estimates on the fundamental solution to heat flows with uniformly elliptic coefficients we obtain estimates from above and below on the fundamental solution to the cauchy initial value problem for parabolic equations having uniformly elliptic coefficients we impose minimal regularity requirements on the coefficients the estimates are expressed in terms of an energy functional associated with the coefficients

integral inequalities for the fundamental solutions of diffusions on manifolds with divergence free drift the main purpose of this article is to present uniform integral inequalities for the fundamental solutions of diffusions on compact manifolds with divergence free drift vector fields the method relies on the fact that the heat flow depends on the isoperimetric function the isoperimetric function is used to construct a suitable comparison manifold the heat kernel of this comparison manifold gives uniform bounds for the fundamental solutions of the original diffusion problem the results presented here can be used to solve some open problem of bhattacharya and gotze about diffusions with periodic divergence free drift vector fields see bhagot1 and bhagot2

dissipation statistics of a passive scalar in a multidimensional smooth flow we compute analytically the probability distribution function p e of the dissipation field e 2 of a passive scalar advected by a d dimensional random flow in the limit of large peclet and prandtl numbers batchelor kraichnan regime the tail of the distribution is a stretched exponential for e ln p e d2e 1 3

modular elliptic curves and fermat s last theorem when andrew john wiles was 10 years old he read eric temple bell s the last problem and was so impressed by it that he decided that he would be the first person to prove fermat s last theorem this theorem states that there are no nonzero integers a b c n with n 2 such that an bn cn the object of this paper is to prove that all semistable elliptic curves over the set of rational numbers are modular fermat s last theorem follows as a corollary by virtue of previous work by frey serre and ribet

a path integral approach to asset liability management functional integrals constitute a powerful tool in the investigation of financial models in the recent econophysics literature this technique was successfully used for the pricing of a number of derivative securities in the present contribution we introduce this approach to the field of asset liability management we work with a representation of cash flows by means of a two dimensional delta function perturbation in the case of a brownian model and a geometric brownian model we derive closed form solutions for a finite horizon alm policy the results are numerically and graphically illustrated

instanton for random advection a path integral over trajectories of 2n fluid particles is identified with a 2nth order correlation function of a passive scalar convected by d dimensional short correlated multiscale incompressible random velocity flow strong intermittency of the scalar is described by means of an instanton calculus saddle point plus fluctuations about it in the path integral at n ensuremath gg d the anomalous scaling exponent of the 2nth scalar s structural function is found analytically

decay of passive scalars under the action of single scale smooth velocity fields in bounded two dimensional domains from non self similar probability distribution functions to self similar eigenmodes we examine the decay of passive scalars with small but nonzero diffusivity in bounded two dimensional 2d domains the velocity fields responsible for advection are smooth i e they have bounded gradients and of a single large scale moreover the scale of the velocity field is taken to be similar to the size of the entire domain the importance of the initial scale of variation of the scalar field with respect to that of the velocity field is strongly emphasized if these scales are comparable and the velocity field is time periodic we see the formation of a periodic scalar eigenmode the eigenmode is numerically realized by means of a deterministic 2d map on a lattice analytical justification for the eigenmode is available from theorems in the dynamo literature weakening the notion of an eigenmode to mean statistical stationarity we provide numerical evidence that the eigenmode solution also holds for aperiodic flows represented by random maps turning to the evolution of an initially small scale scalar field we demonstrate the transition from an evolving i e non self similar probability distribution function pdf to a stationary self similar pdf as the scale of variation of the scalar field progresses from being small to being comparable to that of the velocity field and of the domain furthermore the non self similar regime itself consists of two stages both stages are examined and the coupling between diffusion and the distribution of the finite time lyapunov exponents is shown to be responsible for the pdf evolution

rigidity and a riemann hilbert correspondence for p adic local systems we construct a functor from the category of p adic etale local systems on a smooth rigid analytic variety x over a p adic field to the category of vector bundles with an integrable connection on its base change to mathrm b text dr which can be regarded as a first step towards the sought after p adic riemann hilbert correspondence as a consequence we obtain the following rigidity theorem for p adic local systems on a connected rigid analytic variety if the stalk of such a local system at one point regarded as a p adic galois representation is de rham in the sense of fontaine then the stalk at every point is de rham along the way we also establish some basic properties of the p adic simpson correspondence finally we give an application of our results to shimura varieties

