representations versus numberings on the relationship of two computability notions abstract this paper gives an answer to weihrauch s computability springer berlin 1987 question whether and if not always when an effective map between the computable elements of two represented sets can be extended to a partial computable map between the represented sets examples are known showing that this is not possible in general a condition is introduced and for countably based topological t 0 spaces it is shown that exactly the partial effective maps meeting the requirement are extendable for total effective maps the extra condition is satisfied in the standard cases of effectively given separable metric spaces and continuous directed complete partial orders in which the extendability is already known in the first case a similar result holds also for partial effective maps but not in the second

integrable symplectic maps abstract in this paper we first give a terse survey of symplectic maps their canonical formulation and integrability then we introduce a rigorous procedure to construct integrable symplectic maps starting from integrable evolution equations on lattices a number of illustrative examples are provided

parametrised second order complexity theory with applications to the study of interval computation abstract we extend the framework for complexity of operators in analysis devised by kawamura and cook 2012 to allow for the treatment of a wider class of representations the main novelty is to endow represented spaces of interest with an additional function on names called a parameter which measures the complexity of a given name this parameter generalises the size function which is usually used in second order complexity theory and therefore also central to the framework of kawamura and cook the complexity of an algorithm is measured in terms of its running time as a second order function in the parameter as well as in terms of how much it increases the complexity of a given name as measured by the parameters on the input and output side as an application we develop a rigorous computational complexity theory for interval computation in the framework of kawamura and cook the representation of real numbers based on nested interval enclosures does not yield a reasonable complexity theory in our new framework this representation is polytime equivalent to the usual cauchy representation based on dyadic rational approximation by contrast the representation of continuous real functions based on interval enclosures is strictly smaller in the polytime reducibility lattice than the usual representation which encodes a modulus of continuity furthermore the function space representation based on interval enclosures is optimal in the sense that it contains the minimal amount of information amongst those representations which render evaluation polytime computable

multifractality in forgetful memories learning rules of a forgetful memory generate their synaptic efficacies through iterative procedures that operate on the input data random patterns we analyse invariant distributions of the synaptic couplings as they arise asymptotically and show that they exhibit fractal or multifractal properties we also discuss their dependence upon the learning rule and the parameters specifying it and indicate how the nature of the invariant distribution is related to the network performance

effectiveness of the global modulus of continuity on metric spaces abstract in the e definition of continuity of a function between metric spaces the value of depends on e and function as well this kind of dependence can be described by a function so called global modulus of continuity which maps the triple e to corresponding by a recent result of repovs and semenov proc int conf topol trieste 1993 g gentili ed rent ist mat univ trieste vol 25 1993 pp 441 446 there is a continuous global modulus of continuity for the function space c x y of all continuous functions from a locally compact metric space x to an arbitrary metric space y based on weihrauch s framework on computable metric spaces k weihrauch theoret comput sci 113 1993 191 210 we show that there is a computable global modulus of continuity for c x y if x is an effectively locally compact metric space and y is a computable metric space the proof is a direct construction not depending on the proof of repovs and semenov

a computational model for metric spaces for every metric space x we define a continuous poset bx such that x is homeomorphic to the set of maximal elements of bx with the relative scott topology the poset bx is a dcpo iff x is complete and continuous iff x is separable the computational model bx is used to give domain theoretic proofs of banach s fixed point theorem and of two classical results of hutchinson on a complete metric space every hyperbolic iterated function system has a unique non empty compact attractor and every iterated function system with probabilities has a unique invariant measure with bounded support we also show that the probabilistic power domain of bx provides an continuous computational model for measure theory on a separable complete metric space x

computability on the probability measures on the borel sets of the unit interval while computability theory on many countable sets is well established and for computability on the real numbers several mutually non equivalent definitions are applied for most other uncountable sets in particular for measures no generally accepted computability concepts at all ha ve been available until now in this contribution we introduce computability on the set m of probability measures on the borel subsets of the unit interval 0 1 its main purpose is to demonstrate that this concept of computability is not merely an ad hoc definition but has very natural properties although the definitions and many results can of course be transferred to more general spaces of measures we restrict our attention to m in order to keep the technical details simple and concentrate on the central ideas in particular we show that simple obvious reqirements exclude a number of similar definitions that the definition leads to the expected computability results that there are other natural definitions inducing the same computability theory and that the theory is embedded smoothly into classical measure theory as background we consider tte type 2 theory of effectivity kw84 kw85 which provides a frame for very realistic computability definitions in this approach computability is defined on finite and infinite sequences of symbols explicitly by turing machines and on other sets by means of notations and representations canonical representations are derived from information structures wei97 we introduce a standard representation delta m subseteq sum omega to m via some natural information structure defined by a subbase the atomic properties of some topology on m and a standard notation of while several modifications of m suggesting themselves at first glance violate simple and obvious requirements m has several very natural properties and hence should induce an important computability theory many interesting functions on measures turn out to be computable in particular linear combination integration of continuous functions and any transformation defined by a computable iterated function system with probabilities some other natural representations of m are introduced among them a cauchy representation associated with the hutchinson metric and proved to be equivalent to m as a corollary the final topology of m is the well known weak topology on m

