countable nondeterminism and random assignment four semantics for a small programming language involving unbounded but countable nondeterminism are provided these comprise an operational semantics two state transformation semantics based on the egli milner and smyth orders respectively and a weakest precondition semantics their equivalence is proved a hoare like proof system for total correctness is also introduced and its soundness and completeness in an appropriate sense are shown finally the recursion theoretic complexity of the notions introduced is studied admission of countable nondeterminism results in a lack of continuity of various semantic functions and this is shown to be necessary for any semantics satisfying appropriate conditions in proofs of total correctness one resorts to the use of countable ordinals and it is shown that all recursive ordinals are needed

the m calculus as an assertion language for fairness arguments abstract various principles of proof have been proposed to reason about fairness this paper addresses for the first time the question in what formalism such fairness arguments can be couched to wit we prove that park s monotone first order calculus augmented with constants for all recursive ordinals can serve as an assertion language for proving fair termination of do loops in particular the weakest precondition for fair termination of a loop w r t some postcondition is definable in it the relevance of this result to proving eventualities in the temporal logic formalism of manna and pnuelis in foundations of computer science iv part 2 math centre tracts vol 159 math centrum amsterdam 1983 is discussed

fair derivations in context free grammars this paper connects between notions of pure formal language theory and nondeterministic programming the notion of a fair derivation in a context free grammar is defined whereby for every variable appearing infinitely often in sentential forms of an infinite derivation each of its rules is used infinitely often a context free language is fairly generated if it has a grammar all of whose fair derivations are finite it is proved that a context free grammar is fairly terminating iff it is non expansive

a concave convex elliptic problem involving the fractional laplacian we study a nonlinear elliptic problem defined in a bounded domain involving fractional powers of the laplacian operator together with a concave convex term we completely characterize the range of parameters for which solutions of the problem exist and prove a multiplicity result we also prove an associated trace inequality and some liouville type results

alternating states for dual nondeterminism in imperative programming the refinement calculus of back morgan morris and others is based on monotone predicate transformers weakest preconditions where conjunctions stand for demonic choices between commands and disjunctions for angelic choices arbitrary monotone predicate transformers cannot be modelled by relational semantics but can be modelled by so called multirelations results of morris indicate however that the natural domain for the combination of demonic and angelic choice is the free distributive completion fdc of the state space the present paper provides a new axiomatization and more explicit construction of the fdc of an arbitrary ordered set the fdc concept is self dual but the construction is not we therefore determine the duality function from the fdc to the dual of the fdc of the dual ordered set the elements of the fdc are classified according to their conjunctivity and disjunctivity the theory is applied to imperative programming with operators for sequential composition and demonic and angelic choice the theory based on the fdc is shown to be equivalent to a weakest precondition theory for up closed predicates if the order is discrete i e the equality relation the fdc turns out to be the domain of the choice semantics of back and von wright whereas up closed multirelations are functions towards this domain

simulations in coalgebra a new approach to simulations is proposed within the theory of coalgebras by taking a notion of order on a functor as primitive such an order forms a basic building block for a lax relation lifting or relator as used by other authors simulations appear as coalgebras of this lifted functor and similarity as greatest simulation two way similarity is then similarity in both directions in general it is different from bisimilarity in the usual coalgebraic sense but a sufficient condition is formulated and illustrated to ensure that bisimilarity and two way similarity coincide also suitable conditions are identified which ensures that similarity on a final coalgebra forms an algebraic dcpo structure this involves a close investigation of the iterated applications fn 0 and fn 1 of a functor f with an order to the initial algebras and final objects

ground state solutions of pohozaev type for fractional choquard equations with general nonlinearities abstract in this paper we study the fractional choquard equation s u u x f u f u in r n where n 3 0 s 1 0 min n 4 s and f c r r satisfies the general berestycki lions conditions combining constrained variational method with deformation lemma we obtain a ground state solution of pohozaev type for the above equation the result improves some ones in shen et al 2016

meromorphic extension of the resolvent on complete spaces with asymptotically constant negative curvature abstract we give a detailed description of the schwartz kernel of the resolvent of the laplacian on a certain class of complete riemannian manifolds with negative sectional curvature near infinity these manifolds have compactifications as c manifolds with boundary on which the metric is conformai to a c metric up to the boundary with conformal factor 2 where is a defining function for the boundary the modified resolvent r s 2 n 1 defined and analytic in r n is shown to extend to be meromorphic in the whole complex plane when s 2 is equal to the sectional curvature at infinity for hyperbolic spaces obtained as quotients hn 1 where is a discrete group of fractional linear transformations having a geometrically finite fundamental domain without cusps at infinity s is constant we then deduce from the existence of this meromorphic extension of the resolvent the existence of a similar extension for the eisenstein series the basic method used to produce a parametrix for the laplacian is part of a more general construction by g mendoza and the second author but since this is not yet available the present paper has been made essentially self contained