renormalization group operator product expansion and anomalous scaling in models of turbulent advection recent progress on the anomalous scaling in models of turbulent heat and mass transport is reviewed with the emphasis on the approach based on the field theoretic renormalization group rg and operator product expansion ope in that approach the anomalous scaling is established as a consequence of the existence in the corresponding field theoretic models of an infinite number of dangerous composite fields operators with negative critical dimensions which are identified with the anomalous exponents this allows one to calculate the exponents in a systematic perturbation expansion similar to the e expansion in the theory of critical phenomena the rg and ope approach is presented in a self contained way for the example of a passive scalar field temperature concentration of an impurity etc advected by a self similar gaussian velocity ensemble with vanishing correlation time the so called kraichnan s rapid change model where the anomalous exponents are known up to order o e3 effects of anisotropy compressibility and the correlation time of the velocity field are discussed passive advection by non gaussian velocity field governed by the stochastic navier stokes equation and passively advected vector e g magnetic fields are considered

the functional integral for a free particle on a half plane a free nonrelativistic particle moving in two dimensions on a half plane can be described by self adjoint hamiltonians characterized by boundary conditions imposed on the systems the most general boundary condition is parameterized in terms of the elements of an infinite dimensional matrix we construct the brownian functional integral for each of these self adjoint hamiltonians nonlocal boundary conditions are implemented by allowing the paths striking the boundary to jump to other locations on the boundary analytic continuation in time results in the green s functions of the schrodinger equation satisfying the boundary condition characterizing the self adjoint hamiltonian

optimal investment consumption strategy with liability and regime switching model under value at risk constraint abstract this paper considers the optimal investment consumption problem with liability subject to a maximum value at risk denoted by mvar constraint the model also contains regime switching market modes whose states are interpreted as the states of the economy in each state of economy state we constrain a var value for the portfolio in a short time duration and mvar is defined as the maximum value of the vars in all economy states we suppose that both the price dynamics of the risky asset and the liability value process are governed by a markov modulated geometric brownian motion with the objective of maximizing the discounted utility of consumption we obtain a system of hjb equations corresponding to the economy states by using the dynamic programming principle then by adopting the techniques of chen et al 2010 we get explicit expressions of value functions moreover with the help of lagrange multiplier method we derive the optimal investment and the optimal consumption finally a numerical example is investigated and the effects of many parameters on the optimal investment on the optimal consumption and on var value are studied furthermore we also explore how var value affects the optimal investment and the optimal consumption

viscoelastic behavior of stock indices abstract the scattering diagram of a stock index results in a complex network structure which can be used to analyze the viscoelastic properties of the index the change along x or y direction of the diagram corresponds to purely elastic or spring like movement whereas the diagonal change at an angle of 45 corresponds to purely viscous or dashpot like movement the viscous component pushes the price from its current value to any other value while the elastic component acts like a restoring force four indices namely dji s p 500 nasdaq 100 and nasdaq composite were studied for the period of 2001 2009 nasdaq composite displayed very high elasticity while nasdaq 100 displayed the highest fluidity in the time period considered the fluidity of dji and s p 500 came out to be close to each other and they are almost the same in the second half of the period

a multiscale measure for mixing abstract we present a multiscale measure for mixing that is based on the concept of weak convergence and averages the mixedness of an advected scalar field at various scales this new measure referred to as the mix norm resolves the inability of the l 2 variance of the scalar density field to capture small scale variations when advected by chaotic maps or flows in addition the mix norm succeeds in capturing the efficiency of a mixing protocol in the context of a particular initial scalar field wherein lyapunov exponent based measures fail to do so we relate the mix norm to the classical ergodic theoretic notion of mixing and present its formulation in terms of the power spectrum of the scalar field we demonstrate the utility of the mix norm by showing how it measures the efficiency of mixing due to various discrete dynamical systems and to diffusion in particular we show that the mix norm can capture known exponential and algebraic mixing properties of certain maps we also analyze numerically the behaviour of scalar fields evolved by the standard map using the mix norm

de rham cohomology of rigid spaces we define de rham cohomology groups for rigid spaces over non archimedean fields of characteristic zero based on the notion of dagger space introduced in 12 we establish some functorial properties and a finiteness result and discuss the relation to the rigid cohomology as defined by p berthelot 2

