singularities in anisotropic steady state heat conduction using a boundary element method in many heat conduction problems boundaries with sharp corners or abrupt changes in the boundary conditions give rise to singularities of various types which tend to slow down the rate of convergence with decreasing mesh size of any standard numerical method used for obtaining the solution in this paper it is shown how this difficulty may be overcome in the case of an anisotropic medium by a modified boundary element method the standard boundary element method is modified to take account of the form of the singularity without appreciably increasing the amount of computation involved two test examples the first with a singularity caused by an abrupt change in a boundary condition and the second with a singularity caused by a sharp re entrant corner are investigated and numerical results are presented copyright 2002 john wiley sons ltd

the p t degrees of the recursive sets lattice embeddings extensions of embeddings and the two quantifier theory abstract ambos spies 1984a showed that the two basic nondistributive lattices can be embedded in rp t the polynomial time turing degrees of the recursive sets we introduce more general techniques to extend his results to show that every recursive lattice can be embedded in rp t in addition to lattice theoretic representation theorems we use the scheme of priority style arguments coupled with looking back techniques presented in shinoda and slaman 1988 1990 we also generalize the density type results of ladner 1975 and many others to settle the full extension of the embedding problem for rp t combined with the logical analysis of sentences with one alternation of quantifiers shore 1978 lerman 1983 these results suffice to decide the full theory of rp t they also give a strong nonhomogeneity result the p time degrees of the sets recursive in and if desired p time above two distinct sets a and b are almost never isomorphic the situation for the p time many one degrees is quite different we decide the extension of the embedding problem differently than for rp t but not the theory

a uniform approach to obtain diagonal sets in complexity classes abstract a uniform method for constructing sets which diagonalize over certain complexity classes while preserving other complexity properties is given we obtain some known results as well as some new ones as corollaries of our main theorem the new results concern the complexity classes p np co np pspace apt almost polynomial time r random polynomial time and the polynomial hierarchy

sublattices of the polynomial time degrees we show that any countable distributive lattice can be embedded in any interval of polynomial time degrees furthermore the embeddings can be chosen to preserve the least or the greatest element this holds for both polynomial time bounded many one and turing reducibilities as well as for all of the common intermediate reducibilities

a uniform approach to define complexity classes abstract complexity classes are usually defined by referring to computation models and by putting suitable restrictions on them following this approach many proofs of results are tightly bound to the characteristics of the computation model and of its restrictions and therefore they sometimes hide the essential properties which insure the obtained results in order to obtain more general results a uniform family of computation models which encompasses most of the complexity classes of interest is introduced as a first initial set of results derivable from the proposed approach we will give a sufficient and necessary condition for proving separations of relativized complexity classes a characterization of complexity classes with complete languages and a sufficient condition for proving strong separations of relativized complexity classes examples of applications of these results to some specific complexity classes are then given additional results related to separations by sparse oracles can be found in bovet 1991

index sets and presentations of complexity classes this paper draws close connections between the ease of presenting a given complexity class b and the position of the index sets ib lcub i l mi b rcub and jb lcub i miis total l mi b rcub in the arithmetical hierarchy for virtually all classes b studied in the literature the lowest levels attainable are ib 03 and jb 02 the first holds iff b is 02 presentable and the second iff b is recursively presentable a general kind of priority argument is formulated and it is shown that every property enforcible by it is not recursively presentable it follows that the classes of p immune and p biimmune languages in exponential time are not recursively presentable it is shown that for all b with ib 03 many members of b do not provably in true 2 arithmetic belong to b a class h is exhibited such that ih 03 is open and ih 03 implies that the polynomial hierarchy is infinite

the boundary element solution of the cauchy steady heat conduction problem in an anisotropic medium in this paper the iterative algorithm proposed by kozlov et al for the cauchy problem for the laplace equation is extended in order to solve the cauchy steady state heat conduction problem in an anisotropic medium the iterative algorithm is numerically implemented using the boundary element method bem the convergence and the stability of the numerical method as well as various types of accuracy convergence and stopping criteria are investigated the numerical results obtained confirm that provided an appropriate stopping regularization criterion is imposed then the iterative bem produces a convergent and stable numerical solution with respect to increasing the number of boundary elements and decreasing the amount of noise added into the input data an efficient stopping regularization criterion to cease the iterative process is proposed and the rate of convergence of the algorithm is improved by using various relaxation procedures between iterations a new concept of a variable relaxation factor is proposed analytical formulae for the coefficients of the matrices resulting from the direct application of the bem in an anisotropic medium are also presented copyright 2000 john wiley sons ltd

