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

weak values as context dependent values of observables and born s rule we characterize a value of an observable by a sum rule for generally non commuting observables and a product rule when restricted to a maximal commuting subalgebra of observables together with the requirement that the value is unity for the projection operator of the prepared state and the values are zero for the projection operators of the states which are orthogonal to the prepared state the crucial requirement is that the expectation value and the variance of an observable should be independent of the way of measurement i e the choice of the maximal commuting subalgebra of observables we shall call the value a contextual value we show that the contextual value of an observable coincides with the weak value advocated by aharonov and his colleagues by demanding the consistency of quantum mechanics with kolmogorov s measure theory of probability this also gives a derivation of born s rule which is one of the axioms of conventional quantum mechanics

the dynamics of a quantum bertrand duopoly with differentiated products and heterogeneous expectations abstract based on the bounded rationality and naive expectation the paper sets up a dynamic model of quantum bertrand duopoly game with differentiated products by applying quantum game theory we analyze the effect of quantum entanglement on stability of the quantum equilibrium points and complex dynamic behaviors the result shows that the quantum entanglement can enhance the stability of the system the adjustment speeds of bounded rational player can lead to chaotic characteristics of the system the quantum entanglement can control the system s chaos numerical simulations demonstrate the chaotic features via bifurcation largest lyapunov exponent strange attractors fractal dimensions and sensitivity to initial conditions

implicit schemes and lu decompositions implicit methods for hyperbolic equations are analyzed by constructing lu factorizations it is shown that the solution of the resulting tridiagonal systems in one dimension is well conditioned if and only if the lu factors are diagonally dominant stable implicit methods that have diagonally dominant factors are constructed for hyperbolic equations in n space dimesnions only two factors are required even in three space dimensions acceleration to a steady state is analyzed when the multidimensional backward euler method is used with large time steps it is shown that the scheme approximates a newton raphson iteration procedure

a qualitative approach to quantum probability a likelihood order is defined over linear subspaces of a finite dimensional hilbert space the question arises as to when such an order can be represented by a quantum probability we introduce a few behaviorally plausible axioms that provide the answer in two cases pure state and uniform measure the general problem is answered by using duality like conditions the general problem of characterizing the partial orders that admit a quantum representation by behaviorally justified axioms remains open

various pointer states approaches to polar modular values we theoretically analyze the polar decomposition for quantum modular values under various pointer states approaches we consider both the finite dimensional discrete pointer state and continuous pointer state cases in contrast to that a weak value of an observable is usually divided into its real and imaginary parts here we show that separation from the modulus and phase is necessary to a modular value we show that the modulus of the modular value is related to the pointer post selection conditional probability and the phase of the modular value is connected to the summation of a geometric phase and an intrinsic phase we also discuss a relationship between the modulus and phase and therein the derivative of the phase is related to the derivative of the logarithm of the modulus via a berry simon like connection which is in the form of a weak value as a consequence the modulus phase relation allows us to obtain these polar components whenever the connection is specified one of the possible appli

investigation of the forest fire model on a small world network it is shown that the forest fire model of bak et al run on a square lattice network with additional long range interactions in the spirit of a small world network results in a scale free system reminiscent of self organized criticality without recourse to fine tuning as the number of these long range interactions is increased the cluster size distribution exponent is found to decrease in magnitude as the small world regime is entered indicating a change in its universality class it is suggested that such a model could have applicability in the study of disease spreading in human populations

network induced nonequilibrium phase transition in the game of life a cellular automation model of the game of life on a two dimensional small world network is presented in order to count in long range interactions among living individuals in social or biological systems the density of the life and its fluctuation are calculated respectively the present model exhibits a nonequilibrium phase transition from an inactive sparse state to an active dense one at a certain intermediate value of the network disorder employing finite size scaling analysis we estimate the location of the critical point with p c ensuremath infty ensuremath simeq 0 3685 the transition is of the second order type with power law diverging length we obtain the critical exponents 1 ensuremath nu ensuremath simeq 1 70 ensuremath beta ensuremath simeq 0 50 and ensuremath beta ensuremath nu ensuremath simeq 0 85 the calculated results indicate that the present model may belong to the universality class of directed percolation

derivation of the born rule from operational assumptions the born rule is derived from operational assumptions together with assumptions of quantum mechanics that concern only the deterministic development of the state unlike gleason s theorem the argument applies even if probabilities are defined for only a single resolution of the identity so it applies to a variety of foundational approaches to quantum mechanics it also provides a probability rule for state spaces that are not hilbert spaces

