a topos foundation for theories of physics iv categories of systems this paper is the fourth in a series whose goal is to develop a fundamentally new way of building theories of physics the motivation comes from a desire to address certain deep issues that arise in the quantum theory of gravity our basic contention is that constructing a theory of physics is equivalent to finding a representation in a topos of a certain formal language that is attached to the system classical physics arises when the topos is the category of sets other types of theory employ a different topos the previous papers in this series are concerned with implementing this program for a single system in the present paper we turn to considering a collection of systems in particular we are interested in the relation between the topos representation for a composite system and the representations for its constituents we also study this problem for the disjoint sum of two systems our approach to these matters is to construct a category of systems and to find a topos representation of the entire category

fractional laplacian phase transitions and boundary reactions a geometric inequality and a symmetry result we deal with symmetry properties for solutions of nonlocal equations of the type sv f v in rn where s 0 1 and the operator s is the so called fractional laplacian the study of this nonlocal equation is made via a careful analysis of the following degenerate elliptic equation div x u 0on rn 0 x ux f u on rn 0 where 1 1 y rn x 0 and u u y x this equation is related to the fractional laplacian since the dirichlet to neumann operator u r n 1 x ux r n 1 is 1 2 more generally we study the so called boundary reaction equations given by div x u g x u 0on rn 0 x ux f u on rn 0 under some natural assumptions on the diffusion coefficient and on the nonlinearities f and g we prove a geometric formula of poincare type for stable solutions from which we derive a symmetry result in the spirit of a conjecture of de giorgi

covering moving points with anchored disks consider a set of mobile clients represented by n points in the plane moving at constant speed along n different straight lines we study the problem of covering all mobile clients using a set of k disks centered at k fixed centers each disk exists only at one instant and while it does covers any client within its coverage radius the task is to select an activation time and a radius for each disk such that every mobile client is covered by at least one disk in particular we study the optimization problem of minimizing the maximum coverage radius first we prove that although the static version of the problem is polynomial the kinetic version is np hard moreover we show that the problem is not approximable by a constant factor unless p np we then present a generic framework to solve it for fixed values of k which in turn allows us to solve more general optimization problems our algorithms are efficient for constant values of k

semilinear fractional elliptic equations involving measures abstract we study the existence of weak solutions to e u g u in a bounded regular domain in r n n 2 which vanish in r n where denotes the fractional laplacian with 0 1 is a radon measure and g is a nondecreasing function satisfying some extra hypotheses when g satisfies a subcritical integrability condition we prove the existence and uniqueness of weak solution for problem e for any measure in the case where is a dirac measure we characterize the asymptotic behavior of the solution when g r r k 1 r with k supercritical we show that a condition of absolute continuity of the measure with respect to some bessel capacity is a necessary and sufficient condition in order e to be solved

nonlinear boundary value problems relative to harmonic functions abstract we study the problem of finding a function u verifying u 0 in under the boundary condition u n g u on where r n is a smooth domain n is the normal unit outward vector to is a measure on and g a continuous nondecreasing function we give sufficient condition on g for this problem to be solvable for any measure when g r r p 1 r p 1 we give conditions in order an isolated singularity on to be removable we also give capacitary conditions on a measure in order the problem with g r r p 1 r to be solvable for some we also study the isolated singularities of functions satisfying u 0 in and u n g u 0 on 0

estimates on green functions and poisson kernels for symmetric stable processes one of the most basic and most important subfamily of levy processes is symmetric stable processes a symmetric stable process x on rn is a levy process whose transition density p t x y relative to the lebesgue measure is uniquely determined by its fourier transform rn e ix p t x dx e t here must be in the interval 0 2 when 2 we get a brownian motion running with a time clock twice as fast as the standard one brownian motion plays a central role in modern probability theory and has numerous important applications in other scientific areas as well as in many other branches of mathematics thus it has been intensively studied in this paper symmetric stable processes are referred to the case when 0 2 unless otherwise specified in the last few years there has been an explosive growth in the study of physical and economic systems that can be successfully modeled with the use of stable processes stable processes are now widely used in physics operations research queuing theory mathematical finance and risk estimation in some physics literatures symmetric stable processes are called levy flights and they have been applied to a wide range of very complex physics issues such as turbulent diffusion vortex dynamics anomalous diffusion in rotating flows and molecular spectral fluctuations in mathematical finance stable processes can be used to model stock returns in incomplete market for these and more applications of stable processes please see the interesting book 14 by janicki and weron and the references therein and the recent article 15 by klafter shlesinger and zuomofen in order to make precise predictions about natural phenomena and to better cope with these

