social balance as a satisfiability problem of computer science reduction of frustration was the driving force in an approach to social balance as it was recently considered by antal t antal p l krapivsky and s redner phys rev e 72 036121 2005 we generalize their triad dynamics to k cycle dynamics for arbitrary integer k we derive the phase structure determine the stationary solutions and calculate the time it takes to reach a frozen state the main difference in the phase structure as a function of k is related to k being even or odd as a second generalization we dilute the all to all coupling as considered by antal to a random network with connection probability w 1 interestingly this model can be mapped to a satisfiability problem of computer science the phase of social balance in our original interpretation then becomes the phase of satisfaction of all logical clauses in the satisfiability problem in common to the cases we study the ideal solution without any frustration always exists but the question actually is as to whether this solution can be found by means of a local stochastic algorithm within a finite time the answer depends on the choice of parameters after establishing the mapping between the two classes of models we generalize the social balance problem to a diluted network topology for which the satisfiability problem is usually studied on the other hand in connection with the satisfiability problem we generalize the random local algorithm to a p random local algorithm including a parameter p that corresponds to the propensity parameter in the social balance problem the qualitative effect of the inclusion of this parameter is a bias towards the optimal solution and a reduction of the needed simulation time

the ito clifford integral abstract we present a theory of non commutative stochastic integration analogous to the ito theory it is shown that wick products of fermi fields define martingales and that stochastic integrals with respect to these are defined for adapted operatorvalued square integrable integrands for square integrable martingales associated with an arbitrary probability gage space a stochastic integral is defined and a doob meyer decomposition for supermartingales obtained

opinion dynamics in social networks with hostile camps consensus vs polarization most of the distributed protocols for multi agent consensus assume that the agents are mutually cooperative and trustful and so the couplings among the agents bring the values of their states closer opinion dynamics in social groups however require beyond these conventional models due to ubiquitous competition and distrust between some pairs of agents which are usually characterized by repulsive couplings and may lead to clustering of the opinions a simple yet insightful model of opinion dynamics with both attractive and repulsive couplings was proposed recently by c altafini who examined first order consensus algorithms over static signed graphs this protocol establishes modulus consensus where the opinions become the same in modulus but may differ in signs in this paper we extend the modulus consensus model to the case where the network topology is an arbitrary time varying signed graph and prove reaching modulus consensus under mild sufficient conditions of uniform connectivity of the graph for cut balanced graphs not only sufficient but also necessary conditions for modulus consensus are given

quantum risk sensitive estimation and robustness this paper studies a quantum risk sensitive estimation problem and investigates robustness properties of the filter this is a direct extension to the quantum case of analogous classical results all investigations are based on a discrete approximation model of the quantum system under consideration this allows us to study the problem in a simple mathematical setting we close the paper with some examples that demonstrate the robustness of the risk sensitive estimator

temperature states on loop groups boson fermion correspondence and a determinant identity the boson fermion correspondence for the representation of the car algebra over l2 s1 c defined by the kms state with chemical potential is considered and the non bijectivity shown using an alternative formulation the correlations are recalculated leading to a determinant identity reminiscent of szego s theorem

finding community structure in networks using the eigenvectors of matrices we consider the problem of detecting communities or modules in networks groups of vertices with a higher than average density of edges connecting them previous work indicates that a robust approach to this problem is the maximization of the benefit function known as modularity over possible divisions of a network here we show that this maximization process can be written in terms of the eigenspectrum of a matrix we call the modularity matrix which plays a role in community detection similar to that played by the graph laplacian in graph partitioning calculations this result leads us to a number of possible algorithms for detecting community structure as well as several other results including a spectral measure of bipartite structure in networks and a centrality measure that identifies vertices that occupy central positions within the communities to which they belong the algorithms and measures proposed are illustrated with applications to a variety of real world complex networks