upwind relaxation algorithms for the navier stokes equations analyse des erreurs de troncature de la differenciation spatiale et analyse de la stabilite de l algorithme de relaxation calcul pour une serie d ecoulements a l aide de la forme couche mince des equations calcul du decollement de couche limite induit par chocs

parallel processing for implicit solutions of the navier stokes equations an implicit scheme employing superposition of two three one dimensional solutions in two three dimensions is proposed for the solution of navier stokes equations is proposed boundary conditions are implemented in the discrete governing equations before the super position takes place hence no boundary conditions are needed for the one dimensional solutions details of the method as well as numerical confirmation of the analytical developments are discussed 9 refs

emerging properties of financial time series in the game of life we explore the spatial complexity of conway s game of life a prototypical cellular automaton by means of a geometrical procedure generating a two dimensional random walk from a bidimensional lattice with periodical boundaries the one dimensional projection of this process is analyzed and it turns out that some of its statistical properties resemble the so called stylized facts observed in financial time series the scope and meaning of this result are discussed from the viewpoint of complex systems in particular we stress how the supposed peculiarities of financial time series are often overrated in their importance

quantum probability from decision theory in a recent paper deutsch 1 claims to derive the probabilistic predictions of quantum theory from the non probabilistic axioms of quantum theory and the non probabilistic part of classical decision theory we show that his derivation fails because it includes hidden probabilistic assumptions

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

lower upper implicit schemes with multiple grids for the euler equations on montre qu une methode de surrelaxation successive pour la resolution du schema implicite non factorise est equivalente a une forme modifiee du schema implicite inferieur superieur

weak values with decoherence the weak value of an observable is experimentally accessible by weak measurements as theoretically analyzed by aharonov et al and recently experimentally demonstrated we introduce a w operator associated with the weak values and give a general framework of quantum operations to the w operator in parallel with the kraus representation of the completely positive map for the density operator the decoherence effect is also investigated in terms of the weak measurement by a shift of a probe wavefunction of continuous variable as an application we demonstrate how the geometric phase is affected by the bit flip noise

quantum computing postselection and probabilistic polynomial time i study the class of problems efficiently solvable by a quantum computer given the ability to postselect on the outcomes of measurements i prove that this class coincides with a classical complexity class called pp or probabilistic polynomial time using this result i show that several simple changes to the axioms of quantum mechanics would let us solve pp complete problems efficiently the result also implies as an easy corollary a celebrated theorem of beigel reingold and spielman that pp is closed under intersection as well as a generalization of that theorem due to fortnow and reingold this illustrates that quantum computing can yield new and simpler proofs of major results about classical computation

derivation of the quantum probability law from minimal non demolition measurement one more derivation of the quantum probability rule is presented in order to shed more light on the versatile aspects of this fundamental law it is shown that the change of state in minimal quantum non demolition measurement also known as ideal measurement implies the probability law in a simple way namely the very requirement of minimal change of state put in proper mathematical form gives the well known luders formula which contains the probability rule

blackmail propagation on small world networks abstract the dynamics of the blackmail propagation model based on small world networks is investigated it is found that for a given transmitting probability the dynamical behavior of blackmail propagation transits from linear growth type to logistical growth one with the network randomness p increases the transition takes place at the critical network randomness p c 1 n where n is the total number of nodes in the network for a given network randomness p the dynamical behavior of blackmail propagation transits from exponential decrease type to logistical growth one with the transmitting probability increases the transition occurs at the critical transmitting probability c 1 k where k is the average number of the nearest neighbors the present work will be useful for understanding computer virus epidemics and other spreading phenomena on communication and social networks

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

parallel processing scheme for the navier stokes equations part 1 scheme development using a lower and upper decomposition procedure an implicit parallel processing scheme for the compressible navier stokes equations is developed in this scheme the solution for a multidimensional problem is obtained by the superposition of two independent solution cycles because of the independence of these two solutions they can be performed on separate processors simultaneously to confirm the analytical developments numerical results are obtained using the proposed scheme and compared with other available solutions and experimental data a study of other characteristics of the proposed scheme such as stability symmetry preserving and effects of domain decomposition on convergence and stability is also conducted