the generally covariant locality principle a new paradigm for local quantum field theory a new approach to the model independent description of quantum field theories will be introduced in the present work the main feature of this new approach is to incorporate in a local sense the principle of general covariance of general relativity thus giving rise to the concept of a locally covariant quantum field theory such locally covariant quantum field theories will be described mathematically in terms of covariant functors between the categories on one side of globally hyperbolic spacetimes with isometric embeddings as morphisms and on the other side of algebras with unital injective monomorphisms as morphisms moreover locally covariant quantum fields can be described in this framework as natural transformations between certain functors the usual haag kastler framework of nets of operator algebras over a fixed spacetime background manifold together with covariant automorphic actions of the isometry group of the background spacetime can be re gained from this new approach as a special case examples of this new approach are also outlined in case that a locally covariant quantum field theory obeys the time slice axiom one can naturally associate to it certain automorphic actions called relative cauchy evolutions which describe the dynamical reaction of the quantum field theory to a local change of spacetime background metrics the functional derivative of a relative cauchy evolution with respect to the spacetime metric is found to be a divergence free quantity which has as will be demonstrated in an example the significance of an energy momentum tensor up to addition of scalar functions for the locally covariant quantum field theory furthermore we discuss the functorial properties of state spaces of locally covariant quantum field theories that entail the validity of the principle of local definiteness

regularity theory for fully nonlinear integro differential equations we consider nonlinear integro differential equations like the ones that arise from stochastic control problems with purely jump levy processes we obtain a nonlocal version of the abp estimate harnack inequality and interior c 1 regularity for general fully nonlinear integro differential equations our estimates remain uniform as the degree of the equation approaches two so they can be seen as a natural extension of the regularity theory for elliptic partial differential equations

a topos foundation for theories of physics i formal languages for physics this paper is the first in a series whose goal is to develop a fundamentally new way of constructing theories of physics the motivation comes from a desire to address certain deep issues that arise when contemplating quantum theories of space and time our basic contention is that constructing a theory of physics is equivalent to finding a representation in a topos of a certain formal language that is attached to the system classical physics arises when the topos is the category of sets other types of theory employ a different topos in this paper we discuss two different types of language that can be attached to a system s the first is a propositional language pl s the second is a higher order typed language l s both languages provide deductive systems with an intuitionistic logic the reason for introducing pl s is that as shown in paper ii of the series it is the easiest way of understanding and expanding on the earlier work on topos theory and quantum physics however the main thrust o

approximation schemes for covering and packing problems in image processing and vlsi a unified and powerful approach is presented for devising polynomial approximation schemes for many strongly np complete problems such schemes consist of families of approximation algorithms for each desired performance bound on the relative error e o with running time that is polynomial when e is fixed though the polynomiality of these algorithms depends on the degree of approximation e being fixed they cannot be improved owing to a negative result stating that there are no fully polynomial approximation schemes for strongly np complete problems unless np p the unified technique that is introduced here referred to as the shifting strategy is applicable to numerous geometric covering and packing problems the method of using the technique and how it varies with problem parameters are illustrated a similar technique independently devised by b s baker was shown to be applicable for covering and packing problems on planar graphs

a comparison of two topos theoretic approaches to quantum theory the aim of this paper is to compare the two topos theoretic approaches to quantum mechanics that may be found in the literature to date the first approach which we will call the contravariant approach was originally proposed by isham and butterfield and was later extended by doring and isham the second approach which we will call the covariant approach was developed by heunen landsman and spitters