trace formula of linear continuous time periodic systems via the harmonic lyapunov equation in the former part of this paper a trace formula is established for the h 2 norms of a class of finite dimensional linear continuous time periodic fdlcp systems based on the solution of the harmonic lyapunov equations this trace formula is quite similar in form to what we have for the h 2 norm of an lti continuous time system apart from the fact that infinite dimensional matrices are involved in the fdlcp setting based on this formula in the latter part of this paper some trace formulas are developed via the approximate modelling and truncating approach which are numerically implementable in most practical fdlcp systems there are numerical examples to illustrate the efficacy of the trace formulas suggested

network trending leadership followership and neutrality among companies a random matrix approach abstract in this article we analyze the cross correlation between returns of different stocks to answer the following important questions the first one is if there exists collective behavior in a financial market how could we detect it and the second question is is there a particular company among the companies of a market as the leader of the collective behavior or is there no specified leadership governing the system similar to some complex systems we use the method of random matrix theory to answer the mentioned questions cross correlation matrix of index returns of four different markets is analyzed the participation ratio quantity related to each matrices eigenvectors and the eigenvalue spectrum is calculated we introduce shuffled matrix created of cross correlation matrix in such a way that the elements of the later one are displaced randomly comparing the participation ratio quantities obtained from a correlation matrix of a market and its related shuffled one on the bulk distribution region of the eigenvalues we detect a meaningful deviation between the mentioned quantities indicating the collective behavior of the companies forming the market by calculating the relative deviation of participation ratios we obtain a measure to compare the markets according to their collective behavior answering the second question we show there are three groups of companies the first group having higher impact on the market trend called leaders the second group is followers and the third one is the companies who have not a considerable role in the trend the results can be utilized in portfolio construction

brief h2 and h norm computations of linear continuous time periodic systems via the skew analysis of frequency response operators the h 2 and h norm computations of finite dimensional linear continuous time periodic fdlcp systems through the frequency response operators defined by steady state analysis are discussed by the skew truncation the h 2 norm can be reached to any degree of accuracy by that of an asymptotically equivalent linear time invariant lti continuous time system the h norm can be approximated by the maximum singular value of the frequency response of an asymptotically equivalent lti continuous time system over a certain frequency range via the modified skew truncation by the latter result a hamiltonian test is proved for fdlcp systems in an lti fashion based on which a modified bisection algorithm is developed

epidemic spreading on evolving signed networks most studies of disease spreading consider the underlying social network as obtained without the contagion though epidemic influences people s willingness to contact others a friendly contact may be turned to unfriendly to avoid infection we study the susceptible infected disease spreading model on signed networks in which each edge is associated with a positive or negative sign representing the friendly or unfriendly relation between its end nodes in a signed network according to heider s theory edge signs evolve such that finally a state of structural balance is achieved corresponding to no frustration in physics terms however the danger of infection affects the evolution of its edge signs to describe the coupled problem of the sign evolution and disease spreading we generalize the notion of structural balance by taking into account the state of the nodes we introduce an energy function and carry out monte carlo simulations on complete networks to test the energy landscape where we find local minima corresponding to the so called jammed states we study the effect of the ratio of initial friendly to unfriendly connections on the propagation of disease the steady state can be balanced or a jammed state such that a coexistence occurs between susceptible and infected nodes in the system

representations of lie superalgebras and generalized boson fermion equivalence in quantum stochastic calculus the boson fermion equivalence scheme of 5 can be generalized ton dimensions withr boson andn r fermion creation and annihilation fields the same stochastic integral prescription replaces then 2 generalized number processes ai j which form representations of the lie algebragl n by processes forming representations of the lie superalgebragl n r