random perturbations of 2 dimensional hamiltonian flows we consider the motion of a particle in a periodic two dimensional flow perturbed by small molecular diffusion the flow is generated by a divergence free zero mean vector field the long time behavior corresponds to the behavior of the homogenized process that is diffusion process with the constant diffusion matrix effective diffusivity we obtain the asymptotics of the effective diffusivity when the molecular diffusion tends to zero

mixing in fully chaotic flows passive scalar mixing in fully chaotic flows is usually explained in terms of lyapunov exponents i e rates of particle pair separations we present a unified review of this approach which encapsulates also other nonchaotic flows and investigate its limitations during the final stage of mixing when the scalar variance decays exponentially lyapunov exponents can fail to describe the mixing process the failure occurs when another mixing mechanism first introduced by fereday et al phys rev e 65 035301 2002 leads to a slower decay than the mechanism based on lyapunov exponents here we show that this mechanism is governed by the large scale nonuniformities of the flow which are different from the small scale stretching properties of the flow that are captured by the lyapunov exponents however during the initial stage of mixing i e the stage when most of the scalar variance decays lyapunov exponents describe well the mixing process we develop our theory for the incompressible and diffusive baker map a simple example of a chaotic flow nevertheless our results should be applicable to all chaotic flows

serre s conjecture over f9 in this paper we show that an odd galois representation p i gal pq q gl2 f9 having nonsolvable image and satisfying certain local conditions at 3 and 5 is modular our main tools are ideas of taylor 21 and khare 10 which reduce the problem to that of exhibiting points on a hilbert modular surface which are defined over a solvable extension of q and which satisfy certain reduction properties as a corollary we show that hilbert blumenthal abelian surfaces with ordinary reduction at 3 and 5 are modular

brownian motion reflected on brownian motion we study brownian motion reflected on an independent brownian path we prove results on the joint distribution of both processes and the support of the parabolic measure in the space time domain bounded by a brownian path we show that there exist two different natural local times for a brownian path reflected on a brownian path

sailing the deep blue sea of decaying burgers turbulence we study lagrangian trajectories and scalar transport statistics in decaying burgers turbulence we choose velocity fields solutions of the inviscid burgers equation whose probability distributions are specified by kida s statistics they are time correlated and neither time reversal invariant nor gaussian we discuss in some detail the effect of shocks on trajectories and transport equations we derive the inviscid limit of these equations using a formalism of operators localized on shocks we compute the probability distribution functions of the trajectories although they do not define markov processes as physically expected these trajectories are statistically well defined but collapse with probability one at infinite time we point out that the advected scalars enjoy inverse energy cascades we also make a few comments on the connection between our computations and persistence problems

local constancy in families of non abelian galois representations if mathcal s is a a scheme of finite type over a local field f and mathcal x longrightarrow mathcal s is a proper smooth family then to each rational point s in mathcal s one can assign an extension of the absolute galois group of f by the geometric fundamental group g of the fibre mathcal x s if f has uniformiser pi and residue characteristic p we show that the corresponding extension of the absolute galois group of mathcal s by the maximal prime to p quotient of g is locally constant in the pi adic topology of mathcal s we give a similar result in the case of non proper families and families over pi adic analytic spaces

the mixing efficiency of open flows abstract mixing in open incompressible flows is studied in a model problem with inhomogeneous passive scalar injection on an inlet boundary as a measure of the efficiency of stirring the bulk scalar concentration variance is bounded and the bound is shown to be sharp at low peclet number although no specific flow saturating the bound at high peclet number is produced here the estimate is conjectured to be approached for flows possessing sufficiently sustained chaotic regions

diffusion and mixing in fluid flow we study enhancement of diffusive mixing on a compact riemannian manifold by a fast incompressible flow our main result is a sharp description of the class of flows that make the deviation of the solution from its average arbitrarily small in an arbitrarily short time provided that the flow amplitude is large enough the necessary and suficient condition on such flows is expressed naturally in terms of the spectral properties of the dynamical system associated with the flow in particular we find that weakly mixing flows always enhance dissipation in this sense the proofs are based on a general criterion for the decay of the semigroup generated by an operator of the form g ial with a negative unbounded self adjoint operator g a self adjoint operator l and parameter a 1 in particular they employ the rage theorem describing evolution of a quantum state belonging to the continuous spectral subspace of the hamiltonian related to a classical theorem of wiener on fourier transforms of measures applications to quenching in reaction diffusion equations are also considered