existence of minimizers of functionals involving the fractional gradient in the absence of compactness symmetry and monotonicity abstract we establish general assumptions under which a constrained variational problem involving the fractional gradient and a local nonlinearity admits minimizers

the dirichlet elliptic problem involving regional fractional laplacian in this paper we study the solutions of elliptic equations involving regional fractional laplacian e u f in a bounded regular domain in rn n 2 with c2 boundary subject to dirichlet boundary g on where 12 1 and the operator denotes the regional fractional laplacian we prove that when g 0 problem e admits a unique weak solution under the hypotheses that f l2 f l1 dx and f m where x dist x 2 1 and m is a space of all radon measures satisfying d finally we provide an integration by parts formula for the classical solution of e with boundary data g

singular solutions of some nonlinear elliptic equations abstract we give a classification of isolated singularities of any solution of the equation u u u q 1 0 if 0 is the singular point we prove that there exist two types of singularities when 1 n n 2 and that we have essentially the following in 0 i either u x l q n x 2 q 1 with l q n 2 q 1 2q q 1 n 1 q 1 ii or u x c x where c is any constant and the fundamental solution of the laplacian in r n

a topos for algebraic quantum theory the aim of this paper is to relate algebraic quantum mechanics to topos theory so as to construct new foundations for quantum logic and quantum spaces moti vated by bohr s idea that the empirical content of quantum physics is accessible only through classical physics we show how a noncommutative c algebra of observables a induces a topos t a in which the amalgamation of all of its commutative subalgebras comprises a single commutative c algebra a according to the constructive gelfand duality theorem of banaschewski and mulvey the latter has an internal spectrum a in t a which in our approach plays the role of the quantum phase space of the sys tem thus we associate a locale which is the topos theoretical notion of a space and which intrinsically carries the intuitionistic logical structure of a heyting algebra to a c algebra which is the noncommutative notion of a space in this setting states on a become probability measures more precisely valuations on and self adjoint elements of a define continuous functions more precisely locale maps fromto scott s interval domain noting that open subsets of a correspond to propositions about the system the pairing map that assigns a generalized truth value to a state and a proposition assumes an extremely simple categorical form formulated in this way the quantum theory defined by a is essentially turned into a classical theory internal to the topos t a these results were inspired by the topos theoretic approach to quantum physics pro posed by butterfield and isham as recently generalized by doring and isham

a topos foundation for theories of physics ii daseinisation and the liberation of quantum theory this paper is the second in a series whose goal is to develop a fundamentally new way of constructing theories of physics the motivation comes from a desire to address certain deep issues that arise when contemplating quantum theories of space and time our basic contention is that constructing a theory of physics is equivalent to finding a representation in a topos of a certain formal language that is attached to the system classical physics arises when the topos is the category of sets other types of theory employ a different topos in this paper we study in depth the topos representation of the propositional language pl s for the case of quantum theory in doing so we make a direct link with and clarify the earlier work on applying topos theory to quantum physics the key step is a process we term daseinisation by which a projection operator is mapped to a subobject of the spectral presheaf the topos quantum analog of a classical state space in the second part of the paper we change gear

the boundary trace and generalized boundary value problem for semilinear elliptic equations with coercive absorption in the first part of the paper we establish the existence of a boundary trace for positive solutions of the equation u g x u 0 in a smooth domain n for a general class of positive nonlinearities this class includes every space independent monotone increasing g which satisfies the keller osserman condition as well as degenerate nonlinearities g q of the form g q x u d x u q 1u with 2 and q 1 the boundary trace is given by a positive regular borel measure which may blow up on compact sets in the second part we concentrate on the family of nonlinearities g q determine the critical value of the exponent q for fixed 2 and discuss a positive solutions with an isolated singularity for subcritical nonlinearities and b the boundary value problem for u g q x u 0 with boundary data given by a positive regular borel measure possibly unbounded we show that in the subcritical case the problem possesses a unique solution for every such measure 2003 wiley periodicals inc

