the painleve property for partial differential equations in this paper we define the painleve property for partial differential equations and show how it determines in a remarkably simple manner the integrability the backlund transforms the linearizing transforms and the lax pairs of three well known partial differential equations burgers equation kdv equation and the modified kdv equation this indicates that the painleve property may provide a unified description of integrable behavior in dynamical systems ordinary and partial differential equations while at the same time providing an efficient method for determining the integrability of particular systems

a general decomposition theory for random cascades this announcement describes a probabilistic approach to cascades which in addition to providing an entirely probabilistic proof of the kahane peyriere theorem for independent cascades readily applies to general dependent cascades moreover this unifies various seemingly disparate cascade decompositions including kahane s t martingale decomposition and dimension disintegration 1 brief history of the problem a theory of positive t martingales was introduced in k3 as the general framework for independent multiplicative cascades and random coverings this includes spatial distributions of interest in both probability theory and in the physical sciences e g ccd ck de dg df dm f gw ms mw tls pw s for basic definitions let t be a locally compact metric space with borel sigmafield b and let f p be a probability space together with an increasing sequence fn n 1 2 of sub sigmafields of f a positive t martingale is a sequence qn of non negative functions on t such that i for each t t qn t n 1 2 is a martingale adapted to fn n 1 2 ii for p a s qn n 1 2 is a sequence of borel measurable nonnegative real valued functions on t let m t denote the space of positive borel measures on t and suppose that qn t is a positive t martingale for m t let qn denote the random measure defined by qn and dqn d t qn t t t denote the space of bounded continuous functions on t by cb t then essentially by the martingale convergence theorem one obtains a random borel measure q such that with probability one k3

group method analysis of steady free convective laminar boundary layer flow on a nonisothermal vertical circular cylinder the transformation group theoretic approach is applied to present an analysis of the problem of unsteady free convection from the outer surface of a vertical circular cylinder the application of two parameter group reduces the number of independent variables by two and consequently the system of the governing partial differential equations with the boundary conditions reduces to a system of ordinary differential equations with the appropriate boundary conditions the ordinary differential equations are solved numerically using a fourth order runge kutta scheme and the gradient method numerical results are obtained for the study of the boundary layer characteristics the general analysis developed in this study corresponds to the case of surface temperature that varies exponentially with time and uniform with respect to the axial coordinate i e in the formt w ae bt wherea andb are constants the effect of prandtl number pr andb on the boundary layer characteristics and the maximum value of the vertical component of the velocity are studied

numerical solution of burgers equation by quadratic b spline finite elements in this paper a galerkin quadratic b spline finite element method is used to solve numerically the one dimensional burgers equation reduced by the hopf cole transformation the performance of the method is tested on two model problems involving moderately reynolds numbers with known exact solutions the obtained numerical results show that the method is efficient robust and reliable for solving burgers equation accurately even involving high reynolds numbers for which the exact solution fails computed results are compared with other numerical results in the literature a stability analysis of the method is also investigated

transfinite multifractal dimension spectrums the first order theory of the decomposition of measures with respect to dimension which has been developed by kahane katznelson cutler and others is extended through transfinite recursion to a 1 order theory necessary and sufficient conditions for a finite regular borel measure on 0 d 1 to be a 1 order multispectrum for a finite borel measure on rd is given

estimation under a general partitioned linear model abstract in this paper we consider a general partitioned linear model and a corresponding reduced model we derive a necessary and sufficient condition for the blue for the expectation of the observable random vector under the reduced model to remain blue in the partitioned model the former is shown to be always an admissible estimator under a mild condition we also regard alternative linear estimators and their coincidence with the blue under the partitioned model

a multifractal formalism let x be a metric space and a borel probability measure on x for q t r and e x write formula then hq t and pq t are borel measures hq t is a multifractal generalization of the centered hausdorff measure and pq t is a multifractal generalization of the packing measure the measures hq t and pq t define for a fixed q in the usual way a generalized hausdorff dimension dimq e and a generalized packing dimension dimq e of subsets e of x we study the functions b q dimq supp b q dimq supp and their relation to the so called multifractal spectra functions of formula we prove among other things that f f is bounded from above by the legendre transform of b b and that equality holds for graph directed self similar measures and cookie cutter measures finally we discuss the connection with generalized renyi dimensions

a direct variational methods applied to burgers equation in this study a compact approximate method in limiting form for calculating the solution of burgers equation with appropriate boundary conditions is presented the results obtained by present method are found to be in good agreement with those due to earlier authors and offers appreciable advantages for burgers like nonlinear problems

large deviations for multiplicative chaos the local singularities for a class of random measures obtained by random iterated multiplications are investigated using the thermodynamic formalism this analysis can be interpreted as a rigorous study of the phase transition of a system with random interactions

on a partitioned linear model and some associated reduced models abstract in this paper we consider a partitioned linear model y x 1 1 x 2 2 2 where is positive definite but x x1 x2 can be rank deficient we also consider some associated reduced models involving only 2 and not 1 and study how far the inferences on the estimable linear functions of 2 in the full model coincide with those in the reduced models the results of aigner and balestra 1988 and nurhonen and puntanen 1992 follow as special cases we also point out a flaw in the paper by nurhonen and puntanen 1992 and correct it suitably finally we apply the results proved to obtain explicitly the statistic for turkey s test of nonadditivity in two way classified data with a block intraclass correlation structure

numerical solutions of the burgers equation by the least squares quadratic b spline finite element method in this study a least squares quadratic b spline finite element method for calculating the numerical solutions of the one dimensional burgers like equations with appropriate boundary and initial conditions is presented three test problems have been studied to demonstrate the accuracy of the present method results obtained by the method have been compared with the exact solution of each problem and are found to be in good agreement with each other a fourier stability analysis of the method is also investigated

