Representation Of Real Numbers Research Articles

From coinductive proofs to exact real arithmetic: theory and applications

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.

Performance Analysis of the Interval Algorithm for Random Number Generation Based on Number Systems

In this paper we analyze the interval algorithm for random number generation proposed by Han and Hoshi using the expression of real numbers on the interval . We first establish an explicit representation of the interval algorithm with the representation of real numbers on the interval based on number systems. Next, using the expression of the interval algorithm, we give a rigorous analysis of the interval algorithm. We discuss the difference between the expected number of the coin tosses in the interval algorithm and their upper bound derived by Han and Hoshi and show that it can be characterized explicitly with the established expression of the interval algorithm.

Solving mixed-integer nonlinear programming problems using improved genetic algorithms

This paper proposes a method for solving mixed-integer nonlinear programming problems to achieve or approach the optimal solution by using modified genetic algorithms. The representation scheme covers both integer and real variables for solving mixed-integer nonlinear programming, nonlinear programming, and nonlinear integer programming. The repairing strategy, a secant method incorporated with a bisection method, plays an important role in converting infeasible chromosomes to feasible chromosomes at the constraint boundary. To prevent premature convergence, the appropriate diversity of the structures in the population must be controlled. A cross-generational probabilistic survival selection method (CPSS) is modified for real number representation corresponding to the representation scheme. The efficiency of the proposed method was validated with several numerical test problems and showed good agreement.

The decimal representation of real numbers

The representation of natural numbers in decimal form is an unequivocal procedure while for the representation of real numbers some ambiguities concerning the existence of infinitely many digits equal to 9 still emerge. One of the most frequently confronted misunderstandings is whether 0.999 … equals 1 or not, and if not what number does this sequence of digits represent. In this review article, the decimal representation of any real number is explicitly presented. In particular, it is investigated whether this representation is unique or not. A condition is given that guarantees the uniqueness of decimal representation for a subset of real numbers, while for the remaining numbers two decimal representations exist. It is also investigated which sequences of digits cannot be accepted as decimal representations of real numbers. Moreover, analogous results are presented in the case where a real number is represented in systems with base n, n being a natural number greater than 1.

Representing real numbers in a generalized numeration system

We show how to represent an interval of real numbers in an abstract numeration system built on a language that is not necessarily regular. As an application, we consider representations of real numbers using the Dyck language. We also show that our framework can be applied to the rational base numeration systems.

What Bandwidth Do I Need for My Image?

ABSTRACTComputer representations of real numbers are necessarily discrete, with some finite resolution, discreteness, quantization, or minimum representable difference. We perform astrometric and photometric measurements on stars and co-add multiple observations of faint sources to demonstrate that essentially all of the scientific information in an optical astronomical image can be preserved or transmitted when the minimum representable difference is a factor of 2 finer than the root variance of the per-pixel noise. Adopting a representation this coarse reduces bandwidth for data acquisition, transmission, or storage, or permits better use of the system dynamic range, without sacrificing any information for downstream data analysis, including information on sources fainter than the minimum representable difference itself.

A method for building a real-time cluster-based continuous mental workload scale

Mental workload is known to significantly influence an operator's task performance. Extreme levels of mental workload can lead to operator monotony, low performance and operation errors. Measurement of mental workload is thus very important–especially real-time measurement that involves the representation of continuous real numbers. Existing mental workload measures include human subjects’ self-reporting, task performance and psychophysiological signals. Subjects’ self-reporting and task performance measures are used posterior–i.e. after the task is performed–and thus they cannot be used for real-time measurement of mental workload. Measures based on psychophysiological signals are suitable for real-time measurement and currently are usually represented as a set of discrete numbers–so-called levels. Mental workload is essentially a quantity of continuous numbers and defining it as a set of discrete levels can introduce unnecessary constraints on the accurate understanding of mental workload. In this paper, a novel methodology is proposed for constructing a measure of mental workload with a continuous real number representation. This methodology is based on the view that mental workload is task dependent and its quantitative representation of high or low mental workload should be dependent on training data. For the purpose of illustrating and validating this methodology, the electrocardiogram (ECG) signal was used. The proposed ECG-based measure was compared with the Rating Scale Mental Effort (RSME) method in a driving application. The result of the validation has shown that the proposed method is in good agreement with the RSME method; however, it is known that RSME cannot be done in real time and with a representation of a set of levels.

