Arithmetical definability of field elements

1951 ◽  
Vol 16 (2) ◽  
pp. 125-126 ◽  
Author(s):  
Raphael M. Robinson
Keyword(s):  
Characteristic Zero ◽  
Rational Numbers ◽  
Free Variable ◽  
The Other ◽  
Necessary Condition ◽  
Algebraic Numbers ◽  
Real Numbers ◽  
Fourth Root ◽  
Arithmetical Condition ◽  
Image Position

If F is a field, and α is an element of F, we say that α is arithmetically definable in F if there is a formula containing one free variable and any number of bound variables, involving only the concepts of elementary logic and the operations of addition and multiplication, which is satisfied by α and by no other element of F. The range of the bound variables is understood to be F. Without changing the sense of the above definition, we can allow in our formulas symbols for specific integers, or even (if F has characteristic zero) symbols for specific rational numbers, since these are arithmetically definable.As an example, consider the field F = R(2¼), obtained by adjoining the positive fourth root of 2 to the field R of rationals. Notice that 2¼ is not defined arithmetically by the formula x2 = 2, since this equation has two roots in F.However, 2¼ may be defined arithmetically by the equivalencewhere we have used the logical symbols ↔ (if and only if), ∨ (there exists), and ∧ (and). For the equation y4 = 2 is satisfied by no elements of F except y = ±2¼, and in both cases y2 = 2¼. On the other hand, 2¼ is not arithmetically definable in F, since there is an automorphism of F which takes 2¼ into −2¼, so that every arithmetical condition satisfied by 2¼ is also satisfied by −2¼.In any field F, a necessary condition for the arithmetical definability of an element α is that α should be fixed for all automorphisms of F. That this condition is not always sufficient is shown by considering the field of real numbers. Here there is no automorphism but the identity, but there can of course be but a denumerable infinity of arithmetically definable real numbers. Tarski has shown that only the algebraic numbers are arithmetically definable.

scholarly journals Ubiquity and large intersections properties under digit frequencies constraints

2008 ◽  
Vol 145 (3) ◽  
pp. 527-548 ◽  
Author(s):  
JULIEN BARRAL ◽  
STÉPHANE SEURET
Keyword(s):  
Hausdorff Dimension ◽  
Rational Numbers ◽  
Intersection Property ◽  
Algebraic Numbers ◽  
Real Numbers ◽  
Countable Intersection ◽  
The Real ◽  
Large Intersection ◽  
Digit Frequencies

AbstractWe are interested in two properties of real numbers: the first one is the property of being well-approximated by some dense family of real numbers {xn}n≥1, such as rational numbers and more generally algebraic numbers, and the second one is the property of having given digit frequencies in some b-adic expansion.We combine these two ways of classifying the real numbers, in order to provide a finer classification. We exhibit sets S of points x which are approximated at a given rate by some of the {xn}n, those xn being selected according to their digit frequencies. We compute the Hausdorff dimension of any countable intersection of such sets S, and prove that these sets enjoy the so-called large intersection property.

scholarly journals The inverse limit of some free algebras

1973 ◽  
Vol 16 (2) ◽  
pp. 129-145
Author(s):  
A. L. Allen ◽  
S. Moran
Keyword(s):  
Characteristic Zero ◽  
Inverse Limit ◽  
The Other ◽  
Natural Projection ◽  
Free Algebras ◽  
Fixed Field ◽  
Degree Function ◽  
Image Position

Let Ω[x1, x2, …, xn] denote the algebra of polynomials in variables x1, x2, …, xn with coefficients from a fixed field Ω of characteristic zero, where n = 1, 2,…. There exists a natural projection which maps xn onto 0 and all the other variables onto themselves, for n = 1, 2, …. This enables one to construct the corresponding inverse limit which we here denote by Ω[x]. The algebra Ω[x] has a natural degree function defined on it.

scholarly journals Counting algebraic numbers in short intervals with rational points