characterization of estimators uniformly shrinking on subspaces abstract shrinkage factors play an important role in the behaviour of biased estimators in this paper we first show that the only way to have bounded shrinkage factors on a subspace is to shrink uniformly on this subspace then we characterize regressions on components that shrink uniformly on the subspaces spanned by their associated weight vectors we show that this problem is equivalent to solving a set of linear equations involving two different projectors we define a class of matrices whose eigen decompositions give the solution

symmetries of differential equations from sophus lie to computer algebra the topic of this article is the symmetry analysis of differential equations and the applications of computer algebra to the extensive analytical calculations which are usually involved in it the whole area naturally decomposes into two parts depending on whether ordinary or partial differential equations are considered we show how a symmetry may be applied to lower the order of an ordinary differential equation or to obtain similarity solutions of partial differential equations the computer algebra packages sods and spde respectively which have been developed to perform almost all algebraic manipulations necessary to determine the symmetry group of a given differential equation are presented furthermore it is argued that the application of computer algebra systems has qualitatively changed this area of applied mathematics

high order transverse schemes for the numerical solution of pdes many existing numerical schemes for the solution of initial boundary value problems for partial differential equations can be derived by the method of lines the pdes are converted into a system of ordinary differential equations either with initial conditions longitudinal scheme or with boundary conditions transverse scheme in particular this paper studies the performance of the transverse scheme in combination with boundary value methods moreover we do not restrict the semi discretization by the usual first or second order finite difference approximations to replace the derivative with respect to time but we use high order formulae

group theoretic methods applied to burgers equation abstract in this study the group theoretic methods for calculating the solution of burgers equation with appropriate boundary and initial conditions is presented the application of a one parameter group reduces the number of independent variables by one and consequently the governing partial differential equation with the boundary and initial conditions to an ordinary differential equation with the appropriate corresponding conditions the obtained differential equation is solved analytically and the solution obtained in closed form for a specific choice of boundary condition

on the equivalence of estimations under a general linear model and its transformed models abstract some necessary and sufficient conditions are given for two equalities of ordinary least squares estimators and best linear unbiased estimators of an estimable vector of parametric functions under a general linear model and its transformed linear model to hold

linear transformations preserving best linear unbiased estimators in a general gauss markoff model under a general gauss markoff model mathbf y mathbf x beta v a necessary and sufficient condition is established for a linear transformation mathbf f of the observable random vector mathbf y to have the property that there exists a linear function of mathscr fy which is a blue of mathbf x beta a method for deriving a required blue from the transformed model mathbf fy fx beta fvf is also indicated

an exponentially fitted method for solving burgers equation in this paper an exponentially fitted method is used to numerically solve the one dimensional burgers equation the performance of the method is tested on the model involving moderately large reynolds numbers the obtained numerical results show that the method is efficient stable and reliable for solving burgers equation accurately even involving high reynolds numbers for which the exact solution fails copyright 2009 john wiley sons ltd

numerical solution of the burgers equation by automatic differentiation abstract we compute the solution of the one dimensional burgers equation by marching the solution in time using a taylor series expansion our approach does not require symbolic manipulation and does not involve the solution of a system of linear or non linear algebraic equations instead we use recursive formulas obtained from the differential equation to calculate exact values of the derivatives needed in the taylor series we illustrate the effectiveness of our method by solving four test problems with known exact solutions the numerical solutions we obtain are in excellent agreement with the exact solutions while being superior to other previously reported numerical solutions

a cascade decomposition theory with applications to markov and exchangeable cascades a multiplicative random cascade refers to a positive t martingale in the sense of kahane on the ultrametric space t 0 1 b 1 a new approach to the study of multiplicative cascades is introduced the methods apply broadly to the problems of i non degeneracy criterion ii dimension spectra of carrying sets and iii divergence of moments criterion specific applications are given to cascades generated by markov and exchangeable processes as well as to homogeneous independent cascades 1 positive t martingales positive t martingales were introduced by jean pierre kahane as the general framework for independent multiplicative cascades and random coverings although originating in statistical theories of turbulence the general framework also includes certain spin glass and random polymer models as well as various other spatial distributions of interest in both probability theory and the physical sciences for basic definitions let t be a compact metric space with borel sigmafield b and let f p be a probability space together with an increasing sequence fn n 1 2 of sub sigmafields of f a positive t martingale is a sequence qn of b f measurable non negative functions on t such that i for each t t qn t n 0 1 is a martingale adapted to fn n 0 1 ii for p a s qn n 0 1 is a sequence of borel measurable non negative real valued functions on t let m t denote the space of positive borel measures on t and suppose that qn t is a positive t martingale for m t such that q t eqn t l let n qn denote the random measure defined by qn and dqn d t qn t t t then essentially by the martingale convergence theorem one obtains a random borel measure q such that for f c t 1 1 lim n t f t qn t dt t f t q dt a s received by the editors august 18 1994 1991 mathematics subject classification primary 60g57 60g30 60g42 secondary 60k35 60d05 60j10 60g09

group theoretic approach for solving the problem of diffusion of a drug through a thin membrane the transformation group theoretic approach is applied to study the diffusion process of a drug through a skinlike membrane which tends to partially absorb the drug two cases are considered for the diffusion coefficient the application of one parameter group reduces the number of independent variables by one and consequently the partial differential equation governing the diffusion process with the boundary and initial conditions is transformed into an ordinary differential equation with the corresponding conditions the obtained differential equation is solved numerically using the shooting method and the results are illustrated graphically and in tables