Beta-Expansions with Negative Bases

Abstract This paper investigates representations of real numbers with an arbitrary negative base –β < –1, which we call the (–β)-expansions. They arise from the orbits of the (–β)-transformation which is a natural modification of the β-transformation. We show some fundamental properties of (–β)-expansions, each of which corresponds to a well-known fact of ordinary β-expansions. In particular, we characterize the admissible sequences of (–β)-expansions, give a necessary and sufficient condition for the (–β)-shift to be sofic, and explicitly determine the invariant measure of the (–β)-transformations.

INTUITIONS OF "INFINITE NUMBERS": INFINITE MAGNITUDE VS. INFINITE REPRESENTATION

This study examines undergraduate students’ emerging conceptions of infinity as manifested in their engagement with geometric tasks. Students’ attempts to reduce the level of abstraction of infinity and properties of infinite quantities are described. Their arguments revealed they perceive infinity as an ongoing process, rather than a completed one, and fail to notice conflicting ideas. In particular, confusion between the infinite magnitude of points on a line segment and the infinite representation of real numbers was observed. Furthermore, students struggled to draw a connection between real numbers and their representation on a number line.

A 2-continued fraction representation of real numbers and its geometry

We study the geometry of representations of numbers by continued fractions whose elements belong to the set A 2 = {α1, α2} (A 2-continued fraction representation). It is shown that, for α1α2 ≤ 1/2 , every point of a certain segment admits an A 2-continued fraction representation. Moreover, for α1α2 = 1/2, this representation is unique with the exception of a countable set of points. For the last case, we find the basic metric relation and describe the metric properties of a set of numbers whose A 2-continued fraction representation does not contain a given combination of two elements. The properties of a random variable for which the elements of its A 2-continued fraction representation form a homogeneous Markov chain are also investigated.

Expansions in non-integer bases: Lower, middle and top orders

Let q ∈ ( 1 , 2 ) ; it is known that each x ∈ [ 0 , 1 / ( q − 1 ) ] has an expansion of the form x = ∑ n = 1 ∞ a n q − n with a n ∈ { 0 , 1 } . It was shown in [P. Erdős, I. Joó, V. Komornik, Characterization of the unique expansions 1 = ∑ i = 1 ∞ q − n i and related problems, Bull. Soc. Math. France 118 (1990) 377–390] that if q < ( 5 + 1 ) / 2 , then each x ∈ ( 0 , 1 / ( q − 1 ) ) has a continuum of such expansions; however, if q > ( 5 + 1 ) / 2 , then there exist infinitely many x having a unique expansion [P. Glendinning, N. Sidorov, Unique representations of real numbers in non-integer bases, Math. Res. Lett. 8 (2001) 535–543]. In the present paper we begin the study of parameters q for which there exists x having a fixed finite number m > 1 of expansions in base q. In particular, we show that if q < q 2 = 1.71 … , then each x has either 1 or infinitely many expansions, i.e., there are no such q in ( ( 5 + 1 ) / 2 , q 2 ) . On the other hand, for each m > 1 there exists γ m > 0 such that for any q ∈ ( 2 − γ m , 2 ) , there exists x which has exactly m expansions in base q.

A Possible Approach to Inclusion of Space and Time in Frame Fields of Quantum Representations of Real and Complex Numbers

This work is based on the field of reference frames based on quantum representations of real and complex numbers described in other work. Here frame domains are expanded to include space and time lattices. Strings of qukits are described as hybrid systems as they are both mathematical and physical systems. As mathematical systems they represent numbers. As physical systems in each frame the strings have a discrete Schrodinger dynamics on the lattices. The frame field has an iterative structure such that the contents of a stage j frame have images in a stage j − 1 (parent) frame. A discussion of parent frame images includes the proposal that points of stage j frame lattices have images as hybrid systems in parent frames. The resulting association of energy with images of lattice point locations, as hybrid systems states, is discussed. Representations and images of other physical systems in the different frames are also described.

Signed-bit representations of real numbers

The signed-bit representation of real numbers is like the binary representation, but in addition to 0 and 1 you can also use -1. It lends itself especially well to the constructive (intuitionistic) theory of the real numbers. The first part of the paper develops and studies the signed-bit equivalents of three common notions of a real number: Dedekind cuts, Cauchy sequences, and regular sequences. This theory is then applied to homomorphisms of Riesz spaces into the reals.

Coinductive Formal Reasoning in Exact Real Arithmetic

In this article we present a method for formally proving the correctness of the lazy algorithms for computing homographic and quadratic transformations -- of which field operations are special cases-- on a representation of real numbers by coinductive streams. The algorithms work on coinductive stream of M\"{o}bius maps and form the basis of the Edalat--Potts exact real arithmetic. We use the machinery of the Coq proof assistant for the coinductive types to present the formalisation. The formalised algorithms are only partially productive, i.e., they do not output provably infinite streams for all possible inputs. We show how to deal with this partiality in the presence of syntactic restrictions posed by the constructive type theory of Coq. Furthermore we show that the type theoretic techniques that we develop are compatible with the semantics of the algorithms as continuous maps on real numbers. The resulting Coq formalisation is available for public download.