dynamical systems measures and fractals via domain theory abstract we introduce domain theory in dynamical systems iterated function systems fractals and measure theory for a discrete dynamical system given by the action of a continuous map f x x on a metric space x we study the extended dynamical systems vx vf ux uf and lx lf where v u and l are respectively the vietoris hyperspace the upper hyperspace and the lower hyperspace functors we show that if x f is chaotic then so is ux uf when x is locally compact ux is a continuous bounded complete dcpo if x is second countable as well then ux will be continuous and can be given an effective structure we show how strange attractors attractors of iterated function systems fractals and julia sets are obtained effectively as fixed points of deterministic functions on ux or fixed points of non deterministic functions on cux where c is the convex plotkin power domain we also show that the set m x of finite borel measures on x can be embedded in pux where p is the probabilistic power domain this provides an effective framework for measure theory we then prove that the invariant measure of an hyperbolic iterated function system with probabilities can he obtained as the unique fixed point of an associated continuous function on pux

approximations of fractal sets abstract five approximations of fractal sets a are compared the first three were described by hutchinson barnsley demko and williams respectively two new other approximations are still efficient in all circumstances for any algorithm error estimates in the approximations are found numerical comparisons are also done through classical examples of attractors in the plane

recursion and topology on 2 ω for possibly infinite computations in the context of possibly infinite computations yielding finite or infinite binary outputs the space 2 2 u 2 appears to be one of the most fundamental spaces in computer science though underconsidered next to 2 this space can be viewed section 3 5 2 as the simplest compact space native to computer science in this paper we study some of its properties involving topology and computability though 2 can be considered as a computable metric space in the sense of computable analysis a direct and self contained study based on its peculiar properties related to words is much illuminating it is well known that computability for maps 2 2 reduces to continuity with recursive modulus of continuity with 2 things get less simple maps 2 2 or 2 2 induced by input output behaviours of turing machines on finite or infinite words which we call semicomputable maps are not necessarily continuous but merely lower semicontinuous with respect to the prefix partial ordering on 2 continuity asks for a stronger notion of computability we prove for semi continuous and semi computable maps f f o with f o 2 2 a detailed representation theorem theorem 81 via functions f 2 2 following two approaches bottom up from f to f and top down from f to f

effective domain representations of h x the space of compact subsets this paper gives effective domain representations of spaces h x of non empty compact subsets of effective complete metric spaces x the domain representation of h x is constructed from a domain representation of x using the plotkin power domain construction as an application of the representation an effective version of a fundamental theorem on ifs iterated function system is shown

type 2 computational complexity of functions on cantor s space abstract continuity and computability on cantor s space c has turned out to be a very natural basis for a type 2 theory of effectivity tte in particular the investigation of type 2 computational complexity e g complexity in analysis requires the study of oracle machines which w l g operate on cantor s space however no general type 2 complexity theory has been developed so far in this paper we lay a foundation for this theory by investigating computational complexity of functions c 0 1 and c c and the related concepts of dependence and input lookahead in the former case results of gordon and shamir are extended and generalized for continuous functions c c the relation between the three concepts is studied in detail compact sets are proved to be natural domains of resource bounded functions on c finally we demonstrate that an optimization of the input lookahead used by a machine may result in a rapid increase of computation time

computability theory of generalized functions the theory of generalized functions is the foundation of the modern theory of partial differential equations pde as computers are playing an ever larger role in solving pdes it is important to know those operations involving generalized functions in analysis and pde that can be computed on digital computers in this article we introduce natural concepts of computability on test functions and generalized functions as well as computability on schwartz test functions and tempered distributions type 2 turing machines are used as the machine model weihrauch 2000 it is shown here that differentiation and integration on distributions are computable operators and various types of fourier transforms and convolutions are also computable operators as an application it is shown that the solution operator of the distributional inhomogeneous three dimensional wave equation is computable

admissible representations of effective cpo s abstract the elements of a cpo complete partial order d are abstract objects in general a concrete machine cannot operate with abstract objects but only with names of objects in this paper the set f of total functions from n to n is suggested as the set of names admissible representations f d for effective cpo s are defined by two axioms which generalize the axioms for godel numberings of the partial recursive functions topological and recursion theoretic considerations show that the definition is very natural it is also proved that p the set of subsets of n is not as suitable as a set of names finally it is proved that for admissible representations the computable functions over cpo s are represented by the computable extensional functions over the names

