an introduction to mathematical models of coagulation fragmentation processes a discrete deterministic mean field approach abstract we summarise the properties and the fundamental mathematical results associated with basic models which describe coagulation and fragmentation processes in a deterministic manner and in which cluster size is a discrete quantity an integer multiple of some basic unit size in particular we discuss smoluchowski s equation for aggregation the becker doring model of simultaneous aggregation and fragmentation and more general models involving coagulation and fragmentation

time fractional wave diffusion equation in an inhomogeneous half space we consider the fundamental solution of the time fractional wave diffusion equation in a three dimensional half space medium which contains an inhomogeneity in form of a plane parallel layer the corresponding green s function which is derived by means of the fourier and laplace transforms can be accurately and efficiently evaluated without recourse to the mittag leffler or the fox h function moreover it is shown that in the one dimensional case the fundamental solution in an inhomogeneous half space is no longer a probability density function in addition we consider the advection equation for the fractional laplacian and the caputo time fractional derivative of orders on a bounded domain simple algorithms for accurate evaluation of the m wright function and the mittag leffler function are enclosed at the end of this article

kinetics of gelation and universality the exact solution size distribution ck t and moments m t of smoluchowski s coagulation equation s model and of a modified equation f model with a coagulation rate k ij for i and j clusters is obtained for arbitrary ck 0 in the sol t phases where tc is the gel point the behaviour of ck t and m t is given for k co t co and t i the critical exponents critical amplitudes and scaling function that characterise the singularities near the non equilibrium phase transition are calculated for short range ck 0 i e all m a the f model belongs to the universality class of classical gelation theories and of bond percolation on cayley trees the s model does not

asymptotic solutions of the becker doring equations with size dependent rate constants we find the large time asymptotic behaviour for a number of physically interesting cases of the becker doring equations allowing both the forward and the backward rates to depend on cluster size in a power law fashion we consider in detail the constant monomer form of the equations in the special case where the powers are equal since the structure of the large time asymptotic behaviour is then richest we then turn to cases in which aggregation and fragmentation have different exponents examining both the fragmentation and coagulation dominated cases again under constant monomer conditions

inversion of seismic reflection data in the acoustic approximation the nonlinear inverse problem for seismic reflection data is solved in the acoustic approximation the method is based on the generalized least squares criterion and it can handle errors in the data set and a priori information on the model multiply reflected energy is naturally taken into account as well as refracted energy or surface waves the inverse problem can be solved using an iterative algorithm which gives at each iteration updated values of bulk modulus density and time source function each step of the iterative algorithm essentially consists of a forward propagation of the actual sources in the current model and a forward propagation backward in time of the data residuals the correlation at each point of the space of the two fields thus obtained yields the corrections of the bulk modulus and density models this shows in particular that the general solution of the inverse problem can be attained by methods strongly related to the methods of migration of unstacked data and commerc

characterizations and simulations of a class of stochastic processes to model anomalous diffusion in this paper we study a parametric class of stochastic processes to model both fast and slow anomalous diffusions this class called generalized grey brownian motion ggbm is made up of self similar with stationary increments processes h sssi and depends on two real parameters 0 2 and 0 1 it includes fractional brownian motion when 0 2 and 1 and time fractional diffusion stochastic processes when 0 1 the latter have a marginal probability density function governed by timefractional diffusion equations of order the ggbm is defined through the explicit construction of the underlying probability space however in this paper we show that it is possible to define it in an unspecified probability space for this purpose we write down explicitly all the finite dimensional probability density functions moreover we provide different ggbm characterizations the role of them wright function which is related to the fundamental solution of the time fractional diffusion equation emerges as a natural generalization of the gaussian distribution furthermore we show that the ggbm can be represented in terms of the product of a random variable which is related to the m wright function and an independent fractional brownian motion this representation highlights the h sssi nature of the ggbm and provides a way to study and simulate the trajectories for this purpose we developed a random walk model based on a finite difference approximation of a partial integro differential equation of a fractional type