Reducibility of domain representations and Cantor–Weihrauch domain representations

We introduce a notion of reducibility of representations of topological spaces and study some basic properties of this notion for domain representations.A representation reduces to another if its representing map factors through the other representation. Reductions form a pre-order on representations. A spectrum is a class of representations divided by the equivalence relation induced by reductions. We establish some basic properties of spectra, such as, non-triviality. Equivalent representations represent the same set of functions on the represented space.Within a class of representations, a representation is universal if all representations in the class reduce to it. We show that notions of admissibility, considered both for domains and within Weihrauch's TTE, are universality concepts in the appropriate spectra. Viewing TTE representations as domain representations, the reduction notion here is a natural generalisation of the one from TTE.To illustrate the framework, we consider some domain representations of real numbers and show that the usual interval domain representation, which is universal among dense representations, does not reduce to various Cantor domain representations. On the other hand, however, we show that a substructure of the interval domain more suitable for efficient computation of operations is equivalent to the usual interval domain with respect to reducibility.

Physical constraints on hypercomputation

Many attempts to transcend the fundamental limitations to computability implied by the Halting Problem for Turing Machines depend on the use of forms of hypercomputation that draw on notions of infinite or continuous, as opposed to bounded or discrete, computation. Thus, such schemes may include the deployment of actualised rather than potential infinities of physical resources, or of physical representations of real numbers to arbitrary precision. Here, we argue that such bases for hypercomputation are not materially realisable and so cannot constitute new forms of effective calculability.

On classification of singular measures and fractal properties of quasi-self-affine measures in R 2

A multidimensional classification of singularly continuous (w.r.t. the Lebesgue measure) probability measures in R2 is introduced and a theorem on canonical representation of such measures is proven. A class of random elements on the unit square which is defined by a system of partitions generated by the Q*-representation of real numbers is introduced and studied in details. Conditions for the discreteness, absolute continuity resp. singular continuity (w.r.t. Lebesgue measure) of the corresponding probability measures are found. Metric, topological and fractal properties of the spectra of the corresponding probability distributions are investigated. A class of transformations preserving the Hausdorff-Besicovitch dimension of any subset of the unit square (DP-transformations) is also studied.

Coinduction for Exact Real Number Computation

This paper studies coinductive representations of real numbers by signed digit streams and fast Cauchy sequences. It is shown how the associated coinductive principle can be used to give straightforward and easily implementable proofs of the equivalence of the two representations as well as the correctness of various corecursive exact real number algorithms. The basic framework is the classical theory of coinductive sets as greatest fixed points of monotone operators and hence is different from (though related to) the type theoretic approach by Ciaffaglione and Gianantonio.

Primitive recursive real numbers

AbstractIn mathematics, various representations of real numbers have been investigated. All these representations are mathematically equivalent because they lead to the same real structure – Dedekind‐complete ordered field. Even the effective versions of these representations are equivalent in the sense that they define the same notion of computable real numbers. Although the computable real numbers can be defined in various equivalent ways, if “computable” is replaced by “primitive recursive” (p. r., for short), these definitions lead to a number of different concepts, which we compare in this article. We summarize the known results and add new ones. In particular we show that there is a proper hierarchy among p. r. real numbers by nested interval representation, Cauchy representation, b ‐adic expansion representation, Dedekind cut representation, and continued fraction expansion representation. Our goal is to clarify systematically how the primitive recursiveness depends on the representations of the real numbers. (© 2007 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

Productivity of Edalat-Potts Exact Arithmetic in Constructive Type Theory

In this work we focus on a formalisation of the algorithms of lazy exact arithmetic a la Edalat-Potts in type theory. We choose the constructive type theory extended with coinductive types as our formal verification tool. We show examples of how infinite objects such as streams and expression trees can be formalised as coinductive types. We study the type theoretic notion of productivity which ensures the infiniteness of the outcome of the algorithms on infinite objects. Syntactical methods are not always strong enough to ensure the productivity. However, if some information about the complexity of a function is provided, one may be able to show the productivity of that function. In the case of the normalisation algorithm we show that such information can be obtained from the choice of real number representation that is used to represent the input and the output. Furthermore, we show how to employ this semantic information to formalise a restricted case of normalisation algorithm in our type theory.