a computable version of the daniell stone theorem on integration and linear functionals for every measure the integral i f f d is a linear functional on the set of real measurable functions by the daniell stone theorem for every abstract integral f r on a stone vector lattice f of real functions f r there is a measure such that f d f for all f f in this paper we prove a computable version of this theorem

computability on subsets of euclidean space i closed and compact subsets in this paper we introduce and compare computability concepts on the set of closed subsets of euclidean space we use the language and framework of type 2 theory of effectivity tte which supplies a concise language for distinguishing a variety of effectivity properties and which admits highly effective versions of classical theorems in particular type 2 theory of effectivity allows to separate topological from computational aspects of effectivity we consider three different computability concepts on the set of closed subsets each of which is characterized by several representations which are proved to be equivalent the three induced types of computable closed sets have already been considered by many authors however under different and partly inconsistent names our characterizations show that they can be regarded as straightforward generalizations of the r e co r e and recursive subsets of natural numbers therefore we suggest to call them the recursively enumerable the co recursively enumerable and the recursive closed subsets of euclidean space open subsets obtain the dual names we extend the investigation by introducing several natural representations of the compact subsets of euclidean space and proving equivalences the paper extends and generalizes earlier definitions adds new ones and compares them in a single framework the resultant canonical computability concepts induce computability of objects as well as computability of operators on the space of closed and compact subsets

a new integrable symplectic map associated with lattice soliton equations a method is developed that extends the nonlinearization technique to the hierarchy of lattice soliton equations associated with a discrete 3 3 matrix spectral problem a new integrable symplectic map and its involutive system of conserved integrals are obtained by the nonlinearization of spatial parts and the time parts of lax pairs and their adjoint lax pairs of the hierarchy moreover the solutions of the typical system of lattice equations in the hierarchy are reduced to the solutions of a system of ordinary differential equations and a simple iterative process of the symplectic map

on the computability of walsh functions the haar and the walsh functions are proved to be computable with respect to the fine metric df which is induced from the infinite product 0 1 1 2 with the weighted product metric dc of the discrete metric on 0 1 although they are discontinuous functions on 0 1 with respect to the euclidean metric they are continuous functions on dc and on 0 1 df on dc computable real valued cylinder functions which include the walsh functions become computable and every computable function can be approximated effectively by a computable sequence of cylinder functions the metric space 0 1 df is separable but not complete nor effectively complete we say that a function on 0 1 is uniformly fine computable if it is sequentially computable and effectively uniformly continuous with respect to the metric df it is proved that a uniformly fine computable function is essentially a computable function on it is also proved that walsh fourier coefficients of a uniformly fine computable function f form a computable sequence of reals and there exists a subsequence of the walsh fourier series which fine converges effectively uniformly to f

equality is a jump abstract we define a notion of degree of unsolvability for subsets of r n where r is a real closed archimedean field and prove that in contrast to type 2 computability the presence of exact equality in the bss model forces exactly one jump of the unsolvability degree of decidable sets

discrete versions of some classical integrable systems and factorization of matrix polynomials discrete versions of several classical integrable systems are investigated such as a discrete analogue of the higher dimensional force free spinning top euler arnold equations the heisenberg chain with classical spins and a new discrete system on the stiefel manifold the integrability is shown with the help of a lax pair representation which is found via a factorization of certain matrix polynomials the complete description of the dynamics is given in terms of abelian functions the flow becomes linear on a prym variety corresponding to a spectral curve the approach is also applied to the billiard problem in the interior of ann dimensional ellipsoid

effective symbolic dynamics random points statistical behavior complexity and entropy we consider the dynamical behavior of martin lof random points in dynamical systems over metric spaces with a computable dynamics and a computable invariant measure we use computable partitions to define a sort of effective symbolic model for the dynamics through this construction we prove that such points have typical statistical behavior the behavior which is typical in the birkhoff ergodic theorem and are recurrent we introduce and compare some notions of complexity for orbits in dynamical systems and prove i that the complexity of the orbits of random points equals the kolmogorov sinai entropy of the system ii that the supremum of the complexity of orbits equals the topological entropy

computability of probability measures and martin lof randomness over metric spaces in this paper we investigate algorithmic randomness on more general spaces than the cantor space namely computable metric spaces to do this we first develop a unified framework allowing computations with probability measures we show that any computable metric space with a computable probability measure is isomorphic to the cantor space in a computable and measure theoretic sense we show that any computable metric space admits a universal uniform randomness test without further assumption

one dimensional markovian field ising model physical properties and characteristics of the discrete stochastic mapping the zero temperature physical properties of an ising chain in a markovian field taking only two values with non zero mean are evaluated and a related discrete stochastic mapping with the theory of finite markov chains is investigated a discontinuous behaviour of the magnetisation and the residual entropy dependent on both mean field and exchange is found which can be related to flips of microscopic spin clusters for non zero temperature the mapping is characterised by the fractal properties of its attractor and by the lyapunov exponent an explicit expression for the measure in the nth iteration of the chapman kolmogorov equation is obtained