quartic balance theory global minimum with imbalanced triangles balance theory proposed by heider for the first time modeled triplet interaction in a signed network stating that relationships between two people friendship or enmity is dependent on a third person the hamiltonian of this model has an implicit assumption that all triads are independent meaning that the type of each triad being balanced or imbalanced determined apart from the state of other triads this independence forces the network to have completely balanced final states however there exists evidence indicating that real networks are partially balanced raising the question of what is the mechanism preventing the system to be perfectly balanced our suggestion is to consider a quartic interaction which dissolves the triad s independence we use the mean field method to study the thermal behavior of such systems where the temperature is a parameter that allows the stochastic behavior of agents we show that under a certain temperature the symmetry between balanced and imbalanced triads will spontaneously break and we have a discrete phase transition as a consequence stability arises where either similar balanced or imbalanced triads dominate hence the system obtains two new imbalanced stable states in this model the critical temperature depends on the second power of the number of nodes which was a linear dependence in thermal balance theory our simulations are in good agreement with the results obtained by the mean field method

pseudo paths towards minimum energy states in network dynamics abstract the dynamics of networks forming on heider balance theory moves towards lower tension states the condition derived from this theory enforces agents to reevaluate and modify their interactions to achieve equilibrium these possible changes in network s topology can be considered as various paths that guide systems to minimum energy states based on this theory the final destination of a system could reside on a local minimum energy jammed state or the global minimum energy balanced states the question we would like to address is whether jammed states just appear by chance or there exist some pseudo paths that bound a system towards a jammed state we introduce an indicator to suspect the location of a jammed state based on the inverse participation ratio method ipr we provide a margin before a local minimum where the number of possible paths dramatically drastically decreases this is a condition that proves adequate for ending up on a jammed states

first order continuous models of opinion formation we study certain nonlinear continuous models of opinion formation derived from a kinetic description involving exchanges of opinion between individual agents these models imply that the only possible final opinions are the extremal ones and they are similar to models of pure drift in magnetization both analytical and numerical methods allow us to recover the final distribution of opinion between the two extremal ones

h 2 performance of continuous time periodically time varying controllers abstract this paper uses a frequency domain approach to the analysis of h 2 performance of continuous time periodically time varying controllers for control of linear time invariant plants it is shown that the time varying dynamics of the periodic controller deteriorates the closed loop system h 2 performance and a linear time invariant controller can be found to provide strictly better h 2 control of the system

on martingale inequalities in non commutative stochastic analysis we develop a non commutativelpstochastic calculus for the clifford stochastic integral anl2theory of which has been developed by barnett streater and wilde the main results are certain non commutativelpinequalities relating clifford integrals and their integrands these results are applied to extend the domain of the clifford integral froml2tol1integrands and we give applications to optional stopping of clifford martingales proving an analog of a theorem of burkholder the stopped clifford processfthas zero expectation providedet in proving these results we establish a number of results relating the clifford integral to the differential calculus in the clifford algebra in particular we show that the clifford integral is given by the divergence operator and we prove an explicit martingale representation theorem both of these results correspond closely to basic results for stochastic analysis on wiener space thus furthering the analogy between the clifford process and brownian motion

a martingale characterization of quantum poisson processes we study the characterization problem of quantum poisson processes using the martingale property and some conditions on moments up to fourth order these conditions are inspired by an assumption on the quadratic variation that is equivalent in classical cases to the fact that the process has purely discontinuous sample paths with unit jumps

naming games in two dimensional and small world connected random geometric networks we investigate a prototypical agent based model the naming game on two dimensional random geometric networks the naming game a baronchelli et al j stat mech theory exp 2006 p06014 is a minimal model employing local communications that captures the emergence of shared communication schemes languages in a population of autonomous semiotic agents implementing the naming games with local broadcasts on random geometric graphs serves as a model for agreement dynamics in large scale autonomously operating wireless sensor networks further it captures essential features of the scaling properties of the agreement process for spatiallyembedded autonomous agents among the relevant observables capturing the temporal properties of the agreement process we investigate the cluster size distribution and the distribution of the agreement times both exhibiting dynamic scaling we also present results for the case when a small density of long range communication links are added on top of the random geometric graph resulting in a small world like network and yielding a significantly reduced time to reach global agreement we construct a finite size scaling analysis for the agreement times in this case