multiscale mixing efficiencies for steady sources multiscale mixing efficiencies for passive scalar advection are defined in terms of the suppression of variance weighted at various length scales we consider scalars maintained by temporally steady but spatially inhomogeneous sources stirred by statistically homogeneous and isotropic incompressible flows including fully developed turbulence the mixing efficiencies are rigorously bounded in terms of the p eclet number and specific quantitative features of the source scaling exponents for the bounds at high p eclet number depend on the spectrum of length scales in the source indicating that molecular diffusion plays a more important quantitative role than that implied by classical eddy diffusion theories

boundary layers for cellular flows at high peclet numbers we analyze the behavior of solutions of steady advection diffusion problems in bounded domains with prescribed dirichlet data when the peclet number pe 1 is large we show that the solution converges to a constant in each flow cell outside a boundary layer of width o 1 2 pe 1 around the flow separatrices we construct an dependent approximate water pipe problem purely inside the boundary layer that provides a good approximation of the solution of the full problem but has independent computational cost we also define an asymptotic problem on the graph of streamline separatrices and show that solution of the water pipe problem itself may be approximated by an asymptotic independent problem on this graph finally we show that the dirichlet to neumann map of the water pipe problem approximates the dirichlet to neumann map of the separatrix problem with an error independent of the flow outside the boundary layers 2004 wiley periodicals inc

diffusion advection in cellular flows with large peclet numbers for periodic two dimensional cellular flows in incompressible fluids we provide explicit upper and lower estimates of the effective diffusivity which have the correct scaling behavior for large peclet numbers we demonstrate that all allowed scaling laws can occur the bounds prove that there is no residual diffusion in the infinite peclet number limit for holder continuous flows answering a problem posed by kozlov

inverse cascade and intermittency of passive scalar in one dimensional smooth flow random advection of a lagrangian tracer scalar field ensuremath theta t x by a one dimensional spatially smooth and short correlated in time velocity field is considered scalar fluctuations are maintained by a source concentrated at the integral scale l the statistical properties of both scalar differences and the dissipation field are analytically determined exploiting the dynamical formulation of the model the gaussian statistics known to be present at small scales for incompressible velocity fields emerges here at large scales x ensuremath gg l these scales are shown to be excited by an inverse cascade of ensuremath theta 2 and the probability distribution function pdf of the corresponding scalar differences to approach the gaussian form as larger and larger scales are considered small scale x ensuremath ll l statistics is shown to be strongly non gaussian a collapse of scaling exponents for scalar structure functions takes place moments of order pg 1 all scale linearly independently of the order p smooth scaling x p is found for ensuremath 1lpl1 tails of the scalar difference pdf are exponential while at the center a cusped shape tends to develop when smaller and smaller ratios x l are considered the same tendency is present for the scalar gradient pdf with respect to the inverse of the p eclet number the pumping to diffusion scale ratio the tails of the latter pdf are however much more extended decaying as a stretched exponential of exponent 2 3 smaller than unity this slower decay is physically associated with the strong fluctuations of the dynamical dissipative scale

statistics of a passive scalar advected by a large scale two dimensional velocity field analytic solution steady statistics of a passive scalar advected by a random two dimensional flow of an incompressible fluid is described in the range of scales between the correlation length of the flow and the diffusion scale this corresponds to the so called batchelor regime where the velocity is replaced by its large scale gradient the probability distribution of the scalar in the locally comoving reference frame is expressed via the probability distribution of the line stretching rate the description of line stretching can be reduced to a classical problem of the product of many random matrices with a unit determinant we have found the change of variables that allows one to map the matrix problem onto a scalar one and to thereby prove the central limit theorem for the stretching rate statistics the proof is valid for any finite correlation time of the velocity field whatever the statistics of the velocity field the statistics of the passive scalar averaged over time locally in space is shown to approach gaussian statistics with increase in the p eclet number pe the pumping to diffusion scale ratio the first nsimultaneous correlation functions are expressed via the flux of the square of the scalar and only one factor depending on the velocity field the mean stretching rate which can be calculated analytically in limiting cases non gaussian tails of the probability distributions at finite pe are found to be exponential

closed form approximations for diffusion densities a path integral approach in this paper we investigate the transition probabilities for diffusion processes in a first part we show how transition probabilities for rather general diffusion processes can always be expressed by means of a path integral for several classical models an exact calculation is possible leading to analytical expressions for the transition probabilities and for the maximum probability paths a second part consists of the derivation of an analytical approximation for the transition probability which is useful in case the path integral is too complex to be calculated the approximation we present is based on a convex combination of a new analytical upper and lower bound for the transition probabilities the fact that the approximation is analytical has some important advantages e g for the investigation of asian options finally we demonstrate the accuracy of the approximation by means of some graphical illustrations