generalized zalcman conjecture for starlike and typically real functions abstract zalcman conjectured that a2n a2n 1 n 1 2 n 2 3 forf z z a2z2 a3z3 s the class of normalized holomorphic and univalent functionsf z in the unit disk d we propose a generalized conjecture as follows forf z z a2z2 a3z3 s anam an m 1 n 1 m 1 n m 2 3 we prove that this conjecture holds for starlike functions and univalent functions with real coefficients we also obtain sharp bounds on anam an m 1 for typically real functions

computability on random events and variables in a computable probability space we study aspects of computability concerning random events and variables in a computable probability space which fulfills certain computability axioms to this end we introduce two multirepresentations of random events and random variables respectively employing the frechet nikodym metric and the ky fan metric they are shown to be recursively complete in guaranteeing computability of basic operations on random events and random variables some natural variations of the multirepresentation of random variables are defined for the integrable variables to explore computability of integration

towards a descriptive set theory for domain like structures this is a survey of results in descriptive set theory for domains and similar spaces with the emphasis on the algebraic domains we try to demonstrate that the subject is interesting in its own right and is closely related to some areas of theoretical computer science since the subject is still in its beginning we discuss in detail several open questions and possible future development we also mention some relevant facts of effective descriptive set theory

the zalcman conjecture for certain analytic and univalent functions abstract let a denote the class of analytic functions defined for z d given by f z z n 2 a n z n and let s denote the subclass of a consisting of univalent i e one to one functions in 1960s l zalcman conjectured that if f s then a n 2 a 2 n 1 n 1 2 for n 2 which implies the famous bieberbach conjecture a n n for n 2 for f s ma 19 proposed a generalized zalcman conjecture a n a m a n m 1 n 1 m 1 for n 2 m 2 let u be the class of functions f a satisfying f z z f z 2 1 1 for z d and c r denote the class of functions f a satisfying re 1 z 2 f z 0 for z d in the present paper we prove the zalcman conjecture and the generalized zalcman conjecture for the class u using extreme point theory we also prove the generalized zalcman conjecture for the class c r for the initial coefficients

iterated function systems and the global construction of fractals iterated function systems i f ss are introduced as a unified way of generating a broad class of fractals these fractals are often attractors for i f ss and occur as the supports of probability measures associated with functional equations the existence of certain p balanced measures for i f ss is established and these measures are uniquely characterized for hyperbolic i f ss the hausdorff besicovitch dimension for some attrac tors of hyperbolic i f ss is estimated with the aid of p balanced measures what appears to be the broadest framework for the exactly computable moment theory of p balanced measures that of linear i f ss and of probabilistic mixtures of iterated riemann surfaces is presented this extensively generalizes earlier work on orthogonal polynomials on julia sets an example is given of fractal reconstruction with the use of linear i f ss and moment theory

the local structure of poisson manifolds varietes de poisson et applications decomposition structures de poisson lineaires approximation lineaire systemes hamiltoniens le probleme de linearisation groupes de fonction realisations et applications impulsion paires duales et groupes de jauge existence des realisations unicite des realisations les problemes des 3 corps restreints et autres exemples

computation by while programs on topological partial algebras the language of while programs is a fundamental model for imperative programming on any data type it leads to a generalisation of the theory of computable functions on the natural numbers to the theory of computable functions on any many sorted algebra the language is used to express many algorithms in scientific computing where while programs are applied to continuous data in the theory of data continuous data types are modelled by topological many sorted algebras we study both exact and approximate computations by while programs and while programs with arrays over topological many sorted algebras with partial operations first we establish that partial operations are necessary in order to compute a wide range of continuous functions we prove basic continuity properties of our abstract computability any partial function computable over a partial topological algebra by a while array program is continuous any set semicomputable or computable over a partial topological algebra by a while array program is open or clopen respectively secondly we contrast exact and approximate computations the class of functions that can be computed exactly can be quite limited we show that on connected total algebras the while and while array computable functions are precisely those that are explicitly definable by terms we show that for certain general classes of topological algebras the total functions that can be approximated by while programs are precisely those that can be effectively approximated by terms this property we call generalised weierstrass approximation an application of this result is that a function on the set r of reals is computable in the sense of computable analysis if and only if it is while program approximable on a simple algebra based on r

on the relation of the stationary toda equation and the symplectic maps it is proved that every solution of the stationary toda equation in the generic case is represented as a discrete orbit of the symplectic map obtained through nonlinearization of the toda eigenvalue problem as an application the calculation of the finite band solution of the toda lattice equation is reduced to the solution of a system of odes plus a simple iterative process of the symplectic map

