graphical models for marked point processes based on local independence summary a new class of graphical models capturing the dependence structure of events that occur in time is proposed the graphs represent so called local independences meaning that the intensities of certain types of events are independent of some but not necessarilly all events in the past this dynamic concept of independence is asymmetric similar to granger non causality so the corresponding local independence graphs differ considerably from classical graphical models hence a new notion of graph separation which is called separation is introduced and implications for the underlying model as well as for likelihood inference are explored benefits regarding facilitation of reasoning about and understanding of dynamic dependences as well as computational simplifications are discussed

iterative bregman projections for regularized transportation problems this article details a general numerical framework to approximate so lutions to linear programs related to optimal transport the general idea is to introduce an entropic regularization of the initial linear program this regularized problem corresponds to a kullback leibler bregman di vergence projection of a vector representing some initial joint distribu tion on the polytope of constraints we show that for many problems related to optimal transport the set of linear constraints can be split in an intersection of a few simple constraints for which the projections can be computed in closed form this allows us to make use of iterative bregman projections when there are only equality constraints or more generally bregman dykstra iterations when inequality constraints are in volved we illustrate the usefulness of this approach to several variational problems related to optimal transport barycenters for the optimal trans port metric tomographic reconstruction multi marginal optimal trans port and in particular its application to brenier s relaxed solutions of in compressible euler equations partial un balanced optimal transport and optimal transport with capacity constraints

generalized incompressible flows multi marginal transport and sinkhorn algorithm starting from brenier s relaxed formulation of the incompress ible euler equation in terms of geodesics in the group of measure preserving diffeomorphisms we propose a numerical method based on sinkhorn s algorithm for the entropic regularization of optimal transport we also make a detailed comparison of this entropic regular ization with the so called bredinger entropic interpolation problem see 1 numerical results in dimension one and two illustrate the feasibility of the method

strong convergence for split step methods in stochastic jump kinetics mesoscopic models in the reaction diffusion framework have gained recognition as a viable approach to describing chemical processes in cell biology the resulting computational problem is a continuous time markov chain on a discrete and typically very large state space due to the many temporal and spatial scales involved many different types of computationally more effective multiscale models have been proposed typically coupling different types of descriptions within the markov chain framework in this work we look at the strong convergence properties of the basic first order strang or lie trotter split step method which is formed by decoupling the dynamics in finite time steps thanks to its simplicity and flexibility this approach has been tried in many different combinations we develop explicit sufficient conditions for pathwise well posedness and convergence of the method including error estimates and we illustrate our findings with numerical examples in doing so we also suggest a certain

generalized solutions and hydrostatic approximation of the euler equations abstract solutions to the euler equations on a 3d domain d 3 typically the unit cube or the periodic unit cube can be formally obtained by minimizing the action of an incompressible fluid moving inside d 3 between two given configurations when these two configurations are very close to each other classical solutions do exist as shown by ebin and marsden however shnirelman found a class of data essentially 2d in the sense that they trivially depend on the vertical coordinate for which there cannot be any classical minimizer for such data generalized solutions can be shown to exist as a substitute for classical solutions these generalized solutions have unusual features that look highly unphysical in particular different fluid parcels can cross at the same point and at the same time but the pressure field which does not depend on the vertical coordinate is well and uniquely defined in the present paper we show that these generalized solutions are actually quite conventional in the sense they obey up to a suitable change of variable a well known variant widely used for geophysical flows of the 3d euler equations for which the vertical acceleration is neglected according to the so called hydrostatic approximation

on the h s theory of hydrostatic euler equations in this paper we study the two dimensional hydrostatic euler equations in a periodic channel we prove the local existence and uniqueness of hs solutions under the local rayleigh condition this extends brenier s nonlinearity 12 3 495 512 1999 existence result by removing an artificial condition and proving uniqueness in addition we prove weak strong uniqueness mathematical justification of the formal derivation and stability of the hydrostatic euler equations these results are based on weighted hs a priori estimates which come from a new type of nonlinear cancellation between velocity and vorticity

the entropy production paradox for fractional master equations abstract time fractional evolution equations for probability distributions provide a means to describe an important class of stochastic processes their solutions show features which are essential in modeling a variety of phenomena in real world applications one aspect which has been observed in time fractional diffusion equations shows a surprising and unexpected behavior of the entropy production rate induced by these equations the entropy production rate increases as one moves away from the fully irreversible case corresponding to classical diffusion this rate is analyzed for a new class of systems with state spaces that are finite and denumerable we find that the entropy production paradox reemerges nonetheless but in a new and unexpected form

solving the master equation without kinetic monte carlo in multiscale modeling of heterogeneous catalytic processes one crucial point is the solution of a markovian master equation describing the stochastic reaction kinetics usually this is too high dimensional to be solved with standard numerical techniques and one has to rely on sampling approaches based on the kinetic monte carlo method in this study we break the curse of dimensionality for the direct solution of the markovian master equation by exploiting the tensor train format for this purpose the performance of the approach is demonstrated on a first principles based reduced model for the co oxidation on the ruo2 110 surface we investigate the complexity for increasing system size and for various reaction conditions the advantage over the stochastic simulation approach is illustrated by a problem with increased stiffness

influence of high order nonlinear fluctuations in the multivariate susceptible infectious recovered master equation we perform a high order analytical expansion of the epidemiological susceptible infectious recovered multivariate master equation and include terms up to and beyond single particle fluctuations it is shown that higher order approximations yield qualitatively different results than low order approximations which is incident to the influence of additional nonlinear fluctuations the fluctuations can be related to a meaningful physical parameter the basic reproductive number which is shown to dictate the rate of divergence in absolute terms from the ordinary differential equations more so than the total number of persons in the system in epidemiological terms the effect of single particle fluctuations ought to be taken into account as the reproductive number approaches unity

reversible equivalent monomolecular tau leaping methods provide for efficient and approximate time stepping of chemical reaction systems modeled by continuous time discrete state stochastic dynamics we investigate the application of leaping methods for small number and stiff systems i e systems whose dynamics involve different time scales and have some molecular species present in very small numbers specifically in the range 0 to 10 we propose a new explicit leaping scheme reversible equivalent monomolecular tau remm which shows considerable promise in the simulation of such systems the remm scheme is based on the fact that the exact solution of the two prototypical monomolecular reversible reactions s1 s2 and s 0 as a function of time takes a simple form involving binomial and or poisson random variables the remm method involves approximating bimolecular reversible reactions by suitable monomolecular reversible reactions as well as considering each reversible pair of reactions in the system to be operating in isolation during the time step we illustrate the use of the remm method through a number of biologically motivated examples and compare its performance to those of the implicit and trapezoidal implicit algorithms in most cases considered remm appears to perform better than these two methods while having the important advantage of being computationally faster due to the explicit nature of the method furthermore when stepsize is increased the remm exhibits a more robust performance than the implicit or the trapezoidal implicit for small number stiff problems

axisymmetric flow of ideal fluid moving in a narrow domain a study of the axisymmetric hydrostatic euler equations abstract in this article we will introduce a new model to describe the leading order behavior of an ideal and axisymmetric fluid moving in a very narrow domain after providing a formal derivation of the model we will prove the well posedness and provide a rigorous mathematical justification for the formal derivation under a new sign condition finally a blowup result regarding this model will be discussed as well

stochastic kinetic models dynamic independence modularity and graphs the dynamic properties and independence structure of stochastic kinetic models skms are analyzed an skm is a highly multivariate jump process used to model chemical reaction networks particularly those in biochemical and cellular systems we identify skm subprocesses with the corresponding counting processes and propose a directed cyclic graph the kinetic independence graph or kig that encodes the local independence structure of their conditional intensities given a partition a d b of the vertices the graphical separation a b d in the undirected kig has an intuitive chemical interpretation and implies that a is locally independent of b given a d it is proved that this separation also results in global independence of the internal histories of a and b conditional on a history of the jumps in d which under conditions we derive corresponds to the internal history of d the results enable mathematical definition of a modularization of an skm using its implied dynamics graphical decomposition methods are developed for the identification and efficient computation of nested modularizations application to an skm of the red blood cell advances understanding of this biochemical system

generalized solution for the demodulation of reaction shift keying signals in molecular communication networks this paper considers a diffusion based molecular communication system where the transmitter uses reaction shift keying rsk as the modulation scheme we focus on the demodulation of rsk signal at the receiver the receiver consists of a front end molecular circuit and a back end demodulator the front end molecular circuit is a set of chemical reactions consisting of multiple chemical species the optimal demodulator computes the posteriori probability of the transmitted symbols given the history of the observation the derivation of the optimal demodulator requires the solution to a specific bayesian filtering problem the solution to this bayesian filtering problem had been derived for a few specific molecular circuits and specific choice s of observed chemical species the derivation of such solution is also lengthy the key contribution of this paper is to present a general solution to this bayesian filtering problem which can be applied to any molecular circuit and any choice of observed species

representation of solutions of n order riccati equation via generalized trigonometric functions abstract in this work we suggest a systematic method of construction of solutions of the n order riccati equation with constant coefficients in a field from the set of generalized trigonometric functions the generalized trigonometric functions satisfy the system of evolution equations generated by the companion matrix of n order polynomial the set of trigonometric functions depend of n 1 variables formally expressed by series of exponential functions in a particular case the second order riccati equation with constant coefficients is isomorphic to the evolution equation generated by the companion matrix of the associated quadratic polynomial it is shown that the n 2 order riccati equation with coefficients in a field is derived from a linear system of evolution equations generated by companion matrix of the associated n order polynomial under n 2 constraints

bayesian inference for stochastic kinetic models using a diffusion approximation summary this article is concerned with the bayesian estimation of stochastic rate constants in the context of dynamic models of intracellular processes the underlying discrete stochastic kinetic model is replaced b ya diffusion approximation or stochastic differential equation approach where a white noise term models stochastic behavior and the model is identified using equispaced time course data the estimation framework involves the introduction of m 1 latent data points between every pair of observations mcmc methods are then used to sample the posterior distribution of the latent process and the model parameters the methodology is applied to the estimation of parameters in a prokaryotic autoregulatory gene network

a general dynamical statistical model with causal interpretation summary we develop a general dynamical model as a framework for causal interpretation we first state a criterion of local independence in terms of measurability of processes that are involved in the doob meyer decomposition of stochastic processes then we define direct and indirect influence we propose a definition of causal influence using the concepts of a physical system this framework makes it possible to link descriptive and explicative statistical models and encompasses quantitative processes and events one of the features of the paper is the clear distinction between the model for the system and the model for the observation we give a dynamical representation of a conventional joint model for human immunodeficiency virus load and cd4 cell counts we show its inadequacy to capture causal influences whereas in contrast known mechanisms of infection by the human immunodeficiency virus can be expressed directly through a system of differential equations

chemical langevin equation a path integral view of gillespie s derivation in 2000 gillespie rehabilitated the chemical langevin equation cle by describing two conditions that must be satisfied for it to yield a valid approximation of the chemical master equation cme in this work we construct an original path integral description of the cme and show how applying gillespie s two conditions to it directly leads to a path integral equivalent to the cle we compare this approach to the path integral equivalent of a large system size derivation and show that they are qualitatively different in particular both approaches involve converting many sums into many integrals and the difference between the two methods is essentially the difference between using the euler maclaurin formula and using riemann sums our results shed light on how path integrals can be used to conceptualize coarse graining biochemical systems and are readily generalizable

on stochastic formalisms in transition state theory abstract the simulation of elementary gas phase reactions in a concentrated bath may be accomplished by a markoff chain model the concept of the state of secular equilibrium a necessary condition for the existence of rate coefficients local in time demands that the markoffian process be modelled using an inhomogeneous poisson process such a poisson process will possess microcanonical rates which are functions of time its simulation can be accomplished by quasi monte carlo schemes based on low discrepancy sequences which are also argued here for the computation of the flux integrals over the configurational space within the framework of variational microcanonical transition state theory

expectation propagation for continuous time stochastic processes we consider the inverse problem of reconstructing the posterior measure over the trajectories of a diffusion process from discrete time observations and continuous time constraints we cast the problem in a bayesian framework and derive approximations to the posterior distributions of single time marginals using variational approximate inference giving rise to an expectation propagation type algorithm for non linear diffusion processes this is achieved by leveraging moment closure approximations we then show how the approximation can be extended to a wide class of discrete state markov jump processes by making use of the chemical langevin equation our empirical results show that the proposed method is computationally efficient and provides good approximations for these classes of inverse problems

system size expansion using feynman rules and diagrams few analytical methods exist for quantitative studies of large fluctuations in stochastic systems in this article we develop a simple diagrammatic approach to the chemical master equation that allows us to calculate multi time correlation functions which are accurate to any desired order in van kampen s system size expansion specifically we present a set of feynman rules from which this diagrammatic perturbation expansion can be constructed algorithmically we then apply the methodology to derive in closed form the leading order corrections to the linear noise approximation of the intrinsic noise power spectrum for general biochemical reaction networks finally we illustrate our results by describing noise induced oscillations in the brusselator reaction scheme which are not captured by the common linear noise approximation

hybrid framework for the simulation of stochastic chemical kinetics abstract stochasticity plays a fundamental role in various biochemical processes such as cell regulatory networks and enzyme cascades isothermal well mixed systems can be modelled as markov processes typically simulated using the gillespie stochastic simulation algorithm ssa 25 while easy to implement and exact the computational cost of using the gillespie ssa to simulate such systems can become prohibitive as the frequency of reaction events increases this has motivated numerous coarse grained schemes where the fast reactions are approximated either using langevin dynamics or deterministically while such approaches provide a good approximation when all reactants are abundant the approximation breaks down when one or more species exist only in small concentrations and the fluctuations arising from the discrete nature of the reactions become significant this is particularly problematic when using such methods to compute statistics of extinction times for chemical species as well as simulating non equilibrium systems such as cell cycle models in which a single species can cycle between abundance and scarcity in this paper a hybrid jump diffusion model for simulating well mixed stochastic kinetics is derived it acts as a bridge between the gillespie ssa and the chemical langevin equation for low reactant reactions the underlying behaviour is purely discrete while purely diffusive when the concentrations of all species are large with the two different behaviours coexisting in the intermediate region a bound on the weak error in the classical large volume scaling limit is obtained and three different numerical discretisations of the jump diffusion model are described the benefits of such a formalism are illustrated using computational examples

a rigorous derivation of the chemical master equation it is widely believed that the chemical master equation has no rigorous microphysical basis and hence no a priori claim to validity this view is challenged here through arguments purporting to show that the chemical master equation is exact for any gas phase chemical system that is kept well stirred and thermally equilibrated

minimal geodesics on groups of volume preserving maps and generalized solutions of the euler equations the three dimensional motion of an incompressible inviscid fluid is classically described by the euler equations but can also be seen following arnold 1 as a geodesic on a group of volume preserving maps local existence and uniqueness of minimal geodesics have been established by ebin and marsden 16 in the large for a large class of data the existence of minimal geodesics may fail as shown by shnirelman 26 for such data we show that the limits of approximate solutions are solutions of a suitable extension of the euler equations or equivalently are sharp measure valued solutions to the euler equations in the sense of diperna and majda 14 1999 john wiley sons inc

the least action principle and the related concept of generalized flows for incompressible perfect fluids the link between the euler equations of perfect incompressible flows and the least action principle has been known for a long time i solu tions can be considered as geodesic curves along the manifold of volume pre serving mappings here the shortest path problem is investigated given two different volume preserving mappings at two different times find for the inter mediate times an incompressible flow map that minimizes the kinetic energy or more generally the action in its classical formulation this problem has been solved 7 only when the two different mappings are sufficiently close in some very strong sense in this paper a new framework is introduced where gen eralized flows are defined in the spirit of l c young as probability measures on the set of all possible trajectories in the physical space then the minimiza tion problem is generalized as the continuous linear programming problem that is much easier to handle the existence problem is completely solved in this content downloaded from 157 55 39 159 on sun 18 sep 2016 05 02 20 utc all use subject to http about jstor org terms the least action principle 255 the case of the d dimensional torus it is also shown that under natural re strictions a classical solution to the euler equations is the unique optimal flow in the generalized framework finally a link is established with the concept of measure valued solutions to the euler equations 6 and an example is provided where the unique generalized solution can be explicitly computed and turns out to be genuinely probabilistic inria bp 105 78153 le chesnay cedex france this content downloaded from 157 55 39 159 on sun 18 sep 2016 05 02 20 utc all use subject to http about jstor org terms

hybrid discrete continuum algorithms for stochastic reaction networks direct solutions of the chemical master equation cme governing stochastic reaction networks srns are generally prohibitively expensive due to excessive numbers of possible discrete states in such systems to enhance computational efficiency we develop a hybrid approach where the evolution of states with low molecule counts is treated with the discrete cme model while that of states with large molecule counts is modeled by the continuum fokker planck equation the fokker planck equation is discretized using a 2nd order finite volume approach with appropriate treatment of flux components the numerical construction at the interface between the discrete and continuum regions implements the transfer of probability reaction by reaction according to the stoichiometry of the system the performance of this novel hybrid approach is explored for a two species circadian model with computational efficiency gains of about one order of magnitude

multicomplex algebras on polynomials and generalized hamilton dynamics abstract generator of the complex algebra within the framework of general formulation obeys the quadratic equation in this paper we explore multicomplex algebra with the generator obeying n order polynomial equation with real coefficients this algebra induces generalized trigonometry n 1 gonometry underlies of the nth order oscillator model and nth order hamilton equations the solution of an evolution equation generated by n n matrix is represented via the set of n 1 gonometric functions the general form of the first constant of motion of the evolution equation is established

