodesics and parallels respectively on a normal congruence of surfaces. A significant subclass is shown to be helical in character.

The general theory of space tensors is applied to the study of a space-time manifoldsV 4 carrying a distinguished time-like congruence Γ. The problem is to determine a physically relevant spatial tensor analysis $$\left( {\tilde \nabla ,\tilde \nabla _T } \right)$$ over (V 4, Γ), in order to proceed to a correct formulation of Relative Kinematics and Dynamics. This is achieved by showing that each choice of $$\left( {\tilde \nabla ,\tilde \nabla _T } \right)$$ gives rise to a corresponding notion of ‘frame of reference’ associated with the congruence Γ. In particular, the frame of reference (Γ, ∇*) determined by the standard spatial tensor analysis $$\left( {\tilde \nabla *,\tilde \nabla *_T } \right)$$ is shown to provide the most natural generalization of the concept of frame of reference in Classical Physics. The previous arguments are finally applied to the study of geodesic motion inV 4. As a result, the general structure of the gravitational fields in the frame of reference (Γ, ∇*) is established.

In this paper we generalize a result of Tamura on .S-indecomposable semigroups. Based on this, the concept of a minimal sequence between two points, and from a point to another, is introduced. The relationship between two minimal sequences between the same points is studied. The rank of a semigroup S is defined to be the supremum of the lengths of the minimal sequences between points in S. The semirank of a semigroup S is defined to be the supremum of the lengths of the minimal sequences from a point to another in S. Rank and semirank are further studied. Introduction. Semilattice decompositions of semigroups were first defined and studied by Clifford [1]. Since then several people have worked on this topic, notably Tamura (5H{9]. The author's work on the subject can be found in (3], (4]. In this paper, we start by generalizing a result of Tamura (8] (or (9]) on Sindecomposable semigroups. Based on this, the concept of a minimal sequence between two points, and from a point to another, is introduced. The relationship between two minimal sequences between the same points is studied. The rank of a semigroup is defined to be the supremum of the lengths of the minimal sequences between points in the semigroup. The semirank of a semigroup is defined to be the supremum of the lengths of the minimal sequences from a point to another in the semigroup. Rank and semirank are further studied. To understand this paper, the reader need only be aware of the first few chapters of Clifford and Preston [2] and Tamura's decomposition theorem. (See any of (5], [6], [8] or (9]. It was rediscovered by Petrich [(0].) I Preliminaries. Throughout, S will denote a semigroup and Z+ the set of positive integers. A congruence a on S is called a semilattice congruence if S/a is a semilattice. S x S is the universal congruence on S. S is S-indecomposable if S x S is the only semilattice congruence on S. Definition. Let a, b E S. Then (1) a I b if and only if b E SaS1. I is transitive and reflexive. (2) is defined as a > b iff a I bi for some i E Z'; let -0 denote --, i.e., = _-. (3) a nI b iff there exists x E S such that a x b. (4)a biffa bforsome Z+. (5) -is defined as a b iff a a; let -? denote -, i.e., -? Received by the editors May 9, 1972 AMS (MOS) subject classfications (1970). Primary 20M10.

For integral domains of characteristic not 2, we prove here that the symplectic and unitary congruence groups are not isomorphic if the Witt indices are at least 3. This is Theorem 2.1; Theorem 3.3 describes the isomorphisms of unitary congruence groups.

Excess enthalpies and excess volumes at 298.15 K and atmospheric pressure have been measured for mixtures of n-hexane + (an equimolar mixture of n-tridecane + n-nonadecane). The agreement of the results with the corresponding results for mixtures of n-hexane + n-hexadecane is interpreted as confirmation of the applicability of the principle of congruence to multicomponent mixtures.

This paper gives some indication of the structure of the orthogonal group of a unimodular quadratic form over the 2-adic integers. Various congruence subgroups are defined and their interrelationships investigated. Figure 1 gives the lattice pattern of the subgroups considered when the rank is even. The subgroups normalized by the commutator subgroup are also determined.

A pair (M, Γ) is defined as a Riemannian manifold M of normal hyperbolic type carrying a distinguished time-like congruence Γ. The spatial tensor algebraD associated with the pair (M, Γ) is discussed. A general definition of the concept of spatial tensor analysis over (M, Γ) is then proposed. Basically, this includes a spatial covariant differentiation\(\tilde \nabla \) and a time-derivative\(\tilde \nabla _T \), both acting onD and commuting with the process of raising and lowering the tensor indices. The torsion tensor fields of the pair\(\left( {\tilde \nabla ,\tilde \nabla _T } \right)\) are discussed, as well as the corresponding structural equations. The existence of a distinguished spatial tensor analysis over (M, Γ) is finally established, and the resulting mathematical structure is examined in detail.