embedding metric spaces into cpo s abstract we show that metric spaces with polish topology mp can be represented very well by bounded complete algebraic cpo s with continuous weight and distance cwd an operator cwd mp is defined such that 1 and 2 hold 1 1 d can be embedded topologically into d such that a partial function on d is metrically continuous if it can be extended to a total continuous function on d 2 2 every m mp is isometric to some d therefore the continuous cpo functions are exactly adequate for describing the continuous functions on the corresponding metric spaces the approach could be refined in order to study computability on metric spaces with polish topology examples of application are the euclidean space the space of total functions on the natural numbers baire space and spaces of infinite trees

from axiomatics to intrinsic characterization some open problems in computable analysis abstract five open problems are presented the framework for these problems is the notion of a computability structure on a banach space which is discussed below the study of banach spaces in particular the study of hilbert space is an important topic in mathematics and its applications problems a and b are concerned with transfer principles for obtaining effective versions of classical theorems of mathematics problems c and d deal with separability and effective separability problem e is concerned with computability for classes of mathematical structures other than banach spaces

domain representations of topological spaces a domain representation of a topological space x is a function usually a quotient map from a subset of a domain onto x several dierent classes of domain representations are introduced and studied it is investigated when it is possible to build domain representations from existing ones it is for example discussed whether there exists a natural way to build a domain representation of a product of topological spaces from given domain representations of the factors it is shown that any t0 topological space has a domain representation these domain representations are very large however smaller domain representations are also constructed for large classes of spaces for example each second countable regular hausdor space has a domain representation with a countable base domain representations of functions and function spaces are also studied c 2000 elsevier science b v all rights reserved

concrete models of computation for topological algebras a concrete model of computation for a topological algebra is based on a representation of the algebra made from functions on the natural numbers the functions computable in a concrete model are computable in the representation in the classical sense of the chruch turing thesis moreover the functions turn out to be continuous in the topology of the algebra in this paper we consider different concrete models for computing in topological algebras and prove their mutual equivalence in certain commonly occurring circumstances for topological algebras the concrete models we use are effective representation by algebraic domains stoltenberg hansen and tucker effective representation by continuous domains edelat effective representation by type two recursion on baire space weihrauch and for metric and normed algebras we use effective metric spaces moschovakis and computability structures pour el and richards the result are evidence that computability theory for topological algebras is a stable theory independent of the specific models of computation just as classical computability theory for discrete algebras is stable

random elements in effective topological spaces with measure following a suggestion of zvonkin and levin we generalize martin lof s definition of infinite random sequences over a finite alphabet via randomness tests to effective topological spaces with a measure we show that under weak computability conditions there is a universal randomness test we prove a theorem on randomness preserving functions which corrects and extends a result by schnorr and apply it to a number of examples in particular we show that a real number is random if and only if it has a random b ary representation for any b 2 we show that many computable continuously differentiable real functions preserve randomness especially all computable analytic functions which are not constant on any open subset of their domain preserve randomness finally we introduce a new randomness concept for subsets of natural numbers which we characterize in terms of random sequences surprisingly it turns out that there are infinite co r e random sets

on the empirical validity of the bayesian method the ideas in kolmogorov s programme for algorithmc substantiation of applications of probability make it possible to define a measure of disagreement between the probability distribution representing the attitude of a coherent individual towards a random experiment and the outcome of the experiment when there is agreement we say that the probability distribution is empirically valid we prove quantitatively that formulae of bayesian statistics transform empirically valid probability distributions into other empirically valid distributions

the nonlinearized eigenvalue problem of the toda hierarchy in the lie poisson framework abstract a 3 3 discrete eigenvalue problem associated with toda hierarchy is presented after the nonlinearization procedure the 3 3 discrete eigenvalue problem is turned into an integrable poisson map on the poisson manifold r 3 n with a lie poisson structure as a reduction of the lie poisson structure on the co adjoint orbit the standard symplectic structure on the symplectic manifold r 2 n is obtained the poisson map restricted on the leaves of the symplectic foliation is reduced to a usual symplectic map which is exactly the nonlinearized 2 2 eigenvalue problem

a constructive borel cantelli lemma constructing orbits with required statistical properties in the general context of computable metric spaces and computable measures we prove a kind of constructive borel cantelli lemma given a sequence constructive in some way of sets a i with effectively summable measures there are computable points which are not contained in infinitely many a i as a consequence of this we obtain the existence of computable points which follow the typical statistical behavior of a dynamical system they satisfy the birkhoff theorem for a large class of systems having computable invariant measure and a certain logarithmic speed of convergence of birkhoff averages over lipschitz observables this is applied to uniformly hyperbolic systems piecewise expanding maps systems on the interval with an indifferent fixed point and it directly implies the existence of computable numbers which are normal with respect to any base

