computability and the morphological complexity of some dynamics on continuous domains the partially ordered set of compact intervals provides a convenient embedding space for the analysis of some dynamical systems crucial dynamical properties are transferred to it while allowing an investigation of stability and chaoticity in terms of computability in particular in the presence of singularities we will survey some results which display the connections between the geometric complexity of the dynamics and computability issues as well as new relations between dynamic predictability and effective decidability

prequential randomness and probability this paper studies dawid s prequential framework from the point of view of the algorithmic theory of randomness our first main result is that two natural notions of randomness coincide one notion is the prequential version of the measure theoretic definition due to martin lof and the other is the prequential version of the game theoretic definition due to schnorr and levin this is another manifestation of the close relation between the two main paradigms of randomness the algorithmic theory of randomness can be stripped of its algorithmic aspect and still give meaningful results the measure theoretic paradigm then corresponds to kolmogorov s measure theoretic probability and the game theoretic paradigm corresponds to game theoretic probability our second main result is that measure theoretic probability coincides with game theoretic probability on all analytic in particular borel sets

topological properties of concept spaces full version based on the observation that the category of concept spaces with the positive information topology is equivalent to the category of countably based t 0 topological spaces we investigate further connections between the learning in the limit model of inductive inference and topology in particular we show that the texts or positive presentations of concepts in inductive inference can be viewed as special cases of the admissible representations of computable analysis we also show that several structural properties of concept spaces have well known topological equivalents in addition to topological methods we use algebraic closure operators to analyze the structure of concept spaces and we show the connection between these two approaches the goal of this paper is not only to introduce new perspectives to learning theorists but also to present the field of inductive inference in a way more accessible to domain theorists and topologists

long time behavior of solutions of fisher kpp equation with advection and free boundaries we consider fisher kpp equation with advection ut uxx ux f u for x g t h t where g t and h t are two free boundaries satisfying stefan conditions this equation is used to describe the population dynamics in advective environments we study the influence of the advection coefficient on the long time behavior of the solutions we find two parameters c0 and with c0 0 which play key roles in the dynamics here c0 is the minimal speed of the traveling waves of fisher kpp equation more precisely by studying a family of the initial data 0 where is some compactly supported positive function we show that 1 in case 0 c0 there exists 0 such that spreading happens when i e u t 1 locally uniformly in r and vanishing happens when 0 i e g t h t remains bounded and u t 0 uniformly in g t h t 2 in case c0 there exists 0 such that virtual spreading happens when i e u t 0 locally uniformly in g t and u t ct 1 locally uniformly in r for some c c0 vanishing happens when 0 and in the transition case u t o t v c0 t uniformly the latter is a traveling wave with a big head near the free boundary x c0 t and with an infinite long tail on the left 3 in case c0 there exists 0 such that virtual spreading happens when and u t 0 uniformly in g t h t when 0 4 in case vanishing happens for any solution

on the stability of bilinear constant velocity pressure finite elements an analysis of the babuska stability of bilinear constant finite element pairs for viscous flow calculations is given an unstable mode not of the checkerboard type is given for which the stability constant turns out to beo h thus the indicated spaces are not stable in general for numerical calculation

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

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

error analysis of some finite element methods for the stokes problem we prove the optimal order of convergence for some two dimensional finite element methods for the stokes equations first we consider methods of the taylor hood type the triangular p3 p2 element and the qk qk 1 i k 2 family of quadrilateral elements then we introduce two new low order methods with piecewise constant approximations for the pressure the analysis is performed using our macroelement technique which is reviewed in a slightly altered form

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

when does randomness come from randomness a result of shen says that if f 2 n 2 n is an almost everywhere computable measure preserving transformation and y 2 n is martin lof random then there is a martin lof random x 2 n such that f x y answering a question of bienvenu and porter we show that this property holds for computable randomness but not schnorr randomness these results combined with other known results imply that the set of martin lof randoms is the largest subset of 2 n satisfying this property and also satisfying randomness conservation if f 2 n 2 n is an almost everywhere computable measure preserving map and if x 2 n is random then f x is random

regularity and asymptotic behavior of nonlinear stefan problems we study the following nonlinear stefan problem left begin aligned u t d delta u g u quad rm for x in omega t t 0 u 0 rm and u t mu nabla x u 2 quad rm for x in gamma t t 0 u 0 x u 0 x quad rm for x in omega 0 end aligned right where omega t subset mathbb r n n geqq 2 is bounded by the free boundary gamma t with omega 0 omega 0 and d are given positive constants the initial function u0 is positive in omega 0 and vanishes on partial omega 0 the class of nonlinear functions g u includes the standard monostable bistable and combustion type nonlinearities we show that the free boundary gamma t is smooth outside the closed convex hull of omega 0 and as t to infty either omega t expands to the entire mathbb r n or it stays bounded moreover in the former case gamma t converges to the unit sphere when normalized and in the latter case u to 0 uniformly when g u au bu 2 we further prove that in the case omega t expands to mathbb r n u to a b as t to infty and the spreading speed of the free boundary converges to a positive constant moreover there exists mu geqq 0 such that omega t expands to mathbb r n exactly when mu mu