multidimensional spectral order for selfadjoint operators abstract the notion of the spectral order is extended to finite families of pairwise commuting bounded and unbounded selfadjoint operators in hilbert space it is shown that the multidimensional spectral order is preserved by transformations represented by spectral integrals of separately increasing borel functions on r in particular the dimensional spectral order is the restriction of product of spectral orders for selfadjoint operators in the context of positive tuples of pairwise commuting selfadjoint operator the relation a b holds if and only if a b for every tuples of nonnegative integer numbers

vigier s theorem for the spectral order and its applications abstract the paper mainly deals with suprema and infima of self adjoint operators in a von neumann algebra m with respect to the spectral order let m s a be the self adjoint part of m and let be the spectral order on m s a we show that a decreasing net in m s a with a lower bound has the infimum equal to the strong operator limit the similar statement is proved for an increasing net bounded above in m s a this version of vigier s theorem for the spectral order is used to describe suprema and infima of nonempty bounded sets of self adjoint operators in terms of the strong operator limit and operator means as an application of our results on suprema and infima we study the order topology on m s a with respect to the spectral order we show that it is finer than the restriction of the mackey topology

classical and quantum probabilities as truth values we show how probabilities can be treated as truth values in suitable sheaf topoi the scheme developed in this paper is very general and applies both in classical and quantum physics on the quantum side the results naturally tie in with the topos approach to quantum theory that has been developed in the last 14 years by the authors and others earlier results on the representation of arbitrary quantum states are complemented with a purely logical perspective

nonlinear fractional laplacian problems with nonlocal gradient terms let n 2 be a smooth bounded domain for s 1 2 1 we consider a problem of the form where 0 is a real parameter f belongs to a suitable lebesgue space and is a nonlocal gradient square term given by depending on the real parameter 0 we derive existence and non existence results the proof of our existence result relies on sharp calderon zygmund type regularity results for the fractional poisson equation with low integrability data we also obtain existence results for related problems involving different nonlocal diffusion terms

clustering to minimize the maximum intercluster distance the problem of clustering a set of points so as to minimize the maximum intercluster distance is studied an o kn approximation algorithm where n is the number of points and k is the number of clusters that guarantees solutions with an objective function value within two times the optimal solution value is presented this approximation algorithm succeeds as long as the set of points satisfies the triangular inequality we also show that our approximation algorithm is best possible with respect to the approximation bound if pz np

a topos foundation for theories of physics iii the representation of physical quantities with arrows this paper is the third in a series whose goal is to develop a fundamentally new way of viewing theories of physics our basic contention is that constructing a theory of physics is equivalent to finding a representation in a topos of a certain formal language that is attached to the system in paper ii we studied the topos representations of the propositional language pl s for the case of quantum theory and in the present paper we do the same thing for the more extensive local language l s one of the main achievements is to find a topos representation for self adjoint operators this involves showing that for any physical quantity a there is an arrow o a r where r is the quantity value object for this theory the construction of o a is an extension of the daseinisation of projection operators that was discussed in paper ii the object r is a monoid object only in the topos setsv h op of the theory and to enhance the applicability of the formalism we apply to r a topos a

generalized p center problems complexity results and approximation algorithms abstract in an earlier paper two alternative p center problems where the centers serving costumers must be chosen so that exactly one node from each of p prespecified disjoint pairs of nodes is selected were shown to be np complete this paper considers a generalized version of these problems in which the nodes from which the p servers are to be selected are partitioned into k sets and the number of servers selected from each set must be within a prespecified range we refer to these problems as the set p center problems we establish that the triangle inequality inequality versions of these problems in which the edge weights are assumed to satisfy the triangle inequality are also np complete we also provide a polynomial time approximation algorithm for the two inequality set p center problems that is optimal for one of the problems in the sense that no algorithm with polynomial running time can provide a better constant factor performance guarantee unless p np for the special case alternative p center problems which we refer to as the pair p center problems we extend the previous results in several ways for example the results mentioned above for the set p center problems also apply to the pair p center problems furthermore we establish and exploit a correspondence between satisfiability and the dominating set type of problems that naturally arise when considering the decision versions of the pair p center problems