1. In his book, A Theory of Time and Space, the writer showed that, contrary to the generally accepted view, the ideas of Congruence can be built up from purely ordinal considerations, provided that we do not confine ourselves to linear order, but admit what, for convenience, he has called Conical Order. This Conical Order is built up from the purely abstract asymmetrical relations of before and after and does not pre-suppose the existence of the cones which are used to illustrate it.

If b(m;n) denotes the number of partitions of n into powers of m, then b(m; mr+1n) ≡ b(m; mrn) (mod μr) where μ = m if m is odd and μ = m2 if m is even. The existence of such a congruence was conjectured by R. F. Churchhouse and its truth for m a prime was proved by O. Rödseth.

This paper studies conditions under which every congruence relation θ \theta in a direct product A × B A \times B of algebras is of the form θ 1 × θ 2 {\theta _1} \times {\theta _2} , where θ 1 {\theta _1} and θ 2 {\theta _2} are congruence relations in A A and B B respectively. It is shown that for any equational class K K , every such θ \theta in every A × B A \times B in K K has this property if and only if K K satisfies certain identities.

We denote by b(n) the number of ways of expressing the positive integer n as the sum of powers of 2 and we call b(n) ‘the binary partition function’. This function has been studied by Euler (1), Tanturri (2–4), Mahler (5), de Bruijn(6) and Pennington (7). Euler and Tanturri were primarily concerned with deriving formulae for the precise calculation of b(n), whereas Mahler deduced an asymptotic formula for log b(n) from his analysis of the functions satisfying a certain class of functional equations. De Bruijn and Pennington extended Mahler's work and obtained more precise results.

Since closed cones C in finite dimensional real vector spaces V play the role of “Lie algebras” for finite dimensional compact abelian semigroups, we discuss the structure of congruence relations on such cones. We show in particular that a monotone closed congruence on such a cone C is entirely determined by a countable collection of closed ideals of C and a countable collection of linear subspaces of V (see theorem 3.3).

Let Г(1) denote the homogeneous modular group of 2 × 2 matrices with integral entries and determinant 1. Let (1) be the inhomogeneous modular group of 2 × 2 integral matrices of determinant 1 in which a matrix is identified with its negative. (N), the principal congruence subgroup of level N, is the subgroup of (1) consisting of all T ∈ (1) for which T ≡ ± I (mod N), where N is a positive integer and I is the identity matrix. A subgroup of (1) is said to be a congruence group of level N if contains (N) and N is the least such integer. Similarly, we denote by Г(N) the principal congruence subgroup of level N of Г(1), consisting of those T∈(1) for which T ≡ I (mod N), and we say that a sub group of Г(1) is a congruence group of level N if contains Г (N) and N is minimal with respect to this property. In a recent paper [9] Rankin considered lattice subgroups of a free congruence subgroup of rank n of (1). By a lattice subgroup of we mean a subgroup of which contains the commutator group . In particular, he showed that, if is a congruence group of level N and if is a lattice congruence subgroup of of level qr, where r is the largest divisor of qr prime to N, then N divides q and r divides 12. He then posed the problem of finding an upper bound for the factor q. It is the purpose of this paper to find such an upper bound for q. We also consider bounds for the factor r.

The paper deals with the problem of the primitive period of a sequence satisfying a linear recursion congruence, modulo a power of a prime, with periodic coefficients.

Let S be a finite semigroup and let K be an algebraically closed field of characteristic zero. Herein, we derive a formula for the congruence ≡ induced on S by the direct sum of all the irreducible representations of S over K . This congruence ≡ is proved to be the same as the congruence induced by the minimal homomorphic image of S , which is one-to-one on the subgroups of S and such that two distinct principal ideals of S , each generated by an idempotent, have distinct images. Assume ψ is a faithful finite dimensional representation of S . Let R(ψ) be the associated completely reducible representation having the same character as ψ . That is R(ψ) is the direct sum of the Jordan-Holder factors of ψ . Then by applying the Burnside-Steinberg theorem we prove that the congruence induced by R(ψ) on S equals ≡. That is, the operator R preserves “one-to-oneness as much as possible”. We next apply these results to compute the complexity, # G ( S ), of S when S is a union of groups. A linear transformation T=B(S) is defined on the character ring of S into itself. Then by a theorem of Krohn-Rhodes (which determines the complexity of S in terms of its homomorphic images), together with the previous character results, we prove that T is nilpotent and index( T )=# G ( S ). Finally the character results proved here imply that, if the Fundamental Lemma on Complexity is valid, then the complexity of S is the maximum of the images of all its irreducible representations. This is known to be the case for all regular semigroups.