long time estimates for the heat kernel associated with a uniformly subelliptic symmetric second order operator abstract second order subelliptic operators have been the subject of a considerable amount of research in recent years starting with the paper rothschild and stein in which the sharp form of hormander s famous subellipticity theorem is proved and continuing through the work of fefferman and phong and sanchez calle it has become increasingly clear that precise regularity estimates for these operators depend intimately on the geometry associated with the operator under consideration the main purpose of this article is to obtain bounds from above and below on p t x y t epsilon 1 infinity in terms of standard heat kernels

turbulence without pressure we develop exact field theoretic methods to treat turbulence when the effect of pressure is negligible we find explicit forms of certain probability distributions demonstrate that the breakdown of galilean invariance is responsible for intermittency and establish the operator product expansion we also indicate how the effects of pressure can be turned on perturbatively

turbulence of polymer solutions we investigate high reynolds number turbulence in dilute polymer solutions we show the existence of a critical value of the reynolds number which separates two different regimes in the first regime below the transition the influence of the polymer molecules on the flow is negligible so they can be regarded as passively embedded in the flow this case admits a detailed investigation of the statistics of the polymer elongations the second state is realized when the reynolds number is larger than the critical value this regime is characterized by the strong back reaction of polymers on the flow we establish some properties of the statistics of the stress and velocity in this regime and discuss its relation to the drag reduction phenomenon

canonical desingularization in characteristic zero by blowing up the maximum strata of a local invariant this article contains an elementary constructive proof of resolution of singularities in characteristic zero our proof applies in particular to schemes of finite type and to analytic spaces so we recover the great theorems of hironaka we introduce a discrete local invariant rm inv x a whose maximum locus determines a smooth centre of blowing up leading to desingularization to define rm inv x we need only to work with a category of local ringed spaces x x cal o x satisfying certain natural conditions if a in x then rm inv x a depends only on widehat cal o x a more generally rm inv x is defined inductively after any sequence of blowings up whose centres have only normal crossings with respect to the exceptional divisors and lie in the constant loci of rm inv x cdot the paper is self contained and includes detailed examples one of our goals is that the reader understand the desingularization theorem rather than simply know it is true

anomalous diffusion in fast cellular flows at intermediate time scales it is well known that on long time scales the behaviour of tracer particles diffusing in a cellular flow is effectively that of a brownian motion this paper studies the behaviour on intermediate time scales before diffusion sets in various heuristics suggest that an anomalous diffusive behaviour should be observed we prove that the variance on intermediate time scales grows like o sqrt t hence on these time scales the effective behaviour can not be purely diffusive and is consistent with an anomalous diffusive behaviour

bounds on the mixing enhancement for a stirred binary fluid abstract the cahn hilliard equation describes phase separation in binary liquids here we study this equation with spatially varying sources and stirring or advection we specialize to symmetric mixtures and time independent sources and discuss stirring strategies that homogenize the binary fluid by measuring fluctuations of the composition away from its mean value we quantify the amount of homogenization achievable we find upper and lower bounds on our measure of homogenization using only the cahn hilliard equation and the incompressibility of the advecting flow we compare these theoretical bounds with numerical simulations for two model flows the constant flow and the random phase sine flow using the sine flow as an example we show how our bounds on composition fluctuations provide a measure of the effectiveness of a given stirring protocol in homogenizing a phase separating binary fluid

overconvergent hilbert modular forms we generalize the construction of the eigencurve by coleman mazur to the setting of totally real fields and show that a finite slope hilbert modular eigenform can be deformed into a one parameter family of finite slope eigenforms the key point is to show the overconvergence of the canonical subgroup and the complete continuity of the u p operator we deduce this form some general considerations in rigid analytic geometry

option pricing from path integral for non gaussian fluctuations natural martingale and application to truncated levy distributions within a path integral formalism for non gaussian price fluctuations we set up a simple stochastic calculus and derive a natural martingale for option pricing from the wealth balance of options stocks and bonds the resulting formula is evaluated for truncated levy distributions