the series product and its application to quantum feedforward and feedback networks the purpose of this paper is to present simple and general algebraic methods for describing series connections in quantum networks these methods build on and generalize existing methods for series or cascade connections by allowing for more general interfaces and by introducing an efficient algebraic tool the series product we also introduce another product which we call the concatenation product that is useful for assembling and representing systems without necessarily having connections we show how the concatenation and series products can be used to describe feedforward and feedback networks a selection of examples from the quantum control literature are analyzed to illustrate the utility of our network modeling methodology

coherent quantum lqg control based on a recently developed notion of physical realizability for quantum linear stochastic systems we formulate a quantum lqg optimal control problem for quantum linear stochastic systems where the controller itself may also be a quantum system and the plant output signal can be fully quantum such a control scheme is often referred to in the quantum control literature as coherent feedback control it distinguishes the present work from previous works on the quantum lqg problem where measurement is performed on the plant and the measurement signals are used as the input to a fully classical controller with no quantum degrees of freedom the difference in our formulation is the presence of additional non linear and linear constraints on the coefficients of the sought after controller rendering the problem as a type of constrained controller design problem due to the presence of these constraints our problem is inherently computationally hard and this also distinguishes it in an important way from the standard lqg problem we propose a numerical procedure for solving this problem based on an alternating projections algorithm and as an initial demonstration of the feasibility of this approach we provide fully quantum controller design examples in which numerical solutions to the problem were successfully obtained for comparison we also consider the case of classical linear controllers that use direct or indirect measurements and show that there exists a fully quantum linear controller which offers an improvement in performance over the classical ones

some self adjoint quantum semimartingales it is proved that the quantum stochastic gauge integral preserves self adjointness of vacuum adapted processes this fact together with bounded perturbations and the link between the hudson parthasarathy calculus and vacuum adapted theory is used to produce many self adjoint quantum semimartingales

comparing the structure of an emerging market with a mature one under global perturbation a b s t r a c t in this paper we investigate the tehran stock exchange tse and dow jones industrial average djia in terms of perturbed correlation matrices to perturb a stock market there are two methods namely local and global perturbation in the local method we replace a correlation coefficient of the cross correlation matrix with one calculated from two gaussian distributed time series whereas in the global method we reconstruct the correlation matrix after replacing the original return series with gaussian distributed time series the local perturbation is just a technical study we analyze these markets through two statistical approaches random matrix theory rmt and the correlation coefficient distribution by using rmt we find that the largest eigenvalue is an influence that is common to all stocks and this eigenvalue has a peak during financial shocks we find there are a few correlated stocks that make the essential robustness of the stock market but we see that by replacing these return time series with gaussian distributed time series the mean values of correlation coefficients the largest eigenvalues of the stock markets and the fraction of eigenvalues that deviate from the rmt prediction fall sharply in both markets by comparing these two markets we can see that the djia is more sensitive to global perturbations these findings are crucial for risk management and portfolio selection

optimizing transformations of structural balance in signed networks with potential relationships abstract a signed network includes positive edges negative edges and 0 edges the last of which denote potential relationships however 0 edges are commonly ignored in transformations of unbalanced networks in this paper we take 0 edges into account and solve the optimization problem in a more comprehensive way we transform the unbalanced network by not only changing signs of edges but also changing edges into potential the experimental results show that our method can solve this problem efficiently and that our solutions are cost saving

network of likes and dislikes conflict and membership abstract we all have friends and foes in the study of complex networks such a pairwise interaction is described by a directed link since the relation is not necessarily symmetric we study a real network constructed from a survey in which each individual chooses five members s he wants to work with and other five s he does not like to work together although everyone s outdegrees for such like and dislike links are fixed to five respectively it is found that indegree sequence for each type of links exhibits very different behaviors we also pursue to answer the question of proper divisions of the organization based on the concept of happiness defined for each directed relation for example two individuals connected by like dislike links in both directions are happy if they belong to the same different group s we then adopt the framework of the q state potts model with long ranged ferromagnetic and antiferromagnetic interactions and discuss the group structure in the organization that minimizes a suitably defined unhappiness

