On the real cycle class map for singular varieties

Computational Complexity of Real Functions

We propose a new framework for discussing computational complexity of problems involving uncountably many objects, such as real numbers, sets and functions, that can be represented only through approximation. The key idea is to use a certain class of string functions, which we call regular functions, as names representing these objects. These are more expressive than infinite sequences, which served as names in prior work that formulated complexity in more restricted settings. An important advantage of using regular functions is that we can define their size in the way inspired by higher-type complexity theory. This enables us to talk about computation on regular functions whose time or space is bounded polynomially in the input size, giving rise to more general analogues of the classes P, NP, and PSPACE. We also define NP- and PSPACE-completeness under suitable many-one reductions.Because our framework separates machine computation and semantics, it can be applied to problems on sets of interest in analysis once we specify a suitable representation (encoding). As prototype applications, we consider the complexity of functions (operators) on real numbers, real sets, and real functions. The latter two cannot be represented succinctly using existing approaches based on infinite sequences, so ours is the first treatment of them. As an interesting example, the task of numerical algorithms for solving the initial value problem of differential equations is naturally viewed as an operator taking real functions to real functions. As there was no complexity theory for operators, previous results could only state how complex the solution can be. We now reformulate them to show that the operator itself is polynomial-space complete.

We propose an extension of the framework for discussing the computational complexity of problems involving uncountably many objects, such as real numbers, sets and functions, that can be represented only through approximation. The key idea is to use a certain class of string functions as names representing these objects. These are more expressive than infinite sequences, which served as names in prior work that formulated complexity in more restricted settings. An advantage of using string functions is that we can define their size in a way inspired by higher-type complexity theory. This enables us to talk about computation on string functions whose time or space is bounded polynomially in the input size, giving rise to more general analogues of the classes P, NP, and PSPACE. We also define NP- and PSPACE-completeness under suitable many-one reductions. Because our framework separates machine computation and semantics, it can be applied to problems on sets of interest in analysis once we specify a suitable representation (encoding). As prototype applications, we consider the complexity of functions (operators) on real numbers, real sets, and real functions. For example, the task of numerical algorithms for solving a certain class of differential equations is naturally viewed as an operator taking real functions to real functions. As there was no complexity theory for operators, previous results only stated how complex the solution can be. We now reformulate them and show that the operator itself is polynomial-space complete.

Real functions and numbers defined by Turing machines

Computational complexity of real functions

We present a new approach to computability of real numbers in which real functions have type-1 representations, which also includes the ability to reason about the complexity of real numbers and functions. We discuss how this allows efficient implementations of exact real numbers and also present a new real number system that is based on it.

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.

CHAPTER 5 - FUNCTIONS OF A REAL VARIABLE

Turing machines, in particular, finite Mill automata can be considered as a means for giving corresponding subclasses of the class of real functions. From this point of view, finite Mili automata and generalized sequences of machines were considered in [I]. Here a real function f is defined by a finite automaton A in the following way. To determine the value f(x) of the automaton A one successively transforms the input object, the expansion of the real number x in the binary (decimal) system, into the output object, which is the binary (decimal) expansion of the number y, where y = f(x). Analogously for determining real functions one can use Turing machines. The subclass of the class of real functions arising here is actually a proper subclass of the class of C-computable functions, defined below. The latter is strictly included in the class of R-functions, or real functions, and that, in its own right, is contained in the class of V-functions. Actually, each continuous real functions turns out to be a real function. V-functions are given by indeterminate R-transformers, transforming binary representations of real numbers. R-transformers with finite memory, defining real functions are said to be finite reals. The latter are richer than Mill automata, defining real functions according to [I], in two respects: they can be asynchronous and can define functions which are not strictly real. For example, the function f(x) = 3x is defined by a finitely real, but is not strictly real. For finite reals, on the basis of [2] the solvability of the equivalence problem and of some other algorithmic questions is proved. This generalizes the corresponding results found in [i] for finite Mili automata and generalized succcession machines preserving length. The solvability is also established of the equivalence problem for single-valued indeterminate finite reals with finitely turning accumulators. An algebraic characterization is given of the class of partial C-computable functions. We note that in the papers of Red'ko [3, 4] the problem of describing the algebra of computable (partially computable) functions of a real variable was formulated as open. Bui effected the construction of the algebra of computable operations on the set of rational numbers [5].