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

topology regulates pattern formation capacity of binary cellular automata on graphs abstract we study the effect of topology variation on the dynamic behavior of a system with local update rules we implement one dimensional binary cellular automata on graphs with various topologies by formulating two sets of degree dependent rules each containing a single parameter we observe that changes in graph topology induce transitions between different dynamic domains wolfram classes without a formal change in the update rule along with topological variations we study the pattern formation capacities of regular random small world and scale free graphs pattern formation capacity is quantified in terms of two entropy measures which for standard cellular automata allow a qualitative distinction between the four wolfram classes a mean field model explains the dynamic behavior of random graphs implications for our understanding of information transport through complex network based systems are discussed

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

quantum measuring processes of continuous observables the purpose of this paper is to provide a basis of theory of measurements of continuous observables we generalize von neumann s description of measuring processes of discrete quantum observables in terms of interaction between the measured system and the apparatus to continuous observables and show how every such measuring process determines the state change caused by the measurement we establish a one to one correspondence between completely positive instruments in the sense of davies and lewis and the state changes determined by the measuring processes we also prove that there are no weakly repeatable completely positive instruments of nondiscrete observables in the standard formulation of quantum mechanics so that there are no measuring processes of nondiscrete observables whose state changes satisfy the repeatability hypothesis a proof of the wigner araki yanase theorem on the nonexistence of repeatable measurements of observables not commuting conserved quantities is given in our framework we also discuss the implication of these results for the recent results due to srinivas and due to mercer on measurements of continuous observables

weak measurements of trajectories in quantum systems classical bohmian and sum over paths weak values obtained from weak measurements attempt to describe the properties of a quantum system as it evolves from an initial to a final state without practically altering this evolution trajectories can be defined from weak measurements of the position or inferred from weak values of the momentum operator the former can be connected to the asymptotic form of the feynman propagator and the latter to bohmian trajectories employing a time dependent oscillator as a model this work analyzes to what extent weak measurements can shed light on the underlying dynamics of a quantum system expressed in terms of trajectories in particular by comparing the two approaches

parallel processing scheme for the navier stokes equations part 2 parallel implementation parallel performance of two implicit schemes for solving the compressible navier stokes equations on two parallel computational platforms using the standard portable message passing interface is studied these two schemes are a lower upper cycle independent scheme developed by the authors and the symmetric successive overrelaxation scheme representative of cycle dependent schemes the parallel computational platforms used here are the lewis advanced cluster environment which is an example of a network of workstations and the ohio supercomputer cray t3d massive parallel computing environment two versions of the lower upper cycleindependent scheme are studied version i where one processor per block is used and version ii where two processors per block are used however only version i is possible with the symmetric successive overrelaxation scheme thecompetitivenessoftheworkstationclustertothecrayt3disdemonstrated moreover parallelperformance results indicate that version i of the lower upper cycle independent scheme is superior to version ii in the range of a moderate number of processors however in the future as the number of processors increases and the communication cost decreases this scheme will provide a means for using larger computational power relative to version i that would otherwise be idle

strange weak values we develop a formal theory of the weak values with emphasis on the consistency conditions and a probabilistic interpretation in the counter factual processes we present the condition for the choice of the post selected state to give a negative weak value of a given projection operator and strange values of an observable in general the general framework is applied to hardy s paradox and the spin 1 2 system to explicitly address the issues of counter factuality and strange weak values the counter factual arguments which characterize the paradox specify the pre selected state and a complete set of the post selected states clarifies how the strange weak values emerge

complete description of a quantum system at a given time a generalization of the description of a quantum system in the time interval between two measurements is presented a new concept of a generalized state is introduced generalized states yield a complete description of a quantum system when information about the system is available both from the past snd from the future the formalism of generalized states provides a natural language for describing many peculiar situations in particular situations in which one can ascertain the result of a measurement of any one of several non commuting variables are analysed weak measurements on quantum systems described by generalized states are discussed the relation belween weak and strong measurements is investigated

quantum theory of probability and decisions the probabilistic predictions of quantum theory are conventionally obtained from a special probabilistic axiom but that is unnecessary because all the practical consequences of such predictions fo

game of life on the equal degree random lattice an effective matrix method is performed to build the equal degree random edr lattice and then a cellular automaton game of life on the edr lattice is studied by monte carlo mc simulation the standard mean field approximation mfa is applied and then the density of live cells is given 0 37017 by mfa which is consistent with the result 0 37 0 003 by mc simulation