noncommutative stochastic processes with independent and stationary increments satisfy quantum stochastic differential equations the notion of a unitary noncommutative stochastic process with independent and stationary increments is introduced and it is proved that such a process under a continuity assumption can be embedded into the solution of a quantum stochastic differential equation in the sense of hudson and parthasarathy 8

fermion ito s formula and stochastic evolutions an ito product formula is proved for stochastic integrals against fermion brownian motion and used to construct unitary processes satisfying stochastic differential equations as in the corresponding boson theory 10 11 these give rise to stochastic dilations of completely positive semigroups

mayer problem for quantum stochastic control this paper is concerned with the value function for optimal quantum stochastic control we established the lipschitz continuity of the value function which is inherent from the lipschitz property of the set valued map describing the quantum stochastic control by using a feedback multivalued map we also prove a characterization of optimal solutions relating to a noncommutative generalization of mayer problem

localization of quantum bernoulli noises the family k k k 0 of annihilation and creation operators acting on square integrable functionals of a bernoulli process z zk k 0 can be interpreted as quantum bernoulli noises in this note we consider the operator family lk lk k 0 where lk kek with ek being the conditional expectation operator given field zj 0 j k we show that lk resp lk is essentially a kind of localization of the annihilation operator k resp creation operator k we examine properties of the family lk lk k 0 and prove among other things that lk and lk satisfy a local canonical anti communication relation and lk k 0 forms a mutually orthogonal operator sequence although each lk is not a projection operator we find that the operator series k 0 lk xlk converges in the strong operator topology for each bounded operator x acting on square integrable functionals of z in particular we get an explicit sum of the operator series k 0 lk lk a useful norm estimate on k 0 lk xlk is also obtained final

frequency response of sampled data systems this paper develops a frequency domain theory that provides a method to analyze and design sampled data control systems including their intersample behaviors the key idea is to consider the signal space x x t x t n x xn exp j t jn st n x xn 2 where s is the sampling angular frequency it is shown that a stable sampled data system equipped with a strictly proper pre filter before the sampler maps x into x l2 in the steady state that mapping is denoted by q j and is referred to as an fr operator it is proved that the norm of the sampled data system as an operator from l2 to l2 is given by max q j 2 2 where 12 s 12 s a set of equations relating outputs of a closed loop system to its inputs in the frequency domain is derived and their solution is given in an explicit form based on that solution the sensitivity fr operator y j and the complementary sensitivity fr operator t j are defined for feedback control systems and it is shown that y j gives the improvement of the sensitivity of the transfer characteristics from the reference to the controlled output and also represents the ability of rejecting disturbances and that t j represents the degree of robust stability and at the same time gives the effect of detection noises it is also shown that y j t j i and thus a frequency domain paradigm for the design of sampled data control systems which is exactly parallel to the continuous time case is established

dynamics of social balance on networks we study the evolution of social networks that contain both friendly and unfriendly pairwise links between individual nodes the network is endowed with dynamics in which the sense of a link in an imbalanced triad a triangular loop with one or three unfriendly links is reversed to make the triad balanced with this dynamics an infinite network undergoes a dynamic phase transition from a steady state to paradise all links are friendly as the propensity p for friendly links in an update event passes through 1 2 a finite network always falls into a socially balanced absorbing state where no imbalanced triads remain if the additional constraint that the number of imbalanced triads in the network not increase in an update is imposed then the network quickly reaches a balanced final state i introduction in this work we investigate the role of friends and enemies on the evolution of social networks we represent individuals as nodes in a graph and a relationship between individuals as a link that joins the corresponding nodes to quantify a relationship we assign the binary variable sij 1 to link ij with sij 1 if nodes i and j are friends and sij 1 ifi and j are enemies fig 1 a basic characterization of relationships between mutual acquaintances is the notion of social balance 1 2 the triad ijk is defined as balanced if the sign of the product of the links in the triad sijsjkski equals 1 while the triad is imbalanced otherwise we define a triad to be of type k if it contains k negative links thus 0 and 2 are balanced while 1 and 3 are imbalanced a balanced triad therefore fulfills the adage that i a friend of my friend is my friend ii an enemy of my friend