penalty approximation of stokes flow abstract the particular class of problems considered is stokes flow in which the constraint is the incompressibility condition u 0 on the velocity field u in part i we present first of all a simple constructive method for determining the spurious modes this is followed by an analysis of the discrete stability condition for the penalty method showing that this condition holds with a parameter that is o h we develop improved estimates of the dependence of the stability parameter on mesh size h in a more general proof than previously available in part ii we prove error estimates for the velocity and computed pressure approximations and use an aubin nitsche technique to get l 2 estimates for the velocity in the concluding numerical studies we examine the stability of the computed pressure and convergence rates with uniform mesh refinement for problems having data with varying degree of regularity

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

the minimal habitat size for spreading in a weak competition system with two free boundaries abstract in this paper we focus on the dynamics for a lotka volterra type weak competition system with two free boundaries where free boundaries which may intersect each other as time evolves are used to describe the spreading of two competing species respectively in the weak competition case the dynamics of this model can be classified into four cases which forms a spreading vanishing quartering the notion of the minimal habitat size for spreading is introduced to determine if species can always spread some sufficient conditions for spreading and vanishing are established also when spreading occurs some rough estimates for spreading speed and the long time behavior of solutions are established

set systems by reformulating a learning process of a set system l as a game between teacher and learner we define the order type of l to be the order type of the game tree if the tree is well founded the features of the order type of l diml in symbol are 1 we can represent any well quasi order wqo for short by the set system l of the upper closed sets of the wqo such that the maximal order type of the wqo is equal to diml 2 diml is an upper bound of the mind change complexity of l diml is defined iff l has a finite elasticity fe for short where according to computational learning theory if an indexed family of recursive languages has fe then it is learnable by an algorithm from positive data regarding set systems as subspaces of cantor spaces we prove that fe of set systems is preserved by any continuous function which is monotone with respect to the set inclusion by it we prove that finite elasticity is preserved by various nondeterministic language operators kleene closure shuffle closure union product intersection the monotone continuous functions represent nondeterministic computations if a monotone continuous function has a computation tree with each node followed by at most n immediate successors and the order type of a set system l is then the direct image of l is a set system of order type at most n adic diagonal ramsey number of furthermore we provide an order type preserving contravariant embedding from the category of quasi orders and finitely branching simulations between them into the complete category of subspaces of cantor spaces and monotone continuous functions having girard s linearity between them

the iterative penalty method for stokes equations using q1 p0 element abstract the iterative penalty method for stokes equations is an effective algorithm in this paper we discuss the bilinear constant velocity pressure element for the stokes equations and present the error estimates for the iterative penalty method using this element the key technique we use is the local checkerboard pattern numerical experiments are given which are consistent with our analysis

the diffusive logistic model with a free boundary in heterogeneous environment abstract in this paper we study the population dynamics of an invasive species in heterogeneous environment which is modeled by a diffusive logistic equation with free boundary condition to understand the effect of the dispersal rate d and the parameter the ratio of the expansion speed of the free boundary and the population gradient at the expanding front on the dynamics of this model we divide the heterogeneous environment into two cases strong heterogeneous environment and weak heterogeneous environment by choosing d and as variable parameters we derive sufficient conditions for species spreading resp vanishing in the strong heterogeneous environment while in the weak heterogeneous environment we obtain sharp criteria for the spreading and vanishing moreover when spreading happens we give an estimate for the asymptotic spreading speed of the free boundary these theoretical results may have important implications for prediction and prevention of biological invasions

on a free boundary problem for a reaction diffusion advection logistic model in heterogeneous environment abstract in this paper we investigate a reaction diffusion advection equation with a free boundary which models the spreading of an invasive species in one dimensional heterogeneous environments we assume that the species has a tendency to move upward along the resource gradient in addition to random dispersal and the spreading mechanism of species is determined by a stefan type condition investigating the sign of the principal eigenvalue of the associated linearized eigenvalue problem under certain conditions we obtain the sharp criteria for spreading and vanishing via system parameters also we establish the long time behavior of the solution and the asymptotic spreading speed finally some biological implications are discussed

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

the diffusive logistic equation with a free boundary and sign changing coefficient abstract this short paper concerns a diffusive logistic equation with a free boundary and sign changing coefficient which is formulated to study the spread of an invasive species where the free boundary represents the expanding front a spreading vanishing dichotomy is derived namely the species either successfully spreads to the right half space as time t and survives persists in the new environment or it fails to establish itself and will extinct in the long run the sharp criteria for spreading and vanishing are also obtained when spreading happens we estimate the asymptotic spreading speed of the free boundary

a diffusive logistic equation with a free boundary and sign changing coefficient in time periodic environment abstract this paper concerns a diffusive logistic equation with a free boundary and sign changing intrinsic growth rate in heterogeneous time periodic environment in which the variable intrinsic growth rate may be very negative in a suitable large region see conditions h1 h2 4 3 such a model can be used to describe the spreading of a new or invasive species with the free boundary representing the expanding front in the case of higher space dimensions with radial symmetry and when the intrinsic growth rate has a positive lower bound this problem has been studied by du guo peng 11 they established a spreading vanishing dichotomy the sharp criteria for spreading and vanishing and estimate of the asymptotic spreading speed in the present paper we show that the above results are retained for our problem