aleksandrov bakelman pucci type estimates for integro differential equations in this work we provide an aleksandrov bakelman pucci type estimate for a certain class of fully nonlinear elliptic integro differential equations the proof of which relies on an appropriate generalization of the convex envelope to a nonlocal fractional order setting and on the use of riesz potentials to interpret second derivatives as fractional order operators this result applies to a family of equations involving some nondegenerate kernels and as a consequence provides some new regularity results for previously untreated equations furthermore this result also gives a new comparison theorem for viscosity solutions of such equations which depends only on the l and ln norms of the right hand side in contrast to previous comparison results which utilize the continuity of the right hand side for their conclusions these results appear to be new even for the linear case of the relevant equations

fractional laplacian in conformal geometry abstract in this note we study the connection between the fractional laplacian operator that appeared in the recent work of caffarelli and silvestre and a class of conformally covariant operators in conformal geometry

infinite trees markings and well foundedness abstract a necessary and sufficient condition for a given marked tree to have no infinite paths satisfying a given formula is presented the formulas are taken from a language introduced by harel covering a wide scale of properties of infinite paths including most of the known notions of fairness this condition underlies a proof rule for proving that a nondeterministic program has no infinite computations satisfying a given formula interpreted over state sequences we also show two different forms of seemingly more natural necessary and sufficient conditions to be inadequate

convergence of nonlocal threshold dynamics approximations to front propagation in this note we prove that appropriately scaled threshold dynamics type algorithms corresponding to the fractional laplacian of order 0 2 converge to moving fronts when 1 the resulting interface moves by weighted mean curvature while for 1 the normal velocity is nonlocal of fractional type the results easily extend to general nonlocal anisotropic threshold dynamics schemes

probabilistic termination versus fair termination note abstract in this note we show that probabilistic termination of concurrent programs is in many cases much simpler than the fair one for a wide class of definitions of probabilistic termination we may express termination by 0 2 arithmetic formula whereas the fair termination can be expressed only by second order arithmetic formula proof of fair termination usually needs induction on recursive ordinals but proof of probabilistic termination has the complexity equivalent to that of deterministic program termination

regularity estimates for the solution and the free boundary of the obstacle problem for the fractional laplacian we use a characterization of the fractional laplacian as a dirichlet to neumann operator for an appropriate differential equation to study its obstacle problem we write an equivalent characterization as a thin obstacle problem in this way we are able to apply local type arguments to obtain sharp regularity estimates for the solution and study the regularity of the free boundary

on the regularity of solutions to the yamabe equation and the existence of smooth hyperboloidal initial data for einstein s field equations the regularity of the solutions to the yamabe problem is considered in the case of conformally compact manifolds and negative scalar curvature the existence of smooth hyperboloidal initial data for einstein s field equations is demonstrated

complete conformal metrics with negative scalar curvature in compact riemannian manifolds soit m g une variete de riemann compacte de dimension n 3 et soit une sous variete lisse fermee de dimension d alors il y a une metrique conforme complete sur m m a courbure scalaire negative constante si et seulement si d n 2 2

on fractional schrodinger systems of choquard type abstract in this article we first employ the concentration compactness techniques to prove existence and stability results of standing waves for nonlinear fractional schrodinger choquard equation i t a s 2 1 x n p p 2 in r n 1 where n 2 0 1 0 n s 2 2 4 n p 2 1 2 n and the constants a are nonnegative satisfying a 0 we then extend the arguments to establish similar results for coupled standing waves of nonlinear fractional schrodinger systems of choquard type the same argument works for equations with an arbitrary number of combined nonlinearities and when x n is replaced by a more general convolution potential k r n 0 under certain assumptions

drift diffusion equations with fractional diffusion and the quasi geostrophic equation motivated by the critical dissipative quasi geostrophic equation we prove that drift diffusion equations with l 2 initial data and minimal assumptions on the drift are locally holder continuous as an application we show that solutions of the quasi geostrophic equation with initial l 2 data and critical diffusion 1 2 are locally smooth for any space dimension

a proof rule for fair termination of guarded commands we present a proof rule for fairly terminating guarded commands based on a well foundedness argument the rule is applied to several examples and proved to be sound and semantically complete w r t an operational semantics of computation trees the rule is related to another rule suggested by lehmann pnueli and stavi in proc internat colloq automata lang and programming 81 acre july 1981 by showing that the semantic completeness of the lps rule follows from the completeness or ours

multi bump solutions for fractional nirenberg problem abstract we consider the multi bump solutions of the following fractional nirenberg problem 0 1 s u k x u n 2 s n 2 s u 0 in r n where s 0 1 and n 2 2 s if k is a periodic function in some k variables with 1 k n 2 s 2 we proved that 0 1 has multi bump solutions with bumps clustered on some lattice points in r k via lyapunov schmidt reduction it is also established that the 0 1 has an infinite many bump solutions with bumps clustered on some lattice points in r n which is isomorphic to z k

regularity properties of nonlocal minimal surfaces via limiting arguments abstract we prove an improvement of flatness result for nonlocal minimal surfaces which is independent of the fractional parameter s when s 1 as a consequence we obtain that all the nonlocal minimal cones are flat and that all the nonlocal minimal surfaces are smooth when the dimension of the ambient space is less or equal than 7 and s is close to 1

