Showing papers on "Algebraic number published in 1977"

Methods of algebraic geometry in the theory of non-linear equations

Abstract: CONTENTSIntroduction § 1. The Akhiezer function and the Zakharov-Shabat equations § 2. Commutative rings of differential operators § 3. The two-dimensional Schrodinger operator and the algebras associated with it § 4. The problem of multi-dimensional -algebraic operators Appendix 1. The Hamiltonian formalism in equations of Lax and Novikov types Appendix 2. Elliptic and rational solutions of the K-dV equations and systems of many particles Concluding Remarks References

An algebraic condition for the approach to equilibrium of an open N-level system

Abstract: We give an algebraic condition in order that a completely positive dynamical semigroup of an N-level system has a unique (invariant) equilibrium state and that every initial state approaches this equilibrium state as t→∞. We apply our result to a semigroup arising in the weak coupling limit.

Knot modules. I

Abstract: For a differentiable knot, i.e. an imbedding SI C S,+2, one can associate a sequence of modules (Aq) over the ring Z [t, I -l], which are the source of many classical knot invariants. If X is the complement of the knot, and X -e X the canonical infinite cyclic covering, then Aq = Hq(X).In this work a complete algebraic characterization of these modules is given, except for the Z-torsion submodule of A 1. In classical knot theory there are many "abelian" invariants which have proved useful in distinguishing knots, e.g. knot-polynomials, "elementary" ideals, homology and linking pairings in the finite cyclic branched coverings, ideal classes (see [F], [FS]). It is known (see [T]) that these invariants can all be extracted from a certain module A over the ring A = Z [t, t -] and a "Hermitian" pairing on A taking values in Q (A)/A (Q (A) is the quotient field of A). The construction of A and carries over to higher-dimensional knots and, in certain cases, are enough to classify the knot up to isotopy (see [L], [T1], [K]). In general, there is a finite collection A1,..., An of such modules associated to an n-dimensional knot in (n + 2)-space, and exists on Ak when n = 2k 1. Our first purpose in this work will be to give an algebraic characterization of these objects. There is already a great deal known in this direction (see [K], [Ke], [L3], [G]). Our results, which will be complete except for some problems with the Z-torsion part of A 1, will extend and reformulate these known results. For this purpose we will find it necessary to define a new pairing [ , ] in the Z-torsion part of Ak, when n = 2k. In the second part of this work, we will make an algebraic study of the modules and forms which have arisen from Part I. Our approach is to consider new modules and forms, derived from the original ones, over rings with a good structure theory: polynomial rings over fields, and rings of algebraic integers. The structure theory then classifies the derived object via invariants in these rings. These invariants include-most of the "classical" knot Received by the editors December 18, 1975. AMS (MOS) subject classifications (1970). Primary 57C45. (l) This research was partially supported by the Science Research Council of Great Britain and NSF Grant GP-38920X1. ? American Mathematical Society 1977 This content downloaded from 129.215.149.96 on Thu, 25 Dec 2014 09:30:47 AM All use subject to JSTOR Terms and Conditions

The structure of skew products with ergodic group automorphisms

Abstract: We prove that ergodic automorphisms of compact groups are Bernoulli shifts, and that skew products with such automorphisms are isomorphic to direct products. We give a simple geometric demonstration of Yuzvinskii’s basic result in the calculation of entropy for group automorphisms, and show that the set of possible values for entropy is one of two alternatives, depending on the answer to an open problem in algebraic number theory. We also classify those algebraic factors of a group automorphism that are complemented.

On the distribution of amicable numbers. II.

Abstract: Let σ (ή) denote the sum of the divisors of n and let s(ri) = a(ri) — n. Two natural numbers n, m are called an amicable pair if s(ri) = m and s (m) —n. The least such pair with n Φ m is n = 220, m = 284. We say a natural number n is an amicable number if it is a member of an amicable pair. An equivalent defmition is s(s(ri)) = n and still another is σ (η) = σ($(ιϊ)). Amicable numbers have a very long history; they were mentioned (in fact, defined) by Pythagoras, they were investigated by the Arabs during the European Dark Ages, and they were studied by Fermat, Descartes, and Euler.