community detection based on the clumpiness matrix in complex networks abstract the clumpiness matrix of a network is used to develop a method to identify its community structure a projection space is constructed from the eigenvectors of the clumpiness matrix and a border line is defined using some kind of angular distance in this space the community structure of the network is identified using this borderline and or hierarchical clustering methods the performance of our algorithm is tested on some computer generated and real world networks the accuracy of the results is checked using normalized mutual information the effect of community size heterogeneity on the accuracy of the method is also discussed

statistical analysis of stable fdlcp systems by parametric transfer matrices this paper considers the problem of determining the statistical characteristics of the response of internally stable finite dimensional continuous time periodic fdlcp systems to stationary stochastic input signals closed formulae for calculating the variance matrix of the state and the output vectors are derived as well as for the mean variance of the output and the 2 norm of the systems where only matrices of finite dimension are used the method bases on application of the parametric transfer matrix ptm and needs a precalculation of the transition matrix of the system on the interval 0 t t where t is the period of the system

gain and loss of esteem direct reciprocity and heider balance abstract the effect of gain and loss of esteem is introduced into the equations of time evolution of social relations hostile or friendly in a group of actors the equations allow for asymmetric relations we prove that in the presence of this asymmetry the majority of stable solutions are jammed states i e the heider balance is not attained there a phase diagram is constructed with three phases the jammed phase the balanced phase with two mutually hostile groups and the phase of so called paradise where all relations are friendly

properties of delta functions of a class of observables on white noise functionals abstract let a be the dirac delta function at a r and e l 2 e the canonical framework of white noise analysis over white noise space e where e s r for h h l 2 r with h 0 denote by m h the operator of multiplication by w h h in l 2 in this paper we first show that m h is a composable thus the delta function a m h makes sense as a generalized operator i e a continuous linear operator from e to e we then establish a formula showing an intimate connection between a m h as a generalized operator and a w h as a generalized functional we also obtain the representation of a m h as a series of integral kernel operators finally we prove that a m h depends continuously on a r

quantum q white noise and a q central limit theorem we establish a connection between the azema martingales and certain quantum stochastic processes with increments satisfyingq commutation relations this leads to a theory ofq white noise onq bialgebras and to a generalization of the fock space representation theorem for white noise on bialgebras in particular quantum azema noise q interpolations between fermion and boson quantum brownian motion and unitary evolutions withq independent multiplicative increments are studied it follows from our results that the azema martingales and theq interpolations are central limits of sums ofq independent identically distributed quantum random variables

singular perturbation approximations for a class of linear quantum systems this technical note considers the use of singular perturbation approximations for a class of linear quantum systems arising in the area of linear quantum optics the technical note presents results on the physical realizability properties of the approximate system arising from singular perturbation model reduction

fast computing global structural balance in signed networks based on memetic algorithm abstract structural balance is a large area of study in signed networks and it is intrinsically a global property of the whole network computing global structural balance in signed networks which has attracted some attention in recent years is to measure how unbalanced a signed network is and it is a nondeterministic polynomial time hard problem many approaches are developed to compute global balance however the results obtained by them are partial and unsatisfactory in this study the computation of global structural balance is solved as an optimization problem by using the memetic algorithm the optimization algorithm named meme sb is proposed to optimize an evaluation function energy function which is used to compute a distance to exact balance our proposed algorithm combines genetic algorithm and a greedy strategy as the local search procedure experiments on social and biological networks show the excellent effectiveness and efficiency of the proposed method