computational complexity of real functions in this chapter we introduce the formal definitions of computational complexity of real functions we first review the definitions of computable real numbers and computable real functions and their basic properties then the oracle turing machine is introduced as the formal model for computing real functions this model allows us to define the complexity measures for computing real functions in a natural way the class of polynomial time computable real functions is then defined and several characterizations of this class will be given other complexity classes of real numbers and real functions such as np real functions and log space computable real functions will be defined in later chapters

uniform test of algorithmic randomness over a general space the algorithmic theory of randomness is well developed when the underlying space is the set of finite or infinite sequences and the underlying probability distribution is the uniform distribution or a computable distribution these restrictions seem artificial some progress has been made to extend the theory to arbitrary bernoulli distributions by martin lof and to arbitrary distributions by levin we recall the main ideas and problems of levin s theory and report further progress in the same framework the issues are the following allow non compact spaces like the space of continuous functions underlying the brownian motion the uniform test deficiency of randomness dp x depending both on the outcome x and the measure p should be defined in a general and natural way see which of the old results survive existence of universal tests conservation of randomness expression of tests in terms of description complexity existence of a universal measure expression of mutual information as deficiency of independence the negative of the new randomness test is shown to be a generalization of complexity in continuous spaces we show that the addition theorem survives the paper s main contribution is introducing an appropriate framework for studying these questions and related ones like statistics for a general family of distributions

theory of representations abstract an approach for a simple general and unified theory of effectivity on sets with cardinality not greater than that of the continuum is presented a standard theory of effectivity on f f n n has been developed in a previous paper by representations f m this theory is extended to other sets m topological and recursion theoretical properties of representations are studied where the final topology of a representation plays an essential role it is shown that for any separable t0 space an up to equivalence unique admissible representation can be defined which reflects the topological properties correctly

recursive quasi metric spaces in computable analysis recursive metric spaces play an important role since these are roughly speaking spaces with computable metric and limit operation unfortunately the concept of a metric space is not powerful enough to capture all interesting phenomena which occur in computable analysis some computable objects are naturally considered as elements of asymmetric spaces which are not metrizable nevertheless most of these spaces are t0 spaces with countable bases and thus at least quasi metrizable we introduce a definition of recursive quasi metric spaces in analogy to recursive metric spaces we show that this concept leads to similar results as in the metric case and we prove that the most important spaces of computable analysis can be naturally considered as recursive quasi metric spaces especially we discuss some hyper and function spaces

topological properties of real number representations we prove three results about representations of real numbers or elements of other topological spaces by infinite strings such representations are useful for the description of real number computations performed by digital computers or by turing machines first we show that the so called admissible representations a topologically natural class of representations introduced by kreitz and weihrauch are essentially the continuous extensions with a well behaved domain of continuous and open representations second we show that there is no admissible representation of the real numbers such that every real number has only finitely many names third we show that a rather interesting property of admissible real number representations holds true also for a certain non admissible representation namely for the naive cauchy representation the property that continuity is equivalent to relative continuity with respect to the representation

computable banach spaces via domain theory this paper extends the order theoretic approach to computable analysis via continuous domains to complete metric spaces and banach spaces we employ the domain of formal balls to define a computability theory for complete metric spaces for banach spaces the domain specialises to the domain of closed balls ordered by reversed inclusion we characterise computable linear operators as those which map computable sequences to computable sequences and are effectively bounded we show that the domain theoretic computability theory is equivalent to the well established approach by pour el and richards

computability on continuous lower semi continuous and upper semi continuous real functions in this paper we extend computability theory to the spaces of continuous upper and lower semi continuous real functions we apply the framework of tte type 2 theory of effectivity where not only computable objects but also computable functions on the spaces can be considered first some basic facts about tte are summarized for each of the function spaces we introduce several natural representations based on different intiuitive concepts of effectivity and prove their equivalence computability of several operations on the function spaces is investigated among others limits mappings to open sets images of compact sets and preimages of open sets maximum and minimum values the positive results usually show computability in all arguments negative results usually express non continuity several of the problems have computable but not extensional solutions since computable functions map computable elements to computable elements many previously known results on computability are obtained as simple corollaries

finite dimensional discrete systems and integrable systems through nonlinearization of the discrete eigenvalue problem a method is developed that extends the nonlinearization procedure of eigenvalue problems to the discrete case from which the finite dimensional discrete systems and finite dimensional completely integrable systems associated with the toda hierarchy are obtained a new model for solving discrete soliton equations is proposed which enables one to get the solutions of the toda lattice equations by solving the compatible systems of discrete equations and nonlinear ordinary differential equations