interpretations of recursion under unbounded nondeterminacy abstract a language is constructed that supports arbitrary atomic statements composition alternatives and mutual recursion in the presence of unbounded nondeterminacy the concept of interpretation is defined axiomatically by operational means a standard interpretation is constructed which is proved to be the smallest interpretation with respect to the egli milner ordering the corresponding weakest precondition and weakest liberal precondition are characterized as smallest and largest preparator functions we characterize the interpretations which upon divergence cannot deliver unjustified meaningful values these interpretations form a spectrum that ranges from standard semantics over various forms of fair semantics to the so called friendly semantics where error is signalled only if divergence can become inevitable after finitely many steps we determine the corresponding preparator functions

fixpoint semantics and simulation abstract a general functorial framework for recursive definitions is presented in which simulation of a definition scheme by another one implies an ordering between the values defined by these schemes in an arbitrary model under mild conditions on the functor involved the converse implication also holds a model is constructed such that if the values defined are ordered there is a simulation between the definition schemes the theory is illustrated by applications to context free grammars recursive procedures in imperative languages and simulation and bisimulation of processes

deadlock and fairness in morphisms of transition systems in order to investigate deadlock free implementations of processes a formalization is proposed in which processes are elements of transition systems morphisms of transition systems are introduced as an approximate formalization of implementation of processes a morphism f x y shows deadlock if there is an implementing process x x such that x has no transitions and that the implemented process f x y has transitions the morphism is called strict if the implementing processes x provide all transitions of the implemented processes f x strict morphisms are used to clarify the problem of a universal domain of processes in which bisimilar processes are identified to talk of fairness the alphabet of actions is structured by a family bj j j of subsets each subset bj formalizes one component process which is to be treated fairly infinitely many components are admitted a morphism f x y is said to be fair if every execution sequence in the transition system x such that set bj is enabled infinitely many times by the image sequence in y activates set bj infinitely many times for a given transition system y we construct a family of fair and deadlock free morphisms to y which is universal in a well defined sense this construction is applied to the fair communicating merge of a family of processes

effective transformations on infinite trees with applications to high undecidability dominoes and fairness elementary translations between various kinds of recursive trees are presented it is shown that trees of either finite or countably infinite branching can be effectively put into one one correspondence with infinitely branching trees in such a way that the infinite paths of the latter correspond to the p abiding infinite paths of the former here p can be any member of a very wide class of properties of infinite paths for many properties the converse holds too two of the applications involve a the formulation of large classes of highly undecidable variants of classical computational problems and in particular easily describable domino problems that are iii 1 1 complete and b the existence of a general method for proving termination of nondeterministic or concurrent programs under any reasonable notion of fairness

a modular approach to defining and characterising notions of simulation we propose a modular approach to defining notions of simulation and modal logics which characterise them we use coalgebras to model state based systems relators to define notions of simulation for such systems and inductive techniques to define the syntax and semantics of modal logics for coalgebras we show that the expressiveness of an inductively defined logic for coalgebras w r t a notion of simulation follows from an expressivity condition involving one step in the definition of the logic and the relator inducing that notion of simulation moreover we show that notions of simulation and associated characterising logics for increasingly complex system types can be derived by lifting the operations used to combine system types to a relational level as well as to a logical level we use these results to obtain baltag s logic for coalgebraic simulation as well as notions of simulation and associated logics for a large class of non deterministic and probabilistic systems

ten years of hoare s logic a survey part ii nondeterminism a survey of various results concerning the use of hoare s logic in proving correctness of nondeterministic programs is presented various proof systems together with the example proofs are given and the corresponding soundness and completeness proofs of the systems are discussed programs allowing bounded and countable nondeterminism are studied proof systems deal with partial and total correctness freedom of failure and the issue of fairness the paper is a continuation of part i by apt 1981 where various results concerning hoare s approach to proving correctness of sequential programs are presented

on higher dimensional singularities for the fractional yamabe problem a nonlocal mazzeo pacard program we consider the problem of constructing solutions to the fractional yamabe problem which are singular at a given smooth submanifold for which we establish the classical gluing method of mazzeo and pacard j differential geom 1996 for the scalar curvature in the fractional setting this proof is based on the analysis of the model linearized operator which amounts to the study of a fractional order ordinary differential equation ode thus our main contribution here is the development of new methods coming from conformal geometry and scattering theory for the study of nonlocal odes note however that no traditional phase plane analysis is available here instead we first provide a rigorous construction of radial fast decaying solutions by a blowup argument and a bifurcation method then second we use conformal geometry to rewrite this nonlocal ode giving a hint of what a nonlocal phase plane analysis should be third for the linear theory we use complex analysis and some non euclidean harmonic analysis to examine a fractional schrodinger equation with a hardy type critical potential we construct its green s function deduce fredholm properties and analyze its asymptotics at the singular points in the spirit of frobenius method surprisingly enough a fractional linear ode may still have a 2 dimensional kernel as in the second order case