the derivative of a waveguide acoustic field with respect to a three dimensional sound speed perturbation semianalytic expressions are derived for the first order derivative of a pressure field in a laterally homogeneous waveguide with respect to an arbitrary three dimensional refractive index perturbation in either the water column or ocean bottom these expressions for the environmental derivative derived using an adjoint method require a three dimensional spatial correlation between two green s functions weighted by an environmental parameter basis function with the green s functions expressed in terms of normal modes when a particular set of orthogonal spatial basis functions is chosen the three dimensional spatial integral can be converted into a set of one dimensional integrations over depth and azimuth the use of the orthogonal basis permits environmental derivatives to be computed for an arbitrary sound speed perturbation to illustrate the formulas a simple sensitivity study is presented that explores under what circumstances three dimensional plane wave and cylindrical perturbations produ

differentiation to fractional orders and the fractional telegraph equation using methods of differential and integral calculus we present and discuss the calculation of a fractional green function associated with the one dimensional case of the so called general fractional telegraph equation with one space variable this is a fractional partial differential equation with constant coefficients the equation is solved by means of juxtaposition of transforms i e we introduce the laplace transform in the time variable and the fourier transform in the space variable several particular cases are discussed in terms of the parameters some known results are recovered as a by product of our main result we obtain two new relations involving the two parameter mittag leffler function

the becker doring equations with exponentially size dependent rate coefficients this paper is concerned with an analysis of the becker doring equations which lie at the heart of a number of descriptions of non equilibrium phase transitions and related complex dynamical processes the becker doring theory describes growth and fragmentation in terms of stepwise addition or removal of single particles to or from clusters of similar particles and has been applied to a wide range of problems of physicochemical and biological interest within recent years here we consider the case where the aggregation and fragmentation rates depend exponentially on cluster size these choices of rate coefficients at least qualitatively correspond to physically realistic molecular clustering scenarios such as occur in for example simulations of simple fluids new similarity solutions for the constant monomer becker doring system are identified and shown to be generic in the case of aggregation and fragmentation rates that depend exponentially on cluster size

response characteristics of a fractional oscillator abstract the integral equation of motion of a driven fractional oscillator is obtained by generalizing the corresponding equation of motion of a driven harmonic oscillator to include integrals of arbitrary order according to the methods of fractional calculus the green s function solution for the fractional oscillator is obtained in terms of mittag leffler functions using laplace transforms the response and resonance characteristics of the fractional oscillator are studied for several cases of forcing function

on some fractional green s functions in this paper we discuss some fractional green s functions associated with the fractional differential equations which appear in several fields of science more precisely the so called wave reaction diffusion equation and some of its particular cases the methodology presented is the juxtaposition of integral transforms in particular the laplace and the fourier integral transforms some recent results involving the reaction diffusion equation are pointed out

the modified becker doring system with aggregation dominated rate coefficients we consider a system of becker doring equations modified in the manner of dreyer duderstadt 2006 j stat phys 123 55 we analyse the case of size independent rate coefficients in the case of weak fragmentation aggregation dominated we use matched asymptotic expansions to construct approximations to the cluster size distribution as it evolves from purely monomeric initial data to the equilibrium solution in which the typical cluster size is extremely large we compare the results with earlier calculations of a similar limit in the standard becker doring system and find a similar sequence of timescales with similar profile shapes however there are notable differences in the kinetic behaviour within timescales due to the modified system having a fragmentation rate which depends on the total number of clusters present in the system

