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

velocity model estimation with data derived wavefront attributes kinematic information for constructing velocity models can be extracted in a robust way from seismic prestack data with the common reflection surface crs stack this data driven process results in addition to a simulated zero offset section in a number of wavefront attributes wavefront curvatures and normal ray emergence angles associated with each simulated zero offset sample a tomographic inversion method is presented that uses this kinematic information to determine smooth laterally heterogeneous isotropic subsurface velocity models for depth imaging the input for the inversion consists of wavefront attributes picked at a number of locations in the simulated zero offset section the smooth velocity model is described by b splines an optimum model is found iteratively by minimizing the misfit between the picked data and the corresponding modeled values the required forward modeled quantities are obtained during each iteration by dynamic ray tracing along normal rays pertaining to the input dat

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

computing true amplitude reflections in a laterally inhomogeneous earth recently bortfeld 1982 gave a cursory nonmathematical introduction to a procedure for computing the geometrical spreading factor of a primary zero offset reflection from the common datum point traveltime measurements of the event to underline the significance and consequences of this method a derivation and discussion of geometrical spreading factors is now given for two and three dimensional earth models with curved reflecting boundaries the spreading factors can be used easily to transform primary reflections in a zero offset seismic section to true amplitude reflections these permit an estimation of interface reflection coefficients either directly or in connection with a true amplitude migration a seismic section with true amplitude reflections can be described by one physical experiment the tuned reflector model hence the application of the wave equation in connection with a migration after stack is justified on such a seismic section also the geometrical spreading factors that are deri

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

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

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

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

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

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

sign changing solutions for elliptic problems with critical sobolev hardy exponents abstract let r n be a smooth bounded domain such that 0 n 7 0 s 2 s 2 n s n 2 we prove the existence of sign changing solutions for the singular critical problem u u x 2 u 2 s 2 x s u u with dirichlet boundary condition on for suitable positive parameters and

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

perturbation results of critical elliptic equations of caffarelli kohn nirenberg type abstract we find for small e positive solutions to the equation div x 2a u x 2 1 a u 1 ek x u p 1 x bp in r n which branch off from the manifold of minimizers in the class of radial functions of the corresponding caffarelli kohn nirenberg type inequality moreover our analysis highlights the symmetry breaking phenomenon in these inequalities namely the existence of non radial minimizers

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

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

subcritical perturbations of a singular quasilinear elliptic equation involving the critical hardy sobolev exponent abstract in this work we improve some known results for a singular operator and also for a wide class of lower order terms by proving a multiplicity result the proof is made by applying the generalized mountain pass theorem due to ambrosetti and rabinowitz to do this we show that the minimax levels are in a convenient range by combining a special class of approximating functions due to gazzola and ruf with the concentrating functions of the best sobolev constant

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

a note on the sign changing solutions to elliptic problems with critical sobolev and hardy terms let rn be a bounded domain such that 0 n 7 2 2nn 2 we obtain existence of sign changing solutions for the dirichlet problem u u x 2 u 2 2u u on u 0 on for suitable positive numbers and

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

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

multiple positive solutions for robin problem involving critical weighted hardy sobolev exponents with boundary singularities abstract this paper deals with the existence of positive solutions for robin elliptic problems involving critical weighted hardy sobolev exponents with boundary singularities using the caffarelli kohn nirenberg inequalities and variational methods we prove the existence and multiplicity of positive solutions

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

multiple positive solutions for p laplace elliptic equations involving concave convex nonlinearities and a hardy type term abstract in this paper the following problem is considered p u u p 2 u x p f x u q 2 u g x u p 2 u x u 0 x where r n is a bounded domain such that 0 1 q p 0 f and g are nonnegative functions n p p p is the best hardy constant and p n p n p is the critical sobolev exponent by extracting the palais smale sequence in the nehari manifold the existence of multiple positive solutions to this equation is verified

positive solutions to the weighted critical quasilinear problems in a bounded domain with smooth boundary we consider a kind of weighted quasilinear elliptic problem which satisfies dirichlet boundary condition and involves the hardy sobolev inequality by the analytic techniques we first get the properties of the extremal functions by which the best hardy sobolev constant is achieved then by the variational methods the existence of positive solutions to the problem is verified by careful estimates and computations

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

quasilinear elliptic problems with critical exponents and hardy terms abstract let 0 be an open bounded domain r n n p 2 we are concerned with the multiplicity of positive solutions of p u u p 2 u x p u p 2 u q x u p 2 u u w 0 1 p where p u div u p 2 u 1 p n p np n p 0 n p p p 0 and q x is a nonnegative function on by investigating the effect of the coefficient of the critical nonlinearity we by means of variational method prove the existence of multiple positive solutions

existence and multiplicity of symmetric solutions for the weighted critical quasilinear problems this paper deals with the weighted critical quasilinear problem div u p 2 u x a p m u p 2u x p a 1 q x u p a b 2u x b p a b h x u in w u 0on w where w r n is a smooth bounded domain 0 w and w is g symmetric with respect to a subgroup g of o n 1

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

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

existence and multiplicity of positive solutions for a class of semilinear elliptic equations involving hardy term and hardy sobolev critical exponents abstract existence and multiplicity of positive solutions are studied for a class of semilinear elliptic equations with hardy term hardy sobolev critical exponents and sublinear nonlinearity by variational methods and some analysis techniques

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

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

some properties of solutions to the singular quasilinear problems abstract in this paper a kind of quasilinear elliptic problem is studied which involves the critical exponent and singular potentials by the caffarelli kohn nirenberg inequality and variational methods some important properties of the positive solution to the problem are established