global weak solution to the inhomogeneous navier stokes vlasov equations abstract the inhomogeneous navier stokes vlasov equations for fluid particle flows are considered in the three dimensional space the coupling in the fluid particle system arises from the drag force in the fluid equations and the acceleration in the vlasov equation an initial boundary value problem is studied in a bounded domain with large initial data the existence of global weak solution is established through an approximation scheme a fixed point argument energy estimates and a weak convergence method

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

eigenvalues of elliptic boundary value problems with an indefinite weight function on etudie les problemes aux valeurs propres elliptiques autoadjoints au r x u dans un ensemble ouvert r k a est un operateur positif d ordre 2m et le poids r est une fonction qui change de signe dans et peut etre discontinue on determine le comportement asymptotique des valeurs propres

diffusion approximation for linear transport with multiplying boundary conditions we consider the diffusion limit of a suitably rescaled model transport equation in a slab with multiplying boundary conditions as the scaling parameter varepsilon tends to zero we show that for sufficiently smooth data the solution converges in the l2 norm for each t 0 to the solution of a diffusion equation with robin boundary conditions corresponding to an incoming flux the derivation of the diffusive limit is based on an asymptotic expansion which is rigorously justified

the similarity problem for j nonnegative sturm liouville operators abstract sufficient conditions for the similarity of the operator a 1 r x d 2 d x 2 q x with an indefinite weight r x sgn x r x are obtained these conditions are formulated in terms of titchmarsh weyl m coefficients sufficient conditions for the regularity of the critical points 0 and of j nonnegative sturm liouville operators are also obtained this result is exploited to prove the regularity of 0 for various classes of sturm liouville operators this implies the similarity of the considered operators to self adjoint ones in particular in the case r x sgn x and q l 1 r 1 x d x we prove that a is similar to a self adjoint operator if and only if a is j nonnegative the latter condition on q is sharp i e we construct q 1 l 1 r 1 x d x such that a is j nonnegative with the singular critical point 0 hence a is not similar to a self adjoint operator for periodic and infinite zone potentials we show that j positivity is sufficient for the similarity of a to a self adjoint operator in the case q 0 we prove the regularity of the critical point 0 for a wide class of weights r this yields new results for forward backward diffusion equations

a primal formulation for imposing periodic boundary conditions on conforming and nonconforming meshes abstract a variational multiscale discontinuous galerkin vmdg method is developed for microscale modeling of domains containing conforming and non conforming meshes essentially the product of the applied volume average strain or macro strain and the domain diameter acts as an imposed displacement jump within the vmdg terms hence the method is suitable for modeling deformation of both block and truly self periodic representative volume elements rves the primal displacement field and macro strain are the only unknowns because the method eliminates the lagrange multiplier lm enforcement of the kinematic constraint rigorous derivation of the method provides a framework to accommodate either the macro stress or macro strain as the driver of the microscale boundary value problem the method is developed first for finite deformations and then specialized to small deformation kinematics algorithmic modifications to the method are also studied for their effects on tangent symmetry and convergence rate the results from numerical studies for isotropic and anisotropic materials show that the proposed method is robust accurate stable and variationally consistent for modeling complicated conforming and nonconforming rves

on the application of substochastic semigroup theory to fragmentation models with mass loss a linear integro differential equation modelling multiple fragmentation with inherent mass loss is investigated by means of substochastic semigroup theory the existence of a semigroup is established and under natural conditions on certain coefficients the generator of this semigroup is identified this yields in particular a validation of the formal mass loss rate equation for the model

inverse spectral problems for differential pencils on the half line with turning points the inverse spectral problem of recovering pencils of second order differential operators on the half line with turning points is studied we establish properties of the spectral characteristics give a formulation of the inverse problem prove a uniqueness theorem and provide a constructive procedure for the solution of the inverse problem

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

asymptotic spectrum of the linear boltzmann equation with general boundary conditions in finite bodies abstract the purpose of this paper is the study of the spectral properties of both streaming operator and transport operator with general boundary conditions in multidimensional bounded geometry we discuss the asymptotic spectrum existence and nonexistence results of eigenvalues in the half plane c re s t h where s t h stands for the spectral bound of the streaming operator t h next we discuss the irreducibility of the transport semigroup in particular we establish that the transport semigroup is irreducible if the boundary operator is strictly positive afterwards we discuss the strict monotonicity of the leading eigenvalue when it exists of the transport operator with respect to different parameters of the equation our analysis is based essentially on results from the theory of positive linear operators