2019 ◽  
pp. 4-11
Author(s):  
Vasily I. Bernik ◽  
Friedrich Götze ◽  
Nikolai I. Kalosha
Keyword(s):  
Uniform Distribution ◽  
Rational Numbers ◽  
Algebraic Numbers ◽  
Small Interval ◽  
Real Numbers ◽  
Algebraic Integers ◽  
Small Denominators ◽  
Short Intervals ◽  
Regular Distribution ◽  
Regularity Properties

In 2012 it was proved that real algebraic numbers follow a non­uniform but regular distribution, where the respective definitions go back to H. Weyl (1916) and A. Baker and W. Schmidt (1970). The largest deviations from the uniform distribution occur in neighborhoods of rational numbers with small denominators. In this article the authors are first to specify a gene ral condition that guarantees the presence of a large quantity of real algebraic numbers in a small interval. Under this condition, the distribution of real algebraic numbers attains even stronger regularity properties, indicating that there is a chance of proving Wirsing’s conjecture on approximation of real numbers by algebraic numbers and algebraic integers.

On Some Twisted Chevalley Groups Over Laurent Polynomial Rings

1981 ◽  
Vol 33 (5) ◽  
pp. 1182-1201 ◽  
Author(s):  
Jun Morita
Keyword(s):  
Lie Algebra ◽  
Rational Numbers ◽  
Laurent Polynomial ◽  
Cartan Matrix ◽  
Polynomial Rings ◽  
Complex Numbers ◽  
Real Numbers ◽  
Defining Relations ◽  
Image Position ◽  
Generalized Cartan Matrix

We let Z denote the ring of rational integers, Q the field of rational numbers, R the field of real numbers, and C the field of complex numbers.For elements e and f of a Lie algebra, [e,f] denotes the bracket of e and f. A generalized Cartan matrix C = (cij) is a square matrix of integers satisfying cii = 2, cij ≦ 0 if i ≠ j, cij = 0 if and only if cji = 0. For any generalized Cartan matrix C = (cij) of size l × l and for any field F of characteristic zero, denotes the Lie algebra over F generated by 3l generators e1, …, el, h1, …, hl, f1, …, fl with the defining relationsfor all i, j,for distinct i, j.

An application of the Thue–Siegel–Roth theorem to elliptic functions

1971 ◽  
Vol 69 (1) ◽  
pp. 157-161 ◽  
Author(s):  
J. Coates
Keyword(s):  
Number Theory ◽  
Diophantine Equations ◽  
Elliptic Functions ◽  
Rational Numbers ◽  
Algebraic Numbers ◽  
Number Of Solutions ◽  
Transcendental Numbers ◽  
Linearly Independent ◽  
Image Position ◽  
Effective 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).

Irregularity of the Rate of Decrease of Sequences of Powers in the Volterra Algebra

1981 ◽  
Vol 33 (2) ◽  
pp. 320-324 ◽  
Author(s):  
J. Esterle
Keyword(s):  
Banach Algebra ◽  
The Other ◽  
Real Numbers ◽  
Approximate Identities ◽  
Image Position

G. R. Allan and A. M. Sinclair proved in [1] that if a commutative radical Banach algebra possesses bounded approximate identities then for every sequence (αn) of real numbers such that limn→∞αn = 0 there exists such thatIn the other direction it is shown in [6] that if is separable and if the nilpotents are dense in then for every sequence (βn) of positive reals there exists such that(This result was given in [2] for the Volterra algebra.)

On derivability

1937 ◽  
Vol 2 (3) ◽  
pp. 113-119 ◽  
Author(s):  
W. V. Quine
Keyword(s):  
Formal System ◽  
Free Variable ◽  
The Other ◽  
Free Variables ◽  
Image Position ◽  
Positive Integers