extended admissibility we give a new definition of admissible representations which allows to handle also non countably based topological spaces in the framework of type 2 theory of effectivity we prove that admissible representations x y of topological spaces x y have the desirable property that every partial function f x y is continuously realizable with respect to x y if and only if f is sequentially continuous furthermore we characterize the class of the spaces having an admissible representation many interesting operators creating new topological spaces from old ones are shown to preserve the property of having an admissible representation in particular the class of sequential spaces with admissible representations turns out to be cartesian closed thus a reasonable computability theory is possible on important non countably based spaces

generalized zalcman conjecture for some classes of analytic functions abstract for functions f z z a 2 z 2 a 3 z 3 in various subclasses of normalized analytic functions we consider the problem of estimating the generalized zalcman coefficient functional f n m a n a m a n m 1 for all real parameters and 1 we provide the sharp upper bound of f n m for functions f satisfying re f z and hence settle the open problem of estimating f n m recently proposed by agrawal and sahoo 2016 1 for all real values of the estimations of f n m are provided for starlike and convex functions of order 1 which are sharp for 0 or for certain positive values of moreover for certain positive the sharp estimation of f n m is given when f is a typically real function or a univalent function with real coefficients or is in some subclasses of close to convex functions

effective properties of sets and functions in metric spaces with computability structure we consider an abstract metric space with a computability structure and an effective separating set in this article we also introduce an effectively compact space the computability of real valued functions on such a space is defined it is shown that some of typical propositions in a metric space namely baire category theorem tietze s extension theorem and decomposition of unity can be effectivized it is also proved that computable functions are dense in continuous functions

an integrable poisson map generated from the eigenvalue problem of the lotka volterra hierarchy a 3 3 discrete eigenvalue problem associated with the lotka volterra hierarchy is studied and the corresponding nonlinearized one an integrable poisson map with a lie poisson structure is also presented moreover a 2 2 nonlinearized eigenvalue problem which also begets the lotka volterra hierarchy is proved to be a reduction of the poisson map on the leaves of the symplectic foliation

backlund transformation and nonlinear superposition formula of an extended lotka volterra equation an extended lotka volterra equation is considered and a backlund transformation and a nonlinear superposition formula given for it multiple soliton solutions are derived step by step as an application of the obtained results thus the n soliton conjecture by narita is confirmed

pcf extended with real numbers we extend the programming language pcf with a type for total and partial real numbers by a partial real number we mean an element of a cpo of intervals whose subspace of maximal elements single point intervals is homeomorphic to the euclidean real line we show that partial real numbers can be considered as continuous words concatenation of continuous words corresponds to refinement of partial information the usual basic operations cons head and tail used to explicitly or recursively define functions on words generalize to partial real numbers we use this fact to give an operational semantics to the above referred extension of pcf we prove that the operational semantics is sound and complete with respect to the denotational semantics a program of real number type evaluates to a head normal form iff its value is different from b if its value is different from b then it successively evaluates to head normal forms giving better and better partial results converging to its value author s abstract

recursive characterization of computable real valued functions and relations abstract corresponding to the definition of recursive functions we introduce a class of recursive relations in metric spaces such that each relation is generated from a class of basic relations by a finite number of applications of some specified operators we prove that our class of recursive relations essentially coincides with our class of densely computable relations defined via turing machines in the special case of the real numbers our subclass of recursive functions coincides with the classical class of computable real valued functions defined via turing machines by grzegorczyk lacombe and others

computability on computable metric spaces in previous papers e g weihrauch 1987 type 2 theory of effectivity tte has been shown to be a very general and powerful computer oriented theory for studying effectivity in mathematics in this contribution computability on certain computable separable metric spaces is studied in detail by applying the framework of tte computationally admissible representations of metric spaces are introduced after showing that four different effective representations are computationally equivalent for extending computability to the set of continuous functions several effective namings which are related to definitions of continuity are introduced and compared the definitions are used to prove effective g extension theorems for continuous functions and an effective g characterization of the domains of strongly continous functions which generalize the known properties of real functions kreitz 1984

a 3 sx 3 matrix spectral problem for akns hierarchy and its binary nonlinearization abstract a 3 3 matrix spectral problem for akns soliton hierarchy is introduced and the corresponding bargmann symmetry constraint involving lax pairs and adjoint lax pairs is discussed an explicit new poisson algebra is proposed and thus the liouville integrability is established for the nonlinearized spatial system ind a hierarchy of nonlinearized temporal systems under the control of the nonlinearized spatial system the obtained nonlinearized lax systems in which the nonlinearized spatial system is intimately related to stationary akns flows lead to a sort of new involutive solutions to each akns soliton equation therefore the binary nonlinearization theory is successfully extended to a case of 3 3 matrix spectral problem for akns hierarchy