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

boundary and initial boundary value problems for separable backward forward parabolic problems a backward forward parabolic equation is considered in a strip which is infinite in the timelike direction with boundary conditions on the sides of the strip the unique bounded solution of the problem is given explicitly by separation of variables in addition a similar problem in a semi infinite strip is treated with initial data at the end of the strip it is shown that the solution can be represented arbitrarily closely in the maximum norm by a sum of functions obtain by separation of variables

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

computability 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

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

global weak solutions of the vlasov maxwell system with boundary conditions boundaries occur naturally in physucal systems which satisfy the vlasov maxwell system assume perfect conductor boundary conditions for maxwell and either specular reflection or partial absorption for vlasov then weak solutions with finite energy exist for all time

asymptotic stability of the boltzmann equation with maxwell boundary conditions abstract in a general c 1 domain we study the perturbative cauchy theory for the boltzmann equation with maxwell boundary conditions with an accommodation coefficient in 2 3 1 and discuss this threshold we consider polynomial or stretched exponential weights m v and prove existence uniqueness and exponential trend to equilibrium around a global maxwellian in l x v m of important note is the fact that the methods do not involve contradiction arguments

elliptic eigenvalue problems with eigenparameter dependent boundary conditions abstract let a be a uniformly elliptic second order linear operator on a smooth bounded domain r n we study the eigenvalue problem au u subject to boundary conditions b0u b1u on where bj are linear boundary operators the problem is recast in the form a u u in a hilbert or krein space and results are given on the location and type of the spectrum full and half range completeness and regularity of critical points

indefinite sturm liouville problems and half range completeness where the function h changes sign on i such problems occur in certain physical models particularly in transport theory and statistical physics see 2 9 11 131 and references therein as for the standard eigenvalue problem it is true under quite general conditions and not difficult to prove that there is a set of eigenfunctions un which is complete in an appropriate hilbert space of functions this is known as full range completeness for the physical problems the question of interest is generally that of half range completeness to formulate this we set

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

densities for piecewise deterministic markov processes with boundary abstract we study the existence of densities for distributions of piecewise deterministic markov processes we also obtain relationships between invariant densities of the continuous time process and that of the process observed at jump times in our approach we use functional analytic methods and the theory of linear operator semigroups by imposing general conditions on the characteristics of a given markov process we show the existence of a substochastic semigroup describing the evolution of densities for the process and we identify its generator our main tool is a new perturbation theorem for substochastic semigroups where we perturb both the action of the generator and of its domain allowing to treat general transport type equations with non local boundary conditions a couple of particular examples illustrate our general results

abstract time dependent transport equations abstract we consider the abstract time dependent linear transport equation as an initial boundary value evolution problem in the banach spaces lp 1 p

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

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

half range solutions of indefinite sturm liouville problems abstract for diffusion equations of indefinite sturm liouville type we develop two equivalent methods of constructing explicit representations for the solutions the first method is based on an eigenfunction expansion whereas the second uses a wiener hopf type integral equation and factorization some illustrative examples are worked out

on the classical solutions of the boltzmann equation on etudie le probleme aux valeurs initiales pour l equation de boltzmann on montre que des solutions au sens classique existent pour tout temps si la donnee initiale est assez proche de l equilibre on se restreint a un domaine cubique et on considere seulement la condition aux limites periodique

stability of stationary transport equations with accretive collision operators abstract in this paper we consider transport equations with accretive collision operators we characterize when the equation has a unique solution and show that in this case the solution is stable under small perturbations of the collision operator and the initial value in one case in which there is more than one solution we show how to make a special selection of a solution which is then stable again under small perturbations of both the collision operator and the initial value the results obtained here parallel those obtained earlier for the case where the collision operator is positive semidefinite

half range completeness for the fokker planck equation half range completeness theorems are proved for eigenfunctions associated to the one dimensional fokker planck equation in a semi infinite medium existence and uniqueness results for perfectly absorbing partially absorbing and purely specularly reflecting boundary conditions are deduced for the stationary and time dependent problems similar results are obtained for a slab geometry

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

existence and convergence to equilibrium of a kinetic model for cometary flows a kinetic equation with a relaxation time model for wave particle collisions is considered similarly to the bgk model of gas dynamics it involves a projection onto the set of equilibrium distributions nonlinearly dependent on moments of the distribution function an earlier existence result is extended to bounded domains with reflecting boundaries and to initial conditions permitting vacuum regions the long time behaviour is investigated convergence on compact time intervals shifted to infinity to the set of equilibrium solutions is proven the set of smooth equilibrium solutions is computed