For the solubility of an inhomogeneous polynomial Diophantine equation, there is one well-known necessary, but not sufficient condition; namely the necessary congruence condition (NCC) explained in §2, below. Till recently, no progress had been made with the general cubic equation, because no one knew what else to assume. Examples given here, see (4.3), (5.4), indicate that some rather subtle hypothesis is needed. The first such hypothesis, see Davenport and Lewis [1], was very far from being necessary for the solubility of the equation. It would seem that any supplementary hypothesis which (loosely) is somewhere near necessary and also (together with the NCC) somewhere near sufficient deserves separate detailed investigation before one proceeds to use it.

This note is intended to give a rough survey of the results obtained in the study of twist-free pure radiation fields in general relativity theory. Here we are using the following Definition. A space-time ( V 4 of signature +2) is called a pure radiation field if it contains a distortion-free geodetic null congruence (a so-called ray congruence ), and if it satisfies certain field equations which we will specify below (e.g. Einstein’s vacuum-field equations). A (null) congruence is called twist-free if it is hypersurface-orthogonal (or ‘normal’). The results listed below were obtained by introducing special (‘canonical’) co-ordinates adapted to the ray congruence. Detailed proofs were given by Robinson & Trautman (1962) and by Jordan, Kundt & Ehlers (1961) (see also Kundt 1961). For the sake of completeness we include in our survey the subclass of expanding fields, and make use of some formulae first obtained by Robinson & Trautman.

Heat-of-mixing data of binary mixtures of liquid normal alkanes in a wide range of temperatures and chain lengths are used for testing the validity of the principle of congruence and the principle of corresponding states for excess functions. Deviations from the principle of congruence are found, particularly for systems differing in their volatile components. The temperature dependence of the heat of mixing is found to be qualitatively in agreement with the principle of corresponding states. The possible origin of the deviations from the principle of congruence is discussed.

The analytic basis for any projective theory of congruences, in which the lines of the congruence are defined either by a pair of points or a pair of planes, consists of the invariant theory of a completely integrable system of four homogeneous linear partial differential equations with two dependent and two independent variables, two of the equations of the system being of the first order, and two of the second order. If the developables of the congruence are known, this system of differential equations can be written in a very simple form.t But the determination of the developables of a congruence requires the integration of two partial differential equations of the first order, and it seems highly desirable to possess a theory which will be immediately applicable to any congruence, whether its developables can actually be found explicitly or not. The considerations made by G. M. Greent show that the existing theory can actually be modified so as to cover all such cases. However there are various ways in which this can be done and the various methods which might be used are not all equally desirable. Green himself has indicated one such method for the theory of congruences.? But Green only indicated in a general way what was to be done without actually working out a complete theory. The great disadvantage which this particular theory would have, as compared with the one to which we are devoting this paper, is that only the final results would be of interest. The various types of invariants corresponding to the transformations of certain subgroups would

If A \mathfrak {A} and B \mathfrak {B} are algebras, where A ⊆ B \mathfrak {A} \subseteq \mathfrak {B} , and θ \theta is a congruence relation on A \mathfrak {A} , let θ B {\theta ^\mathfrak {B}} be the smallest congruence relation on B \mathfrak {B} containing θ \theta . A \mathfrak {A} is called a congruence subalgebra of B \mathfrak {B} if A ⊆ B \mathfrak {A} \subseteq \mathfrak {B} and, for every congruence relation θ \theta on A , θ B ∩ | A | 2 = θ \mathfrak {A},{\theta ^\mathfrak {B}} \cap |\mathfrak {A}{|^2} = \theta . Elementary subalgebras are congruence subalgebras, and there are Directed Union and Loewenheim-Skolem Theorems for congruence subalgebras analogous to those for elementary subalgebras. Consequently we obtain full analogues of the Jónsson-Morley-Vaught results concerning homogeneous-universal algebras, where the notion of “subalgebra” is everywhere replaced by “congruence subalgebra".

Measured viscosities of mixtures of hexane and tetradecane are used to predict viscosities in three binary and one ternary system containing hexadecane. Predictions are based upon extrapolation, by means of the restricted congruence principle, of the excess Cibbs free energy of activation for flow. The errors in that function thereby range from 2.3 to 14.8 cal/mole for the several systems, and the corresponding errors in viscosity from 0.5 to 2.5%. On a determine les viscosites de melanges d'hexane et de tetradecane pour predire celles qu'on rencontre dans trois systemes binaires et un systeme ternaire contenant de l'hexadecane. Les predictions sont basees sur l'extrapolation, au moyen du principe de congruence restreinte, de l'exces d'energie libre d'activation de Gibbs pour l'ecoulement. Les erreurs dans la fonction varient entre 2.3 et 14.8 calories par molecule pour les differents systemes, ce qui donne une erreur correspondante dans la viscosite de 0.5 a 2.5%.