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

coloring trees in reverse mathematics abstract the tree theorem for pairs tt 2 2 first introduced by chubb hirst and mcnicholl asserts that given a finite coloring of pairs of comparable nodes in the full binary tree 2 there is a set of nodes isomorphic to 2 which is homogeneous for the coloring this is a generalization of the more familiar ramsey s theorem for pairs rt 2 2 which has been studied extensively in computability theory and reverse mathematics we answer a longstanding open question about the strength of tt 2 2 by showing that this principle does not imply the arithmetic comprehension axiom aca 0 over the base system recursive comprehension axiom rca 0 of second order arithmetic combined with a recent result of patey s that tt 2 2 is strictly stronger than rt 2 2 this establishes tt 2 2 as the first known example of a natural combinatorial principle to occupy the interval strictly between aca 0 and rt 2 2 the proof of this fact uses an extension of the bushy tree forcing method and develops new techniques for dealing with combinatorial statements formulated on trees rather than on

plottable real number functions and the computable graph theorem the graph theorem of classical recursion theory states that a total function on the natural numbers is computable if and only if its graph is recursive it is known that this result can be generalized to real number functions where it has an important practical interpretation the total computable real number functions are precisely those which can be effectively plotted with any given resolution we generalize the graph theorem to appropriate partial real number functions and even further to functions defined on certain computable metric spaces besides the nonuniform version of the graph theorem which logically relates computability properties of the function and computability properties of its graph we also discuss the uniform version given a program of a function can we algorithmically derive a description of its graph and vice versa given a description of the graph can we derive a program of the function while the passage from functions to graphs is always computable the inverse direction from graphs to functions is problematic and it turns out that the answers to the uniform and the nonuniform questions do not coincide we prove that in both cases certain topological and computational properties such as compactness or effective local connectedness are sufficient for a positive answer and we provide counterexamples which show that the corresponding properties are not superfluous additionally we briefly discuss the special situation of the linear case

randomness for non computable measures different approaches have been taken to defining randomness for non computable probability measures we will explain the approach of reimann and slaman along with the uniform test approach first introduced by levin and also used by gacs hoyrup and rojas we will show that these approaches are fundamentally equivalent having clarified what it means to be random for a non computable probability measure we turn our attention to levin s neutral measures for which all sequences are random we show that every pa degree computes a neutral measure we also show that a neutral measure has no least turing degree representation and explain why the framework of the continuous degrees a substructure of the enumeration degrees studied by miller can be used to determine the computational complexity of neutral measures this allows us to show that the turing ideals below neutral measures are exactly the scott ideals since x 2 is an atom of a neutral measure if and only if it is computable from every representation of we have a complete understanding of the possible sets of atoms of a neutral measure one simple consequence is that every neutral measure has a martin lof random atom 1 defining randomness let x be an element of cantor space and a borel probability measure on cantor space what should it mean for x to be random with respect to in the case that is the lebesgue measure then the theory of randomness is well developed for recent treatises on the subject the reader is referred to downey and hirschfeldt and nies 2 13 in fact if is a computable measure then early work of levin showed that randomness can be seen as essentially a variant on randomness for lebesgue measure 10 this leaves the question of how to define randomness if is non computable we will show that the two approaches that have previously been used to define randomness for non computable are equivalent later in theorem 4 12 we will provide another characterization of randomness using the enumeration degrees last compilation september 8 2011 last time the following date was changed november 22 2010 2010 mathematics subject classification primary 03d32 secondary 68q30 03d30 the second author was supported by the national science foundation under grants dms 0945187 and dms 0946325 the latter being part of a focused research group in algorithmic randomness

on the role of procrastination in machine learning the traditional model of inductive inference is enhanced to allow learning machines to procrastinate about how many trials they will need to complete an inference or about how accurate their solution will be hierarchies of classes of learnable phenomena represented by sets of recursive functions isomorphic to the structure of the constructive ordinals are revealed the existence of such hierarchies indicates a potential advantage of procrastination as a learning technique tradeoffs between the two types of procrastination are shown not to exist in general one of the new classes of sets of inferrible functions introduced in this paper turns out to have the somewhat unusual property of being closed under finite unions a new technique based on a hardest to lear set relative to a class is used

