On the Weak Computability of Continuous Real Functions

In computable analysis, sequences of rational numbers which effectively converge to a real number x are used as the (rho-) names of x. A real number x is computable if it has a computable name, and a real function f is computable if there is a Turing machine M which computes f in the sense that, M accepts any rho-name of x as input and outputs a rho-name of f(x) for any x in the domain of f. By weakening the effectiveness requirement of the convergence and classifying the converging speeds of rational sequences, several interesting classes of real numbers of weak computability have been introduced in literature, e.g., in addition to the class of computable real numbers (EC), we have the classes of semi-computable (SC), weakly computable (WC), divergence bounded computable (DBC) and computably approximable real numbers (CA). In this paper, we are interested in the weak computability of continuous real functions and try to introduce an analogous classification of weakly computable real functions. We present definitions of these functions by Turing machines as well as by sequences of rational polygons and prove these two definitions are not equivalent. Furthermore, we explore the properties of these functions, and among others, show their closure properties under arithmetic operations and composition

The descriptive set-theoretic complexity of the set of points of continuity of a multi-valued function (Extended Abstract)

In this article we treat a notion of continuity for a multi-valued function F and we compute the descriptive set-theoretic complexity of the set of all x for which F is continuous at x. We give conditions under which the latter set is either a G_\delta set or the countable union of G_\delta sets. Also we provide a counterexample which shows that the latter result is optimum under the same conditions. Moreover we prove that those conditions are necessary in order to obtain that the set of points of continuity of F is Borel i.e., we show that if we drop some of the previous conditions then there is a multi-valued function F whose graph is a Borel set and the set of points of continuity of F is not a Borel set. Finally we give some analogue results regarding a stronger notion of continuity for a multi-valued function. This article is motivated by a question of M. Ziegler in "Real Computation with Least Discrete Advice: A Complexity Theory of Nonuniform Computability with Applications to Linear Algebra", (submitted)

A domain-theoretic investigation of posets of sub-sigma-algebras (extended abstract)

Given a measurable space (X, M) there is a (Galois) connection between sub-sigma-algebras of M and equivalence relations on X. On the other hand equivalence relations on X are closely related to congruences on stochastic relations. In recent work, Doberkat has examined lattice properties of posets of congruences on a stochastic relation and motivated a domain-theoretic investigation of these ordered sets. Here we show that the posets of sub-sigma-algebras of a measurable space do not enjoy desired domain-theoretic properties and that our counterexamples can be applied to the set of smooth equivalence relations on an analytic space, thus giving a rather unsatisfactory answer to Doberkat's question

Effective Capacity and Randomness of Closed Sets

We investigate the connection between measure and capacity for the space of nonempty closed subsets of {0,1}*. For any computable measure, a computable capacity T may be defined by letting T(Q) be the measure of the family of closed sets which have nonempty intersection with Q. We prove an effective version of Choquet's capacity theorem by showing that every computable capacity may be obtained from a computable measure in this way. We establish conditions that characterize when the capacity of a random closed set equals zero or is >0. We construct for certain measures an effectively closed set with positive capacity and with Lebesgue measure zero

A hierarchy of Turing degrees of divergence bounded computable real numbers

AbstractA real number x is f-bounded computable (f-bc, for short) for a function f if there is a computable sequence (xs) of rational numbers which converges to x f-bounded effectively in the sense that, for any natural number n, the sequence (xs) has at most f(n) non-overlapping jumps of size larger than 2-n. f-bc reals are called divergence bounded computable if f is computable. In this paper we give a hierarchy theorem for Turing degrees of different classes of f-bc reals. More precisely, we will show that, for any computable functions f and g, if there exists a constant γ>1 such that, for any constant c, f(nγ)+n+c⩽g(n) holds for almost all n, then the classes of Turing degrees given by f-bc and g-bc reals are different. As a corollary this implies immediately the result of [R. Rettinger, X. Zheng, On the Turing degrees of the divergence bounded computable reals, in: CiE 2005, June 8–15, Amsterdam, The Netherlands, Lecture Notes in Computer Science, vol. 3526, 2005, Springer, Berlin, pp. 418–428.] that the classes of Turing degrees of d-c.e. reals and divergence bounded computable reals are different

Noncomputable functions in the Blum-Shub-Smale model

Working in the Blum-Shub-Smale model of computation on the real numbers, we answer several questions of Meer and Ziegler. First, we show that, for each natural number d, an oracle for the set of algebraic real numbers of degree at most d is insufficient to allow an oracle BSS-machine to decide membership in the set of algebraic numbers of degree d + 1. We add a number of further results on relative computability of these sets and their unions. Then we show that the halting problem for BSS-computation is not decidable below any countable oracle set, and give a more specific condition, related to the cardinalities of the sets, necessary for relative BSS-computability. Most of our results involve the technique of using as input a tuple of real numbers which is algebraically independent over both the parameters and the oracle of the machine

Computing the speed of convergence of ergodic averages and pseudorandom points in computable dynamical systems

A pseudorandom point in an ergodic dynamical system over a computable metric space is a point which is computable but its dynamics has the same statistical behavior as a typical point of the system. It was proved in [Avigad et al. 2010, Local stability of ergodic averages] that in a system whose dynamics is computable the ergodic averages of computable observables converge effectively. We give an alternative, simpler proof of this result. This implies that if also the invariant measure is computable then the pseudorandom points are a set which is dense (hence nonempty) on the support of the invariant measure

Point-Separable Classes of Simple Computable Planar Curves

In mathematics curves are typically defined as the images of continuous real functions (parametrizations) defined on a closed interval. They can also be defined as connected one-dimensional compact subsets of points. For simple curves of finite lengths, parametrizations can be further required to be injective or even length-normalized. All of these four approaches to curves are classically equivalent. In this paper we investigate four different versions of computable curves based on these four approaches. It turns out that they are all different, and hence, we get four different classes of computable curves. More interestingly, these four classes are even point-separable in the sense that the sets of points covered by computable curves of different versions are also different. However, if we consider only computable curves of computable lengths, then all four versions of computable curves become equivalent. This shows that the definition of computable curves is robust, at least for those of computable lengths. In addition, we show that the class of computable curves of computable lengths is point-separable from the other four classes of computable curves

Divergence bounded computable real numbers

AbstractA real x is called h-bounded computable, for some function h:N→N, if there is a computable sequence (xs) of rational numbers which converges to x such that, for any n∈N, at most h(n) non-overlapping pairs of its members are separated by a distance larger than 2-n. In this paper we discuss properties of h-bounded computable reals for various functions h. We will show a simple sufficient condition for a class of functions h such that the corresponding h-bounded computable reals form an algebraic field. A hierarchy theorem for h-bounded computable reals is also shown. Besides we compare semi-computability and weak computability with the h-bounded computability for special functions h

Impact of social influence in English proficiency and performance in English examinations of mathematics students from a Sino-US undergraduate education program

This study examines the influence of certain academic and demographic variables upon the academic performance of Chinese students enrolled in a cooperative Bachelor’s degree program in Pure and Applied Mathematics. The program is English taught and jointly organised by Jiangsu University, China and Arcadia University, USA. Data from a sample of 166 students is processed using inferential and path analysis, as well as mathematical modelling. As evidenced by the inferential and path analysis, no steady improvement in the English proficiency of students has been observed, while the latter has been found to be influenced by gender and to strongly influence academic performance in Mathematics courses. The effects of negative social influences are assessed via a qualitative analysis of the mathematical model. Threshold quantities similar to the basic reproduction number of mathematical epidemiology have been found to be stability triggers. Possible interventional measures are discussed based on these findings