on the structure of cal np_ bbb c this paper deals with complexity classes cal p bbb c and cal np bbb c as they were introduced over the complex numbers by blum shub and smale bull amer math soc 21 1989 p 1 under the assumption cal p bbb c ne cal np bbb c the existence of noncomplete problems in cal np bbb c not belonging to cal p bbb c is established

a fourth order limit 3 expression with nonempty essential spectrum on montre que meme dans le cas du quatrieme ordre il existe des expressions differentielles ordinaires symetriques reelles avec un spectre essentiel non vide qui ne sont pas dans le cas du point limite

structural properties of bounded relations with an application to np optimization problems we introduce the notion bounded relation which comprises most resource bounded reducibilities which can be found in the literature including non uniform bounded reducibilities such as tp poly we state conditions on bounded relations which are again satisfied for most bounded reducibilities and which imply that every countable partial ordering can be embedded into every proper interval of the recursive degrees as corollaries we obtain that every countable partial ordering can be embedded into every proper interval of rec tp poly as well as into every proper interval between either two maximization or two minimization problems in the structures npo e and npo l for the two latter structures we show further that the result about embeddings of partial orderings extends to embeddings of arbitrary countable distributive lattices where in addition the least or the greatest element of the lattice can be preserved among other corollaries we obtain that for both structures every non trivial np optimization problem bounds a minimal pair in connection with embeddings into npo we introduce a representation of maximization or minimization problems by polynomial time computable functions we consider decidability issues w r t this representation and show for example that there is no effective procedure which decides for a subrecursive index of a polynomial time computable function whether the corresponding maximization problem is approximable within a constant factor

minimal pairs and complete problems two sets are said to form a minimal pair for polynomial many one reductions if neither set is in p and the only sets which are polynomially reducible via polynomial many one reductions to both sets are in p we consider the question of which complete sets can be the top that is the supremum of a minimal pair our main result is that any elementary recursive set which is hard for deterministic exponential time cannot be the top of a minimal pair hence in particular any complete set for an elementary recursive class containing exponential time cannot be such a supremum for np complete sets the problem remains open we show here that it is oracle dependent this is the first example of an oracle dependent property which is completely degree theoretic

diagonalization uniformity and fixed point theorems abstract we derive new fixed point theorems for subrecursive classes together with a theorem on the uniformity of certain reductions from a general formulation of the technique of delayed diagonalization this formulation extends the main theorem of u schoning theoret comput sci 18 1982 95 103 to cases which involve infinitely many diagonal classes c k and which allow each c k to contain uncountably many members the main technical work ties the familiar concept of a witness function directly to the often studied cantor set topology on languages and provides a delay construction which refines those due to schoning s breidtbart and d schmidt our a e fixed point theorems do not require that the programming system for the subrecursive class in question be well behaved we compare them to results which do the other theorem is similar to the uniform boundedness theorem of classical analysis and extends work of j grollmann and a selman

the use of neural network approximation models to speed up the optimisation process in electrical impedance tomography a reduced approximation model technique based on neural networks is developed in order to increase the rate of convergence of an evolution strategy es used for solving a non destructive evaluation problem the inverse problem investigated consists of identifying the geometry of discontinuities in a conductive material from cauchy data measurements taken on the boundary in this study we use neural network nn approximation models in order to increase the rate of convergence of the optimisation algorithm and to efficiently detect from a computational time point of view a subsurface cavity such as a circle the algorithm developed by combining evolution strategies and neural networks is found to be a robust fast and efficient method for detecting the size and location of subsurface cavities

diagonalizations over polynomial time computable sets abstract a formal notion of diagonalization is developed which allows to enforce properties that are related to the class of polynomial time computable sets the class of polynomial time computable functions respectively like e g p immunity it is shown that there are sets called p generic which have all properties enforceable by such diagonalizations we study the behaviour and the complexity of p generic sets in particular we show that the existence of p generic sets in np is oracle dependent even if we assume p np

odd order differential expressions with positive supporting coefficients on determine les indices de defaut et le spectre essentiel pour une classe d expressions differentielles d ordre impair

