Rational approximations to algebraic numbers

It was proved in a recent paper that if α is any algebraic number, not rational, then for any ζ > 0 the inequalityhas only a finite number of solutions in relatively prime integers h, q. Our main purpose in the present note is to deduce, from the results of that paper, an explicit estimate for the number of solutions.

It was proved by Roth in a recent paper that if α is any real algebraic number, and if K > 2, then the inequalityhas only a finite number of solutions in relatively prime integers p, q (q > 0) The object of the present paper is to prove that the lower bound for κ can be reduced if conditions are imposed on p and q. The result obtained is as follows.

only a finite number of solutions. Unfortunately, the underlying method of Thue–Siegel–Roth is ineffective in the sense that it does not provide upper bounds for y or H0(β) respectively. However, it allows giving an explicit upper bound for the number of x/y ∈ Q satisfying (1.1). A first result was proved by Davenport and Roth ([3], 1955). This bound was improved by Bombieri and van der Poorten ([1], 1987) and independently by Luckhardt ([10], 1989) using the modified proof

Let α1, …, αn be n ≥ 2 algebraic numbers such that log α1,…, log αn and 2πi are linearly independent over the field of rational numbers Q. It is well known (see (6), Ch. 1) that the Thue–Siegel–Roth theorem implies that, for each positive number δ, there are only finitely many integers b1,…, bn satisfyingwhere H denotes the maximum of the absolute values of b1, …, bn. However, such an argument cannot provide an explicit upper bound for the solutions of (1), because of the non-effective nature of the theorem of Thue–Siegel–Roth. An effective proof that (1) has only a finite number of solutions was given by Gelfond (6) in the case n = 2, and by Baker(1) for arbitrary n. The work of both these authors is based on arguments from the theory of transcendental numbers. Baker's effective proof of (1) has important applications to other problems in number theory; in particular, it provides an algorithm for solving a wide class of diophantine equations in two variables (2).

Given an unstable finite-dimensional linear system, one can relate the existence of a memoryless feedback law stabilizing the system to the existence of a real solution of a set of multivariable polynomial inequalities. From these inequalities, a set of equalities may be constructed with two properties: the equality set has a real solution precisely when the inequality set does; generically the equality set has a finite number of solutions. Multivariable polynomial resultants provide a method of solving the equalities subject to the condition that the equalities have a finite number of solutions. The property that there is a finite number of solutions is established using some results of algebraic geometry.

An eigenvalue problem for Maxwell’s equations with anisotropic cubic nonlinearity is studied. The problem describes propagation of transverse magnetic waves in a dielectric layer filled with (nonlinear) anisotropic Kerr medium. The nonlinearity involves two non-negative parameters a, b that are usually small. In the case a = b = 0 one arrives at a linear problem that has a finite number of solutions (eigenvalues and eigenwaves). If a > 0, b ⩾ 0, then the nonlinear problem has infinitely many solutions; only a finite number of these solutions have linear counterparts. This shows that perturbation theory methods are inapplicable to study the problem in this case. For a = 0, b > 0 the nonlinear problem has a finite number of solutions; in this case each solution has a linear counterpart. Asymptotic distribution of the eigenvalues is found, periodicity of the eigenfunctions is proved and exact formula for the period is found, zeros of the eigenfunctions are determined, and a (nonlinear) eigenvalue comparison theorem is proved. Numerical experiments are presented.

1. Introduction. The principle of the Conservation of Number is concerned with the following situation. One starts with a system of algebraic equations having only a finite number of solutions and then applies a homomorphism whose domain contains the coefficients of those equations. This produces a new system. Let us suppose that the new system of equations also has only a finite number of solutions. The question then arises as to how the number of solutions before specialization compares with the number present afterwards. In a typical geometrical situation, one usually wishes to assert that the two systems have equally many solutions. However, it is easy to construct algebraic situations where the number changes,† and where the change is not to be explained away through the confluence of solutions or by their slipping off to infinity. At first sight this represents a breakdown of the conservation principle, but this principle has proved so useful in the past that one has a natural reluctance to discard it. The alternative is to attempt a reformulation and in (l) the present author gave such a reformulation for the case in which the specialization consisted in mapping a regular local ring on to its residue field. The modified theory requires that we take account of systems of equations which arise in connexion with the homology modules of a certain complex. The system associated with the homology module of degree zero is found to be the same as the one that arises in the naive theory, and usually this is the only one that makes a contribution. However, in cases where the number of solutions appears to change, the other systems become active and act in such a way that the balance is restored. For an amplification of these remarks we must refer the reader to (l). They are made here to indicate how the relevance of homological concepts first became clear in any detail. In the present paper these ideas are taken further, the principal gain being that it is no longer necessary to restrict the type of specialization to that which consists in mapping a regular local ring on to its residue field. Indeed one can use very general specializations provided that one transfers the homological requirement from the ring to the system of equations under consideration. In this way, one obtains a theory which is more general and, in some of its aspects, simpler as well.