spreading vanishing dichotomy in a diffusive logistic model with a free boundary ii abstract we study the diffusive logistic equation with a free boundary in higher space dimensions and heterogeneous environment such a model may be used to describe the spreading of a new or invasive species with the free boundary representing the expanding front for simplicity we assume that the environment and the solution are radially symmetric in the special case of one space dimension and homogeneous environment this free boundary problem was investigated in du and lin 2010 10 we prove that the spreading vanishing dichotomy established in du and lin 2010 10 still holds in the more general and ecologically realistic setting considered here moreover when spreading occurs we obtain best possible upper and lower bounds for the spreading speed of the expanding front when the environment is asymptotically homogeneous at infinity these two bounds coincide our results indicate that the asymptotic spreading speed determined by this model does not depend on the spatial dimension

randomness and differentiability we characterize some major algorithmic randomness notions via differentiability of effective functions 1 we show that a real number z 0 1 is computably random if and only if every nondecreasing computable function 0 1 r is differentiable at z 2 we prove that a real number z 0 1 is weakly 2 random if and only if every almost everywhere differentiable computable function 0 1 r is differentiable at z 3 recasting results of the constructivist demuth from 1975 in classical language we show that a real z is martin lof random if and only if every computable function of bounded variation is differentiable at z and similarly for absolutely continuous functions we also use the analytic methods to show that computable randomness of a real is base invariant and to derive preservation results for randomness notions

a constructive version of birkhoff s ergodic theorem for martin lof random points we prove the effective version of email protected s ergodic theorem for martin lof random points and effectively open sets improving the results previously obtained in this direction in particular those of vyugin nandakumar and hoyrup rojas the proof consists of two steps first we prove a generalization of email protected s theorem which is a particular case of effective ergodic theorem a trajectory of a computable ergodic mapping that starts from a random point cannot remain inside an effectively open set of measure less than 1 second we show that the full statement of the effective ergodic theorem can be reduced to this special case both steps use the statement of classical ergodic theorem but not its usual classical proof therefore we get a new simple proof of the effective ergodic theorem with weaker assumptions than before this result was recently obtained independently by franklin greenberg miller and ng

a diffusive logistic model with a free boundary in time periodic environment abstract we study the diffusive logistic equation with a free boundary in time periodic environment such a model may be used to describe the spreading of a new or invasive species with the free boundary representing the expanding front for time independent environment in the cases of one space dimension and higher space dimensions with radial symmetry this free boundary problem has been studied in du and lin 2010 12 du and guo 2011 9 in both cases a spreading vanishing dichotomy was established and when spreading occurs the asymptotic spreading speed was determined in this paper we show that the spreading vanishing dichotomy is retained in time periodic environment and we also determine the spreading speed the former is achieved by further developing the earlier techniques and the latter is proved by introducing new ideas and methods

schnorr randomness for noncomputable measures abstract this paper explores a novel definition of schnorr randomness for noncomputable measures we say x is uniformly schnorr random if t x for all lower semicomputable functions t x such that t x d x is computable we prove a number of theorems demonstrating that this is the correct definition which enjoys many of the same properties as martin lof randomness for noncomputable measures nonetheless a number of our proofs significantly differ from the martin lof case requiring new ideas from computable analysis

continuity and computability of reachable sets the computation of reachable sets of nonlinear dynamic and control systems is an important problem of systems theory in this paper we consider the computability of reachable sets using turing machines to perform approximate computations we use weihrauch s type two theory of effectivity for computable analysis and topolgy which provides a natural setting for performing computations on sets and maps the main result is that the reachable set is lower computable but is only outer computable if it equals the chain reachable set in the course of the analysis we extend the computable topology theory to locally compact hausdorff spaces and semicontinuous set valued maps and provide a framework for computing approximations

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

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

the symmetry of positive solutions of periodic parabolic problems abstract in this paper we generalized a result of gidas ni and nirenberg 1979 on the forced symmetry of positive solutions of nonlinear elliptic problems to the periodic in time problem for nonlinear parabolic equations with dirichlet boundary conditions

spreading in space time periodic media governed by a monostable equation with free boundaries part 1 continuous initial functions abstract we aim to classify the long time behavior of the solution to a free boundary problem with monostable reaction term in space time periodic media such a model may be used to describe the spreading of a new or invasive species with the free boundary representing the expanding front in time periodic and space homogeneous environment as well as in space periodic and time autonomous environment such a problem has been studied recently in 11 12 in both cases a spreading vanishing dichotomy has been established and when spreading happens the asymptotic spreading speed is proved to exist by making use of the corresponding semi wave solutions the approaches in 11 12 seem difficult to apply to the current situation where the environment is periodic in both space and time here we take a different approach based on the methods developed by weinberger 31 32 and others 16 22 23 24 26 which yield the existence of the spreading speed without using traveling wave solutions in part 1 of this work we establish the existence and uniqueness of classical solutions for the free boundary problem with continuous initial data extending the existing theory which was established only for c 2 initial data this will enable us to develop weinberger s method in part 2 to determine the spreading speed without knowing a priori the existence of the corresponding semi wave solutions in part 1 here we also establish a spreading vanishing dichotomy

