A Real Number Structure that is Effectively Categorical

We study Σ-definability of countable models over hereditarily finite {ie193-01} superstructures over the field ℝ of reals, the field ℂ of complex numbers, and over the skew field ℍ of quaternions. In particular, it is shown that each at most countable structure of a finite signature, which is Σ-definable over {ie193-02} with at most countable equivalence classes and without parameters, has a computable isomorphic copy. Moreover, if we lift the requirement on the cardinalities of the classes in a definition then such a model can have an arbitrary hyperarithmetical complexity, but it will be hyperarithmetical in any case. Also it is proved that any countable structure Σ-definable over {ie193-03}, possibly with parameters, has a computable isomorphic copy and that being Σ-definable over {ie193-04} is equivalent to being Σ-definable over {ie193-05}.

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.

This article reports on a study of high-school and of technologist students (prospective engineers and economists) understanding of real numbers. Our study was based on written response to a properly designed questionnaire and on interviews taken from students. The quantitative results of our experiment showed an almost complete failure of the technologist students to deal with processes connected to geometric constructions of incommensurable magnitudes. However, this didn’t prevent them in answering satisfactorily the other questions. The superiority of their correct answers with respect to those of high-school students, although the majority of them correspond to mediocre graduates of the secondary education, was evident in most cases. This is a strong indication that the age and the width of mathematical knowledge of the individual play an important role for the better understanding of the real numbers. The results of our experiment suggest also that the ability to transfer in comfort among several representations of real numbers helps students in obtaining a better understanding of them. A theoretical explanation about this is obtained through the adoption of the conceptual framework of dimensions of knowledge, introduced by Tirosh et al. (1998) for studying the comprehension of rational numbers. Following in part the idea of generic decomposition of the APOS analysis (Weller et al. 2009) we suggest a possible order for development of understanding the real numbers by students when teaching them at school. Some questions open to further research are also mentioned at the end of the paper. Keywords: Real, rational, irrational, algebraic and transcendental numbers, fractions, decimals, representations of real numbers.

In this chapter we discuss the convergence of sequences, the completeness of real numbers in terms of the convergence of Cauchy sequences and the existence of the least upper bound of a set that is bounded above. We also discuss infinite limits.

Strengthening a theorem of D. W. Kueker, this paper completely charaterizes which countable structures do not admit uncountable Lω1ω-elementarily equivalent models. In particular, it is shown that if the automorphism group of a countable structure M is abelian, or even just solvable, then there is no uncountable model of the Scott sentence of M. These results arise as part of a study of Polish groups with compatible left-invariant complete metrics.

Topological properties of real number representations

Arbitrary precision real arithmetic: design and algorithms

A real number x is called semi-computable if it is a limit of an increasing or decreasing computable sequence (x n ), n∈ℕ of ratio­nal numbers. In this case, a later member of the sequence is always a better approximation to x in the sense that |x − x n | ≥ |x − x m |, if n ≤ m. As a natural generalization, we call a real number x k-monotone computable (k-mc, for short), for any real number k >0,, if there is a computable sequence (xn),n∈ℕ of rational numbers which converges to x k-monotonically in the sense that k · |x − x n | ≥ |x − x m | for any n ≤ m and x is monotonically computable (mc, for short) if it is k-mc for some k >0. Various properties of k-mc real numbers are discussed in this pa­per. Among others we show that a real number is computable if it is k-mc for some k <1; the 1-mc real numbers are just the semi-computable real numbers and the set of all mc real numbers is contained properly in the set of weakly computable real numbers, where x is weakly computable if there are semi-computable real numbers y, z such that x = y + z. Furthermore, we show also an infinite hierarchy of mc real numbers.

Based on a new coinductive characterization of continuous functions we extract certified programs for exact real number computation from constructive proofs. The extracted programs construct and combine exact real number algorithms with respect to the binary signed digit representation of real numbers. The data type corresponding to the coinductive definition of continuous functions consists of finitely branching non-wellfounded trees describing when the algorithm writes and reads digits. We discuss several examples including the extraction of programs for polynomials up to degree two and the definite integral of continuous maps.

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.

This paper investigates the effectiveness of cellular automata to compute arithmetic functions over the set of real numbers. Since cellular automata (CA) have parallel computational ability in conjunction with the capacity to accept an infinite length input, the CA should be able to effectively compute real number arithmetic. This paper shows that typical representations of real numbers do not lend themselves to effective computation by a CA and that a nonstandard representation must be employed in order to accomplish this.

Several applications have problems with the representation of the real numbers because of its drawbacks like the propagation and the accumulation of errors. These numbers have a fixed length format representation that provides a large dynamic range, but on the other hand it causes truncation of some parts of the numbers in case of a number that needs to be represented by a long stream of bits. Researchers suggested many solutions for these errors, one of these solutions is the Multi-Number (MN) system. MN system represents the real number as a vector of floating-point numbers with controlled accuracy by adjusting the length of the vector to accumulate the non-overlapping real number sequences. MN system main drawback is the MN computations that are iterative and time consuming, making it unsuitable for real time applications. In this work, the Single Instruction Multiple Data (SIMD) model supported in modern CPUs is exploited to accelerate the MN Computations. The basic arithmetic operation algorithms had been adjusted to make use of the SIMD architecture and support both single and double precision operations. The new architecture maintains the same accuracy of the original one, when was implemented for both single and double precision. Also, in this paper the normal Gaussian Jordan Elimination algorithm was proposed and used to get the inverse of the Hilbert Matrix, as an example of ill-conditioned matrices, instead of using iterative and time-consuming methods. The accuracy of the operations was proved by getting the inverse of the Hilbert Matrix and verify that the multiplication of the inverse and the original matrix producing the unity matrix. Hilbert Matrix inverse execution time was accelerated and achieved a speedup 3x, compared to the original NM operations. In addition to the previous, the accelerated MN system version was used to solve the polynomial regression problem.

This paper presents a metod of inductive inference of realvalued functions from given pairs of observed data of (x,h(x)), where h is a target function to be inferred. Eac of suc observed data inevitably involves some ranges of errors, and hence it is usually represented by a pair of rational numbers sow te approximate value and te error bound, respectively. On the other hand, a real number called a recursive real number can be represented by a pair of two of rational numbers, which converges to the real number and converges to zero, respectively. These sequeces show an approximate value of te real number and an error bound at each point. Such a real number can also be represented by a sequence of closed intervals wit rational end points converges to a singleton interval with the real number as both end points.In this paper, we propose a notion of recursive real-valued functions that can enjoy the merits of the both representations of the recursive real numbers. Then we present an algorithm which approximately infers real-valued functions from numerical data with some error bounds, and show that there exists a rich set of real-valued functions which is approximately inferable in the limit from such numerical data. We also discuss the precision of the guesses from the machine when sufficient data have not yet given.

Exact real number computation is a fast growing field with applications varying from debugging to specification of numerical to program. We present a specification of the real numbers represented as infinite lists of signed digits in Z. The expressive power and closeness to usual set theoretical mathematical notation gives us a clean and readable specification which is further directly implementable. A comparison with other formal methods is given together with a partial proof that the object being specified is actually the real numbers.

Reasoning about real number expressions in a proof assistant is challenging. Several problems in theorem proving can be solved by using exact real number computation. I have implemented a library for reasoning and computing with complete metric spaces in the Coq proof assistant and used this library to build a constructive real number implementation including elementary real number functions and proofs of correctness. Using this library, I have created a tactic that automatically proves strict inequalities over closed elementary real number expressions by computation.KeywordsReal NumberRational NumberTheorem ProveRegular FunctionStrict InequalityThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.