1. The notion of derivability. Italic capitals, with or without subscripts, will be used as variables. They are to take as values some manner of elements which may for the present be left undetermined. Now let us consider abstractly the notion of the derivability of an element X from one or more specified elements by a series of steps of a specified kind. This involves reference to two conditions upon elements. One of these conditions, expressible by some statement form containing a single free variable, determines the elements from which X is said to be derivable. The other condition, expressed say by a statement form containing k + 1 free variables, determines the kind of steps by which the derivation is to proceed; it is the condition which any elements Z1, … Zk, Y must fulfill if progress from Zi, …, Zk to Y is to constitute a step of derivation in the intended sense. Supposing “f(Y)” supplanted by the first of these statement forms, whatever it may be, and “g(Z1, …, Zk, Y)” supplanted by the other, let us adopt the form of notationto express derivability of X in the suggested sense. The meaning of (1) can be formulated more accurately as follows:(i) There are elements Y1 to Ym (for some m) such that Ym = X and, for each i≦m, either f(Yi) or else there are numbers j1 to Ym, each less than i, for which g(Yj1, …, Yjk, Yi).(Variable subscripts are to be understood, here and throughout the paper, as referring only to positive integers.)The notion (1) is illustrated in the ancestral R* of a relation R;1 for,Another illustration is afforded by metamathematics. Suppose our elements are the expressions used in some formal system; suppose we have defined “Post(Y)”, meaning that Y is a postulate of that system; and suppose we have defined “Inf(ZI, …, Zk, Y)” (for some fixed k large enough for all purposes of the system in question), meaning that Y proceeds from Z1, …, Zk by one application of one or another of the rules of inference of the system. Thenwould mean that X is a theorem of the system.

Non-homogeneous binary cubic forms

1951 ◽  
Vol 47 (3) ◽  
pp. 457-460 ◽  
Author(s):  
R. P. Bambah
Keyword(s):  
Lower Bound ◽  
Finite Volume ◽  
Upper Bound ◽  
The Other ◽  
Real Numbers ◽  
Cubic Form ◽  
Homogeneous Form ◽  
Other Hand ◽  
Image Position ◽  
Cubic Forms

1. Let f(x1, x2, …, xn) be a homogeneous form with real coefficients in n variables x1, x2, …, xn. Let a1, a2, …, an be n real numbers. Define mf(a1, …, an) to be the lower bound of | f(x1 + a1, …, xn + an) | for integers x1, …, xn. Let mf be the upper bound of mf(a1, …, an) for all choices of a1, …, an. For many forms f it is known that there exist estimates for mf in terms of the invariants alone of f. On the other hand, it follows from a theorem of Macbeath* that no such estimates exist if the regionhas a finite volume. However, for such forms there may be simple estimates for mf dependent on the coefficients of f; for example, Chalk has conjectured that:If f(x,y) is reduced binary cubic form with negative discriminant, then for any real a, b there exist integers x, y such that

On calculating extinction probabilities for branching processes in random environments

1969 ◽  
Vol 6 (03) ◽  
pp. 478-492 ◽  
Author(s):  
William E. Wilkinson
Keyword(s):  
Generating Functions ◽  
Infinite Sequence ◽  
Branching Processes ◽  
Transition Probability ◽  
Transition Probability Matrix ◽  
Random Environments ◽  
Real Numbers ◽  
Discrete Time Markov Chain ◽  
Extinction Probabilities ◽  
Image Position

Consider a discrete time Markov chain {Zn } whose state space is the non-negative integers and whose transition probability matrix ║Pij ║ possesses the representation where {Pr }, r = 1,2,…, is a finite or denumerably infinite sequence of non-negative real numbers satisfying , and , is a corresponding sequence of probability generating functions. It is assumed that Z 0 = k, a finite positive integer.

scholarly journals An Inhomogeneous Minimum For Non-Convex Star-Regions With Hexagonal Symmetry

1955 ◽  
Vol 7 ◽  
pp. 337-346 ◽  
Author(s):  
R. P. Bambah ◽  
K. Rogers
Keyword(s):  
Quadratic Form ◽  
Binary Quadratic Form ◽  
Elementary Geometry ◽  
Hexagonal Symmetry ◽  
Real Numbers ◽  
Inhomogeneous Minimum ◽  
Image Position ◽  
Indefinite Binary Quadratic Form

1. Introduction. Several authors have proved theorems of the following type:Let x0, y0 be any real numbers. Then for certain functions f(x, y), there exist numbers x, y such that1.1 x ≡ x0, y ≡ y0 (mod 1),and1.2 .The first result of this type, but with replaced by min , was given by Barnes (3) for the case when the function is an indefinite binary quadratic form. A generalisation of this was proved by elementary geometry by K. Rogers (6).

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