singularities in 2d anisotropic potential problems in multi material corners real variable approach an analysis of singular solutions at corners consisting of several different homogeneous wedges is presented for anisotropic potential theory in plane the concept of transfer matrix is applied for a singularity analysis first of single wedge problems and then of multi material corner problems explicit forms of eigenequations for evaluation of singularity exponent in the case of multi material corners are derived both for all combinations of homogeneous neumann and dirichlet boundary conditions at faces of open corners and for multi material planes with singular interior points perfect transmission conditions at wedge interfaces are considered in both cases it is proved that singularity exponents are real for open anisotropic multi material corners and a sufficient condition for the singularity exponents to be real for anisotropic multi material planes is deduced a case of a complex singularity exponent for an anisotropic multi material plane is reported apparently for the first time in potential theory simple expressions of eigenequations are presented first for open bi material corners and bi material planes and second for a crack terminating at a bi material interface as examples of application of the theory developed here analytical solutions of these eigenequations are presented for interface cracks with any combination of homogeneous boundary conditions along the interface crack faces and also for a special case of a crack perpendicular to a bi material interface a numerical study of variation of the singularity exponent as a function of inclination of a crack terminating at a bi material interface is presented

the density and complexity of polynomial cores for intractable sets let a be a recursive problem not in p lynch has shown that a then has an infinite recursive polynomial complexity core this is a collection c of instances of a such that every algorithm deciding a needs more than polynomial time almost everywhere on c we investigate the complexity of recognizing the instances in such a core and show that every recursive problem a not in p has an infinite core recognizable in subexponential time we further study how dense the core sets for a can be under various assumptions about the structure of a our main results in this direction are that if p np then np complete problems have polynomially nonsparse cores recognizable in subexponential time and that exptime complete problems have cores of exponential density recognizable in exponential time

canonical disjoint np pairs of propositional proof systems we prove that every disjoint np pair is polynomial time many one equivalent to the canonical disjoint np pair of some propositional proof system therefore the degree structure of the class of disjoint np pairs and of all canonical pairs is identical we show that this degree structure is not superficial assuming there exist p inseparable disjoint np pairs every countable distributive lattice can be embedded into every interval of polynomial np degrees of disjoint pairs by maps that preserve the least and greatest element respectively as one consequence of this embedding under the same assumption there exist intermediate disjoint np pairs that is if a b is a p separable disjoint np pair and c d is a p inseparable disjoint np pair then there exist p inseparable incomparable np pairs e f and g h whose degrees lie strictly between a b and c d furthermore between any two disjoint np pairs that are comparable and inequivalent such a diamond exists

the numerical solution of plane potential problems by improved boundary integral equation methods abstract a modified boundary integral bie method which facilitates accurate solution of laplacian boundary value problems is presented this method is designed specifically for treatment of problems in which singularities occur on the interface between two regions with different physical properties and is illustrated by application to two physical problems analytic expressions for the integrals arising in the piecew se linear and piecewise quadratic bie approximations are also presented these analytical expressions afford an appreciable reduction in computational time when compared with previously employed quadrature formulae

minimal pairs for p abstract we prove a general minimal pair theorem which yields as corollaries many results about minimal pairs for p obtained previously by various authors furthermore it yields the new counterintuitive result that there exist arbitrarily complex minimal pairs we also investigate the question of whether minimal pairs in np are low

some remarks on witness functions for nonpolynomial and noncomplete sets in np abstract we present two results about witness functions for sets in np and conp first any set that has a polynomially computable function which witnesses that it is not in conp must be at least np hard it follows from this result that any set in np conp that has a polynomially computable function which witnesses this fact must already be complete for np second if b is any set for which there is a polynomially computable function which witnesses that it is not complete for np by witnessing that some fixed set in np is not in p b then b must already be in np conp thus for two sets in np conp there are no polynomially computable functions which witness that one is not polynomially reducible to the other in proving the first result we introduce the notion of a k creative set and prove that all k creative sets are np complete since these sets seem not to be all polynomially isomorphic we counter the conjecture of berman and hartmanis that all np complete sets are isomorphic to sat with our own conjecture that not all k creative sets are isomorphic to sat the proofs we give are recursion theoretic in style but straightforward

a note on deterministic and nondeterministic time complexity it is proved that there exists for every set b e np co np a set a np co np such that a and b have the same deterministic time complexity note that b is easier nondeterministically than both a and a in that b e np while a np co np we also prove there exists for each b np co np a set a np co np such that the deterministic time complexity of a is less than that of b on some infinite set in this case then a is easier than b deterministically while b is easier than a nondeterministically

a construction of symmetric differential expressions with non empty essential spectrum on presente une methode relativement simple de construction des expressions differentielles symetriques reelles qui ne sont pas dans le cas du point limite mais ont pour spectre essentiel la demi droite reelle