probabilistic computability and choice we study the computational power of randomized computations on infinite objects such as real numbers in particular we introduce the concept of a las vegas computable multi valued function which is a function that can be computed on a probabilistic turing machine that receives a random binary sequence as auxiliary input the machine can take advantage of this random sequence but it always has to produce a correct result or to stop the computation after finite time if the random advice is not successful with positive probability the random advice has to be successful we characterize the class of las vegas computable functions in the weihrauch lattice with the help of probabilistic choice principles and weak weak k nig s lemma among other things we prove an independent choice theorem that implies that las vegas computable functions are closed under composition in a case study we show that nash equilibria are las vegas computable while zeros of continuous functions with sign changes cannot be computed on las vegas machines however we show that the latter problem admits randomized algorithms with weaker failure recognition mechanisms the last mentioned results can be interpreted such that the intermediate value theorem is reducible to the jump of weak weak k nig s lemma but not to weak weak k nig s lemma itself these examples also demonstrate that las vegas computable functions form a proper superclass of the class of computable functions and a proper subclass of the class of non deterministically computable functions we also study the impact of specific lower bounds on the success probabilities which leads to a strict hierarchy of classes in particular the classical technique of probability amplification fails for computations on infinite objects we also investigate the dependency on the underlying probability space besides cantor space we study the natural numbers the euclidean space and baire space

asymptotic spreading in heterogeneous diffusive excitable media abstract we establish propagation and spreading properties for nonnegative solutions of nonhomogeneous reaction diffusion equations of the type t u a t x u q t x u f t x u with compactly supported initial conditions at t 0 here a q f have a general dependence in t r and x r n we establish properties of families of propagation sets which are defined as families of subsets s t t 0 of r n such that lim inf t inf x s t u t x 0 the aim is to characterize such families as sharply as possible in particular we give some conditions under which 1 a given path t t 0 where t r n forms a family of propagation sets or 2 one can find such a family with s t x r n x r t and lim t r t this second property is called here complete spreading furthermore in the case q 0 and inf t x r r n f u t x 0 0 as well as under some more general assumptions we show that there is a positive spreading speed that is r t can be chosen so that lim inf t r t t 0 in the general case we also show the existence of an explicit upper bound c 0 such that lim sup t r t t c on the other hand we provide explicit examples of reaction diffusion equations such that for an arbitrary e 0 any family of propagation sets s t t 0 has to satisfy s t x r n x e t for large t in connection with spreading properties we derive some new uniqueness results for the entire solutions of this type of equations lastly in the case of space time periodic media we develop a new approach to characterize the largest propagation sets in terms of eigenvalues associated with the linearized equation in the neighborhood of zero

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

the diffusive logistic model with a free boundary in a heterogeneous time periodic environment abstract this paper is concerned with a diffusive logistic model with advection and a free boundary in a spatially heterogeneous and time periodic environment such a model may be used to describe the spreading of a new or invasive species with the free boundary representing the expanding front under more general assumptions on the initial data and the function standing for the intrinsic growth rate of the species sharp criteria for spreading and vanishing are established and estimates for spreading speed when spreading occurs are also derived the obtained results considerably improve and complement the existing ones especially those of 11 25

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

analysis of some mixed finite element methods for plane elasticity equations we analyze some mixed finite element methods based on rectangular elements for solving the two dimensional elasticity equations we prove error estimates for a method proposed by taylor and zienkiewicz and for some new variants of the known equilibrium methods a numerical example is given demonstrating the performance of the various algorithms considered 1 introduction in the numerical solution of problems of continuum mechanics the stresses are normally of primary interest in the elastic region it is therefore natural to design the numerical algorithms so that the stresses can be obtained directly without first computing the displacements such methods can be derived from the dual variational formulation of the elasticity problem the corresponding finite element algorithms are usually formulated as mixed methods where both the displacements and the stresses are first approximated and the displacements are then eliminated from the discrete equations in many cases the elimination can be rather effectively done using penalty perturbation techniques or their iterative variants cf 3 11 12 the best known finite element methods of the above type are the so called equilibrium methods first proposed by fraejis de veubeke 17 cf also 14 16 18 and analyzed theoretically by johnson and mercier 9 cf also 8 in these methods one uses specific composite elements which allow the equilibrium condi tion between the stresses and the volume load to be satisfied exactly in the case where the volume load is zero

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

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

dynamics of a generic brownian motion recursive aspects we study the local fluctuations of brownian motions which are represented by infinite binary strings which are random in the sense of kolmogorov chaitin we show how the dynamical properties of such a brownian motion at a point depend on its recursive properties

