high order finite difference techniques for linear singular perturbation boundary value problems abstract three high order finite difference techniques are developed for the second order singularity perturbed linear boundary value problem in one dimension taylor series expansions and error conversions are used for the development of the techniques convergence and stability conditions of these techniques are proved and some numerical results are given

duals of compact lie groups realized in the cuntz algebras and their actions on c algebras abstract as a first step towards a new duality theorem for compact groups we consider a representation category t g of a compact lie group g u d c whose objects are the tensor powers of the defining representation and whose arrows are the intertwiners we associate to t g in a natural way a c algebra og which can be identified with the fixed points of the cuntz algebra od under the natural action of g og carries a canonical endomorphism g and t g can be reconstructed from og g if g su d then og is simple we give conditions for an endomorphism of a unital c algebra a to determine an action of t su d on a such actions correspond to monomorphisms of osu d into a with natural intertwining properties

an efficient numerical method for singular perturbation problems in this paper we propose a method for the numerical solution of singularly perturbed two point boundary value problems bvps first we develop two schemes to integrate initial value problem ivp for system of two first order differential equations and then by using these schemes we solve the bvp precisely we convert the second order bvp into a system of first order differential equations and then apply the numerical schemes to obtain the solution in order to get an initial condition for the system we use the asymptotic approximate solution error estimates are derived and numerical examples are provided to illustrate the present method

symmetries local names and dynamic de allocation of names the semantics of name passing calculi is often defined employing coalgebraic models over presheaf categories this elegant theory lacks finiteness properties hence it is not apt to implementation coalgebras over named sets called history dependent automata are better suited for the purpose due to locality of names a theory of behavioural functors for named sets is still lacking the semantics of each language has been given in an ad hoc way and algorithms were implemented only for the p calculus existence of the final coalgebra for the p calculus was never proved we introduce a language of accessible functors to specify history dependent automata in a modular way leading to a clean formulation and a generalisation of previous results and to the proof of existence of a final coalgebra in a wide range of cases

solving singularly perturbed problems by a weak form integral equation with exponential trial functions abstract the second order singularly perturbed problem is transformed to a singularly perturbed parabolic type partial differential equation by using a fictitious time technique then we use green s second identity to derive a boundary integral equation in terms of the adjoint trefftz test functions namely a weak form integral equation method wfiem it accompanying with the exponential trial functions which are designed to satisfy the boundary conditions automatically can provide very accurate numerical solutions of linear and nonlinear singularly perturbed problems for the latter problem the iterative procedure is convergent very fast

stochastic integration on the cuntz algebra o abstract we develop a theory of non commutative stochastic integration with respect to the creation and annihilation process on the full fock space over l2 r our theory largely parallels the theories of non commutative stochastic ito integration on boson and fermion fock space as developed by r hudson and k r parthasarathy it provides the first example of a non commutative stochastic calculus which does not depend on the quantum mechanical commutation or anticommutation relations but it is based on the theory of reduced free products of c algebras by d voiculescu this theory shows that the creation and annihilation processes on the full fock space over l2 r which generate the cuntz algebra o can be interpreted as a generalized brownian motion we should stress the fact that in contrast to the other theories of stochastic integration our integrals converge in the c norm on o i e uniformly rather than in some state dependent strong operator topology or non commutative l2 norm

an exponentially fitted special second order finite difference method for solving singular perturbation problems an exponentially fitted special second order finite difference method is presented for solving singularly perturbed two point boundary value problems with the boundary layer at one end left or right point a fitting factor is introduced in a tri diagonal finite difference scheme and is obtained from the theory of singular perturbations thomas algorithm is used to solve the system and its stability is investigated to demonstrate the applicability of the method we have solved several linear and non linear problems from the results it is observed that the present method approximates the exact solution very well