universal mixers in all dimensions abstract we construct universal mixers incompressible flows that mix arbitrarily well general solutions to the corresponding transport equation in all dimensions this mixing is exponential in time i e essentially optimal for any initial condition with at least some regularity and we also show that a uniform mixing rate for all initial conditions cannot be achieved the flows are time periodic and uniformly in time bounded in spaces w s p for a range of s p that includes points with s 1 and p 2

instantons and intermittency we describe the method for finding the non gaussian tails of the probability distribution function pdf for solutions of a stochastic differential equation such as the convection equation for a passive scalar the random driven navier stokes equation etc the existence of such tails is generally regarded as a manifestation of the intermittency phenomenon our formalism is based on the wkb approximation in the functional integral for the conditional probability of large fluctuation we argue that the main contribution to the functional integral is given by a coupled field force configuration char22 the instanton as an example we examine the correlation functions of the passive scalar u advected by a large scale velocity field ensuremath delta correlated in time we find the instanton determining the tails of the generating functional and show that it is different from the instanton that determines the probability distribution function of high powers of u we discuss the simplest instantons for the navier stokes equation textcopyright 1996 the american physical society

advected fields in maps i magnetic flux growth in the stretch fold shear map abstract the behaviour of magnetic field in the stretch fold shear sfs dynamo map is considered for zero magnetic diffusion it is shown by a mixture of analytical and numerical approaches that the sfs map is a perfect dynamo for sufficiently large shear the adjoint operator has smooth growing eigenfunctions and so smooth flux averages grow exponentially with time for zero diffusion in the paper first a number of numerical discretisations are presented that give differing results for growth rates and indicate the need to develop systematic theory then magnetic fields that are only required to be square integrable are considered and the spectral properties of the sfs dynamo operator and its adjoint are discussed as operators in l 2 adjoint eigenfunctions are typically not smooth however to obtain smooth growing adjoint eigenfunctions attention is restricted to a subset of magnetic fields that are analytic in a disc in the complex plane restricted to this subset and using a supremum norm the sfs adjoint operator is compact and this allows a numerical treatment of eigenvalues and eigenfunctions with systematic error estimates these estimates show that for sufficiently large shear there are smooth growing adjoint eigenfunctions and so perfect dynamo action is established

path integrals for potential problems with δ function perturbation the author presents several examples of potential problems with a function perturbation by means of path integrals the idea is to sum a perturbation series expansion resulting in an energy dependent green function g e the energy levels en of the perturbed problem are determined by the equation one dimensional case ig v a a en gamma where g v is the green function of the unperturbed problem gamma is the strength of the potential and a its location in r in d dimensional radial problems with a spherically shaped delta function located at r a this equation changes into ig v l a a en ad 1 where l denotes the angular momentum number

long time behaviour of heat flow global estimates and exact asymptotics we obtain global upper and lower bounds on the heat kernel of an elliptic second order differential operator which become sharp in certain long time and large space asymptotics we prove a generalization of aronson s gaussian bounds which identifies correctly an effective drift for heat flow in the case of periodic coefficients we give variational characterizations of the effective conductivity which is then made to appear in heat kernel bounds these results are for heat kernels with measurable coefficients for differentiable coefficients we prove tighter estimates in which the rate of homogenization is known to be optimal

stirring up trouble multi scale mixing measures for steady scalar sources abstract the mixing efficiency of a flow advecting a passive scalar sustained by steady sources and sinks is naturally defined in terms of the suppression of bulk scalar variance in the presence of stirring relative to the variance in the absence of stirring these variances can be weighted at various spatial scales leading to a family of multi scale mixing measures and efficiencies we derive a priori estimates on these efficiencies from the advection diffusion partial differential equation focusing on a broad class of statistically homogeneous and isotropic incompressible flows the analysis produces bounds on the mixing efficiencies in terms of the peclet number a measure of the strength of the stirring relative to molecular diffusion we show by example that the estimates are sharp for particular source sink and flow combinations in general the high peclet number behavior of the bounds scaling exponents as well as prefactors depends on the structure and smoothness properties of and length scales in the scalar source and sink distribution the fundamental model of the stirring of a monochromatic source sink combination by the random sine flow is investigated in detail via direct numerical simulation and analysis the large scale mixing efficiency follows the upper bound scaling within a logarithm at high peclet number but the intermediate and small scale efficiencies are qualitatively less than optimal the peclet number scaling exponents of the efficiencies observed in the simulations are deduced theoretically from the asymptotic solution of an internal layer problem arising in a quasi static model