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

stabilization of low order cross grid pkql mixed finite elements abstract in this paper we analyze a low order family of mixed finite element methods for the numerical solution of the stokes problem and a second order elliptic problem in two space dimensions in these schemes the pressure is interpolated on a mesh of rectangular elements while the velocity is approximated on a triangular mesh obtained by dividing each rectangle into four triangles by its diagonals for the lowest order p 1 q 0 a global spurious pressure mode is shown to exist and so this element as p 1 q 1 case analyzed in armentano and blasco 2010 is unstable however following the ideas given in bochev et al 2006 a simple stabilization procedure can be applied when we approximate the solution of the stokes problem such that the new p 1 q 0 and p 1 q 1 methods are unconditionally stable and achieve optimal accuracy with respect to solution regularity with simple and straightforward implementations moreover we analyze the application of our p 1 q 1 element to the mixed formulation of the elliptic problem in this case by introducing the modified mixed weak form proposed in brezzi et al 1993 optimal order of accuracy can be obtained with our stabilized p 1 q 1 elements numerical results are also presented which confirm the existence of the spurious pressure mode for the p 1 q 0 element and the excellent stability and accuracy of the new stabilized methods

traveling wave solutions of a time periodic competitive system with a free boundary abstract we consider a time periodic competition system with a free boundary which can be used to describe two species that are struggling on the free boundary to obtain their own habitats and the final competition result is the regional partition of two species in 15 we have studied the classical traveling wave solutions of the temporal homogeneous problems in this paper we consider the time periodic case and study periodic traveling wave solutions of the problem more precisely we will prove the existence and uniqueness of the periodic traveling wave solution and show that the speed of the periodic traveling wave is uniquely determined by and monotonically dependent on the time periodic terms in the system moreover we will present some sufficient conditions to guarantee that the wave travels with a positive speed

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 notions of mind change complexity gold introduced the notion of learning in the limit where a class s is learnable iff there is a recursive machine m which reads the course of values of a function f and converges to a program for f whenever f is in s an important measure for the speed of convergence in this model is the quantity of mind changes before the onset of convergence the oldest model is to consider a constant bound on the number of mind changes m makes on any input function such a bound is referred here as type 1 later this was generalized to a bound of type 2 where a counter ranges over constructive ordinals and is counted down at every mind change although ordinal bounds permit the inference of richer concept classes than constant bounds they still are a severe restriction therefore the present work introduces two more general approaches to bounding mind changes these are based on counting by going down in a linearly ordered set type 3 and on counting by going down in a partially ordered set type 4 in both cases the set must not contain infinite descending recursive sequences these four types of mind changes yield a hierarchy and there are identifiable classes that cannot be learned with the most general mind change bound of type 4 it is shown that existence of type 2 bound is equivalent to the existence of a learning algorithm which converges on every also nonrecursive input function and the existence of type 4 is shown to be equivalent to the existence of a learning algorithm which converges on every recursive function a partial characterization of type 3 yields a result of independent interest in recursion theory the interplay between mind change complexity and choice of hypothesis space is investigated it is established that for certain concept classes a more expressive hypothesis space can sometimes reduce mind change complexity of learning these classes the notion of mind change bound for behaviourally correct learning is indirectly addressed by employing the above four types to restrict the number of predictive errors of commission in finite error next value learning nv a model equivalent to behaviourally correct learning again natural characterizations for type 2 and type 4 bounds are derived their naturalness is further illustrated by characterizing them in terms of branches of uniformly recursive families of binary trees

mind change complexity of inferring unbounded unions of restricted pattern languages from positive data this paper shows that the mind change complexity of inferring from positive data the class of unbounded unions of languages of regular patterns with constant segment length bound is of the form w w a b assuming that the patterns are defined over a finite alphabet containing at least two elements here a and b are natural numbers and we give tight bounds on their values based on the length of the constant segments and the size of the alphabet of the pattern languages this is to the authors knowledge the first time a natural class of languages has been shown to be inferable with mind change complexity above w w the proof uses the notion of closure operators on a class of languages and also uses the order type of well partial orderings to obtain a mind change bound the inference algorithm presented can be easily applied to a wide range of classes of languages finally we show an interesting connection between proof theory and mind change complexity

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

existence and uniqueness of the solution of a space time periodic reaction diffusion equation this paper is concerned with the study of the periodic solutions and the entire solutions of the equation 1 tu a t x u q t x u f t x u where the diffusion matrix a the advection term q and the reaction term f are periodic in t and x we prove that the sign of the periodic principal eigenvalue associated with the linearized problem determines the existence and the uniqueness of the periodic solution introducing another eigenvalue we are able to state uniqueness conditions for the entire solution and to derive the asymptotic behavior of the solutions of the associated cauchy problem

thin set theorems and cone avoidance the thin set theorem mathsf rt n infty ell asserts the existence for every k coloring of the subsets of natural numbers of size n of an infinite set of natural numbers all of whose subsets of size n use at most ell colors whenever ell 1 the statement corresponds to ramsey s theorem from a computational viewpoint the thin set theorem admits a threshold phenomenon in that whenever the number of colors ell is sufficiently large with respect to the size n of the tuples then the thin set theorem admits strong cone avoidance let d 0 d 1 dots be the sequence of catalan numbers for n geq 1 mathsf rt n infty ell admits strong cone avoidance if and only if ell geq d n and cone avoidance if and only if ell geq d n 1 we say that a set a is mathsf rt n infty ell encodable if there is an instance of mathsf rt n infty ell such that every solution computes a the mathsf rt n infty ell encodable sets are precisely the hyperarithmetic sets if and only if ell 2 n 1 the arithmetic sets if and only if 2 n 1 leq ell d n and the computable sets if and only if d n leq ell