Algebras Having Linear Multiplicative Complexities

Abstract: The foundations are laid for a theory of multiplicative complexity of algebras and it is shown how “multiplication problems” such as multiplication of matrices, polynomials, quaternions, etc., are instances of this theory. The usefulness of the theory is then demonstrated by utilizing algebraic ideas and results to derive complexity bounds. In particular linear upper and lower bounds for the complexity of certain types of algebras are established.

Induced representations and Banach *-algebraic bundles

Abstract: The abstract inducing process and abstract imprimitivity.- Banach *-algebraic bundles.- Induced representations and imprimitivity for Banach *-algebraic bundles.

Problems and results on combinatorial number theory III

Abstract: Publisher Summary This chapter discusses the problems and results on combinatorial number theory. It discusses number theoretic problems that are of combinatorial nature. Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers along with the properties of objects made out of integers. Integers can be considered either in themselves or as solutions to equations. Questions in number theory are often understood through the study of analytical objects that encode properties of the integers, primes or other number-theoretic objects in some fashion. Real numbers can also be studied in relation to rational numbers.

Arithmetic properties of certain functions in several variables III

Abstract: We obtain a general transcendence theorem for the solutions of a certain type of functional equation. A particular and striking consequence of the general result is that, for any irrational number w, the functiontakes transcendental values at all algebraic points α with 0 < |α| < 1.

Superfield Equations of Motion

Abstract: A general method for deriving the superfield equations of motion is proposed. These equations contain the supplementary conditions of irreducibility with respect to the supersymmetry group. The method uses projection operators which single out the irreducible representations and especially the algebraic roots of these operators. It is found that the standard equations of motion for spin-vector and symmetric tensor fields can in fact be obtained by extracting the square roots of the projection operators for spin-3/2 and spin-2, respectively. The spinor superfield equation is deduced and discussed in detail.

A Block–Stodola eigensolution technique for large algebraic systems with non‐symmetrical matrices

Abstract: A Block–Stodola eigensolution method is presented for large algebraic eigensystems of the form AU = λBU where A is real but non-symmetric. The steps in this method parallel those of a previous technique for the case when both A and B were real and symmetric. The essence of the technique is simultaneous iteration using a group of trial vectors instead of only one vector as is the case in the classical Stodola–Vianello iteration method. The problem is then transformed into a subspace where a direct solution of the reduced algebraic eigenvalue problem is sought. The main advantage is the significant reduction of computational effort in extracting a subset of eigenvalues and corresponding eigenvectors. Theorems from linear algebra serve to underlie the basis of the present technique. Complex eigendata that emerge during iteration can be handled without doubling the size of the problem. Higher order eigenvalue problems are reducible to first order form for which this technique is applicable. The treatment of the quadratic eigenvalue problem illustrates the details of this extension.

Multiplicative relations in number fields

Abstract: In this paper, we obtain an explicit form of the currently best known inequality for linear forms in the logarithms of algebraic numbers. The results complete our previous investigations (Bull. Austral. Math. Soc. 15 (1976), 33–57) which were conditional on a certain independence condition on the algebraic numbers. The extra work needed to obtain unconditional results centres on the properties of multiplicative relations in number fields. In particular, we show that a set of multiplicatively dependent algebraic numbers always satisfies a relation with relatively small exponents.

Matroids as the Study of Geometrical Configurations

Abstract: Matroids arise in a variety of combinatorial and algebraic contexts. The ideas of independence and bases as in vector spaces dependence as in algebraic dependence circuits as in graphs flats as in projective geometries atomic semimodular lattices, all come down to the same underlying structure. The essential feature of matroids is that they extract from a situation the basic incidence properties of points, lines and planes which we associate with geometry. That they show up in so many places and disguises suggests them as worthwhile objects of study. How that study is carried out is affected to some extent by the direction of approach. Graph theory suggests certain ideas to be generalised, lattice theory suggests others, and vector spaces still others. Often it seems that the geometrical aspect is lost in a profusion of algebraic notation.

