Real Numbers In Terms Research Articles

Galois cohomology of reductive algebraic groups over the field of real numbers

We describe functorially the first Galois cohomology set $H^1({\mathbb R},G)$ of a connected reductive algebraic group $G$ over the field $\mathbb R$ of real numbers in terms of a certain action of the Weyl group on the real points of order dividing 2 of the maximal torus containing a maximal compact torus. This result was announced with a sketch of proof in the author's 1988 note. Here we give a detailed proof.

Galois cohomology of real quasi-connected reductive groups

By a quasi-connected reductive group (a term of Labesse) over an arbitrary field we mean an almost direct product of a connected semisimple group and a quasi-torus (a smooth group of multiplicative type). We show that a linear algebraic group is quasi-connected reductive if and only if it is isomorphic to a smooth normal subgroup of a connected reductive group. We compute the first Galois cohomology set \(H^1({\mathbb {R}},G)\) of a quasi-connected reductive group G over the field \({\mathbb {R}}\) of real numbers in terms of a certain action of a subgroup of the Weyl group on the Galois cohomology of a fundamental quasi-torus of G.

Randomness and uniform distribution modulo one

We elaborate the notions of Martin-Löf and Schnorr randomness for real numbers in terms of uniform distribution of sequences. We give a necessary condition for a real number to be Schnorr random expressed in terms of classical uniform distribution of sequences. This extends the result proved by Avigad for sequences of linear functions with integer coefficients to the wider classical class of Koksma sequences of functions. And, by requiring equidistribution with respect to every computably enumerable open set (respectively, computably enumerable open set with computable measure) in the unit interval, we give a sufficient condition for Martin-Löf (respectively Schnorr) randomness.

Arithmetic representations of real numbers in terms of self-similar sets

Suppose $n\geq 2$ and $\mathcal{A}_{i}\subset \{0,1,\cdots ,(n-1)\}$ for $ i=1,\cdots ,l,$ let $K_{i}=\bigcup\nolimits_{a\in \mathcal{A}_{i}}n^{-1}(K_{i}+a)$ be self-similar sets contained in $[0,1].$ Given $ m_{1},\cdots ,m_{l}\in \mathbb{Z}$ with $\prod\nolimits_{i}m_{i}\neq 0,$ we let \begin{equation*} S_{x}=\left\{ \mathbf{(}y_{1},\cdots ,y_{l}\mathbf{)}:m_{1}y_{1}+\cdots +m_{l}y_{l}=x\text{ with }y_{i}\in K_{i}\text{ }\forall i\right\} . \end{equation*} In this paper, we analyze the Hausdorff dimension and Hausdorff measure of the following set \begin{equation*} U_{r}=\{x:\mathbf{Card}(S_{x})=r\}, \end{equation*} where $\mathbf{Card}(S_{x})$ denotes the cardinality of $S_{x}$, and $r\in \mathbb{N}^{+}$. We prove under the so-called covering condition that the Hausdorff dimension of $U_{1}$ can be calculated in terms of some matrix. Moreover, if $r\geq 2$, we also give some sufficient conditions such that the Hausdorff dimension of $U_{r}$ takes only finite values, and these values can be calculated explicitly. Furthermore, we come up with some sufficient conditions such that the dimensional Hausdorff measure of $U_{r}$ is infinity. Various examples are provided. Our results can be viewed as the exceptional results for the classical slicing problem in geometric measure theory.

Commensurable continued fractions

We compare two families of continued fractions algorithms, the symmetrized Rosen algorithm and the Veech algorithm. Each of these algorithms expands real numbers in terms of certain algebraic integers. We give explicit models of the natural extension of the maps associated with these algorithms; prove that these natural extensions are in fact conjugate to the first return map of the geodesic flow on a related surface; and, deduce that, up to a conjugacy, almost every real number has an infinite number of common approximants for both algorithms.

A combinatorial approach to binary positional number systems

Although the representation of the real numbers in terms of a base and a set of digits has a long history, new questions arise even in the binary case – digits 0 and 1. A binary positional number system (binary radix system) with base equal to the golden ratio \((1+\sqrt{5}\,)/2\) is fairly well known. The main result of this paper is a construction of infinitely many binary radix systems, each one constructed combinatorially from a single pair of binary strings. Every binary radix system that satisfies even a minimal set of conditions that would be expected of a positional number system, can be constructed in this way.

Transcendence with Rosen continued fractions

We give the first transcendence results for the Rosen continued fractions. Introduced over half a century ago, these fractions expand real numbers in terms of certain algebraic numbers.

On some properties of the alternating Sylvester series and alternating Engel series representation of real numbers

We investigate some properties connected with the alternating Sylvester series and alternating Engel Series representation for real numbers, in terms of the integer digits involved. In particular, we look at an algorithm that leads to a general alternating series expansion for real numbers in terms of rationals and deduce the alternating Sylvester and alternating Engel series from this general series.KEY WORDS: alternating Sylvester series, alternating Engel series, rational numbers, alternating series expansion.