biased movement and the ideal free distribution in some free boundary problems abstract some invasive species can persist in their habitat but eventually spread very slow in a nonlinear fashion to expand their habitat range in order to capture this phenomenon we consider reaction diffusion advection models with a free boundary modeling the spreading and the biased movement of species in one dimensional spatially heterogeneous environments under a condition of low resource quality we find that large advection can lead to the spreading of the species but the spreading speed goes asymptotically to zero moreover we investigate the effect of the resource on the dynamics of the current problem finally we bring the notion of an ideal free distribution ifd into free boundary problems to understand the mechanism such that the species can eventually match the environmental quality perfectly under the current problem setting the ifd may not hold even if the population plays an ideal free strategy we then provide a sufficient condition for the ifd to be reached when using an ideal free strategy

dynamics for a diffusive prey predator model with different free boundaries abstract to understand the spreading and interaction of prey and predator in this paper we study the dynamics of the diffusive lotka volterra type prey predator model with different free boundaries these two free boundaries which may intersect each other as time evolves are used to describe the spreading of prey and predator we investigate the existence and uniqueness regularity and uniform estimates and long time behaviors of global solution some sufficient conditions for spreading and vanishing are established when spreading occurs we provide the more accurate limits of u v as t and give some estimates of asymptotic spreading speeds of u v and asymptotic speeds of g h some realistic and significant spreading phenomena are found

a free boundary problem for the diffusive intraguild predation model with intraspecific competition abstract in this paper we investigate a free boundary problem for the diffusive intraguild predation model with intraspecific competition in one dimensional space the main objective is to portray the asymptotic behavior of spread of an invasive or new predator species via a free boundary in this model we assume both the ig prey and the ig invasive new predator species with intraspecific competition and the ig predator species can only expand further into the new environment from the right end of the initial region in both cases we prove a spreading vanishing dichotomy for this model specifically the ig predator species either successfully spreads to infinity as t at the front and survives in the new environment or spreads within a bounded area and dies out in the long run the long time behavior of r n p and criteria for spreading and vanishing are also obtained

stable and unstable cross grid pkql mixed finite elements for the stokes problem in this paper we develop and analyze a family of mixed finite element methods for the numerical solution of the stokes problem in two space dimensions in these schemes the pressure is interpolated on a mesh of rectangular elements while the velocity is approximated on a triangular mesh obtained by dividing each rectangle into four triangles by its diagonals continuous interpolations of degrees k for the velocity and l for the pressure are considered so the new finite elements are called cross grid p kq l a stability analysis of these approximations is provided based on the macroelement technique of stenberg the lowest order p 1q 1 and p 2q 1 cases are analyzed in detail in the first case a global spurious pressure mode is shown to exist so this element is unstable in the second case however stability is rigorously proved numerical results obtained in these two cases are also presented which confirm the existence of the spurious pressure mode for the p 1q 1 element and the stability of the p 2q 1 element

the metamathematics of stable ramsey s theorem for pairs we show that over the base theory rca0 stable ramsey s the orem for pairs implies neither ramsey s theorem for pairs nor 0 induction

analysis of some mixed elements for the stokes problem abstract in this paper we discuss some mixed finite element methods related to the reduced integration penalty method for solving the stokes problem we prove optimal order error estimates for bilinear constant and biquadratic bilinear velocity pressure finite element solutions the result for the biquadratic bilinear element is new while that for the bilinear constant element improves the convergence analysis of johnson and pitkaranta 1982 in the degenerate case when the penalty parameter is set to be zero our results reduce to some related known results proved in by brezzi and fortin 1991 for the bilinear constant element and bercovier and pironneau 1979 for the biquadratic bilinear element our theoretical results are consistent with the numerical results reported by carey and krishnan 1982 and oden et al 1982

mixed finite element methods reduced and selective integration techniques a unification of concepts the equivalence of certain classes of mixed finite element methods with displacement methods which employ reduced and selective integration techniques is established this enables one to obtain the accuracy of the mixed formulation without incurring the additional computational expense engendered by the auxiliary field of the mixed method applications and numerical examples are presented for problems with constraints which can be difficult to enforce in finite element approximations and have often dictated the use of mixed principles these include thin beams and plates and linear and nonlinear incompressible and nearly incompressible continuum problems in solid and fluid mechanics