necessary conditions for a maximum likelihood estimate to become asymptotically unbiased and attain the cramer rao lower bound ii range and depth localization of a sound source in an ocean waveguide analytic expressions for the first order bias and second order covariance of a maximum likelihood estimate mle are applied to the problem of localizing an acoustic source in range and depth in a shallow water waveguide with a vertical hydrophone array these expressions are then used to determine necessary conditions on sample size or equivalently signal to noise ratio snr for the localization mle to become asymptotically unbiased and attain minimum variance as expressed by the cramer rao lower bound crlb these analytic expressions can be applied in a similar fashion to any ocean acoustic inverse problem involving random data both deterministic and completely randomized signals embedded in independent and additive waveguide noise are investigated as the energy ratio of received signal to additive noise sanr descends to the lower operational range of a typical passive localization system source range and depth estimates exhibit significant biases and have variances that can exceed the crlb by orders of magnitude the spatial structure of the bias suggests that acoustic range and depth estimates tend to converge around particular range and depth cells for moderate sanr values

a three dimensional propagation algorithm using finite azimuthal aperture the use of three dimensional 3 d propagation models is becoming increasingly common in current underwater acoustics applications such models typically treat the propagation as a function of range depth and bearing consistent with previous multibearing two dimensional n 2 d models however to obtain accurate 3 d solutions at long ranges many bearings must be computed in order to maintain the necessary cross range resolution between bearings in this paper a 3 d parabolic equation pe model is developed with a marching algorithm based on the split step fourier technique in both depth and azimuthal bearing the algorithm includes a scheme to compute the solution for only a finite azimuthal aperture rather than the full 360 of bearing the success of this algorithm allows for increased cross range resolution at long range without increasing the number of bearings needed in the calculation results of this technique are compared to results from a suggested benchmark case

a variational data assimilation system for the range dependent acoustic model using the representer method theoretical derivations this study presents the theoretical framework for variational data assimilation of acoustic pressure observations into an acoustic propagation model namely the range dependent acoustic model ram ram uses the split step pade algorithm to solve the parabolic equation the assimilation consists of minimizing a weighted least squares cost function that includes discrepancies between the model solution and the observations the minimization process which uses the principle of variations requires the derivation of the tangent linear and adjoint models of the ram the mathematical derivations are presented here and for the sake of brevity a companion study presents the numerical implementation and results from the assimilation simulated acoustic pressure observations

adjoint modeling for acoustic inversion the use of adjoint modeling for acoustic inversion is investigated an adjoint model is derived from a linearized forward propagation model to propagate data model misfit at the observation points back through the medium to the medium perturbations not being accounted for in the model this adjoint model can be used to aid in inverting for these unaccounted medium perturbations adjoint methods are being applied to a variety of inversion problems but have not drawn much attention from the underwater acoustic community this paper presents an application of adjoint methods to acoustic inversion inversions are demonstrated in simulation for both range independent and range dependent sound speed profiles using the adjoint of a parabolic equation model sensitivity and error analyses are discussed showing how the adjoint model enables calculations to be performed in the space of observations rather than the often much larger space of model parameters using an adjoint model enables directions of steepest descent in the model parameters what we invert for to be calculated using far fewer modeling runs than if a forward model only were used

kinetics of helium bubble formation in nuclear materials the formation and growth of helium bubbles due to self irradiation in plutonium has been modelled by discrete kinetic equations for the number densities of bubbles having k atoms analysis of these equations shows that the bubble size distribution function can be approximated by a composite of i the solution of partial differential equations describing the continuum limit of the theory but corrected to take into account the effects of discreteness and ii a local expansion about the advancing leading edge of the distribution function in size space both approximations contribute to the memory term in a close integrodifferential equation for the monomer concentration of single helium atoms the present boundary layer theory for discrete equations is compared to the numerical solution of the full kinetic model and to the previous approximation of schalclach and wolfer involving a truncated system of moment equations

homogeneous fractional embeddings fractional equations appear in the description of the dynamics of various physical systems for lagrangian systems the embedding theory developed by cresson fractional embedding of differential operators and lagrangian systems j math phys 48 033504 2007 provides a univocal way to obtain such equations stemming from a least action principle however no matter how equations are obtained the dimension of the fractional derivative differs from the classical one and may induce problems of temporal homogeneity in fractional objects in this paper we show that it is necessary to introduce an extrinsic constant of time then we use it to construct two equivalent fractional embeddings which retains homogeneity the notion of fractional constant is also discussed through this formalism finally an illustration is given with natural lagrangian systems and the case of the harmonic oscillator is entirely treated