universal coalgebra a theory of systems in the semantics of programming finite data types such as finite lists have traditionally been modelled by initial algebras later final coalgebras were used in order to deal with infinite data types coalgebras which are the dual of algebras turned out to be suited moreover as models for certain types of automata and more generally for transition and dynamical systems an important property of initial algebras is that they satisfy the familiar principle of induction such a principle was missing for coalgebras until the work of aczel 1988 on a theory of non wellfounded sets in which he introduced a proof principle nowadays called coinduction it was formulated in terms of bisimulation a notion originally stemming from the world of concurrent programming languages milner 1980 park 1981 using the notion of coalgebra homomorphism the definition of bisimulation on coalgebras can be shown to be formally dual to that of congruence on algebras aczel and mendler 1989 thus the three basic notions of universal algebra algebra homomorphism of algebras and congruence turn out to correspond to coalgebra homomorphism of coalgebras and bisimulation respectively in this paper the latter are taken as the basic ingredients of a theory called universal coalgebra some standard results from universal algebra are reformulated using the afore mentioned correspondence and proved for a large class of coalgebras leading to a series of results on e g the lattices of subcoalgebras and bisimulations simple coalgebras and coinduction and a covariety theorem for coalgebras similar to birkhoff s variety theorem

bisimulation can t be traced in the concurrent language ccs two programs are considered the same if they are bisimilar several years and many researchers have demonstrated that the theory of bisimulation is mathematically appealing and useful in practice however bisimulation makes too many distinctions between programs we consider the problem of adding operations to ccs to make bisimulation fully abstract we define the class of gsos operations generalizing the style and technical advantages of ccs operations we characterize gsos congruence in as a bisimulation like relation called ready simulation bisimulation is strictly finer than ready simulation and hence not a congruence for any gsos language

fitted mesh b spline collocation method for solving self adjoint singularly perturbed boundary value problems in this paper we develop b spline collocation method for solving a class of self adjoint singularly perturbed equations given by 0 1 email protected a x y b x y x g x a x a 0 a x 0 b x b 0 y 0 email protected y 1 email protected use the fitted mesh technique to generate piecewise uniform mesh and use b spline method which leads to a tridiagonal linear system the convergence analysis is given and the method is shown to have uniform convergence of second order numerical illustrations are given in the end to demonstrate the efficiency of our method

endomorphisms of c algebras cross products and duality for compact groups on definit une nouvelle sorte de produit croise d une c algebre par des semigroupes d endomorphismes avec des proprietes reliees a la symetrie de permutation et a l existence de conjugues un groupe compact est alors defini intrinsequement par son action duale sur le produit croise cette construction donne une caracterisation des duaux de groupes compacts abstraits qui est hors d atteinte de la theorie de dualite de taunaka kein et est independante de cette theorie ceci resout le probleme de demontrer l existence d un groupe de jauge global compact en physique des particules etant seulement donnees les observables locales

compact lie groups associated with endomorphisms of c algebras on donne des conditions sur un endomorphisme d une c algebre 7b a avec unite et centre trivial en termes d operateurs d entrelacements pour des puissances de cet endomorphisme qui suffisent a determiner un groupe de lie compact g et une action d un g dual sur 7b a

petri nets are moniods abstract petri nets are widely used to model concurrent systems however their composition and abstraction mechanisms are inadequate we solve this problem in a satisfactory way we start by remarking that place transition petri nets can be viewed as ordinary directed graphs equipped with two algebraic operations corresponding to parallell and sequential composition of transitions a distributive law between the two operations captures a basic fact about concurrency new morphisms are defined mapping single atomic transitions into whole computations thus relating system descriptions at different levels of abstraction categories equipped with products and coproducts corresponding to parallel and nondeterministic compositions are introduced for petri nets with and without initial markings petri net duality is expressed as a duality functor and several new invariants are introduced a tensor product is defined on nets and their category is proved to be symmetric monoidal closed this construction is generalized to a large class of algebraic theories on graphs these results provide a formal basis for expressing the semantics of concurrent languages in terms of petri nets they also provide a new understanding of concurrency in terms of algebraic structures over graphs and categories that should apply to other models besides petri nets and thus contribute to the conceptual unification of concurrency

bialgebras for structural operational semantics bialgebras and distributive laws are an abstract categorical framework to study various flavors of structural operational semantics this paper aims to introduce the reader to the basics of bialgebras for operational semantics and to sketch the state of the art in this research area