computability on subsets of metric spaces the notions recursively enumerable and recursive are the basic notions of effectivity in classical recursion theory in computable analysis these notions are generalized to closed subsets of euclidean space using their metric distance functions we study a further generalization of these concepts to subsets of computable metric spaces it appears that different characterizations which coincide in case of euclidean space lead to different notions in the general case however under certain additional conditions such as completeness and effective local compactness the situation is similar to the euclidean case we present all results in the framework of type 2 theory of effectivity which allows to express effectivity properties in a very uniform way instead of comparing properties of single subsets we compare corresponding representations of the hyperspace of closed subsets such representations do not only induce a concept of computability for single subsets but they even yield a concept of computability for operations on hyperspaces such as union intersection etc we complete our investigation by studying the special situation of compact subsets

power domains and iterated function systems we introduce the notion of weakly hyperbolic iterated function system ifs on a compact metric space which generalises that of hyperbolic ifs based on a domain theoretic model which uses the plotkin power domain and the probabilistic power domain respectively we prove the existence and uniqueness of the attractor of a weakly hyperbolic ifs and the invariant measure of a weakly hyperbolic ifs with probabilities extending the classic results of hutchinson for hyperbolic ifss in this more general setting we also present finite algorithms to obtain discrete and digitised approximations to the attractor and the invariant measure extending the corresponding algorithms for hyperbolic ifss we then prove the existence and uniqueness of the invariant distribution of a weakly hyperbolic recurrent ifs and obtain an algorithm to generate the invariant distribution on the digitised screen the generalised riemann integral is used to provide a formula for the expected value of almost everywhere continuous functions with respect to this distribution for hyperbolic recurrent ifss and lipschitz maps one can estimate the integral up to any threshold of accuracy

a computable approach to measure and integration theory we introduce a computable framework for lebesgue s measure and integration theory in the spirit of domain theory for an effectively given second countable locally compact hausdorff space and an effectively given finite borel measure on the space we define a recursive measurable set which extends the corresponding notion due to s anin for the lebesgue measure on the real line we also introduce the stronger notion of a computable measurable set where a measurable set is approximated from inside and outside by sequences of closed and open subsets respectively the more refined property of computable measurable sets give rise to the idea of partial measurable subsets which naturally form a domain for measurable subsets we then introduce interval valued measurable functions and develop the notion of recursive and computable measurable functions using interval valued simple functions this leads us to the interval versions of the main results in classical measure theory the lebesgue integral is shown to be a continuous operator on the domain of interval valued measurable functions and the interval valued lebesgue integral provides a computable framework for integration

an effective riemann mapping theorem the main results of the paper are two effective versions of the riemann mapping theorem the first uniform version is based on the constructive proof of the riemann mapping theorem by cheng and by bishop and bridges and formulated in the computability framework developed by kreitz and weihrauch it states which topological information precisely one needs about a nonempty proper open connected and simply connected subset of the complex plane in order to compute a description of a holomorphic bijection from the unit disk onto this set and vice versa which topological information about the set can be obtained from a description of a holomorphic bijection the second version which is derived from the first by considering the sets and the functions with computable descriptions characterizes the subsets of the complex plane for which there exists a computable holomorphic bijection from the unit disk this gives a partial answer to a problem posed by pour el and richards we also show that this class of sets is strictly larger than a class of sets considered by zhou which solves an open problem posed by him in preparation recursively enumerable open subsets and closed subsets of euclidean spaces are considered and several effective results in complex analysis are proved

integrable maps for the garnier and for the neumann system we construct integrable maps for the garnier and for the neumann system they are related to the toda hierarchy exactly in the same way as the garnier and the neumann systems are related to the kdv hierarchy as restricted flows here we give lax representations for these maps and prove that they are completely integrable

solution of an inverse problem for fractals and other sets abstract the problem of providing succinct approximate descriptions of given bounded subsets of rn can be solved by application of the contraction mapping principle

type 2 recursion theory abstract an attempt is made to lay a basis for a general unified concise and simple theory of computable and continuous functions from f to f or n where f f n n the theoy is formally very similar to ordinary recursion theory it splits into a purely topological version and more special theory of computability the basic definitions are given and fundamental properties are proved as an example it is shown how the theory of recursively enumerable subsets of n can be transferred to a theory of open and a theory of computably open subsets of f

effective metric spaces and representations of the reals based on standard notions of classical recursion theory a natural model of approximate computability for partial functions between effective metric spaces is presented it generalizes the ko friedman approach to computability of real functions by means of oracle turing machines follows the main ideas of weihrauch s type 2 theory of effectivity but it avoids the explicit use of representations the topological arithmetical hierarchy is introduced and shown to be strict if the underlying space contains an effectively discrete sequence the domains of computable functions are exactly the 2 sets of this hierarchy if the space admits a finitary stratification finally this framework is used to investigate and characterize the standard representations of the real numbers they are just those functions from the name space onto the reals which have both computable extensions and inversions that are computable as relations