constructing np intermediate problems by blowing holes with parameters of various properties the search for natural np intermediate problems is one of the holy grails within computational complexity ladner s original diagonalization technique for generating np intermediate problems blowing holes has a serious shortcoming it creates problems with a highly artificial structure by arbitrarily removing certain problem instances in this article we limit this problem by generalizing ladner s method to use parameters with various characteristics this allows one to define more fine grained parameters resulting in np intermediate problems where we only blow holes in a controlled subset of the problem we begin by fully characterizing the problems that admit np intermediate subproblems for a broad and natural class of parameterizations and extend the result further such that structural csp restrictions based on parameters that are hard to compute such as tree width are covered thereby generalizing a result by grohe for studying certain classes of problems including csps parameterized by constraint languages we consider more powerful parameterizations first we identify a new method for obtaining constraint languages such that csp are np intermediate the sets can have very different properties compared to previous constructions by for instance bodirsky grohe and provides insights into the algebraic approach for studying the complexity of infinite domain csps second we prove that the propositional abduction problem parameterized by constraint languages admits np intermediate problems this settles an open question posed by nordh zanuttini we provide a framework to obtain np intermediate problems based on diagonalization we categorize different types of parameters that are usable in our framework we construct np intermediate constraint satisfaction problems we resolve the open question whether boolean abduction admits intermediate problems

lattice embeddings for abstract bounded reducibilities we give an abstract account of resource bounded reducibilities as exemplified by the polynomially time or logarithmically space bounded reducibilities of turing truth table and many one type we introduce a small set of axioms that are satisfied for most of the specific resource bounded reducibilities appearing in the literature some of the axioms are of a more algebraic nature such as the requirement that the reducibility under consideration is a reflexive relation while others are formulated in terms of recursion theory and for example are related to delayed computations of arbitrary recursive sets the main technical result shown is that for any reducibility that satisfies these axioms every countable distributive lattice can be embedded into any proper interval of the structure induced on the recursive sets this extends a corresponding result for polynomially time bounded reducibilities due to ambos spies inform and control 65 1985 pp 63 84 as well as a result on embeddings of partial orderings for axiomatically described reducibilities due to mehlhorn j comput system sci 12 1976 pp 147 178

strong nondeterministic polynomial time reducibilities abstract building on the work of adleman and manders 1 we define a new class of reducibilities the strong nondeterministic polynomial time reducibilities we show that several reducibilities in this class differ over the recursive sets and assuming np conp that they each determine different classes of np hard sets comparisons in strength are also made between the strong nondeterministic polynomial time reducibilities and the classes of deterministic and non deterministic polynomial time reducibilities in ladner lynch and selman 6

uniformly hard languages ladner j assoc comput mach 22 1975 155 showed that there are no minimal recursive sets under polynomial time reductions given any recursive set a ladner constructs a set b such that b strictly reduces to a but b does not lie in p the set b does have very long sequences of input lengths of easily computable instances we examine whether ladner s results hold if we restrict ourselves to uniformly hard languages which have no long sequences of easily computable instances under a hard to disprove assumption we show that there exists a minimal recursive uniformly hard set under honest many one polynomial time reductions

treatment of singularities in helmholtz type equations using the boundary element method abstract in many engineering problems boundaries with sharp corners or abrupt changes in the boundary conditions and or the material properties give rise to singularities of various types which tend to slow down the rate of convergence with respect to decreasing the mesh size of any standard numerical method used for obtaining the solution in this paper in order to develop a method which overcomes this difficulty the singular solutions of isotropic and anisotropic helmholtz type equations with homogeneous dirichlet and or neumann boundary conditions in the neighbourhood of a singular point are derived the standard boundary element method bem is then modified to take account of the form of singularity without an appreciable increase in the computational effort and at the same time keeping a uniform discretization three examples are carefully investigated and the numerical results presented show an excellent performance of the approach developed

the recursion theoretic structure of complexity classes abstract we prove easy recursion theoretic results which have as corollaries generalizations of existing diagonalization theorems on complexity classes roughly speaking almost no reasonable time space or even abstract complexity class can be expressed as the non trivial union of two recursively presentable classes which are closed under finite variations e g unless np p np p np complete languages and consequently the non trivial complement of one complexity class in another e g npp provided np p is almost never recursively presentable

honest polynomial degrees and p np abstract we present a relatively simple proof of a result from homer 1986 showing that if nonrecursive sets cannot be minimal for honest polynomial time turing reducibility h t minimal then p np as a corollary to our proof we strengthen homer s result by showing without assuming that p np that there are h t minimal sets for all tally sets we also consider the converse of homer s result providing some evidence that the nonexistence of h t minimal sets may folow from p np in an interesting way finally we consider structural and or computability properties of sets that cannot be h t minimal