anomalous scaling for passively advected magnetic fields the kraichnan model 1 for passive scalar advection is by now considered a paradigm for intermittency problems a first systematic understanding of the scalar field scaling properties has been attained the crucial property of the model is that equal time correlation functions obey closed equations of motion this stems from the small correlation time of the velocity field advecting the scalar it has been recognized that a general mechanism for intermittency and anomalous scaling is the presence of nontrivial zero modes of the closed equations satisfied by the correlation functions 2 3 4 such mechanism can be illustrated in its simplest form by considering the following linear differential equation frac d 2 y r d r 2 frac a r frac dy r dr b frac y r r 2 f frac r l 1 where r varies on the positive axis y r is the unknown function a and b are two constants and f r l varies over distances of order l and is almost constant for r l in the physical situations l is the integral scale and the mechanisms maintaining the turbulence are supposed to act at this scale as for intermittency in navier stokes turbulence we are interested in the scaling behavior in the inertial range i e for r l note that the l h s of 1 is invariant under scale transformations reflecting the scale invariance of the general solution of 1 is given by a non homogeneous and a linear combination of two homogeneous solutions the combination is fixed by the conditions that y r being a correlation function should be regular and vanish at infinity

evolution of the probability measure for the majda model new invariant measures and breathing pdfs in 1993 majda proposed a simple random shear model from which scalar inter mittency was rigorously predicted for the invariant probability measure of passive tracers in this work we present an integral formulation for the tracer measure which leads to a new comprehensive study on its temporal evolution based on monte carlo simulation and direct numerical integration an interesting non monotonic breathing phenomenon is discov ered from these results and carefully defined with a solid example for special initial data to predict such phenomenon the signature of this phenomenon may persist at long time char acterized by the approach of the pdf core to its infinite time invariant value we find that this approach may be strongly dependent on the non dimensional peclet number of which the invariant measure itself is independent further the breathing pdf is recovered as a new invariant measure in a distinguished time scale in the diffusionless limit rigorous as ymptotic analysis is also performed to identify the gaussian core of the invariant measures and the critical rate at which the heavy stretched exponential regime propagates towards the tail as a function of time is calculated

universal long time properties of lagrangian statistics in the batchelor regime and their application to the passive scalar problem we consider the transport of dynamically passive quantities in the batchelor regime of a smooth in space velocity field for the case of arbitrary temporal correlations of the velocity we formulate the statistics of relevant characteristics of lagrangian motion this allows us to generalize many results obtained previously for strain ensuremath delta correlated in time thus answering a question about the universality of these results

optimizing the source distribution in fluid mixing abstract a passive scalar is advected by a velocity field with a nonuniform spatial source that maintains concentration inhomogeneities for example the scalar could be temperature with a source consisting of hot and cold spots such that the mean temperature is constant which source distributions are best mixed by this velocity field this question has a straightforward yet rich answer that is relevant to real mixing problems we use a multiscale measure of steady state enhancement to mixing and optimize it by a variational approach we then solve the resulting euler lagrange equation for a perturbed uniform flow and for simple cellular flows the optimal source distributions have many broad features that are as expected they avoid stagnation points favour regions of fast flow and their contours are aligned such that the flow blows hot spots onto cold and vice versa however the detailed structure varies widely with diffusivity and other problem parameters though these are model problems the optimization procedure is simple enough to be adapted to more complex situations

applications of δ function perturbation to the pricing of derivative securities in the recent econophysics literature the use of functional integrals is widespread for the calculation of option prices in this paper we extend this approach in several directions by means of function perturbations first we show that results about infinitely repulsive function are applicable to the pricing of barrier options we also introduce functional integrals over skew paths that give rise to a new european option formula when combined with function potential we propose accurate closed form approximations based on the theory of comonotonic risks in case the functional integrals are not analytically computable

normal and anomalous scaling of the fourth order correlation function of a randomly advected passive scalar for a delta correlated velocity field simultaneous correlation functions of a passive scalar satisfy closed equations we analyze the equation for the four point function to describe a solution completely one has to solve the matching problems at the scale of the source and at the diffusion scale we solve both the matching problems and thus find the dependence of the four point correlation function on the diffusion and pumping scale for large space dimensionality d it is shown that anomalous scaling appears in the first order of 1 d perturbation theory anomalous dimensions are found analytically both for the scalar field and for it s derivatives in particular for the dissipation field