bialgebraic methods and modal logic in structural operational semantics bialgebraic semantics invented a decade ago by turi and plotkin is an approach to formal reasoning about well behaved structural operational semantics sos an extension of algebraic and coalgebraic methods it abstracts from concrete notions of syntax and system behaviour thus treating various kinds of operational descriptions in a uniform fashion in this paper bialgebraic semantics is combined with a coalgebraic approach to modal logic in a novel general approach to proving the compositionality of process equivalences for languages defined by structural operational semantics to prove compositionality one provides a notion of behaviour for logical formulas and defines an sos like specification of modal operators which reflects the original sos specification of the language this approach can be used to define sos congruence formats as well as to prove compositionality for specific languages and equivalences

a survey of numerical techniques for solving singularly perturbed ordinary differential equations this survey paper contains a surprisingly large amount of material and indeed can serve as an introduction to some of the ideas and methods of singular perturbation theory starting from prandtl s work a large amount of work has been done in the area of singular perturbations this paper limits its coverage to some standard singular perturbation models considered by various workers and the numerical methods developed by numerous researchers after 1984 2000 the work done in this area during the period 1905 1984 has already been surveyed by the first author of this paper see appl math comput 30 1989 223 for details due to the space constraints we have covered only singularly perturbed one dimensional problems

numerical solutions of linear and nonlinear singular perturbation problems a new method is developed by detecting the boundary layer of the solution of a singular perturbation problem on the non boundary layer domain the singular perturbation problem is dominated by the reduced equation which is solved with standard techniques for initial value problems while on the boundary layer domain it is controlled by the singular perturbation its numerical solution is provided with finite difference methods of which up to sixth order methods are developed the numerical error is maintained at the same level with a constant number of mesh points for a family of singular perturbation problems numerical experiments support the analytical results

non standard difference schemes for singular perturbation problems revisited non standard finite difference techniques for solving various linear two point singularly perturbed boundary value problems are considered they consist of replacing the original differential equation by a related differential equation with a small deviating argument this related problem is then discretized using finite differences this discretized problem is solved using standard matrix methods the accuracy and robustness of these numerical schemes are then carefully investigated and compared to standard centered finite differences in particular earlier claims for the superiority of such schemes over standard finite differences are shown to be invalid

canonical actions on o we show that the group of automorphisms of the cuntz algebra o canonically associated to a group g of unitaries on the generating hilbert space is ergodic if and only if g has no non zero finite dimensional stable subspace it is prime if g is strongly compact and more generally we give necessary and sufficient conditions if the defining representation of g is the direct sum of finite dimensional representations with determinant one the fixed point subalgebra og of o is a simple c algebra these results extend in part to o results previously known for on n

a brief survey on numerical methods for solving singularly perturbed problems in the present paper a brief survey on computational techniques for the different classes of singularly perturbed problems is given this survey is a continuation of work performed earlier by the first author and contains the literature of the work done by the researchers during the years 2000 2009 however some older important relevant papers are also included in this survey we also mentioned those papers which are not surveyed in the previous survey papers by the first author of this paper see appl math comput 30 1989 223 259 130 2002 457 510 134 2003 371 429 for details thus this survey paper contains a surprisingly large amount of literature on singularly perturbed problems and indeed can serve as an introduction to some of the ideas and methods for the singular perturbation problems

a generalization of the cuntz algebras and model actions a cuntz algebra oh is associated functorially with an infinite dimensional hilbert space h it is a simple c algebra distinct from the algebra o introduced by cuntz every locally compact group g acts in a canonical way on oh h l2 g as a galois closed group of automorphisms the fixed point subalgebra og together with the restriction to og of the canonical endomorphism of oh provides an abstract group dual which determines the group if furthermore g is amenable og and oh are isomorphic a result which is in fact valid for finite groups too we also consider a generalization involving a hopf c algebra or more precisely a regular multiplicative unitary