regularity of the obstacle problem for a fractional power of the laplace operator given a function and s 0 1 we will study the solutions of the following obstacle problem u in r n s u 0i nr n s u x 0 for those x such that u x x lim x u x 0 we show that when is c 1 s or smoother the solution u is in the space c 1 for every s in the case where the contact set u is convex we prove the optimal regularity result u c 1 s when is only c 1 for a s we prove that our solution u is c 1 for every c 2006 wiley periodicals inc

nonlinear fractional elliptic systems with boundary measure data existence and a priori estimates abstract we are concerned with positive solutions of the fractional elliptic system e s u f v in s v g u in where is an arbitrary domain in r n n 2 s s 1 2 1 and f g c l o c r for some 0 1 we establish universal a priori estimate for positive solutions of e then for c 2 bounded domain we prove the existence of positive solutions of e with prescribed boundary value u and v where are positive bounded measure on and discuss regularity property of the solutions

topos models for physics and topos theory what is the role of topos theory in the topos models for quantum theory as used by isham butterfield doring heunen landsman spitters and others in other words what is the interplay between physical motivation for the models and the mathematical framework used in these models concretely we show that the presheaf topos model of butterfield isham and doring resembles classical physics when viewed from the internal language of the presheaf topos similar to the copresheaf topos model of heunen landsman and spitters both the presheaf and copresheaf models provide a quantum logic in the form of a complete heyting algebra although these algebras are natural from a topos theoretic stance we seek a physical interpretation for the logical operations finally we investigate dynamics in particular we describe how an automorphism on the operator algebra induces a homeomorphism or isomorphism of locales on the associated state spaces of the topos models and how elementary propositions and truth values transform under the action of this homeomorphism also with dynamics the focus is on the internal perspective of the topos

on the complexity of some common geometric location problems given n demand points in the plane the p center problem is to find p supply points anywhere in the plane so as to minimize the maximum distance from a demand point to its respective nearest supply point the p median problem is to minimize the sum of distances from demand points to their respective nearest supply points we prove that the p center and the p median problems relative to both the euclidean and the rectilinear metrics are np hard in fact we prove that it is np hard even to approximate the p center problems sufficiently closely the reductions are from 3 satisfiability

fractional heat equations with subcritical absorption having a measure as initial data we study existence and uniqueness of weak solutions to f tu u h t u 0 in 0 rn with initial condition u 0 in rn where n 2 the operator is the fractional laplacian with 0 1 is a bounded radon measure and h 0 r r is a continuous function satisfying a subcritical integrability condition in particular if h t u t up with 1 and 0 p p 1 2 1 n we prove that there exists a unique weak solution uk to f with k 0 where 0 is the dirac mass at the origin we obtain that uk in 0 rn as k for p 0 1 and the limit of uk exists as k when 1 p p we denote it by u when 1 2 1 n 2 p p p u is the minimal self similar solution of f tu u t up 0 in 0 rn with the initial condition u 0 0 in rn 0 and it satisfies u 0 x 0 for x 0 while if 1 p p then u up where up is the maximal solution of the differential equation y t yp 0 on r

fractional kpz equations with critical growth in the gradient respect to hardy potential abstract in this work we study the existence of positive solution to the fractional quasilinear problem s u u x 2 s u p f in u 0 in u 0 in r n where is a c 1 1 bounded domain in r n n 2 s 0 1 2 s 1 and 0 n s is defined in 3 we assume that f is a non negative function with additional hypotheses as we will see there are deep differences with respect to the case 0 more precisely if 0 there exists a critical exponent p s such that for p p s there is no positive solution moreover p s is optimal in the sense that if p p s there exists a positive solution for suitable data and sufficiently small