quantum ito s formula and stochastic evolutions using only the boson canonical commutation relations and the riemann lebesgue integral we construct a simple theory of stochastic integrals and differentials with respect to the basic field operator processes this leads to a noncommutative ito product formula a realisation of the classical poisson process in fock space which gives a noncommutative central limit theorem the construction of solutions of certain noncommutative stochastic differential equations and finally to the integration of certain irreversible equations of motion governed by semigroups of completely positive maps the classical ito product formula for stochastic differentials with respect to brownian motion and the poisson process is a special case

using the fredholm resolvent for computing the ℋ 2 norm of linear periodic systems this article presents a method for the 2 norm computation of single loop finite dimensional linear continuous periodic systems which is based on the parametric transfer function concept and the calculus of fredholm integral equations of the second kind general formulae for the computation of the mean output variance are provided as relations of two integral functions of an auxiliary parameter on the basis of the general solution an approximate computation method is derived and the deviation of the approximate solution is estimated in principle this approach allows one to perform the computation with any desired accuracy

reachability and recurrence in a modular generalization of annihilating random walks and lights out games to hypergraphs we study a discrete asynchronous dynamical system on hypergraphs that can be regarded as a natural extension of annihilating walks along two directions first the interaction topology is a hypergraph second the number of particles at a vertex of the hypergraph is an element of a finite ring z p of integers modulo an odd number p 3 equivalently particles move on a hypergraph with a moving particle at a vertex being replaced by one indistinguishable copy at each neighbor in a given hyperedge particles at a vertex collectively annihilate when their number reaches p the boolean version of this system arose in earlier work 22 motivated by the statistical physics of social balance 3 2 generalizes certain lights out games 31 to finite fields and also has some applications to the complexity of local search procedures 23 our result shows that under a liberal sufficient condition on the nature of the interaction hypergraph there exists a polynomial time algorithm based on linear algebra over z p for deciding reachability and recurrence of this dynamical system interestingly we provide a counterexample that shows that this connection does not extend to all graphs

noncommutative brownian motion in monotone fock space an example of noncommutative brownian motion is constructed on the monotone fock space which is a kind of fock space generated by all the decreasing finite sequences of positive real numbers the probability distribution at time associated to this brownian motion is shown to be the arcsine law normalized to mean 0 and variance t

diagonal versions and quantum stochastic integrals on the symmetric fock space with nonadapted integrands we develop a quantum stochastic calculus for nonadapted processes and the three standard integrators on the symmetric fock space over an arbitrary polish space the integrable processes are restricted to a class of operators which possess some kernel description our approach is mainly based on the concept of diagonalized versions 16 in the case of the real line we consider connections to the usual stochastic and quantum stochastic calculi and derive ito formulae besides the standard integrals with two integrands and one integrator these formulae lead to a new type of quantum stochastic integrals which has two integrators and three integrands

quantum stochastic calculus with maximal operator domains quantum stochastic calculus is extended in a new formulation in which its stochastic integrals achieve their natural and maximal domains operator adaptedness conditional expectations and stochastic integrals are all defined simply in terms of the orthogonal projections of the time filtration of fock space together with sections of the adapted gradient operator free from exponential vector domains our stochastic integrals may be satisfactorily composed yielding quantum ito formulas for operator products as sums of stochastic integrals the calculus has seen two reformulations since its discovery one closely related to classical ito calculus the other to noncausal stochastic analysis and malliavin calculus our theory extends both of these approaches and may be viewed as a synthesis of the two the main application given here is existence and uniqueness for the attal meyer equations for implicit definition of quantum stochastic integrals