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

computational homogenization of cellular materials abstract in this work we propose to study the behavior of cellular materials using a second order multi scale computational homogenization approach during the macroscopic loading micro buckling of thin components such as cell walls or cell struts can occur even if the behavior of the materials of which the micro structure is made remains elliptic the homogenized behavior can lose its ellipticity in that case a localization band is formed and propagates at the macro scale when the localization occurs the assumption of local action in the standard approach for which the stress state on a material point depends only on the strain state at that point is no longer suitable which motivates the use of the second order multi scale computational homogenization scheme at the macro scale of this scheme the discontinuous galerkin method is chosen to solve the mindlin strain gradient continuum at the microscopic scale the classical finite element resolutions of representative volume elements are considered since the meshes generated from cellular materials exhibit voids on the boundaries and are not conforming in general the periodic boundary conditions are reformulated and are enforced by a polynomial interpolation method with the presence of instability phenomena at both scales the arc length path following technique is adopted to solve both macroscopic and microscopic problems

half range completeness for the fokker planck equation with an external force this paper deals with a generalization of a classical result obtained by r beals and v protopopescu for the fokker planck equations to the case in which a constant external force is present

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

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

a mortar based approach for the enforcement of periodic boundary conditions on arbitrarily generated meshes a new approach for the enforcement of periodic boundary conditions on representative volume element rve problems is presented the strategy is based on a mortar formulation and considers two and three dimensional problems in the finite deformation frame the mortar discretization is combined with the lagrange multiplier method for the enforcement of periodicity constraints at each pair of corresponding boundaries of the rve in a weak integral sense the interpolation of the lagrange multipliers is undertaken with dual shape functions which can be locally eliminated from the rve system of equations by static condensation avoiding an increase on the system size the numerical treatment within a newton based finite element solution procedure for the rve equilibrium problem is discussed in detail the resulting method is able to efficiently enforce periodic configurations over complex rves at finite strains with arbitrarily generated finite element meshes several numerical examples are presented to demonstrate the robustness of the method and the quality of results

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

the diffusion limit of transport equations derived from velocity jump processes in this paper we study the diffusion approximation to a transport equation that describes the motion of individuals whose velocity changes are governed by a poisson process we show that under an appropriate scaling of space and time the asymptotic behavior of solutions of such equations can be approximated by the solution of a diffusion equation obtained via a regular perturbation expansion in general the resulting diffusion tensor is anisotropic and we give necessary and sufficient conditions under which it is isotropic we also give a method to construct approximations of arbitrary high order for large times in a second paper part ii we use this approach to systematically derive the limiting equation under a variety of external biases imposed on the motion depending on the strength of the bias it may lead to an anisotropic diffusion equation to a drift term in the flux or to both our analysis generalizes and simplifies previous derivations that lead to the classical patlak keller segel alt

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

full and partial range eigenfunction expansions for sturm liouville problems with indefinite weights this article is concerned with eigenvalue problems of the form au tu in a hilbert space h where ais a selfadjoint positive operator generated by a second order sturm liouville differential expression and t a selfadjoint indefinite multiplicative operator which is one to one emphasis is on the full range and partial range expansionproperties of the eigenfunctions

a local finite element implementation for imposing periodic boundary conditions on composite micromechanical models recent advances in computational speed have resulted in the ability to model composite materials using larger representative volume elements rves with greater numbers of inclusions than have been previously studied imposing periodic boundary conditions on very large rves can mean enforcing thousands of constraint equations in addition a periodic mesh is essential for enforcing the constraints the present study investigates a method that uses a local implementation of the constraints that does not adversely affect the computational speed the present study demonstrates the method for two dimensional triangular and square rves of periodically spaced regular hexagonal and square arrays of composite material containing fibers of equal radii to impose the boundary conditions along the edges this study utilizes a cubic interpolant to model the displacement field along the matrix edges and a linear interpolant to model the field along the fiber edges it is shown that the method eliminates the need for the conventional node coupling scheme for imposing periodic boundary conditions consequently reducing the number of unknowns to the interior degrees of freedom of the rve along with a small number of global parameters the method is demonstrated for periodic and non periodic mesh designs