The spectrum of a first-order sentence is the set of natural numbers occurring as the cardinalities of finite models of the sentence. In a recent survey, Durand et al. introduce a new class of real numbers, the spectral reals, induced by spectra and pose two open problems associated to this class. In the present note, we answer these open problems as well as other open problems from an earlier, unpublished version of the survey. Specifically, we prove that (i) every algebraic real is spectral, (ii) every automatic real is spectral, (iii) the subword density of a spectral real is either 0 or 1, and both may occur, and (iv) every right-computable real number between 0 and 1 occurs as the subword entropy of a spectral real. In addition, Durand et al. note that the set of spectral reals is not closed under addition or multiplication. We extend this result by showing that the class of spectral reals is not closed under any computable operation satisfying some mild conditions.

We extend exact deterministic remote state preparation (RSP) with minimal classical communication to quantum systems of continuous variables. We show that, in principle, it is possible to remotely prepare states of an ensemble that is parametrized by infinitely many real numbers---i.e., by a real function---while the classical communication cost is one real number only. We demonstrate continuous variable RSP in three examples using (i) quadrature measurement and phase-space displacement operations, (ii) measurement of the optical phase and unitaries shifting the same, and (iii) photon counting and photon number shift.

CHAPTER II - Functions

Computation properties of spatial dynamics simulation by probabilistic cellular automata

CBV functions are computable real functions of bounded variation. In this paper we investigate the basic properties of CBV functions. We are especially interested in the question of whether a real number class is closed under CBV functions. The real number classes considered here include the classes of computable (EC), semi-computable (SC), weakly computable (WC), divergence bounded computable (DBC) and recursively approximable (RA) real numbers. We show that the classes EC, RA and DBC are closed under CBV functions but SC and WC are not. Furthermore, WC$ is not even closed under computable monotone functions and, finally, the image sets of $\wc$ under computable monotone functions and CBV functions are different.

I mplicit functions arise throughout mathematical analysis. They are especially prominent in applications to economics. From the beginning of mathematical economics one must work with implicit functions, as in analysis of a consumer and his indifference curves, or of a firm and its isoquants. Consequently, in teaching calculus to business students we come early to the challenge of explaining differentiation of implicit functions. The aim of this note is to give a systematic approach to computing the successive derivatives of an implicit function of one real variable, an approach that we think brings students naturally to the main results. The idea is that smooth curves looked at closely enough are straight, so that analysis problems are locally problems of linear algebra. Our method invokes that intuition: we look at curves through (virtual) microscopes—but we pay for this by having to deal with hyperreal numbers. Although Leibniz and Newton, for instance, already worked with ‘‘infinitesimals’’, the rigorous treatment called ‘‘nonstandard analysis’’ was introduced only in 1961 by A. Robinson [4]. An especially simple presentation for didactical purposes was given by Keisler [3]; we shall mostly adopt his definitions and notations. We recall that the hyperreal numbers extend the real ones with the same algebraic rules; technically, the set *R of the hyperreal numbers is a non-archimedean ordered field in which the real line R is embedded. Moreover, *R contains at least one infinitesimal (and then it must contain infinitely many). An infinitesimal is a number e such that its absolute value is less thanevery real number butwhich is unequal to 0; its reciprocal 1e is infinite, that is, is a number whose absolute value is greater than every real number. Clearly, non-zero infinitesimals and infinite numbers are not real. A hyperreal number x, which is not infinite, is of course said to be finite; for any such x there exists one and only one real number r, which is infinitely close to x, that is, such that the difference x r is an infinitesimal: this r is called the standard part of x and is denoted by r = st(x). Formally, st is a ring homomorphism from the set of finite hyperreal numbers to R; and its kernel is the set of infinitesimals. Moreover, a function of one or several real variables may have a natural extension to the hyperreal numbers, with the same definition and the same properties as the original one. Indeed, if a real-valued function is defined by a system of formulas, its extension can be obtained by applying the same formulas to the hyperreal system. In this article, we adopt the same notation for a real function and for its natural extension. The concept of (virtual) microscope is well known (see, for example, [1, 2, 5]). For a point P(a, b) in the hyperreal plane *R and a positive infinite hyperreal number H, a microscope pointed on P and with H as power magnifies the distances from P by a factor H; more explicitly, it is a map, denoted by MPH ; defined on *R as follows: MPH : x; y ð Þ 7! X ;Y ð Þ with X 1⁄4 H x a ð Þ and Y 1⁄4 H y b ð Þ: