thermostochastics heat conduction and temperature fluctuations abstract a stochastic approach to the energy balance equation of nonequilibrium thermodynamics is suggested the stochastic formulation interprets the temperature field as a multivariate stochastic process which is governed by a master equation by means of a systematic expansion in powers of the inverse number of degrees of freedom per volume element it is shown that the expectation value of the stochastic process obeys the macroscopic fourier equation the expansion reveals that in the linear noise approximation the fluctuations superimposed on the macroscopic dynamics are governed by the equations of fluctuating hydrodynamics the master equation formulation gives rise to a new stochastic simulation method which is illustrated by applying it to heat conduction and temperature fluctuations in a fluid between two infinite parallel planes which are kept at different temperatures

fluctuations in the lattice gas for burgers equation the langevin equation for the biased random hopping of particles onto unoccupied sites of a lattice is obtained from the master equation for finite systems by invoking a limit theorem due to kurtz for a weakly asymmetric process with diffusive scaling the continuum limit of this equation yields a deterministic equation that is equivalent to burgers equation the stochastic equation for the leading lattice corrections is derived and the approach to the continuum limit is illustrated with simulations

stationary turbulent closure via the hopf functional equation exact closed form solutions are exhibited for the hopf equation for stationary incompressible 3d navier stokes flow for the cases of homogeneous forced flow including a solution with depleted nonlinearity and inhomogeneous flow with arbitrary boundary conditions this provides an exact method for computing two and higher point moments given the mean flow

numerical methods for high dimensional probability density function equations in this paper we address the problem of computing the numerical solution to kinetic partial differential equations involving many phase variables these types of equations arise naturally in many different areas of mathematical physics e g in particle systems liouville and boltzmann equations stochastic dynamical systems fokker planck and dostupov pugachev equations random wave theory malakhov saichev equations and coarse grained stochastic systems mori zwanzig equations we propose three different classes of new algorithms addressing high dimensionality the first one is based on separated series expansions resulting in a sequence of low dimensional problems that can be solved recursively and in parallel by using alternating direction methods the second class of algorithms relies on truncation of interaction in low orders that resembles the bogoliubov born green kirkwood yvon bbgky framework of kinetic gas theory and it yields a hierarchy of coupled probability density function equations the third class of algorithms is based on high dimensional model representations e g the anova method and probabilistic collocation methods a common feature of all these approaches is that they are reducible to the problem of computing the solution to high dimensional equations via a sequence of low dimensional problems the effectiveness of the new algorithms is demonstrated in numerical examples involving nonlinear stochastic dynamical systems and partial differential equations with up to 120 variables

a classification theorem for albert algebras let k be a field whose characteristic is different om 2 and 3 and let l k be a quadratic extension in this paper we prove that for a fixed degree 3 central simple algebra b over l with an involution ff of the second kind over k the jordan algebra j b a 11 u obtained through tits second co nstruction is determined up to isomorphism by the class of 11 u in h1 k su b a thus settling a question raised by petersson and racine as a consequence we derive a skolem noether type theorem for albert algebras we also show that the cohomological invariants determine the isomorphism class of j b a u u if b cr is fixed introduction let k be a field with characteristic different from 2 and 3 exceptional simple jordan algebras over k are called albert algebras there are rational constructions due to tits of all albert algebras over k referred to as the frst and the second constructions we begin by recalling the second construction briefly let l k be a quadratic extension bar denottng its nontrivial k automorphisme let b be a degree 3 central simple associative algebra over l with an involution a of the second kind over k let u be a unit of b with v u u and n u l i u e l where n denotes the reduced norm on b let h b x e blv x x let j b a u 1 xt ba d3 b be the jordan algebra with the multiplication given by bo b bo bt bo bo b l5 b btu5 b bob bob y ff b x o b u l where x y 2 xy yx z 2 t x x and x x y x y 2 t x y 2 t v x 2 t x t v t x tt t denoting the reduced trace of b the cubic norm of an element aoaa of j b c u is given by 81 n ao a n ao hn a hn ff a t aoau5 a for any unit w e b we have an isomorphism j b c u o j b c wu5 w n w h given by ao a l ao aw the following converse problem is posecs in p r question if j b c u j b c u y then does there e is 1 unit w e b such that u wuv w n w y received by the editors june 12 1996 1991 mathematics subject classification primary 17c40

a nonparametric estimation procedure for a periodically observed three state markov process with application to aids estimation in a three state markov process with irreversible transitions in the presence of interval censored data is considered a nonparametric maximum likelihood procedure for the estimation of the cumulative transition intensities is presented a self consistent estimator of the parameters is defined and it is shown that the maximum likelihood estimator is a self consistent estimator this extends the idea of self consistency introduced by efron to the estimation of more than one parameter an algorithm based on selfconsistency equations is provided for the computation of the estimators

classification of algebras arising from the tits process in the preceding paper lo we generalized the second tits construction of simple exceptional jordan algebras 6 to a construction over an arbitrary ring of scalars which we called the tits process and claimed that it plays for jordan rings a role akin to that of the cayley dickson process for alternative rings in this paper we assume that the scalars form a field f and determine the semisimple jordan algebras arising from the tits process we see that except for predictable problems in low characteristics all simple jordan algebras of degree 3 are obtainable by starting from fl and repeating the tits process review let be a unital commutative associative ring a module recall that jv n 1 is a cubic norm structure on f if 1 n f is cubic form 2 f is a quadratic map 1 is an element of 2 2 is an udjoint for n

cyclic compositions and trisotopies abstract cyclic compositions in the sense of springer t a springer oktaven jordan algebren und ausnahmegruppen universitat gottingen 1963 and knus merkurjev rost and tignol m a knus a merkurjev m rost j p tignol the book of involutions amer math soc colloq publ vol 44 amer math soc providence ri 1998 with a preface in french by j tits are investigated by means of cyclic trisotopies a concept originally due to albert a a albert a construction of exceptional jordan division algebras ann of math 2 67 1958 1 28 using the quadrupling of composition algebras we enumerate cyclic trisotopies and compositions in a rational manner i e without extending the base field we relate cyclic trisotopies explicitly to simple associative algebras of degree 3 with involution and to the tits process of cubic jordan algebras

spline based self controlled case series method the self controlled case series sccs method is an alternative to study designs such as cohort and case control methods and is used to investigate potential associations between the timing of vaccine or other drug exposures and adverse events it requires information only on cases individuals who have experienced the adverse event at least once and automatically controls all fixed confounding variables that could modify the true association between exposure and adverse event time varying confounders such as age on the other hand are not automatically controlled and must be allowed for explicitly the original sccs method used step functions to represent risk periods windows of exposed time and age effects hence exposure risk periods and or age groups have to be prespecified a priori but a poor choice of group boundaries may lead to biased estimates in this paper we propose a nonparametric sccs method in which both age and exposure effects are represented by spline functions at the same time to avoid a numerical integration of the product of these two spline functions in the likelihood function of the sccs method we defined the first second and third integrals of i splines based on the definition of integrals of m splines simulation studies showed that the new method performs well this new method is applied to data on pediatric vaccines copyright 2017 john wiley sons ltd

semiparametric estimation in a three state duration dependent markov model from interval censored observations with application to aids data the semiparametric maximum likelihood estimation is considered in a three state duration dependent markov process when times of the intermediate transition e g onset of a disease are interval censored and the times of transitions to an absorbing state e g death are known exactly or are right censored it is assumed that the intensity of the transition to an absorbing state depends both on chronological time and duration in the intermediate state de gruttola and lagakos 1989 biometrics 45 1 11 and frydman 1992 journal of the royal statistical society series b 54 853 866 discussed non parametric estimation from the same sampling scheme under the assumption that the intensity of transition to an absorbing state depends only on the duration in the intermediate state or only on the chronological time respectively the approach taken here generalizes but in discrete time framework the results from frydman 1992 the distribution of the time to the intermediate transition is modelled nonparametrically and the intensity of onset of terminal condition semiparametrically the algorithm is developed for the computation of the estimators the methods are illustrated with aids data

response to comment on application of the extended lie group analysis to the hopf functional formulation of the burgers equation j math phys 57 034102 2016 we address the criticism of frewer et al concerning the paper application of the extended lie group analysis to the hopf functional formulation of the burgers equation j math phys 54 072901 2013 most importantly we stress that we never claimed that any new statistical symmetries were found in this paper the aim of this paper was to apply the lie group analysis to an equation with functional derivatives and derive invariant solutions for this equation these results still stand as they are most important mathematically correct we address also other critical statements of frewer et al and show that there is a connection between the translational invariance of statistics and transformations of the functional to sum up key ideas and fundamental result in the work of waclawczyk and oberlack are still unaffected

a penalized likelihood approach for arbitrarily censored and truncated data application to age specific incidence of dementia the cos model is the model of choice when analyzing survival data presenting only right censoring and left truncation there is a need for methods that can accommodate more complex observation schemes involving general censoring and truncation in addition it is important in many epidemiological applications to have a smooth estimate of the hazard function we show that the penalized likelihood approach gives a solution to these problems the solution of the maximum of the penalized likelihood is approximated on a basis of splines the smoothing parameter is estimated using approximate cross validation confidence bands can be given a simulation study shows that this approach gives better results than the smoothed nelson aalen estimator we apply this method to the analysis of data from a large cohort study on cerebral aging the age specific incidence of dementia is estimated and risk factors of dementia studied

flexible nonhomogeneous markov models for panel observed data methods for fitting nonhomogeneous markov models to panel observed data using direct numerical solution to the kolmogorov forward equations are developed nonhomogeneous markov models occur most commonly when baseline transition intensities depend on calendar time but may also occur with deterministic time dependent covariates such as age we propose transition intensities based on b splines as a smooth alternative to piecewise constant intensities and also as a generalization of time transformation models an expansion of the system of differential equations allows first derivatives of the likelihood to be obtained which can be used in a fisher scoring algorithm for maximum likelihood estimation the method is evaluated through a small simulation study and demonstrated on data relating to the development of cardiac allograft vasculopathy in post transplantation patients

semi markov models with phase type sojourn distributions continuous time multistate models are widely used for categorical response data particularly in the modeling of chronic diseases however inference is difficult when the process is only observed at discrete time points with no information about the times or types of events between observation times unless a markov assumption is made this assumption can be limiting as rates of transition between disease states might instead depend on the time since entry into the current state such a formulation results in a semi markov model we show that the computational problems associated with fitting semi markov models to panel observed data can be alleviated by considering a class of semi markov models with phase type sojourn distributions this allows methods for hidden markov models to be applied in addition extensions to models where observed states are subject to classification error are given the methodology is demonstrated on a dataset relating to development of bronchiolitis obliterans syndrome in post lung transplantation patients

design and analysis of arm in cage experiments inference for three state progressive disease models with common periodic observation times we develop statistical methods for designing and analyzing arm in cage experiments used to test the efficacy of insect repellents and other topical treatments in these experiments a controlled amount of the treatment is applied to a volunteer s forearm which then is exposed to the insects by being placed into a special cage arms are not kept in the cages continuously but rather placed there periodically for a brief period of time during which it is noted whether an insect lands but does not bite or lands and bites efficacy of a repellent can be described using a progressive three state model in which the first two states represent varying degrees of protection no landing and landing without biting and the third state occurs once protection is completely lost biting because subjects within a treatment group follow the same cage visit schedule transition times between states are interval censored into one of several fixed intervals we develop an approach that uses a mixture of nonparametric and parametric techniques for estimating the parameters of interest when sojourn times are dependent design considerations for arm in cage experiments are addressed and the proposed methods are illustrated on data from a recent arm in cage experiment as well as simulated data

tests for differences in tumor incidence based on animal carcinogenesis experiments abstract in animal carcinogenesis studies statistical methods that compare the production of occult tumors should compare tumor incidence rates in general the incidence rate cannot be identified even approximately without both survival and sacrifice information if information from frequent sacrifices is available however a function that approximates the incidence rate can be identified without assumptions about the relationship between tumor incidence and death some earlier proposals that based comparisons on tumor prevalence rates and other functions also compared tumor incidence rates under special conditions but these tests can be biased when the conditions are violated tests that compare functions approximating tumor incidence rates can be developed using survival sacrifice information but these tests may be seriously inefficient in special cases in which sacrifices are not necessary to identify the tumor incidence rate

a penalized likelihood approach for a progressive three state model with censored and truncated data application to aids summary we consider the estimation of the intensity and survival functions for a continuous time progressive three state semi markov model with intermittently observed data the estimator of the intensity function is defined nonparametrically as the maximum of a penalized likelihood we thus obtain smooth estimates of the intensity and survival functions this approach can accommodate complex observation schemes such as truncation and interval censoring the method is illustrated with a study of hemophiliacs infected by hiv the intensity functions and the cumulative distribution functions for the time to infection and for the time to aids are estimated covariates can easily be incorporated into the model

the structure of strongly prime quadratic jordan algebras abstract we extend the structure theory for strongly prime jordan algebras of arbitrary dimension to quadratic algebras of characteristic 2 we show that such j are either 1 hermitian forms h 0 a j h q a lying between an ample subspace of hermitian elements in a prime associative algebra a and those in its martindale ring of quotients q a 2 clifford forms with a scalar extension j j which is an algebra j j q 1 of a nondegenerate quadratic from q with basepoint 1 or 3 albert forms with a scalar extension j j which is a 27 dimensional split albert algebra j h 3 k of 3 3 hermitian matrices over an 8 dimensional split octonion algebra k as a consequence the simple jordan algebras of arbitrary dimension are either hermitian h 0 a for simple a or clifford forms in j q 1 or are albert algebras in particular our methods give an idempotent free classification of all simple finite dimensional algebras which are i special the key to this characterization is the existence of zel manov polynomials in the free special jordan algebra which are both hermitian polynomials whose values always look like hermitian elements and at the same time clifford polynomials nontrivial on 3 3 hermitian matrices hence on special algebras with 3 interconnected idempotents if a zel manov polynomial does not vanish on j then j is of hermitian type and if it does vanish on j then j is of clifford type one example of such a polynomial is q48 p16 x1 y1 z1 w1 p16 x2 y2 z2 w2 p16 x3 y3 z3 w3 for p16 d2x y z 2 dx y w dx y w dx y z x y z thus the structure of jordan algebras depends very directly on polynomial identities and commutators

on the invariants mod 2 of albert algebras the existence of a new invariant for central simple exceptional jordan algebras nowadays called albert algebras over an arbitrary base field k of characteristic not 2 or 3 following a suggestion of serre se has recently been established by rost ro1 in fact serre loc cit has raised the question as to whether this new invariant called the invariant mod 3 together with the ordinary trace form already employed in the work of springer sp classifies albert algebras over k

lie symmetry analysis of the lundgren monin novikov equations for multi point probability density functions of turbulent flow the problem of turbulence statistics described by the lundgren monin novikov lmn hierarchy of integro differential equations is studied in terms of its group properties for this we perform a lie group analysis of a truncated lmn chain which presents the first two equations in an infinite set of integro differential equations for the multi point probability density functions pdf s of velocity a complete set of point transformations is derived for the one point pdf s and the independent variables sample space of velocity space and time for this purpose we use a direct method based on the canonical lie backlund operator due to the one way coupling of correlation equations the present results are complete in the sense that no additional symmetries exist for the first leading equation even if the full infinite hierarchy is considered

study of the hopf functional equation for turbulence duhamel principle and dynamical scaling we consider a formulation for the hopf functional differential equation which governs statistical solutions of the navier stokes equations by introducing an exponential operator with a functional derivative we recast the hopf equation as an integro differential functional equation by the duhamel principle on this basis we introduce a successive approximation to the hopf equation as an illustration we take the burgers equation and carry out the approximations to the leading order scale invariance of the statistical navier stokes equations in d dimensions is formulated and contrasted with that of the deterministic navier stokes equations for the statistical navier stokes equations critical scale invariance is achieved for the characteristic functional of the d th derivative of the vector potential in d dimensions the deterministic equations corresponding to this choice of the dependent variable acquire the linear fokker planck operator under dynamic scaling in three dimensions it is the vorticity gradient that behaves like a fundamental solution more precisely source type solution of deterministic navier stokes equations in the long time limit physical applications of these ideas include study of a self similar decaying profile of fluid flows moreover we reveal typical physical properties in the late stage evolution by combining statistical scale invariance and the source type solution this yields an asymptotic form of the hopf functional in the long time limit improving the well known hopf titt solution in particular we present analyses for the burgers equations to illustrate the main ideas and indicate a similar analysis for the navier stokes equations

the etale tits process of jordan algebras revisited the etale tits process which was called the toral tits process by petersson racine 12 may be viewed as a jordan theoretical method to construct associative algebras with involution more specifically starting from a cubic etale algebra e and a quadratic etale algebra l both over an arbitrary base field as well as from invertible elements u e b l having the same norms the etale tits process produces an absolutely simple jordan algebra j j e l u b of degree 3 and dimension 9 which by structure theory must be the symmetric elements of a central simple associative algebra of degree 3 with involution of the second kind in addition e identifies canonically with a subalgebra of j

a master equation representation of two dimensional turbulence a stochastic representation of the turbulent dynamics of the two dimensional navier stokes equation is developed this stochastic formulation interprets the fluctuating velocity field as a discrete multivariate stochastic process which is governed by a master equation by deriving the hopf functional equation of statistical fluid mechanics it is shown that this approach yields a complete description of the stochastic properties of the turbulent velocity field on the basis of the multivariate master equation a stochastic simulation method for the two dimensional navier stokes equation is constructed this method is applied to the stochastic simulation of high reynolds number turbulent flows

statistical symmetries of the lundgren monin novikov hierarchy it was shown by oberlack and rosteck discr cont dyn sys s 3 451 2010 that the infinite set of multipoint correlation mpc equations of turbulence admits a considerable extended set of lie point symmetries compared to the galilean group which is implied by the original set of equations of fluid mechanics specifically a new scaling group and an infinite set of translational groups of all multipoint correlation tensors have been discovered these new statistical groups have important consequences for our understanding of turbulent scaling laws as they are essential ingredients of e g the logarithmic law of the wall and other scaling laws which in turn are exact solutions of the mpc equations in this paper we first show that the infinite set of translational groups of all multipoint correlation tensors corresponds to an infinite dimensional set of translations under which the lundgren monin novikov lmn hierarchy of equations for the probability density functions pdf are left invariant second we derive a symmetry for the lmn hierarchy which is analogous to the scaling group of the mpc equations most importantly we show that this symmetry is a measure of the intermittency of the velocity signal and the transformed functions represent pdfs of an intermittent i e turbulent or nonturbulent flow interesting enough the positivity of the pdf puts a constraint on the group parameters of both shape and intermittency symmetry leading to two conclusions first the latter symmetries may no longer be lie group as under certain conditions group properties are violated but still they are symmetries of the lmn equations second as the latter two symmetries in its mpc versions are ingredients of many scaling laws such as the log law the above constraints implicitly put weak conditions on the scaling parameter such as von karman constant ensuremath kappa as they are functions of the group parameters finally let us note that these kind of statistical symmetries are of much more general type i e not limited to mpc or pdf equations emerging from navier stokes but instead they are admitted by other nonlinear partial differential equations like for example the burgers equation when in conservative form and if the nonlinearity is quadratic

application of the extended lie group analysis to the hopf functional formulation of the burgers equation a study concerning the lie group analysis of the functional differential equations has been performed this is a continuation of the previous common work of oberlack and waclawczyk on the extension of lie group analysis to functional differential equations arch mech 58 597 2006 where lie group theory has been extended to functional differential equations and a special solution of the functional formulation of the burgers equation was derived based on the calculated set of infinitesimals here we derive an infinite set of symmetry transformations of this equation and find new and more general invariant solutions with this we get a step closer to solutions of functional differential equations which e g give a complete statistical description of turbulence in case of the famous hopf novikov stokes functional differential equations

an exact representation of the space time characteristic functional of turbulent navier stokes flows with prescribed random initial states and driving forces by adopting a formal operator viewpoint the space time characteristic functional associated with navier stokes turbulence is expressed in terms of a linear operator acting on the space of functionals obtained by a simple similarity transformation of the local translation operator generated by the nonlinear terms in the navier stokes equation this operator is unitary with respect to the formal scalar product of functionals the equivalence of this operator representation to the functional integral representation of rosen is shown and for gaussian initial velocity and external force fields some consequences of this representation are presented

albert algebras over curves of genus zero and one abstract albert algebras and other jordan algebras are constructed over curves of genus zero and one using a generalization of the tits process and the first tits construction due to achhammer

burger s model of turbulence as a stochastic process a recently proposed mesoscopic description of fluid dynamics leads to a new approach to turbulence in contrast to the classical statistical theory of turbulence the new approach introduces a probabilistic time evolution of the random velocity governed by a master equation the mesoscopic approach is explained by means of the 1 1 dimensional burger s model of turbulence by a continuous time stochastic simulation realizations of turbulent velocity fields are generated correlation functions and energy spectra are evaluated from appropriate ensemble averages