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

partial range completeness and existence of solutions to two way diffusion equations separating variables to solve a two way diffusion equation leads to a nonstandard eigenvalue problem in one variable it is shown that the eigenfunctions having negative eigenvalues are complete on the part of the domain where initial conditions are imposed while those with positive eigenvalues are complete where final conditions are imposed the corresponding exponentially growing and exponentially decaying solutions may be used to expand arbitrary solutions on semi infinite intervals in the time variable a natural iterative procedure for obtaining solutions on finite intervals is shown to converge in some cases a linearly growing solution must also be taken into account

estimates on the non real eigenvalues of regular indefinite sturm liouville problems regular sturm liouville problems with indefinite weight functions may possess finitely many non real eigenvalues in this paper we prove explicit bounds on the real and imaginary parts of these eigenvalues in terms of the coefficients of the differential expression

tetrahedral composite finite elements we develop and analyse a composite ct3d tetrahedral element consisting of an ensemble of 12 four node linear tetrahedral elements coupled to a linear assumed deformation defined over the entire domain of the composite element the element is designed to have well defined lumped masses and contact tractions in dynamic contact problems while at the same time minimizing the number of volume constraints per element the relation between displacements and deformations is enforced weakly by recourse to the hu washizu principle the element arrays are formulated in accordance with the assumed strain prescription the formulation of the element accounts for fully non linear kinematics integrals over the domain of the element are computed by a five point quadrature rule the element passes the patch test in arbitrarily distorted configurations our numerical tests demonstrate that ct element has been found to possess a convergence rate comparable to those of linear simplicial elements and that these convergence rates are maintained as the near incompressible limit is approached we have also verified that the element satisfies the babuska brezzi condition for a regular mesh configuration these tests suggest that the ct3d element can indeed be used reliably in calculations involving near incompressible behaviour which arises e g in the presence of unconfined plastic flow

a global existence theorem for the initial boundary value problem for the boltzmann equation when the boundaries are not isothermal we extend the existence theorem recently proved by hamdache for the initial boundary value problem for the nonlinear boltzmann equation in a vessel with isothermal boundaries to more general situations including the case when the boundaries are not isothermal in the latter case a cut off for large speeds is introduced in the collision term of the boltzmann equation

limit sets for multidimensional nonlinear transport equations abstract we propose a hyperbolic generalisation of the well known reaction diffusion equation and study the structure of its limit sets under a dissipativity condition on the nonlinearity we show quasi convergence of the flow saying that the limit set of any orbit is non empty compact connected and contained in the set of equilibrium solutions

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

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

on the initial boundary value problem for the boltzmann equation a theorem on the traces of the solutions of initial boundary value problems for the boltzmann equation is proved this result makes it possible to extend a recent theorem of existence proved by hamdache to more realistic situations

l2 solution of linear non cutoff boltzmann equation with boundary conditions abstract in this paper we are concerned about the existence of solution to linear non cutoff boltzmann equation with boundary conditions in 9 and 18 the existence of the solutions to the kinetic equation and linear boltzmann equation were obtained by characteristic line and iteration methods however their methods are not applicable in the context of non cutoff linear operators to overcome this difficulty we cut off the non cutoff linear operator l to l r the latter is compatible with the method in 9 via this process we are able to use the sequence of solutions associated to l r to approximate the solution associated to l which is corresponding to the original equation in turn obtain the weak solutions of linear non cutoff boltzmann equation with specular and diffusion reflection boundary conditions

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

relatively strictly singular perturbations essential spectra and application to transport operators p strictly singular operator was discussed in a previous paper by k latrach and a jeribi 1998 j math anal appl 225 461485 in the present paper we prove the invariance of the gustafsonweidmann wolf schechter and browder essential spectra of a under relatively strictly singular not necessarily bounded perturba tions on these spaces further a precise characterization of the schechter essential spectrum is given we show that these results are also valid on c where is a compact hausdorff space the results are applied to the one dimensional transport equations with anisotropic scattering and abstract boundary conditions 2000 academic press

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

spectral properties of singular sturm liouville operators with indefinite weight sgn x we consider a singular sturm liouville expression with the indefinite weight sgnx to this expression there is naturally a self adjoint operator in some krein space associated we characterize the local definitizability of this operator in a neighbourhood of 1 moreover in this situation the point 1 is a regular critical point we construct an operator a sgnx id 2 dx 2 q with non real spectrum accumulating to a real point the obtained results are applied to several classes of sturm liouville operators

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