optimal nonlocal boundary control of the wide angle parabolic equation for inversion of a waveguide acoustic field this paper applies the concept of optimal boundary control for solving inverse problems in shallow water acoustics to treat the controllability problem a continuous analytic adjoint model is derived for the claerbout wide angle parabolic equation pe using a generalized nonlocal impedance boundary condition at the water bottom interface while the potential of adjoint methodology has been recently demonstrated for ocean acoustic tomography this approach combines the advantages of exact transparent boundary conditions for the wide angle pe with the concept of adjoint based optimal control in contrast to meta heuristic approaches the inversion procedure itself is directly controlled by the waveguide physics and in a numerical implementation based on conjugate gradient optimization many fewer iterations are required for assessment of an environment that is supported by the underlying subbottom model furthermore since regularization schemes are particularly important to enhance the performance of full field acoustic inversion special attention is devoted to the application of penalization methods to the adjoint optimization formalism regularization incorporates additional information about the desired solution in order to stabilize ill posed inverse problems and identify useful solutions a feature that is of particular importance for inversion of field data sampled on a vertical receiver array in the presence of measurement noise and modeling uncertainty results with test data show that the acoustic field and the bottom properties embedded in the control parameters can be efficiently retrieved

fractional variational calculus in terms of riesz fractional derivatives this paper presents extensions of traditional calculus of variations for systems containing riesz fractional derivatives rfds specifically we present generalized euler lagrange equations and the transversality conditions for fractional variational problems fvps defined in terms of rfds we consider two problems a simple fvp and an fvp of lagrange results of the first problem are extended to problems containing multiple fractional derivatives functions and parameters and to unspecified boundary conditions for the second problem we present lagrange type multiplier rules for both problems we develop the euler lagrange type necessary conditions which must be satisfied for the given functional to be extremum problems are considered to demonstrate applications of the formulations explicitly we introduce fractional momenta fractional hamiltonian fractional hamilton equations of motion fractional field theory and fractional optimal control the formulations presented and the resulting equations are similar to the formulations for fvps given in agrawal 2002 j math anal appl 272 368 2006 j phys a math gen 39 10375 and to those that appear in the field of classical calculus of variations these formulations are simple and can be extended to other problems in the field of fractional calculus of variations

cluster growth poised on the edge of break up size distributions with small exponent power laws abstract in many naturally occurring growth processes cluster size distributions of power law form n s s with small exponents 0 1 are observed we suggest here that such distributions emerge naturally from cluster growth where size dependent aggregation is counterbalanced by size dependent break up the model used in the study is a simple reaction kinetic model including only monomer cluster processes it is shown that under such conditions power law size distributions with small exponents are obtained therefore the results suggest that the ubiquity of small exponent power law distributions is related to the growth process where aggregation driven cluster growth is poised on the edge of cluster break up

necessary conditions for a maximum likelihood estimate to become asymptotically unbiased and attain the cramer rao lower bound part i general approach with an application to time delay and doppler shift estimation analytic expressions for the first order bias and second order covariance of a general maximum likelihood estimate mle are presented these expressions are used to determine general analytic conditions on sample size or signal to noise ratio snr that are necessary for a mle to become asymptotically unbiased and attain minimum variance as expressed by the cramer rao lower bound crlb the expressions are then evaluated for multivariate gaussian data the results can be used to determine asymptotic biases variances and conditions for estimator optimality in a wide range of inverse problems encountered in ocean acoustics and many other disciplines the results are then applied to rigorously determine conditions on snr necessary for the mle to become unbiased and attain minimum variance in the classical active sonar and radar time delay and doppler shift estimation problems the time delay mle is the time lag at the peak value of a matched filter output it is shown that the matched filter estimate a