commutative extended complex numbers and connected trigonometry abstract in order to construct an extension of the complex numbers we consider an n dimensional commutative algebra generated by the n vectors 1 e en 1 where the fundamental element satisfies the basic relation en 1 these spaces can be classified according to the values of n prime number power of a prime number general number the question of the invertibility leads to the definition of a pseudo norm for which the triangle inequality is not satisfied the n 1 2 cases excepted when one tries to pass from the polar form the cartesian one one obtains functions generalizing the usual circular and hyperbolic functions and their inverse the extended sine and hyperbolic sine functions thus constructed satisfy a determinantal type relation and they lay the foundation of a new trigonometry for which summation and derivative formulas are given an extended 2 quantity is defined as the periodicity of the generalized circular functions this formalism is applied to solve the nth order differential equations n 1i 1 n ni 0 as a further application the solutions of the n laplacian operator are derived

geodesics in the space of measure preserving maps and plans we study brenier s variational models for incompressible euler equations these models give rise to a relaxation of the arnold distance in the space of measure preserving maps and more generally measure preserving plans we analyze the properties of the relaxed distance we show a close link between the lagrangian and the eulerian model and we derive necessary and sufficient optimality conditions for minimizers these conditions take into account a modified lagrangian induced by the pressure field moreover adapting some ideas of shnirelman we show that even for non deterministic final conditions generalized flows can be approximated in energy by flows associated to measure preserving maps

spatially adaptive stochastic numerical methods for intrinsic fluctuations in reaction diffusion systems stochastic partial differential equations are introduced for the continuum concentration fields of reaction diffusion systems the stochastic partial differential equations account for fluctuations arising from the finite number of molecules which diffusively migrate and react spatially adaptive stochastic numerical methods are developed for approximation of the stochastic partial differential equations the methods allow for adaptive meshes with multiple levels of resolution neumann and dirichlet boundary conditions and domains having geometries with curved boundaries a key issue addressed by the methods is the formulation of consistent discretizations for the stochastic driving fields at coarse refined interfaces of the mesh and at boundaries methods are also introduced for the efficient generation of the required stochastic driving fields on such meshes as a demonstration of the methods investigations are made of the role of fluctuations in a biological model for microorganism direction sensing based on concentration gradients also investigated a mechanism for spatial pattern formation induced by fluctuations the discretization approaches introduced for spdes have the potential to be widely applicable in the development of numerical methods for the study of spatially extended stochastic systems

young measures generated by ideal incompressible fluid flows in their seminal paper diperna and majda commun math phys 108 4 667 689 1987 introduced the notion of a measure valued solution for the incompressible euler equations in order to capture complex phenomena present in limits of approximate solutions such as persistence of oscillation and development of concentrations furthermore they gave several explicit examples exhibiting such phenomena in this paper we show that any measure valued solution can be generated by a sequence of exact weak solutions in particular this gives rise to a very large arguably too large set of weak solutions of the incompressible euler equations

on complex singularities of the 2d euler equation at short times abstract we present a study of complex singularities of a two parameter family of solutions for the two dimensional euler equation with periodic boundary conditions and initial conditions 0 z 1 z 2 f p cos p z f q cos q z in the short time asymptotic regime as has been shown numerically in pauls et al w pauls t matsumoto u frisch j bec nature of complex singularities for the 2d euler equation physica d 219 2006 40 59 the type of the singularities depends on the angle between the modes p and q thus the fourier coefficients of the solutions decrease as g k c k e k with the exponent depending on here we show for the two particular cases of going to zero and to that the type of the singularities can be determined very accurately being characterised by 5 2 and 3 respectively in these two cases we are also able to determine the subdominant corrections furthermore we find that the geometry of the singularities in these two cases is completely different the singular manifold being located over different points in the real domain

oscillatory expression and variability in p53 regulatory network the effect of delay nonlinearity and noise on oscillatory motion is of permanent interest for theoretical and experimental research here we explore a negative feedback loop between p53 and mdm2 with a time delay which is a key circuit in the response of cells to damage this circuit shows noisy sustained oscillations in individual human cells following dna damage and damped oscillations at the cell population level we demonstrate the effect of delay on the oscillation and the correlation in time course in a multi species system the events at different time points which span a time delay are coupled even when the delay is large compared with the other characteristic times of the system we also clarify that the dynamics at the single cell level appears to be coherent resonance and the origin of the damped oscillation at the macroscopic level out of the sustained ones at the single cell level can be ascribed to the dephasing process which is induced by the interplay between nonlinearity and noise the findings are consistent with experimental observations and advance our understanding of the dynamics of the p53 network