continuous unital dilations of completely positive semigroups abstract since it began in the 1970s the study of completely positive semigroups has included among its central topics the dilation of a completely positive semigroup to an endomorphism semigroup several authors have proved the existence of dilations but have usually only achieved unital dilations of specific examples while general theorems have tended to produce non unital dilations a unique approach due to jean luc sauvageot leads to a unital dilation in a general setting but leaves unclear the continuity of the dilation semigroup the major purpose of this paper therefore is to further develop sauvageot s theory in order to prove the existence of continuous unital dilations the dilation is characterized by a combinatorial property akin to free independence this property is implicit in some of sauvageot s original calculations but a secondary goal of this paper is to present it as its own object of study which is done in section 4 this work is based on the author s ph d thesis

filtered random variables bialgebras and convolutions we introduce the filtered bialgebra which is a multivariate generalization of the unital bialgebra c x x p of polynomials in noncommuting variables x x x x and a projection p p p2 endowed with the coproduct x x 1 1 x x x p p x with p being group like we study the associated convolutions random walks and filtered random variables the gns representations of the limit states lead to filtered fundamental operators which are the ccr fundamental operators on the multiple symmetric fock space h over h l2 r g where g is a separable hilbert space multiplied by appropriate projections the importance of filtered random variables and fundamental operators stems from the fact that by addition and strong limits one obtains from them the main types of noncommutative random variables and fundamental operators respectively regardless of the type of noncommutative independence

universality class of triad dynamics on a triangular lattice we consider triad dynamics as it was recently considered by antal et al phys rev e 72 036121 2005 as an approach to social balance here we generalize the topology from all to all to the regular one of a two dimensional triangular lattice the driving force in this dynamics is the reduction of frustrated triads in order to reach a balanced state the dynamics is parametrized by a so called propensity parameter p that determines the tendency of negative links to become positive as a function of p we find a phase transition between different kinds of absorbing states the phases differ by the existence of an infinitely connected percolated cluster of negative links that forms whenever p ensuremath leqslant p c moreover for p ensuremath leqslant p c the time to reach the absorbing state grows powerlike with the system size l while it increases logarithmically with l for pg p c from a finite size scaling analysis we numerically determine the static critical exponents ensuremath beta and ensuremath nu ensuremath perp together with ensuremath gamma ensuremath tau ensuremath sigma and the dynamical critical exponents ensuremath nu ensuremath vert and ensuremath delta the exponents satisfy the hyperscaling relations we also determine the fractal dimension d f that satisfies a hyperscaling relation as well the transition of triad dynamics between different absorbing states belongs to a universality class with different critical exponents we generalize the triad dynamics to four cycle dynamics on a square lattice in this case again there is a transition between different absorbing states going along with the formation of an infinite cluster of negative links but the usual scaling and hyperscaling relations are violated

quantum and non causal stochastic calculus the quantum stochastic calculus initiated by hudson and parthasarathy and the non causal stochastic calculus originating with the papers of hitsuda and skorohod are two potent extensions of the ito calculus currently enjoying intensive development the former provides a quantum probabilistic extension of schrodinger s equation enabling the construction of a markov process for a quantum dynamical semigroup the latter allows the treatment of stochastic differential equations which involve terms which anticipate the future in this paper the close relationship between these theories is displayed and a noncausal quantum stochastic calculus already in demand from physics is described

solving quantum stochastic differential equations with unbounded coefficients we demonstrate a method for obtaining strong solutions to the right hudson parthasarathy quantum stochastic differential equation dut f utd t u0 1 where u is a contraction operator process and the matrix of coefficients f consists of unbounded operators this is achieved whenever there is a positive self adjoint reference operator c that behaves well with respect to the f allowing us to prove that domc1 2 is left invariant by the operators ut thereby giving rigorous meaning to the formal expression above we give conditions under which the solution u is an isometry or coisometry process and apply these results to construct unital homomorphic dilations of quantum markov semigroups arising in probability and physics