iterative inverse scattering algorithms methods of computing frechet derivatives iterative approaches to the nonlinear inverse scattering problem generally attempt to find the scattering distribution that best predicts the data by minimizing a global error norm e g the mean square error which quantifies the misfit between a set of measured data and data predicted on the basis of a forward calculation a crucial quantity in this minimization is the frechet derivative of the error norm which tells us how to update the current estimate of the scattering distribution to reduce the global error at each iteration this paper demonstrates how to compute the frechet derivative using three different but fundamentally equivalent methods the conventional adjoint method the lagrange multiplier method and the integral equation method the first two begin with the wave equation while the latter method is based on a lippmann schwinger integral equation these techniques are not only far more efficient but also numerically less error prone than brute force methods for computing derivatives based for example on finite differences for simplicity a variational approach is employed in which the fields and scattering distribution are represented by continuous functions but the finite dimensional discretized problem is shown to follow directly from the continuous space results

adjoint based acoustic inversion for the physical characterization of a shallow water environment recently the concept of adjoint modeling has been introduced in shallow water acoustics for solving inverse problems analytical adjoints have been derived for normal modes and for both the standard parabolic equation and claerbout s wide angle approximation wape this paper proposes the application of a semiautomatic adjoint approach that has been successfully applied in the past for multidimensional variational data assimilation in meteorological and climate modeling starting from a modular graph representation of the underlying forward model a programming tool facilitates the generation and coding of both the tangent linear and the adjoint models the potential of this numerical adjoint approach for the physical characterization of a shallow water environment is illustrated with two applications for geoacoustic inversion and ocean acoustic tomography using claerbout s wape in combination with nonlocal boundary conditions furthermore the adjoint optimization is extended to multiple frequencies and

multiple order derivatives of a waveguide acoustic field with respect to sound speed density and frequency an adjoint perturbative method is used to derive expressions for the first through third order derivatives of a pressure field with respect to sound speed density and frequency for the restricted case of a laterally homogenous waveguide in which environmental parameters are only a function of depth by using a normal mode green s function the three dimensional spatial correlation required by the standard acoustic adjoint equation can be reduced to a set of one dimensional depth integrals the resulting expressions for the first order derivative are similar to those obtained by previous perturbative approaches based on the depth separated wave equation but the approach followed here permits straightforward extension to higher order derivatives explicit evaluations of the expressions for a representative shallow water waveguide model are in excellent agreement with numerical finite difference computations an analysis of the expressions as a function of source receiver range finds the contributions t

asymptotic solutions of the becker doring equations we describe the solution of various becker doring models first we analyse equilibrium and steady state solutions together with the large time behaviour of solutions for the case of constant monomer concentration then we solve a weak fragmentation constant mass case by using matched asymptotic expansions the methods are applied both to the full becker doring equations and to a coarse grained or contracted system comparison of the results show good qualitative agreement all the phenomena present in the full model are reproduced in the contracted system and no anomalous effects are introduced however the quantitative agreement depends strongly on the choice of parameter values in the contracted system

the becker doring cluster equations basic properties and asymptotic behaviour of solutions existence and uniqueness results are established for solutions to the becker doring cluster equations the density is shown to be a conserved quantity under hypotheses applying to a model of a quenched binary alloy the asymptotic behaviour of solutions with rapidly decaying initial data is determined denoting the set of equilibrium solutions byc 0 s the principal result is that if the initial density 0 s then the solution converges strongly toc o while if 0 s the solution converges weak toc s in the latter case the excess density 0 s corresponds to the formation of larger and larger clusters i e condensation the main tools for studying the asymptotic behaviour are the use of a lyapunov function with desirable continuity properties obtained from a known lyapunov function by the addition of a special multiple of the density and a maximum principle for solutions