Tong’s spectrum for Rosen continued fractions

In the 1990s, J.C. Tong gave a sharp upper bound on the minimum of k consecutive approximation constants for the nearest integer continued fractions. We generalize this to the case of approximation by Rosen continued fraction expansions. The Rosen fractions are an infinite set of continued fraction algo- rithms, each giving expansions of real numbers in terms of certain algebraic integers. For each, we give a best possible upper bound for the minimum in appropriate consecutive blocks of approxima- tion coecients. We also obtain metrical results for large blocks of bad approximations.

Schnorr randomness

Abstract.Schnorr randomness is a notion of algorithmic randomness for real numbers closely related to Martin-Löf randomness. After its initial development in the 1970s the notion received considerably less attention than Martin-Löf randomness, but recently interest has increased in a range of randomness concepts. In this article, we explore the properties of Schnorr random reals, and in particular the c.e. Schnorr random reals. We show that there are c.e. reals that are Schnorr random but not Martin-Löf random, and provide a new characterization of Schnorr random real numbers in terms of prefix-free machines. We prove that unlike Martin-Löf random c.e. reals, not all Schnorr random c.e. reals are Turing complete, though all are in high Turing degrees. We use the machine characterization to define a notion of “Schnorr reducibility” which allows us to calibrate the Schnorr complexity of reals. We define the class of “Schnorr trivial” reals, which are ones whose initial segment complexity is identical with the computable reals, and demonstrate that this class has non-computable members.

Schnorr Randomness

Schnorr randomness is a notion of algorithmic randomness for real numbers closely related to Martin-Löf randomness. Since its initial development in the 1970s, the notion has received considerably less attention than Martin-Löf randomness. In this article, we explore the properties of Schnorr random reals, and in particular the c.e. Schnorr random reals. We show that there are c.e. reals that are Schnorr random but not Martin-Löf random, and provide a new characterization of Schnorr random real numbers in terms of prefix-free machines. We prove that unlike Martin-Löf random c.e. reals, not all Schnorr random c.e. reals are Turing complete, though all are in high Turing degrees. We use the machine characterization to define a notion of “Schnorr reducibility” which allows us to calibrate the Schnorr complexity of reals. We define the class of “Schnorr trivial” reals, which are ones whose initial segment complexity is identical with the computable reals, and demonstrate that this class has non-computable members.

A Real Number Structure that is Effectively Categorical

AbstractOn countable structures computability is usually introduced via numberings. For uncountable structures whose cardinality does not exceed the cardinality of the continuum the same can be done via representations. Which representations are appropriate for doing real number computations? We show that with respect to computable equivalence there is one and only one equivalence class of representations of the real numbers which make the basic operations and the infinitary normed limit operator computable. This characterizes the real numbers in terms of the theory of effective algebras or computable structures, and is reflected by observations made in real number computer arithmetic. Demanding computability of the normed limit operator turns out to be essential: the basic operations without the normed limit operator can be made computable by more than one class of representations. We also give further evidence for the well‐known non‐appropriateness of the representation to some basebby proving that strictly less functions are computable with respect to these representations than with respect to a standard representation of the real numbers. Furthermore we consider basic constructions of representations and the countable substructure consisting of the computable elements of a represented, possibly uncountable structure. For countable structures we compare effectivity with respect to a numbering and effectivity with respect to a representation. Special attention is paid to the countable structure of the computable real numbers.

Stochastic Dominance and Moments of Distributions

Stochastic dominance orders of all finite degrees are defined on the set of distribution functions on the nonnegative real numbers in terms of integrals of the distributions. It is proved that if F strictly nth-degree stochastically dominates G, and if the moments of F and G through order n are finite with μFk = ∫Xk dF (x), then (μF1, …, μFn) ≠ (μG1, …, μGn) and (−1)k−1 μFk > (−1)k−1 μGk for the smallest k for which μFk ≠ μGk.

Extensions of Samuelson's Inequality

Abstract In this note we obtain upper and lower bounds for the kth largest number in a set of real numbers in terms of their mean and standard deviation. For each inequality necessary and sufficient conditions for equality are given.

A refinement of the arithmetic mean-geometric mean inequality

Upper and lower bounds are given for the difference between the arithmetic and geometric means of n positive real numbers in terms of the variance of these numbers.

Who’s afraid of the measuring numbers?

The term measuring number is acquiring currency as a popular synonym for non-negative real number. This is good. Yet the belief persists that the real numbers, by whatever name, must remain fundamentally inaccessible to school children, obscured behind a veil of Dedekind cuts or Cauchy sequences. (Thus Griffiths and Howson, in what is generally a humane book [1], devote several pages to an ingenious, but ultimately condescending, attempt to foreshadow the mysteries of real numbers in terms of points of a line.)

Best Summation Formulae and Discrete Splines

The problem of obtaining a best summation formula for a finite sequence of real numbers in terms of a fixed number of terms of the sequence is reduced to a solvable linear or quadratic programming problem. This is done by developing the appropriate discrete Taylor and Peano theorems. The best summation formulae are related to discrete splines studied earlier. Discrete monosplines are introduced here and related to best summation formulae. Some numerical results are given.