It is known that Gröbner bases approach can be useful to solve systems of algebraic equations with a finite number of solutions. Nevertheless, as stated in [Bu], numerical accuracy attainable when using floating point arithmetic is not yet studied. In this paper we discuss numerical approach, defining conveniently the condition number of every root, and the numerical g.c.d.'s of Buchberger's algorithm. With these instruments we are able to compute confidence-intervals for the roots. We complete the study with two illustrative examples. We also give some insight about the best order of the variables when pure lexicographical order is concerned.

In this paper, by using variational methods, we study the following elliptic problem [Formula: see text] involving a general operator in divergence form of [Formula: see text]-Laplacian type ([Formula: see text]). In our context, [Formula: see text] is a bounded domain of [Formula: see text], [Formula: see text], with smooth boundary [Formula: see text], [Formula: see text] is a continuous function with potential [Formula: see text], [Formula: see text] is a real parameter, [Formula: see text] is allowed to be indefinite in sign, [Formula: see text] and [Formula: see text] is a continuous function oscillating near the origin or at infinity. Through variational and topological methods, we show that the number of solutions of the problem is influenced by the competition between the power [Formula: see text] and the oscillatory term [Formula: see text]. To be precise, we prove that, when [Formula: see text] oscillates near the origin, the problem admits infinitely many solutions when [Formula: see text] and at least a finite number of solutions when [Formula: see text]. While, when [Formula: see text] oscillates at infinity, the converse holds true, that is, there are infinitely many solutions if [Formula: see text], and at least a finite number of solutions if [Formula: see text]. In all these cases, we also give some estimates for the [Formula: see text] and [Formula: see text]-norm of the solutions. The results presented here extend some recent contributions obtained for equations driven by the Laplace operator, to the case of the [Formula: see text]-Laplacian or even to more general differential operators.

It was proved recently by Roth that if α is any real algebraic number, and κ > 2, then the inequalityhas only a finite number of solutions in integers h and q, where q > 0 and (h, q) = 1. This remarkable result answered finally a question which had been only partially answered by the work of Thue and Siegel.

In 1887 Runge [13] proved that a binary Diophantine equation F ( x , y ) = 0 F(x,y) = 0 , with F F irreducible, in a class including those in which the leading form of F F is not a constant multiple of a power of an irreducible polynomial, has only a finite number of solutions. It follows from Runge’s method of proof that there exists a computable upper bound for the absolute value of each of the integer solutions x x and y y . Runge did not give such a computation. Here we first deduce Runge’s Theorem from a more general theorem on Puiseux series that may be of interest in its own right. Second, we extend the Puiseux series theorem and deduce from the generalized version a generalized form of Runge’s Theorem in which the solutions x x and y y of the polynomial equation F ( x , y ) = 0 F(x,y) = 0 are integers, satisfying certain conditions, of an arbitrary algebraic number field. Third, we compute bounds for the solutions ( x , y ) ∈ Z 2 (x,y) \in {{\mathbf {Z}}^2} in terms of the height of F F and the degrees in x x and y y of F F .

Differentially finite series are solutions of linear differential equations with polynomial coefficients. P-recursive sequences are solutions of linear recurrences with polynomial coefficients. Corresponding notions are obtained by replacing classical differentiation or difference operators by their q-analogues. All these objects share numerous properties that are described in the framework of D-finiteness. Our aim in this area is to enable computer algebra systems to deal in an algorithmic way with a large number of special functions and sequences. Indeed, it can be estimated that approximately 60% of the functions described in Abramowitz & Stegun's handbook [1] fall into this category, as well as 25% of the sequences in Sloane's encyclopedia [20,21]. In a way, D-finite sequences or series are non-commutative analogues of algebraic numbers: the role of the minimal polynomial is played by a linear operator.Ore [14] described a non-commutative version of Euclidean division and extended Euclid algorithm for these linear operators (known as Ore polynomials). In the same way as in the commutative case, these algorithms make several closure properties effective (see[22]). It follows that identities between these functions or sequences can be proved or computed automatically. Part of the success of the gfun package [17] comes from an implementation of these operations. Another part comes from the possibility of discovering such identities empirically, with Pade-Hermite approximants on power series [2] taking the place of the LLL algorithm on floating-point numbers. The discovery that a series is D-finite is also important from the complexity point of view: several operations can be performed on D-finite series at a lower cost than on arbitrary power series. This includes multiplication, but also evaluation at rational points by binary splitting [4]. A typical application is the numerical evaluation of π in computer algebra systems; we give another one in these proceedings [3]. Also, the local behaviour of solutions of linear differential equations in the neighbourhood of their singularities is well understood [9] and implementations of algorithms computing the corresponding expansions are available [24, 13]. This gives access to the asymptotics of numerous sequences or to analytic proofs that sequences or functions cannot satisfy such equations [10]Results of a more algebraic nature are obtained by differential Galois theory [18, 19], which naturally shares many subroutines with algorithms for D-finite series. The truly spectacular applications of D-finiteness come from the multivariate case: instead of series or sequences, one works with multivariate series or sequences, or with sequences of series or polynomials,.... They obey systems of linear operators that may be of differential, difference, q-difference or mixed types, with the extra constraint that a finite number of initial conditions are sufficient to specify the solution. This is a non-commutative analogue of polynomial systems with a finite number of solutions. It turns out that, as in the polynomial case, Grobner bases give algorithmic answers to many decision questions, by providing normal forms in a finite dimensional vector space. This has been observed first in the differential case [11, 23] and then extended to the more general multivariate Ore case [8]. A crucial insight of Zeilberger [27, 15] is that elimination in this non-commutative setting computes definite integrals or sums. This is known as creative telescoping. In thehypergeometric setting (when the quotient is a vector space of dimension1), a fast algorithm for this operation is known as Zeilberger's fast algorithm [26]. In the more general case, Grobner bases are of help in this elimination. This is true in the differential case [16, 25] and to a large extent in the more general multivariate case [8]. Also, Zeilberger's fast algorithm has been generalized to the multivariate Ore case by Chyzak [5, 6]. Still, various efficiency issues remain and phenomena of non-minimality of the eliminated operators are not completely understood. A further generalization of D-finite series is due to Gessel [12] who developed a theory of symmetric series. These series are such than when all but a finite number of their variables (in a certain basis) are specialized to0, the resulting series is D-finite in the previous sense. Closure properties under scalar product lead to proofs of D-finiteness (in the classical sense) for various combinatorial sequences. Again, algorithms based on Grobner bases make these operations effective [7]. The talk will survey the nicest of these algorithms and their applications. I will also indicate where current work is in progress, or where more work is needed.