a new duality theory for compact groups the tannaka krein duality theory characterizes the category g of finite dimensional continuous unitary representations of a compact group as a subcategory of the category of hilbert spaces we prove a more powerful result characterizing g as an abstract category every strict symmetric monoidalc category with conjugates which has subobjects and direct sums and for which thec algebra of endomorphisms of the monoidal unit reduces to the complex numbers is isomorphic to a category g for a compact groupg unique up to isomorphism

completely positive maps of the cuntz algebras we construct a covariant functor from the category whose objects are the complex infinite dimensional separable hilbert spaces and whose morphisms are the contractions into the category whose objects are the unitalc algebras and whose morphisms are the completely positive identity preserving maps thec algebras involved are isomorphic to the cuntz algebra o and our functor extends the canonical action of the group of unitaries on the generating hilbert space our result cannot apply to on n

compositional sos and beyond a coalgebraic view of open systems in this paper we address the issue of providing a structured coalgebra presentation of transition systems with algebraic structure on states determined by an equational specification more precisely we aim at representing such systems as coalgebras for an endofunctor on the category of algebras the systems we consider are specified by using arbitrary sos rules which in general do not guarantee that bisimilarity is a congruence we first show that the structured coalgebra representation works only for systems where transitions out of complex states can be derived from transitions out of corresponding component states this decomposition property of transitions indeed ensures that bisimilarity is a congruence for a system not satisfying this requirement next we propose a closure construction which adds context transitions i e transitions that spontaneously embed a state into a bigger context or vice versa the notion of bisimulation for the enriched system coincides with the notion of dynamic bisimilarity for the original one i e with the coarsest bisimulation which is a congruence this is sufficient to ensure that the structured coalgebra representation works for the systems obtained as result of the closure construction

an initial value approach for solving singularly perturbed two point boundary value problems in this paper an initial value approach is presented for solving singularly perturbed two point boundary value problems with the boundary layer at one end left or right point this approach is based on the boundary layer behavior of the solution the method is distinguished by the following fact the given singularly perturbed two point boundary value problem is replaced by three first order initial value problems several linear and non linear problems are solved to demonstrate the applicability of the method it is observed that the present method approximates the exact solution very well

structured coalgebras and minimal hd automata for the π calculus the coalgebraic framework developed for the classical process algebras and in particular its advantages concerning minimal realizations does not fully apply to the calculus due to the constraints on the freshly generated names that appear in the bisimulation in this paper we propose to model the transition system of the calculus as a coalgebra on a category of name permutation algebras and to define its abstract semantics as the final coalgebra of such a category we show that permutations are sufficient to represent in an explicit way fresh name generation thus allowing for the definition of minimal realizations we also link the coalgebraic semantics with a slightly improved version of history dependent hd automata a model developed for verification purposes where states have local names and transitions are decorated with names and name relations hd automata associated with agents with a bounded number of threads in their derivatives are finite and can be actually minimized we show that the bisimulation relation in the coalgebraic context corresponds to the minimal hd antomaton

processes as formal power series a coinductive approach to denotational semantics we characterize must testing equivalence on csp in terms of the unique homomorphism from the moore automaton of csp processes to the final moore automaton of partial formal power series over a certain semiring the final automaton is then turned into a csp algebra operators and fixpoints are defined respectively via behavioural differential equations and simulation relations this structure is then shown to be preserved by the final homomorphism as a result we obtain a fully abstract compositional model of csp phrased in purely set theoretical terms

a congruence rule format for name passing process calculi abstract we introduce a gsos like rule format for name passing process calculi specifications in this format correspond to theories in nominal logic the intended models of such specifications arise by initiality from a general categorical model theory for operational semantics given in this rule format a natural behavioural equivalence a form of open bisimilarity is a congruence

high order parameter uniform numerical method for singular perturbation problems in this paper we systematically describe how to derive a method of higher order for the numerical solution of singularly perturbed ordinary differential equations first we apply this idea to derive a fourth order method for a self adjoint singularly perturbed two point boundary value problem this method is uniformly convergent on a piecewise uniform mesh of shishkin type after we have developed and analyzed a fourth order method we explain with appropriate details how can one obtain the methods of order higher than four which looks straightforward but has not been seen in the literature so far besides these the fourth order e uniformity in the theoretical estimate has been justified by some numerical experiments