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

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

penalty finite element methods for the analysis of stokesian flows a study of a class of finite element methods for the analysis of stokes problem based on the use of exterior penalty formulations is described the effects of selective reduced integration i e the use of quadrature rules for integrating the penalty terms which are of lower order than that required to integrate polynomial approximations of these terms exactly are investigated error estimates are derived and the numerical stability of these methods as depicted by a special babuska brezzi condition is explored in some detail the results of several numerical experiments with these methods are also given

iterative schemes for mixed finite element methods with applications to elasticity and compressible flow problems iterative schemes for mixed finite element methods are proposed and analyzed in two abstract formulations the first one has applications to elliptic equations and incompressible fluid flow problems while the second has applications to linear elasticity and compressible stokes problems these schemes are constructed through iteratively penalizing the mixed finite element scheme of which iterated penalty method and augmented lagrangian method are special cases convergence theorems are demonstrated in abstract formulations in hilbert spaces and applications to individual physical problems are considered as examples theoretical analysis and computational experiments both show that the proposed schemes have very fast convergence a few iterations are normally enough to reduce the iterative error to a prescribed precision numerical examples with continuous and discontinuous coefficients are presented

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

on uniform relationships between combinatorial problems the enterprise of comparing mathematical theorems according to their logical strength is an active area in mathematical logic with one of the most common frameworks for doing so being reverse mathematics in this setting one investigates which theorems provably imply which others in a weak formal theory roughly corresponding to computable mathematics since the proofs of such implications take place in classical logic they may in principle involve appeals to multiple applications of a particular theorem or to non uniform decisions about how to proceed in a given construction in practice however if a theorem q implies a theorem p it is usually because there is a direct uniform translation of the problems represented by p into the problems represented by q in a precise sense formalized by weihrauch reducibility we study this notion of uniform reducibility in the context of several natural combinatorial problems and compare and contrast it with the traditional notion of implication in reverse mathematics we show for instance that for all n j k 1 if j k then ramsey s theorem for n tuples and k many colors is not uniformly or weihrauch reducible to ramsey s theorem for n tuples and j many colors the two theorems are classically equivalent so our analysis gives a genuinely finer metric by which to gauge the relative strength of mathematical propositions we also study weak konig s lemma the thin set theorem and the rainbow ramsey s theorem along with a number of their variants investigated in the literature weihrauch reducibility turns out to be connected with sequential forms of mathematical principles where one wishes to solve infinitely many instances of a particular problem simultaneously we exploit this connection to uncover new points of difference between combinatorial problems previously thought to be more closely related

the stefan problem for the fisher kpp equation abstract we study the fisher kpp equation with a free boundary governed by a one phase stefan condition such a problem arises in the modeling of the propagation of a new or invasive species with the free boundary representing the propagation front in one space dimension this problem was investigated in du and lin 2010 11 and the radially symmetric case in higher space dimensions was studied in du and guo 2011 10 in both cases a spreading vanishing dichotomy was established namely the species either successfully spreads to all the new environment and stabilizes at a positive equilibrium state or fails to establish and dies out in the long run moreover in the case of spreading the asymptotic spreading speed was determined in this paper we consider the non radially symmetric case in such a situation similar to the classical stefan problem smooth solutions need not exist even if the initial data are smooth we thus introduce and study the weak solution for a class of free boundary problems that include the fisher kpp as a special case we establish the existence and uniqueness of the weak solution and through suitable comparison arguments we extend some of the results obtained earlier in du and lin 2010 11 and du and guo 2011 10 to this general case we also show that the classical aronson weinberger result on the spreading speed obtained through the traveling wave solution approach is a limiting case of our free boundary problem here

some logically weak ramseyan theorems abstract we study four families of consequences of ramsey s theorem from the viewpoint of reverse mathematics the first which we call the achromatic ramsey theorem is from a partition relation introduced by erd s hajnal and rado c d r which asserts that for every f r c there exists an infinite h with f h r d the second and third are the free set theorem and the thin set theorem which were introduced by harvey friedman and the last is the rainbow ramsey theorem we show that most theorems from these families are quite weak i e they are strictly weaker than aca 0 over rca 0 interestingly these families turn out to be closely related we establish the so called strong cone avoidance property of most instances of the achromatic ramsey theorem by an induction of exponents then apply this and a similar induction to obtain the strong cone avoidance property of the free set theorem from the strong cone avoidance property of the achromatic ramsey theorem and the free set theorem we derive the strong cone avoidance property of the thin set theorem and the rainbow ramsey theorem it follows easily that a theorem with the strong cone avoidance property does not imply aca 0 over rca 0

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

analysis of some mixed finite element methods related to reduced integration we prove error estimates for the following two mixed finite element methods related to reduced integration a method for stokes problem using rectangular elements with piecewise bilinear approximations for the velocities and piecewise constants for the pressure and one method for a plate problem using bilinear approximations for transversal displacement and rotations and piecewise constants for the shear stress the main idea of the proof in the case of stokes problem is to combine a weak babuska brezzi type stability estimate for the pressure with a superapproximability property for the velocities a similar technique is used in the case of the plate problem