1. The basics of algebraic number theory. An algebraic number field is a field K = Q(a) where a is a zero of an irreducible (over Q) polynomial f(x) with integral coefficients. The degree of K, which we denote by n = n(K) = [K:Q], is the degree of ƒ(%). We write the roots of /(x) = 0 as a, a, • • • ,a ( n ) m such a way that for l^j^r1 = r1(K), a Q) is real, while for j>ru a 0 ) is complex. If we let n = ri+2r2, then it is customary to order the rr=r2(K) complex conjugate pairs of roots so that for r i+ l ^ j ^ r i+ r 2 , a=a The a ( , ) s are called the conjugates of a and the fields K ( , =Q(a) are called the conjugate fields of K. If r2 = 0, we say K is totally real and if ri=0, we say K is totally complex. The integers of K are those elements of K which are zeros of a polynomial with integer coefficients and leading coefficient 1. The integers of K form a ring which we denote by o. As is well known, factorization of the integers of K into prime integers is not necessarily unique. Various equivalent ways of remedying this have been used; we follow Dedekind's method. If a i , • • • , ak are elements of X, the set

Our overall research interest is in synthesizing human like reaching and grasping using anthropomorphic robot hand-arm systems, as well as understanding the principles underlying human control of these actions. When one needs to define the control and task requirements in the Cartesian space, the problem of inverse kinematics needs to be solved. For non-redundant manipulators, a desired end-effector position and orientation can be achieved by a finite number of solutions. For redundant manipulators however, there are in general infinitely many solutions where the cardinality of the solution set must be made finite by imposing certain constraints. In this paper, we consider the Mitsubishi PA10 manipulator which is similar to the human arm, in the sense that both wrist and shoulder joints can be considered to emulate a 3DOF ball joint. We explicitly derive the analytic solution for the inverse kinematics using quaternions. Then, we derive a parameterization in terms of a pure quaternion called the swivel quaternion. The swivel quaternion is similar to the elbow swivel angle used in most approaches, but avoid the computation of inverse trigonometric functions. This parameterization of the self-motion manifold is continuous with any end-effector motion. Given the pose of the end-effector and the swivel quaternion (or swivel angle), the algorithm derives all solution of the inverse kinematics (finite number). We then show how the parameterization of the elbow self-motion can be used for the real-time control of the PA10 manipulator in the presence of obstacles.

Discrete event simulation has been widely applied to study the behavior of stochastic manufacturing systems. This is due to the fact that manufacturing systems are usually too complex to obtain a closed-form analytical model that can accurately predict their performance. This becomes particularly critical when the optimization of these systems is of concern. In fact, Simulation optimization techniques are employed to identify the manufacturing system configuration which can maximize the expected system performance when this can only be estimated by running a simulator. In this article, we look into simulation-based optimization when a finite number of solutions are available and we have to identify the best. In particular, we propose, for the first time, the integration of Optimal Computing Budget Allocation (OCBA), which is based on independent measures from each simulation experiment, and Time Dilation (TD), which is a single-run simulation optimization algorithm. As a result, the optimization problem is solved when only one experiment of the system is performed by changing the “speed” of the simulation at each configuration in order to control the computational effort. The challenge is how to iteratively select such a speed. We solve this problem by proposing TD-OCBA, which integrates TD and OCBA while relying on standardized time series variance estimators. Numerical experiments have been conducted to study the performance of the algorithm when the response is generated from a time series. This provides the possibility to test the robustness of TD-OCBA. Comparison between TD-OCBA and the original TD method was performed by simulating a job shop system reported in the literature. Finally, an application involving semiconductors remote diagnostics is used to compare the TD-OCBA method and what is known as the equal allocation method.