Error analysis of penalty function techniques for constraint definition in linear algebraic systems

Abstract: The penalty function approach has been recently formalized as a general technique for adjoining constraint conditions to algebraic equation systems resulting from variational discretization of boundary value problems by finite difference or finite element techniques. This paper studies the numerical behaviour of the penalty function method for the special case of individual equation constraints imposed on a symmetric system of linear algebraic equations. Constraint representation and computational roundoff error components are distinguished and asymptotically characterized in terms of the penalty function weight coefficients. On the basis of this study, practical rules for the automatic assignment of values to those coefficients within the linear equation solver are proposed. Numerical problems encountered in the case of more general constraints are briefly discussed, and procedures for circumventing such difficulties are suggested.

On values and strategies for infinite-time linear quadratic games

Abstract: In a recent short paper, Mageirou [1] showed that for infinite-time linear quadratic games an appropriate solution of an algebraic Riccati-type equation determines the value of the game but not necessarily equilibrium strategies. In this note we show that a solution of the algebraic Riccati equation does determine equilibrium strategies if the class of admissible strategies is appropriately defined.

Dense orbits with two ends

Abstract: In this paper almost transitive actions of linear algebraic groups G are studied on complete normal manifolds X defined over the field of complex numbers. Such actions are completely described in cases when the complement in X of an open orbit is disconnected or contains an isolated point. As a preliminary, all homogeneous spaces of G having two Freudenthal ends are found.Bibliography: 16 titles.

Infallible Calculation of Polynomial Zeros to Specified Precision

Abstract: Let D be a Euclidean domain which is a subring of the field of complex numbers and in which the arithmetic operations can be algorithmically performed. Examples of D are Z , the integers, Q , the rational numbers, G , the Gaussian integers, and P , the real algebraic numbers. We are concerned with algorithms which, given any polynomial A(x)∈D[x] and any positive rational number ∈, computes a sequence of disjoint intervals (rectangles) of length (width) ∈ or less, each containing exactly one real (complex) zero of A, and together containing all real (complex) zeros. The algorithms must also compute the multiplicity of each zero. Any algorithm strictly fulfilling all of these specification may truly be described as infallible. While retaining these lofty goals we are nevertheless concerned with algorithms which are as efficient as possible, in practice as well as theory. Such infallible algorithms can be derived quite readily from several diverse mathematical theorems using “exact” arithmetic in the domain D, but these algorithms may differ significantly in their time complexities. Also, one may further improve practical efficiency by appropriate substitution of “approximate” arithmetic (e.g., interval arithmetic) for exact arithmetic in certain contexts without sacrificing infallibility (using exact arithmetic as backup). This paper surveys recent progress on this problem, starting from Heindel's 1970 implementation and analysis of Sturm's theorem, including Pinkert's 1973 application of Sturm sequences to complex zeros, the 1975 Collins-Loos algorithm based on Rolle's theorem, the 1976 Collins-Akritas modification of Uspensky's method based on Descartes' theorem, and concluding with a brief report on current research by Collins and Chou relating to use of approximate arithmetic and the “principle of argument”. Both theoretical time bounds and empirical times are presented.

A Ring of Delay Operators with Applications to Delay-Differential Systems

Abstract: A ring of delay operators is used to obtain a representation of the solution of systems of linear delay-differential equations. With the aid of this representation an algebraic rank-test is obtained for the $R^n $-controllability of the systems.

On general boundary value problems for differential equations with constant coefficients in a half-space

Abstract: Exact classes of uniqueness of the solutions are indicated for general boundary value problems in a half-space. Algebraic conditions sufficient for these problems to be well posed in classes much broader than those considered heretofore are adduced. The results are applied to the study of a mixed problem in a quarter-space in classes of increasing functions.Bibliography: 18 titles.